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+{"text":"\\section{Introduction}\n\nThe analysis of quantum information processing protocols is a challenging task. Let it be a quantum tomography process, transmission of quantum information over a noisy channel or a cryptographic protocol -- all need to be analysed under general conditions. Since one usually has limited information about the actual quantum state given as input, the analysis should be valid for any given quantum state. For example, a cryptographic protocol should be proven secure independently of the input state, which can be chosen by a malicious adversary. As the space of all possible states can be very large and the structure of the states therein might be complicated due to entanglement, this task can be tedious in the good case, and infeasible in the worst. \n\nThe quantum de Finetti theorems \\cite{hudson1976locally, raggio1989quantum, caves2002unknown, renner2007symmetry} and the post selection theorem \\cite{christandl2009postselection} address the above problem, by exploiting the symmetry of the considered states, namely permutation invariance. These mathematical tools allow us to simplify the analysis of quantum information processing tasks by reducing permutation invariant quantum states to a more structured state, called the quantum de Finetti state.\nIn general, we say that a state is of de Finetti-type if it is a convex combination of independent identically distributed (i.i.d.\\@) states. \n\nde Finetti states are usually much easier to handle than general states due to their simple structure. Moreover, most established information-theoretic techniques can be applied only to i.i.d\\ states, and therefore while not applicable when considering a general state, they can be used when considering de Finetti states. Therefore, a reduction to such states can simplify calculations and proofs of various quantum information processing tasks. Indeed, one of the famous applications of reductions to de Finetti states is a proof which states that in order to establish security of quantum key distribution against general attacks it is sufficient to consider attacks on individual signals~\\cite{christandl2009postselection}. Other applications include quantum tomography \\cite{christandl2012reliable} or quantum reverse Shannon coding \\cite{berta2011reverse}. \n\nUnfortunately, the known variants of the quantum de Finetti theorems are not always applicable. A big class of protocols, commonly used in the past several years, to which those theorems are not applicable is the class of protocols in which the dimension of the states is unknown or cannot be bounded, and in particular, the class of device independent protocols (for a review on the topic, see for example \\cite{scarani2012device, brunner2013bell}). The above mentioned theorems cannot be used in such cases for they depend on the dimension of the quantum state. \n\nIn device independent cryptography \\cite{mayers1998quantum,pironio2009device}, for example, one considers the devices as black boxes, about which we know nothing. The security of such protocols can therefore rely only on the observed statistics and not on the specific quantum states and measurements used in the protocol (in some protocols one does not even assume that the underlying physical system is restricted to be quantum! \\cite{barrett2005no,hanggi2009quantum}). In these cases, one possible framework to work with is the framework of conditional probability distributions.  \n\nConditional probability distributions describe the operational behaviour of physical systems under measurements. That is, if we are only interested in modelling the measurement-outcome behaviour of our physical system, then the system can be described by a conditional probability distribution $\\mathrm{P}_{A|X}$ where $X$ is the input, or the measurement performed on the system, and $A$ is the output. $\\mathrm{P}_{A|X}(a|x)$ is the probability for outcome $a$ given that a measurement $x$ was made. We then say that $\\mathrm{P}_{A|X}$ is the state of the system. Note that the state may have as many inputs and outputs as required and therefore we do not restrict the structure of the underlying system by describing it as a conditional probability distribution.\n\nIn quantum physics, for example, $\\mathrm{P}_{A|X}$ is given by Born's rule. However, conditional probability distributions can also be used to describe states that might not conform with the theory of quantum physics, such as non-signalling states. Consider for example a state $\\mathrm{P}_{AB|XY}$ shared by two space-like separated parties, Alice and Bob, each holding a subsystem of the state. $X$ and $A$ are then, respectively, the input and output of Alice, and $Y$ and $B$ of Bob. We then say that the state is non-signalling if it cannot be used to communicate, i.e., the output of one party is independent of the input of the other. The PR-box \\cite{PR-box} is an example for a non-quantum bipartite state which can be written as a (non-signalling) conditional probability distribution. \n\nGiven all the above, it is thus necessary to see whether de Finetti theorems are unique for quantum states or can be also proven on the level of the correlations in the framework of conditional probability distributions. More specifically, we are interested in a theorem that will allow us to reduce permutation invariant conditional probability distributions to a simple de Finetti-type conditional probability distribution, in a way that will be applicable in device independent protocols and, more generally, when the dimension of the underlying quantum states is unknown. Several different non-signalling de Finetti theorems have been established recently \\cite{barrett2009finetti,christandl2009finite,brandao2012quantum}, but it is yet unknown how these can be applied to device independent cryptography\\footnote{In most of these variants of de Finetti theorems, for example, it is assumed that the subsystems cannot signal each other. For current applications this is a too restrictive condition, since it is equivallent to assuming there is no memory in the devices.}. \n\nIn this letter we prove a general de Finetti reduction theorem, from which we can derive several more specialised statements that are of interest for applications. The different reductions differ from one another in two main aspects -- the set of states to which they can be applied and the specific structure of the de Finetti state. Different de Finetti reductions can therefore be useful in different scenarios and under different assumptions. \n\nThe simplest and most straightforward variant is a de Finetti reduction which can be applied to any permutation invariant conditional probability distribution. The second variant is a reduction which can be applied to a family of states which is relevant for cryptographic protocols based on the CHSH inequality \\cite{CHSH} or the chained Bell inequalities \\cite{braunstein1990wringing,Barrett2006Maximally}. There we connect any state $\\mathrm{P}_{AB|XY}$ out of this family of states to a special \\emph{non-signalling} de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$. We do not assume any non-signalling conditions between the subsystems of $\\mathrm{P}_{AB|XY}$ and therefore the use of the de Finetti reduction is not restricted only to scenarios where each of the subsystems cannot signal each other. \n\nUp to date, almost all known device independent cryptographic protocols are based on the CHSH inequality or the more general chained Bell inequalities. For this reason we pay specific attention to states which are relevant for such protocols. However, our theorem can be applied also to other families of states which might be useful in future protocols. As an example of an application of our theorem we prove that for protocols which are based on the violation of the CHSH and chained Bell inequalities it is sufficient to consider the case where Alice and Bob share the de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$. We do this by bounding the distance between two channels which act on conditional probability distributions. \n\nIn the following we start by describing and explaining the different de Finetti reductions. We then illustrate how the reductions can be used in applications. All the proofs are given in the Appendix.\n\n\\section{Results}\n\nFor stating the different de Finetti reductions we will need some basic definitions. $A$ and $X$ denote discrete random variables over $a \\in \\{0,1, ... , l-1\\}^n$ and $x \\in \\{0,1, ... , m-1\\}^n$ respectively. We use $[n]$ to denote the set $\\{1,\\dotsc,n\\}$. An $n$-partite state $\\mathrm{P}_{A|X}$ is a conditional probability distribution if for every $x$, $\\sum_a \\mathrm{P}_{A|X}(a|x)=1$ and for every $a, x$, $\\mathrm{P}_{A|X}(a|x)\\geq 0$. When we consider two different states $\\mathrm{P}_{A|X}$ and $\\mathrm{Q}_{A|X}$ it is understood that both states are over the same random variables $X$ and $A$. The de Finetti reductions deal with permutation invariant states and de Finetti states. Formally we define these as follows.\n\\begin{defn}\\label{def:permutation}\n\tGiven a state $\\mathrm{P}_{A|X}$ and a permutation $\\pi$ of its subsystems\\footnote{Since we permute $a$ and $x$ together this is exactly as permuting the subsystems.} we denote by $\\mathrm{P}_{A|X}\\circ\\pi$ the state which is defined by \n\t\\[\n\t\t\\forall a,x \\quad \\left(\\mathrm{P}_{A|X}\\circ\\pi \\right) (a|x)=\\mathrm{P}_{A|X}(\\pi(a)|\\pi(x)) \\;.\n\t\\]\n\tAn $n$-partite state $\\mathrm{P}_{A|X}$ is permutation invariant if for any permutation $\\pi$, $\\mathrm{P}_{A|X} = \\mathrm{P}_{A|X}\\circ\\pi$.  \n\\end{defn}\nAs mentioned above, we say that a state is a de Finetti state if it is a convex combination of i.i.d.\\ states. Formally, \n\\begin{defn}\n\tA de Finetti state is a state of the form\n\t\\[\n\t\t\\tau_{A|X} = \\int Q_{A_1|X_1}^{\\otimes n} \\mathrm{d}Q_{A_1|X_1}\n\t\\]\n\twhere $x_1\\in \\{0,1, ... , m-1\\}$, $a_1 \\in \\{0,1, ... , l-1\\}$, $\\mathrm{d}Q_{A_1|X_1}$ is some measure on the space of 1-party states and $Q_{A_1|X_1}^{\\otimes n}$ is a product of $n$ identical 1-party states $Q_{A_1|X_1}$, i.e., it is defined according to \n\t\\[\n\t\tQ_{A_1|X_1}^{\\otimes n}(a|x) = \\prod_{i\\in[n]} Q_{A_1|X_1}(a_i|x_i) \\;.\n\t\\]\n\\end{defn}\nAs seen from the above definition, by choosing different measures $\\mathrm{d}Q_{A_1|X_1}$ we define different de Finetti states. \n\nWe are now ready to state the de Finetti reductions. For simplicity we start by giving the first corollary of the more general theorem (Theorem \\ref{thm:post-selection}). This corollary is a reduction for conditional probability distributions, which connects general permutation invariant states to a specific de Finetti state. \n\n\\begin{cor}[de Finetti reduction for conditional probability distributions]\\label{cor:conditional}\n\tThere exists a de Finetti state $\\tau_{A|X}$ where $x \\in \\{ 0,1, ... ,m-1 \\}^n$ and $a \\in \\{ 0,1, ... ,l-1 \\} ^n$ such that for every permutation invariant state $\\mathrm{P}_{A|X}$  \n\t\\[\n\t\t\\forall a,x \\quad \\mathrm{P}_{A|X} (a|x) \\leq (n+1)^{m(l-1)} \\; \\tau_{A|X} (a|x)  \\;.\n\t\\]\n\\end{cor}\n\nThe de Finetti state $\\tau_{A|X}$ is an \\emph{explicit} state that we construct in the proof of the general theorem in Appendix~\\ref{sec:general-proof}. The proof uses mainly combinatoric arguments; we choose $\\tau_{A|X}$ in a specific way, such that a lower bound on $\\tau_{A|X}(a|x)$ for all $a,x$ can be proven. We then use the permutation invariance of $\\mathrm{P}_{A|X}$  to prove an upper bound on $\\mathrm{P}_{A|X}(a|x)$. The result is then derived by combining the two bounds.\n\nCorollary \\ref{cor:conditional} holds for every permutation invariant state $\\mathrm{P}_{A|X}$, not necessarily quantum or non-signalling. At first sight, the generality of the above mathematical statement might seem as a drawback in applications where only a restricted set of correlations is considered (e.g., only non-signalling correlations). Nevertheless, in a following work \\cite{arnon2014nonsignalling} we show that this is not the case and apply this general theorem to prove parallel repetition theorems for non-signalling games. \nNote that according to Definition~\\ref{def:permutation} we consider permutations which permute the 1-party subsystems of $\\mathrm{P}_{A|X}$\\footnote{This is in contrast to states $\\mathrm{P}_{AB|XY}$ which can also be permuted as $\\left(\\mathrm{P}_{AB|XY}\\circ\\pi \\right) (ab|xy)=\\mathrm{P}_{AB|XY}\\left(\\pi(a)\\pi(b)|\\pi(x)\\pi(y)\\right)$, as is usually the case in cryptographic tasks. For dealing with such states we will consider a different reduction, stated as Corollary~\\ref{cor:chsh-post-selection}.}. \n\nThe multiplicative pre-factor of the de Finetti reduction, $(n+1)^{m(l-1)}$ in Corollary \\ref{cor:conditional} for example, is relevant for applications. Intuitively, this is the ``cost'' for using $\\tau_{A|X}$ instead of $\\mathrm{P}_{A|X}$ in the analysis of the considered protocol. We therefore want it to be as small as possible. Nevertheless, as will be explained later, in many cases a pre-factor polynomial in $n$ suffices.  \n\nCorollary \\ref{cor:conditional} is relevant for scenarios in which one considers permutation invariant conditional probability distributions $\\mathrm{P}_{A|X}$. However, if the states one considers have additional symmetries $\\mathcal{S}$ then we can prove a better de Finetti reduction --- a reduction with a smaller pre-factor and a special de Finetti state with the same symmetries~$\\mathcal{S}$.\n\nIn the following we consider a specific family of symmetries --- symmetries between different inputs and outputs of the subsystems of $\\mathrm{P}_{A|X}$. Formally, the types of symmetries that we consider are described, among other things, by a number $d\\leq m(l-1)$ which we call the degrees of freedom of the symmetry (see Appendix~\\ref{sec:general-proof} for details and formal definition of the symmetries). More symmetry implies less degrees of freedom, i.e., smaller $d$, and as shown in the following theorem, this leads to a smaller pre-factor in the reduction. The general theorem then reads:\n\n\\begin{thm}[de Finetti reduction for conditional probability distributions with symmetries]\\label{thm:post-selection}\n\tThere exists a de Finetti state $\\tau^\\mathcal{S}_{A|X}$ where $x \\in \\{ 0,1, ... ,m-1 \\}^n$ and $a \\in \\{ 0,1, ... ,l-1 \\} ^n$ such that for every permutation invariant state $\\mathrm{P}_{A|X}$ with symmetry $\\mathcal{S}$ (with $d$ degrees of freedom) \n\t\\[\n\t\t\\forall a,x \\quad \\mathrm{P}_{A|X} (a|x) \\leq (n+1)^d \\; \\tau^\\mathcal{S}_{A|X} (a|x) \\;. \n\t\\]\n\\end{thm}\nFor the case of no symmetry we have $d=m(l-1)$ from which Corollary  \\ref{cor:conditional} stated before follows. \n\nThe symmetries $\\mathcal{S}$ that we consider are of particular interest when considering cryptographic protocols which are based on non-signalling states. For example, the states which are relevant for protocols which are based on the violation of the CHSH inequality (such as \\cite{masanes2009universally,hanggi2009quantum}) have a great amount of symmetry. The additional symmetry allows us to prove a corollary of Theorem \\ref{thm:post-selection} which can be used to simplify such protocols. \n\nBefore we state the corollary for the CHSH case, let us define what we mean when we say that a state has a CHSH-type symmetry. In cryptographic protocols based on the CHSH inequality the basic states that we consider are bipartite states $\\mathrm{P}_{AB|XY}$ held by Alice and Bob where $a,b,x,y\\in \\{0,1\\}^n$. \n\\begin{defn}[CHSH-type symmetry]\\label{def:chsh-symmetry}\n\tA state $\\mathrm{P}_{AB|XY}$ where $a,b,x,y\\in \\{0,1\\}^n$ has a CHSH-type symmetry if there exist $p_1,\\dotsc,p_n\\in [0,\\frac{1}{2}]$ such that $\\forall i\\in\\{1,\\dotsc,n\\}$,\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t & \\forall a_i, b_i, x_i, y_i \\\\\n\t\t & a_i\\oplus\\ b_i=x_i\\cdot y_i \\rightarrow \\mathrm{P}_{AB|XY}(a_{\\overline{i}}a_ib_{\\overline{i}}b_i|x_{\\overline{i}}x_iy_{\\overline{i}}y_i) = \\frac{1}{2}-p_i \\\\\n\t\t& a_i\\oplus\\ b_i \\neq x_i\\cdot y_i \\rightarrow \\mathrm{P}_{AB|XY}(a_{\\overline{i}}a_ib_{\\overline{i}}b_i|x_{\\overline{i}}x_iy_{\\overline{i}}y_i) = p_i \\;.\n\t\\end{split}\n\t\\end{equation*}\n\twhere $a_{\\overline{i}}=a_1a_2\\dotsc a_{i-1}a_{i+1} \\dotsc a_n$ and $b_{\\overline{i}}, x_{\\overline{i}}, y_{\\overline{i}}$ are defined in a similar way.\n\\end{defn}\nA simple state $\\mathrm{P}_{AB|XY}$ which has this symmetry for example is a product state of 2-partite states as in Figure~\\ref{fig:CHSH_symmetry} with different values of $p$.\n\n\\begin{figure}\n\\begin{centering}\n\t\\begin{tikzpicture}[scale=0.5]\n\n\t\t\\draw[step=2] (-5,-4) grid (4,5);\n\t\t\\draw[ultra thick] (-6,4)--(4,4);\n\t\t\\draw[ultra thick] (-6,-4)--(4,-4);\n\t\t\\draw[ultra thick] (-4,-4)--(-4,6);\n\t\t\\draw[ultra thick] (4,-4)--(4,6);\n\t\t\\draw[ultra thick] (-6,0)--(4,0);\n\t\t\\draw[ultra thick] (0,-4)--(0,6);\n\t\t\\draw (-4,4)--(-6,6);\n\n\t\t\\draw (-3,3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (-1,3) node {$p$};\n\t\t\\draw (1,3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (3,3) node {$p$};\n\n\t\t\\draw[red] (-3,1) node {$p$};\n\t\t\\draw (-1,1) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (1,1) node {$p$};\n\t\t\\draw (3,1) node {$\\frac{1}{2}-p$};\n\n\t\t\\draw (-3,-1) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (-1,-1) node {$p$};\n\t\t\\draw[red] (1,-1) node {$p$};\n\t\t\\draw (3,-1) node {$\\frac{1}{2}-p$};\n\n\t\t\\draw[red] (-3,-3) node {$p$};\n\t\t\\draw (-1,-3) node {$\\frac{1}{2}-p$};\n\t\t\\draw (1,-3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (3,-3) node {$p$};\n\n\t\t\\draw (-5,3) node {0};\n\t\t\\draw (-5,1) node {1};\n\t\t\\draw (-5,-1) node {0};\n\t\t\\draw (-5,-3) node {1};\n\t\t\\draw (-6,2) node {0};\n\t\t\\draw (-6,-2) node {1};\n\n\t\t\\draw (-3,5) node {0};\n\t\t\\draw (-1,5) node {1};\n\t\t\\draw (1,5) node {0};\n\t\t\\draw (3,5) node {1};\n\t\t\\draw (-2,6) node {0};\n\t\t\\draw (2,6) node {1};\n\n\t\t\\draw (-5,4.4) node {$B_1$};\n\t\t\\draw (-4.4,5) node {$A_1$};\n\t\t\\draw (-6,5) node {$Y_1$};\n\t\t\\draw (-5,6) node {$X_1$};\n\t\t\t\n\t\\end{tikzpicture}\n\\end{centering}\n\\caption{A simple 2-partite state $P_{A_1B_1|X_1Y_1}$ with the CHSH symmetry.} \\label{fig:CHSH_symmetry}\n\\end{figure} \n\n\\begin{cor}[de Finetti reduction for states with the CHSH symmetry] \\label{cor:chsh-post-selection}\n\tThere exists a non-signalling de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$ where $a,b,x,y \\in \\{ 0,1 \\}^n$ such that for every permutation invariant\\footnote{Here a permutation acts on the bipartite state as $\\left(\\mathrm{P}_{AB|XY}\\circ\\pi \\right) (ab|xy)=\\mathrm{P}_{AB|XY}\\left(\\pi(a)\\pi(b)|\\pi(x)\\pi(y)\\right)$.} state $\\mathrm{P}_{AB|XY}$ with the CHSH symmetry, for all $a,b,x,y$,\n\t\\[\n\t\t\\mathrm{P}_{AB|XY} (a,b|x,y) \\leq (n+1) \\; \\tau^\\mathcal{CHSH}_{AB|XY} (a,b|x,y)\\;. \n\t\\]\n\\end{cor}\nNote that we do not assume that the state $\\mathrm{P}_{AB|XY}$ satisfies any non-signalling conditions. Our theorem holds even when there is signalling between the subsystems, and therefore can be used in a broad set of applications.\n\nCorollary \\ref{cor:chsh-post-selection} is derived from Theorem \\ref{thm:post-selection} by showing that $d=1$ for the CHSH symmetry\\footnote{Intuitivly, in the CHSH symmetry there is only one degree of freedom, i.e. $d=1$, since we are only free to choose one value $p$ when defining the basic CHSH state given in Figure \\ref{fig:CHSH_symmetry}. Less symmetry implies more degrees of freedom. }. For pedagogical reasons, we also present a self-contained proof including an explicit construction of the state $\\tau^\\mathcal{CHSH}_{AB|XY}$ in Appendix~\\ref{sec:chsh-proof}. \n\nAlthough the assumption about the symmetry of the states in Corollary \\ref{cor:chsh-post-selection} appears to be rather restrictive, the statement turns out to be useful for applications. \n\n\\section{Applications}\n\nTo illustrate the use of the de Finetti reductions, we start by considering the following abstract application. Let $\\mathcal{T}$ be a test which interacts with a state $\\mathrm{P}_{A|X}$ and outputs ``success'' or ``fail'' with some probabilities. One can think about this test, which can be chosen according to the application being considered, as a way to quantify the success probability of the protocol when the state $\\mathrm{P}_{A|X}$ is given as input. For example, if one considers an estimation, or a tomography, protocol a test can be chosen to output ``success'' when the estimated state is close to the actual state \\cite{christandl2009postselection}. \n\nWe denote by $\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X})$ the probability that $\\mathcal{T}$ outputs ``fail'' after interacting with $\\mathrm{P}_{A|X}$. We consider permutation invariant tests, defined as follows. \n\\begin{defn}\\label{def:permutation-invariant-test}\n\tA test $\\mathcal{T}$ is permutation invariant if for all states $\\mathrm{P}_{A|X}$ and all permutations $\\pi$ we have\n\t\\[\n\t \t\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X}) = \\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X}\\circ\\pi) \\;.\n\t\\]\n\\end{defn}\n\nUsing the de Finetti reduction in Corollary \\ref{cor:conditional} we can prove upper bounds of the following type: \n\n\\begin{lem}\\label{lem:test-bound}\n\tLet $\\mathcal{T}$ be a permutation invariant test. Then for every state $\\mathrm{P}_{A|X}$ \n\t\\[\n\t\t\t\\mathrm{Pr}_{\\mathrm{fail}}(\\mathrm{P}_{A|X}) \\leq (n+1)^{m(l-1)} \\mathrm{Pr}_{\\mathrm{fail}}(\\tau_{A|X}) \\;.\n\t\\]\n\\end{lem}\n\nThe importance of the de Finetti reductions is obvious from this abstract example --- if one wishes to prove an upper bound on the failure probability of the test $\\mathcal{T}$, instead of proving it for all states $\\mathrm{P}_{A|X}$ it is sufficient to prove it for the de Finetti state $\\tau_{A|X}$ and ``pay'' for it with the additional polynomial pre-factor of $(n+1)^{m(l-1)}$. Since the de Finetti state has an i.i.d.\\ structure this can highly simplify the calculations of the bound. \n\nMoreover, in many cases one finds that the bound on $\\mathrm{Pr}_{\\text{fail}}(\\tau_{A|X})$ is exponentially small in $n$. For an estimation protocol, the failure probability of the test, when interacting with an i.i.d.\\ state, can be shown to be exponentially small in the number of subsystems used for the estimation, using Chernoff bounds. This is also the case when dealing with security proofs -- the failure probability of a protocol, when a de Finetti state is given as input, is usually exponentially small in the number of subsystems used in the protocol. If this is indeed the case then the polynomial pre-factor of  $(n+1)^{m(l-1)}$ will not affect the bound in the asymptotic limit of large $n$. That is, an exponentially small bound on $\\mathrm{Pr}_{\\text{fail}}(\\tau_{A|X})$ implies an exponentially small bound on $\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X})$.\n\nFor an estimation protocol as mentioned above the notion of the test, combined with the de Finetti reductions, can be used to prove that an estimation procedure of permutation invariant states succeeds with high probability. \n\nFor readers who are interested in cryptography, we show in Appendix~\\ref{sec:diamond-proofs} how to derive a similar result when considering the diamond norm~\\cite{kitaev1997quantum}, i.e., the distance between channels acting on conditional probability distributions, instead of the abstract test $\\mathcal{T}$. The diamond norm is the relevant distance measure when considering cryptographic protocols, and therefore using de Finetti reductions to upper bound the diamond norm can simplify the analysis of device independent  protocols. \n\n\\section{Concluding remarks}\nIn this letter we introduced a general de Finetti-type theorem from which various more specialised variants can be derived. Crucially, such theorems can be formulated even without relying on assumptions regarding the non-signalling conditions between the subsystems or the underlying dimension. In the general theorem, Theorem~\\ref{thm:post-selection}, we can also see how additional symmetries of the states can affect the pre-factor in the de Finetti reduction. This suggests that the same relationship might also exist in the quantum post selection theorem \\cite{christandl2009postselection}, which is the quantum variant of the de Finetti reductions presented here.\n\nAs an example for an application we showed how our theorems can be used to bound the failure probability of a test. In a following work \\cite{arnon2014nonsignalling} we show how to use the concept of the test, together with the de Finetti reduction given in Corollary \\ref{cor:conditional} to prove parallel repetition results for non-local games. Previous de Finetti theorems could have not been used in the setting of non-local games due to their dependency on the dimension of the systems or the strict non-signalling conditions they assume. The new de Finetti theorem presented here therefore opens new possibilities and therefore strictly extends the range of applications to which de Finetti reductions can be applied.\n\nAs an additional example, we explain how our theorem can be used in device independent protocols in which the parties are not assumed to be restricted by quantum theory in Appendix~\\ref{sec:diamond-proofs}. We hope that this approach will also be useful for quantum device independent information processing protocols in the future. One possible direction can be to use a similar de Finetti reduction as in Corollary \\ref{cor:chsh-post-selection}, but for a Bell inequality in which the maximal violation is achieved within quantum theory. This way, the resulting de Finetti state will be not only non-signalling but also quantum.\nDue to the general structure of the de Finetti reductions and the increasing use of conditional probability distributions in quantum information theory, we also hope that the presented reductions will be useful in other applications apart from cryptography, such as quantum tomography, as was the case for the quantum post selection theorem~\\cite{christandl2009postselection}.\n\nThe techniques used to prove our theorems (mainly combinatoric arguments) differ from the techniques used in previous papers to establish general de Finetti theorems. We therefore hope that they will shed new light on de Finetti reductions in general. For example, it might be possible to apply some ideas from the proof in (device dependent) quantum de Finetti reductions. \n\n\\begin{acknowledgments}\nThe authors thank Roger Colbeck and Michael Walter for discussing a preliminary version of this work. This work was supported by the Swiss National Science Foundation (via the National Centre of Competence in Research ``QSIT'' and SNF project No. 200020-135048), by the European Research Council (via project No. 258932), by the CHIST-ERA project ``DIQIP'' and by the EC STREP project ``RAQUEL''. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/overview-apinet.pdf}\n    \\caption{Overview of proposed Multi Channel Xception Attentive Pairwise Interaction (MCX-API) network. Two inputs are first represented in $n$ kinds of color spaces, $CS_1$ to $CS_n$ to obtain a two N-channel input and subsequently feature vectors. We thereafter obtain $x_{i}^{self}$ and $x_{i}^{other}$ by comparison through MCX-API, where $x_{i}^{self}$ is enhanced by its own images and $x_{i}^{other}$ is activated by the other image. $x_{i}$ is therefore improved with discriminative clues that come from both images. By comparison, we can finally distinguish if an image is pristine or fake.}\n    \\label{fig:overview}\n\\end{figure}\nDeepfakes are synthetic media that are generated by deep learning methods to manipulate the content in images and videos. The manipulations include altering people's identities, faces, expressions, speech or bodies to both entertainment and malicious intent (for example pornographic uses). Benefiting from the remarkable advancement in generation models, amateurish individuals are capable of creating Deepfakes using off-the-shelf models~\\cite{DeepFaceLab, fffs, FaceApp} without tedious efforts. \nIn the meantime, channelized efforts have been dedicated to devising Deepfakes detection algorithms using multiple approaches such as by determining unique artifacts~\\cite{matern2019exploiting,ciftci2020fakecatcher,fernandes2019predicting,haliassos2021lips,agarwal2019protecting,li2020face}, utilizing Convolutional Neural Networks (CNNs) based networks~\\cite{marra2019gans,rossler2019faceforensics++,nguyen2019use}, employing frequency domain information~\\cite{durall2019unmasking,chen2021attentive,qian2020thinking} and other clues~\\cite{cozzolino2021id, cozzolino2022audio}.\n\\par With an atomic effort, these methods could perform well with an average of more than 99\\%~\\cite{rossler2019faceforensics++} accuracy in a closed-set problem where the training and testing data are pulled from the same label and feature spaces. For example, the network is trained on attacks A, B and C and tested on images\/videos drawn from attack A or B or C. However, newer DeepFakes generation mechanisms make the detection algorithms untrustworthy and non-generalizable by degrading the performance of the detector~\\cite{zhao2021learning,zhou2021joint} as no exception to those classifiers trained with machine learning methods. In the context of DeepFakes detection, this can be parallel to detecting attack D when the detector is trained on A, B, and C, making it an open-set problem. The reasons behind the collapse of detection models towards unseen contents can, to some degree, be attributed to various generation algorithms, which often result in different data distributions, feature spaces, and appearance properties of images or videos. While one can see the imperative need for a generalizable detection technique to make reliable decisions on unknown\/unseen generation types in addition to known\/seen generation data, we note low performances of networks in this direction \\cite{zhao2021learning,zhou2021joint,xu2022supervised, aneja2020generalized}. \n\\par We thus motivate our work, focusing on both closed-set and open-set detection in this article. We draw our inspiration from how humans tend to detect altered media in a fine-grained manner by comparing one kind of visual content to another. Human decision making relies on detecting an unseen kind of manipulated images\/videos as fake by comparing the unknown generation type to the known generation types, especially the artifacts and clues~\\cite{zhuang2020learning}. Initial work using on pairwise interaction has shown promising directions to capture subtle differences in a pairwise manner with not only principal parts of the image but also distinct details from the other image \\cite{zhuang2020learning}. Using such a paradigm, we propose to learn the known type of generations in a fine-grained pairwise manner explicitly to improve the performance of a Deepfake detector for unknown types. Further, we also note the complementary information an image\/video can exhibit in different color spaces along the same lines. We therefore incorporate information from four color spaces, including RGB, CIELab, HSV, and YCbCr integrating to boost the attentive pairwise learning to guide the detector to classify the non-manipulated images efficiently. Our proposed approach exploits the information from color channels in a pairwise manner using the strengths of the Xception network and we refer to this as the Multi-Channel Xception Attentive Pairwise Interaction (MCX-API) network between non-manipulated images against a set of manipulated images and to try to generalize the detector towards unknown manipulation types or unseen data. \\Cref{fig:overview} shows an overview of the idea presented in this work. \n\\par To validate our idea in this work, we conduct various experiments using FaceForensics++ dataset~\\cite{rossler2019faceforensics++} which consists of four different manipulation classes including DeepFakes (DF)~\\cite{ffdf}, FaceSwap (FS)~\\cite{fffs}, Face2Face (F2F)~\\cite{thies2016face2face} and NeuralTextures (NT)\\cite{thies2019deferred} where we obtain better state-of-the-art (SOTA) performance or at par detection performance to best performing SOTA approaches in closed-set experiments~\\cite{chollet2017xception,afchar2018mesonet,zhao2021learning,li2020face}. Furthermore, we demonstrate the effectiveness of variants of the proposed approach in detecting Deepfakes in open-set scenarios where our approach achieves better results than SOTA models on three other public datasets such as FakeAV~\\cite{khalid2021fakeavceleb}, KoDF~\\cite{kwon2021kodf}, and Celeb-DF~\\cite{li2020celeb}.\n\\par A detailed ablation study is presented on MCX-API to illustrate the variability of performance of the detector to various design choices in the network. Thus, the main contributions of our paper are \\textbf{(1)} We propose a new framework - Multi-Channel Xception Attentive Pairwise Interaction (MCX-API)  for Deepfakes detection by exploiting color space and pairwise interaction simultaneously, bringing a novel fine-grained idea for the Deepfakes detection field. \\textbf{(2)}We report all results by balanced-open-set-classification (BOSC) accuracy to exemplify the generalizability of our proposed approach. \n   \n   \n\\textbf{(3)}We conduct cross-datasets validations with three SOTA Deepfake datasets, Celeb-DF~\\cite{li2020celeb}, KoDF~\\cite{kwon2021kodf} and FakeAVCelebDF~\\cite{khalid2021fakeavceleb}. Furthermore, we compared the results with SOTA Deepfake detection methods. Our MCX-API obtains 98.48\\% BOSC accuracy on the FF++ dataset and 90.87\\% BOSC accuracy on the Celeb-DF dataset, indicating an optimistic direction for the generalization of DeepFake detection.\n\nIn the rest of the paper, we list a set of directly related works in \\cref{sec:related-works} and then present our proposed approach in \\cref{sec:proposed-approach}. \nWe provide an analysis of explainability in \\cref{sec:ExplainableAnalysisofMCX-API} with the set of experiments and results on generalizability detailed in \\cref{sec:Resutls}. We finally conclude the work in \\cref{sec:conclusion}.\n\n\\begin{figure*}[htp]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/API-Net.pdf}\n    \\caption{The architecture of MCX-API network.}\n    \\label{fig:arr_bar}\n\\end{figure*}\n\n\\section{Related Work}\n\\label{sec:related-works}\n\\textbf{Deepfakes detection methods.} A track of Deepfakes detection focuses on the unique artifacts on human faces, such as eye blinking~\\cite{li2018ictu}, different eye colors~\\cite{matern2019exploiting}, abnormal heartbeat rhythms shown on the face~\\cite{ciftci2020fakecatcher,fernandes2019predicting}. LipForensics~\\cite{haliassos2021lips} targets high-level semantic abnormalities in mouth movements, which the authors observe as a common indicator in many generated videos. Some articles are dedicated to finding inconsistencies in images and videos. These inconsistencies arise out of generation process where landmarks, head pose are inconsistent~\\cite{yang2019exposing,agarwal2019protecting} or observable in image blending~\\cite{li2018exposing,li2020face}. Cozzolino \\textit{et. al.}~\\cite{cozzolino2021id} have introduced ID-Reveal, an identity-aware detection approach leveraging a set of reference videos of a target person and trained in an adversarial manner. \nMany papers have utilized CNNs-based methods for detecting features existing in forged images\\cite{marra2019gans,rossler2019faceforensics++,nguyen2019use}. Using high-frequency features~\\cite{durall2019unmasking,chen2021attentive,qian2020thinking} to distinguish Deepfakes are also gaining more popularity on this topic.\nAlthough pairwise learning have been used for Deepfake detection~\\cite{hsu2020deep, xu2022supervised}, they lack the pairwise interactions by using contrastive learning.\n\n\\par \\textbf{Generalization to unseen manipulations.} While many works are proposed for detecting DeepFakes, they have focused on closed-set experiments where the training and testing set distributions are similar. The open-set experiments indicate that they underperform on unseen manipulations. In the meantime, an increasing number of works have tried to address the problem of generalization of DeepFakes detection. These works have focused on domain adaptation and transfer learning to minimize the task of learning parameters in an end-to-end manner~\\cite{aneja2020generalized,kim2021fretal,lee2021tar}. \\textit{Cozzolino et. al.}~\\cite{cozzolino2018forensictransfer} proposed an autoencoder-like structure ForensicTransfer and the generalization aspect was studied using a single detection method for multiple target domains. The follow-up works like Locality-aware AutoEncoder (LAE)~\\cite{du2019towards} and Multi-task Learning were proposed for detecting and segmenting manipulated facial images and videos~\\cite{nguyen2019multi}. Transfer learning-based Autoencoder with Residuals (TAR)~\\cite{lee2021tar} recently proposed uses the residuals from autoencoders to handle generalizability. \\textit{Kim et. al.}~\\cite{kim2021fretal} employed the Representation Learning (ReL) and Knowledge Distillation (KD) paradigms to introduce a transfer learning-based Feature Representation Transfer Adaptation Learning (FReTAL) method. While these transfer learning and zero-shot\/few-shot learning methods could not wholly deal with the Deepfakes detection generalization problem, because the networks have already seen the manipulated image\/videos. Therefore, strictly speaking, it is not an open-set situation.\n\\par In the meantime, some other novel networks have been proposed dealing with the generalization problem of Deepfakes detection. A new method to detect deepfake images using the cue of the source feature inconsistency within the forged images~\\cite{zhao2021learning} is proposed based on the hypothesis that distinct source features can be preserved and extracted through SOTA deepfake generation processes. Joint Audio-Visual Deepfake Detection~\\cite{zhou2021joint} is proposed by jointly modeling video and audio modalities. This novel visual\/auditory deepfake combined detection task shows that exploiting the intrinsic synchronization between the visual and auditory modalities could benefit deepfake detection. \\textit{Xu et. al.}~\\cite{xu2022supervised} proposed a novel method using supervised contrastive learning to deal with the generalization problem in detecting forged visual media.\n\n\\section{Proposed Method}\n\\label{sec:proposed-approach}\nFine-grained method has been widely used for classification problems where the categories are visually very similar~\\cite{zhuang2020learning, xiao2015application, akata2015evaluation}. We draw similar inspiration to our problem of Deepfake detection following the architecture proposed by earlier~\\cite{zhuang2020learning} and build upon with number of improvements. We assert that architecture for fine-grained classification can help in detecting Deepfakes. Unlike the orginal architecture, we introduce Xception~\\cite{chollet2017xception} to extract the embeddings motivated by earlier works in Deepfake detection~\\cite{rossler2019faceforensics++, zhao2021multi, wang2022m2tr, kim2021fretal}. \n\nSecond, to benefit from information from different color spaces, we make the base network to a multi-channel network. Then, we enforce pairwise learning by following the architecture of Attentive Pairwise Learning \\cite{zhuang2020learning}. We propose using the Multi Channel Xception Attentive Pairwise Interaction Network (MCX-API) to deal with the Deepfakes classification problem as detailed further.\n\n\\subsection{Architecture}\n\\par We first utilize MTCNN\\cite{zhang2016joint} to crop and align the face region of a single frame. Two selected face images are further sent to a Multi-Channel Xception backbone, and this backbone network extracts two corresponding $\\mathrm{D}$-dimension feature vectors $x_{1}$ and $x_{2}$ using the face image represented in $N$ different channels that include RGB, CIELab, HSV, and YCbCr. A mutual vector $x_{m}\\in \\mathbb{R}^{D}$ is further generated by concatenating $x_{1}$ and $x_{2}$ and using a Multi-Layer Perceptron (MLP) function for mapping $x_{m}$ to get a $\\mathrm{D}$ dimension. $x_{m}$ is a joint feature that includes high-level contrastive clues of both input images across multiple color channels.\n\n\\par In order to compare $x_{m}$ with $x_{1}$ and $x_{2}$, we need to activate $x_{m}$ using sigmoid function to increase the positive relation with $x_{i}$ and decrease the negative relation against $x_{i}$~\\cite{zhuang2020learning}. Therefore, two gate vectors $g_{1}$ and $g_{2}$ will be generated. $g_{i}$ is calculated by $x_{m}$ and $x_{i}$, thus containing contrastive clues and acting as discriminative attention spots semantic contrasts with a distinct view of each $x_{i}$. The gate vector $g_{i}$ is the sigmoid of the output of channel-wise product between $x_{m}$ and $x_{i}$, whose formula is provided in \\Cref{eqn:gate-vector}. \n\\vspace{-1mm}\n\\begin{equation}\n    g_{i} = sigmoid(x_{m} \\odot x_{i}), \\;\\; i \\in{\\{1,2\\}}\n    \\label{eqn:gate-vector}\n\\end{equation}\n\n\\par A pairwise interaction between input features $x_{i}$ and gate vectors $g_{i}$ is performed to induce residual attention by comparing one image to the other to distinguish the final class. The sequence of interaction can be shown in \\Cref{eqn:pairwise-interaction}.\n\\begin{equation}\n\\centering\n\\begin{split}\n    x_{1}^{pristine}=x_{1}+x_{1}\\odot g_{1} \\\\\n    x_{1}^{fake}=x_{1}+x_{1}\\odot g_{2} \\\\\n    x_{2}^{pristine}=x_{2}+x_{2}\\odot g_{2} \\\\\n    x_{2}^{fake}=x_{2}+x_{2}\\odot g_{1}\n\\end{split}\n\\label{eqn:pairwise-interaction}\n\\end{equation}\nThrough the pairwise interaction of each feature $x_{i}$, two attentive feature vectors $x_{i}^{pristine}\\in \\mathbb{R}^{D}$ and $x_{i}^{fake}\\in \\mathbb{R}^{D}$ are further produced. The former one is highlighted by its gate vector, and the latter is triggered by the gate vector of the compared image. $x_{i}$ is thus enhanced with discriminative clues from both input features through pairwise interaction.\n\n\n\n\\subsection{Loss calculation}\nThe four attentive features $x_{i}^{j}$ where $i \\in {\\{1,2\\}}$ and $j \\in {\\{pristine,fake\\}}$, the pairwise interaction outputs, are fed into a $softmax$ classifier for the loss calculation~\\cite{zhuang2020learning}. The output of $softmax$ denoted by $p_{i}^{j}$ is the probability of a feature belonging to a specific class (i.e., non-manipulated or Deepfake). The main loss in our case is the cross-entropy loss \n\\begin{equation}\n\\mathcal{L}_{ce} = -\\sum_{i \\in \\{ 1,2 \\}} \\sum_{j \\in \\{ pristine,fake \\}} y_{i}^{\\intercal} log(p_{i}^{j})\n\\label{eqn:lce}\n\\end{equation}\nwhere $y_{i}$ is the one-hot label for image $i$ in the pair and $\\intercal$ represents the transpose. MCX-API can be trained to determine all the attentive features $x_{i}^{j}$ under the supervision of the label $y_{i}$ through this loss.\n\n\\par Furthermore, a hinge loss of score ranking regularization\n\\begin{equation}\n    \\mathcal{L}_{rk} = \\sum_{i\\in {\\{1, 2\\}}} max(0, p_{i}^{fake}(c_{i})-p_{i}^{pristine}(c_{i})+\\epsilon )\n\\label{eqn:lrk}\n\\end{equation}\nis also introduced when computing the complete loss~\\cite{zhuang2020learning}. $c_{i}$ is the corresponding index associated with the ground truth label of image $i$. So $p_{i}^{j}(c_{i})$ is a softmax score of $p_{i}^{j}$. Since $x_{i}^{pristine}$ is activated by its gate vector $g_{i}$, it should contain more discriminative features to identify the corresponding image, compared to $x_{i}^{fake}$. $\\mathcal{L}_{rk}$ is utilized to promote the priority of $x_{i}^{pristine}$ where the score difference between $p_{i}^{fake}(c_{i})$ and $p_{i}^{pristine}(c_{i})$ should be greater than a margin.\nThe whole loss for a pair is composed of two losses, cross-entropy loss $\\mathcal{L}_{ce}$ and score ranking regularization $\\mathcal{L}_{rk}$ with coefficient $\\lambda$. \n\\begin{equation}\n    \\mathcal{L} = \\mathcal{L}_{ce} + \\lambda \\mathcal{L}_{rk}\n\\label{eqn:loss}\n\\end{equation}\nIn this way, MCX-API is able to take feature priorities into account adaptively and learns to recognize each image in the pair.\n\n\\section{Experiments and Results}\n\\label{sec:Resutls}\n\n\\subsection{Datasets}\n\\textbf{Training data: }We select FaceForensics++ \\cite{rossler2019faceforensics++} to train the proposed approach. This forensics dataset consists of 1000 original videos and corresponding number of manipulated videos consisting of 1000 videos for each of the subsets - DeepFakes (denoted as DF)~\\cite{ffdf}, Face2Face (denoted as F2F)~\\cite{thies2016face2face}, FaceSwap (denoted as FS)~\\cite{fffs}, and NeuralTextures (denoted as NT)~\\cite{thies2019deferred}.\n\n\\textbf{Cross-dataset Validation: }We also select three other SOTA datasets for generalization test and comparison. \\textbf{Celeb-DF~\\cite{li2020celeb}}: For Celeb-DF, we choose id51-id61 from Celeb-real, Celeb-synthesis and id240-id299 from YouTube-real for the test set.\n   \n\\textbf{KoDF~\\cite{kwon2021kodf}} We randomly selected 265 real videos and 734 fake ones as our test set.\n\\textbf{FakeAV~\\cite{khalid2021fakeavceleb}} We randomly selected 500 videos as our test set.\n\n\n\\begin{table*}[htp]\n\\caption{\\textbf{Frame-level BOSC Accuracy and AUC for our proposed MCX-API networks and SOTA methods on seen data.} We compare the results with the SOTA methods on DF\/F2F\/FS\/NT respectively. All networks are trained on the whole FF++ c23 dataset. The data of the first three methods are adopted from Table 5 in Appendix of FF++~\\cite{chollet2017xception}.}\n\\label{tab:bosc-ffpp}\n\\begin{threeparttable}\n\\centering\n\\begin{tabular}{llllllll}\n\\toprule\nFF++ c23 &  & \\multicolumn{6}{c}{Frame-level (BOSC(\\%)\/AUC)} \\\\ \\cline{1-1} \\cline{3-8} \nMethod &  & \\multicolumn{1}{c}{DF} & \\multicolumn{1}{c}{F2F} & \\multicolumn{1}{c}{FS} & \\multicolumn{1}{c}{NT} &  & Average \\\\ \\cline{1-6} \\cline{8-8} \nCozzolino \\textit{et al.}~\\cite{cozzolino2017recasting} &  & 75.51\/ - & 86.34\/ - & 76.81\/ - & 75.34\/ - &  & 78.50\/ - \\\\\nBayar and Stamm~\\cite{bayar2016deep} &  & 90.25\/ - & 93.96\/ - & 87.74\/ - & 83.69\/ - &  & 88.91\/ - \\\\\nMesoNet~\\cite{afchar2018mesonet} &  & 89.55\/ - & 88.60\/ - & 81.24\/ - & 92.19\/ - &  & 87.90\/ - \\\\\nXception\\tnote{*} \\cite{chollet2017xception} &  & 96.35\/0.9941 & 96.26\/0.9937 & 96.29\/0.9952 & 92.43\/0.9736 &  & 95.33\/0.9892 \\\\\nSupCon\\tnote{*} \\cite{xu2022supervised} &  & 97.18\/0.9984 & 96.88\/0.9978 & 97.05\/0.9980 & 92.92\/0.9846 &  & 96.01\/0.9947 \\\\ \nAPI-Net(ResNet101)\\tnote{*} \\cite{zhuang2020learning} &  & 88.71\/0.9820 & 90.13\/0.9860 & 87.79\/0.9728 & 82.96\/0.9248 &  & 87.40\/0.9664 \\\\ \\hline\nOurs &  &  &  &  &  &  &  \\\\\n\\textbf{MCX-API(RGB)} &  & \\textbf{98.75}\/0.9996 & \\textbf{99.90}\/0.9986 & \\textbf{98.5}\/\\textbf{0.9993} & \\textbf{96.75}\/0.9896 &  & \\textbf{98.48}\/0.9968 \\\\\n\\textbf{MCX-API(RGB+HSV)} &  & \\textbf{98.75}\/0.9988 & 98.50\/0.9979 & 97.75\/0.9978 & 95.75\/0.9829 &  & 97.69\/0.9943 \\\\\n\\textbf{MCX-API(RGB+CIELab)} &  & 97.00\/0.9996 & 96.50\/0.9985 & 96.25\/0.9989 & 95.25\/0.9909 &  & 96.25\/0.9970 \\\\\n\\textbf{MCX-API(RGB+YCbCr)} &  & 98.00\/\\textbf{0.9998} & 98.25\/\\textbf{0.9991} & 97.75\/\\textbf{0.9993} & \\textbf{96.75}\/0.9920 &  & 97.69\/\\textbf{0.9976} \\\\\n\\textbf{MCX-API(RGB+HSV+CIELab)} &  & 96.50\/0.9990 & 95.50\/0.9888 & 96.00\/0.9835 & 95.50\/\\textbf{0.9933} &  & 95.88\/0.9912 \\\\\n\\textbf{MCX-API(RGB+LAB+YCbCr)} &  & 92.00\/0.9963 & 92.25\/0.9972 & 91.50\/0.9960 & 91.00\/0.9870 &  & 91.69\/0.9941 \\\\\n\\bottomrule\n\\end{tabular}\n    \\begin{tablenotes}\n         \\footnotesize\n         \\item [*] Our implementation of the method.\n    \\end{tablenotes}\n\\end{threeparttable}\n\\end{table*}\n\n\\textbf{Implementation details.} \nWe choose uncompressed videos for our experiments in this work using the Pytorch framework~\\cite{pytorch} to develop the models and the experiments are conducted on Python 3.6 environment on NVIDIA Tesla V100 32Gb in IDUN system owned by NTNU~\\cite{sjalander+:2019epic}. \n\nMulti-task Cascade Convolutional Neural Networks (MTCNN)~\\cite{zhang2016joint} is employed for face detection and face alignment since our experiments are focused on detecting the manipulated face region alone. We allow loose cropping of the face region to capture the entire silhouette against tight cropping. The first 30 frames from each video are extracted, resulting in 150000 total images. We use random cropping in the training phase and center cropping during the testing phase ($512^{2}\\to448^{2}$). In all our experiments, we employ Xception as the backbone where we derive the feature vector $x_{i}\\in \\mathbb{R}^{2048}$ after the global average pooling. We use a batch sampler during the training by randomly sampling three categories in each batch. For each category, we randomly choose nine images due to the limitations of the GPU and memory constraints. We further exercise care to have no sample overlap among all batches, as we exclude the selected sample from the dataset. We locate its most similar image from both its own class and the rest classes for each image by calculating the distance between features by utilizing both Euclidean distance and cosine distance. Each image would get one image as its intra- and inter-pair in the batch, respectively. Each pair is used as input $x_{1}$ and $x_{2}$ as well as generating a mutual vector $x_{m}\\in \\mathbb{R}^{2048}$ through the concatenation and the multilayer perceptron (MLP).\n\\par Based on empirical evaluations, we adopt the coefficient $\\lambda$ in \\Cref{eqn:loss} as 1.0, and 0.05 as the margin value in the score-ranking regularization. We use cosine annealing strategy to alter the learning rate starting from 0.01 \\cite{zhao2021learning}. We train the network with 100 epochs and freeze the parameters in the CNN backbone, and further on train only the classifier in the first eight epochs.\n\n\\textbf{Evaluation Metrics.}\nWe adopt Balanced-Open-Set-Classification (BOSC) accuracy and AUC as evaluation metrics.\n$BOSC = \\frac{Sensitivity+Specificity}{2}$, where $Sensitivity=\\frac{TP}{TP + FN}$ and $Specificity = \\frac{TN}{TN + FP}$.\n\n\\begin{table}[htp]\n\\caption{Comparison of the test results on the FF++ dataset with c23 (high-quality compression) settings. Training for all networks is carried out on FF++ c23. The accuracy and AUC score are at frame-level. The best performances are marked in bold. Data for Xception, $F^3$-Net, and EfficientNet-B4 are adopted from Table 2 in MaDD~\\cite{zhao2021multi}.}\n\\label{tab:ff-sota}\n\\centering\n\\begin{tabular}{llcc}\n\\toprule\nMethod          && ACC            & AUC            \\\\ \\cline{1-1} \\cline{3-4} \nXception        && 95.73          & 0.9909          \\\\ \n$F^3$-Net~\\cite{qian2020thinking}          && 97.52          & 0.9810          \\\\ \nEfficientNet-B4~\\cite{tan2019efficientnet} && 96.63          & 0.9918          \\\\ \nDCL~\\cite{sun2022dual}             && 96.74          & 0.9930          \\\\ \nMaDD~\\cite{zhao2021multi}            && 97.60          & 0.9929          \\\\ \nM2TR~\\cite{wang2022m2tr}            && 97.93          & 0.9951          \\\\ \nAPI-Net            && 87.40          & 0.9664          \\\\ \\hline\nOurs            && \\textbf{98.48} & \\textbf{0.9968} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[htp]\n\\caption{\\textbf{Video-level BOSC Accuracy and AUC for our proposed MCX-API networks and SOTA methods on unseen data.} We compare the results with the SOTA methods on FakeAV\/KoDF\/Celeb-DF respectively. All the networks are trained on the whole FF++ c23 dataset. The data of the SOTA methods are adopted from Table 2 from \\cite{cozzolino2022audio}.}\n\\label{tab:cross-all}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{lllll}\n\\toprule\nFF++ c23 &  & \\multicolumn{3}{c}{Video-level (BOSC(\\%)\/AUC)} \\\\ \\cline{1-1} \\cline{3-5} \nMethod &  & \\multicolumn{1}{c}{FakeAV} & \\multicolumn{1}{c}{KoDF} & \\multicolumn{1}{c}{Celeb-DF} \\\\ \\hline\nXception\\tnote{*} &  & 23.99\/0.450 & 25.97\/0.482 & 31.34\/0.505 \\\\\nSeferbekov~\\cite{seferbekov} &  & \\textbf{95.0\/0.986} & 79.2\/0.884 & 75.3\/0.860 \\\\\nFTCN~\\cite{zheng2021exploring} &  & 64.9\/0.840 & 63.0\/0.765 & - \\\\\nLipForensics~\\cite{haliassos2021lips} &  & 83.3\/0.976 & 56.1\/0.929 & -\/0.820 \\\\\nID-reveal~\\cite{cozzolino2021id} &  & 63.7\/0.876 & 60.3\/0.702 & 71.6\/0.840 \\\\\nPOI~\\cite{cozzolino2022audio} &  & 86.6\/0.941 & 81.1\/0.899 & - \\\\ \nAPI-Net(ResNet101)\\tnote{*} &  & 59.99\/0.72 & 66.92\/0.76 & 58.00\/0.76 \\\\\\hline\nOurs &  &  &  &  \\\\\n\\textbf{MCX-API(RGB)} &  & 74.94\/0.95    & 78.09\/\\underline{0.87}  & 77.88\/0.87 \\\\\n\\textbf{MCX-API(HSV)} &  & 74.63\/0.75 & \\underline{80.64}\/0.85  & 75.67\/0.88 \\\\\n\\textbf{MCX-API(CIELab)} &  & 84.28\/0.90 & \\textbf{81.16\/0.90}  & 64.28\/0.81 \\\\\n\\textbf{MCX-API(RGB+HSV)} &  & 71.58\/0.93    & 78.11\/\\underline{0.87}  & \\underline{80.18}\/0.88 \\\\\n\\textbf{MCX-API(RGB+CIELab)} &  & 83.89\/0.93 & 77.93\/0.83  & 68.34\/\\textbf{0.91} \\\\\n\\textbf{MCX-API(RGB+YCbCr)} &  & 70.41\/0.92   & 78.39\/0.85  & \\textbf{90.87}\/\\underline{0.90} \\\\\n\\textbf{MCX-API(RGB+HSV+CIELab)} &  & \\underline{92.38\/0.98} & 78.91\/0.83 & 59.04\/0.89 \\\\\n\\textbf{MCX-API(RGB+LAB+YCbCr)} &  & 82.93\/0.96  & 76.20\/0.80  &  54.92\/0.85 \\\\\n\\bottomrule\n\\end{tabular}\n    \\begin{tablenotes}\n         \\footnotesize\n         \\item [*] Our implementation of the method.\n    \\end{tablenotes}\n\\end{threeparttable}\n\\end{table*}\n\n\\subsection{Experimental Results}\n\\label{ExperimentalResults}\nWe evaluate the effectiveness of the proposed MCX-API network with both seen and unseen data in this section.  \n\n\\subsubsection{Intra-dataset Evaluation (Closed Set Protocol)}\nWe conduct experiments on six networks with different color spaces on MCX-API whose results are presented in \\cref{tab:bosc-ffpp}. All networks are trained with all four manipulated methods along with pristine in FF++ c23 dataset. We test the frame-level detection performance on the test data of FF++ c23 in a non-overlapping manner regarding the ID. \n\nIn \\cref{tab:bosc-ffpp}, the frame-level test results are listed. We observe that our proposed MCX-API network with RGB inputs reaches the highest average accuracy, 98.48\\%. In addition, this setting also gains the highest accuracy on DF, F2F, and FS with 98.87\\%, 99.90\\% and 98.50\\%, respectively. MCX-API with YCbCr achieves the highest accuracy for NT with 97.00\\%. As RGB provides best performance under  3-channel setting, we combine RGB with HSV, CIELab, and YCbCr, respectively, to create three 6-channel MCX-API networks. From the second block in \\cref{tab:bosc-ffpp}, we can see that RGB+YCbCr obtains the highest average AUC score of 0.9976 and the best performance on DF, F2F, and FS regarding AUC score. This indicates better prediction output scores using MCX-API with the combination of RBG and YCbCr color spaces. The 9-channel MCX-API network with RGB, HSV, and CIELab further gains the highest 0.9933 AUC score for NT.\n\nThe results of the comparison with the SOTA methods are reported in \\cref{tab:ff-sota}. All networks are trained on FF++ c23 (high-quality compression). The accuracy and AUC scores are measured at frame level. The results are averaged on all the test sets from FF++ c23, including pristine and all four kinds of manipulated videos. Our proposed method MCX-API with RGB color space obtains the best performance compared to SOTA methods. The best accuracy of the BOSC is 98.48\\%, and the highest AUC score is 0.9968. The result shows that our idea of pairwise learning in a fine-grained manner could work well in inter-class (closed-set) setting of Deepfake detection problem.\n\n\\subsubsection{Cross-dataset Evaluation}\nWe conduct a comparison on cross-dataset validation with SOTA methods to validate the proposed approach.  We employ FakeAV, KoDF, and Celeb-DF to test the generalizability of our MCX-API network. Training for all networks are carried out on the FF++ c23 dataset and tested on FakeAV, KoDF, and Celeb-DF. We note that MCX-API with CIELab color space gets the best scores for KoDF with an accuracy of 81.86\\% and an AUC score of 0.90 as presented in \\cref{tab:cross-all}. MCX-API with RGB+YCbCr wins in the cross-dataset validation for Celeb-DF with an accuracy of 90.87\\% and the second best AUC score 0.90. MCX-API with color space RGB+HSV+CIELab achieves the second best place for FakeAV with 92.38\\% accuracy and 0.98 AUC score. In general, our proposed network gets a relatively better performance than the SOTA methods which indicates the better generalizability of the proposed MCX-API network.\n\n\\section{Explainable Analysis of MCX-API}\n\\label{sec:ExplainableAnalysisofMCX-API}\nWe further analyze the network to understand the performance gain by analyzing embeddings using t-SNE plots~\\cite{van2008visualizing} and class activation maps~\\cite{selvaraju2017grad, chattopadhay2018grad, draelos2020use, jiang2021layercam, fu2020axiom}. While the t-SNE provides topology explanations of the learned features, the activation maps allow for a better visualization of what has been learned by our network. \n\n\\begin{figure*}[ht]\\centering\n\t\\resizebox{2.\\columnwidth}{!}{\n\t\t\\begin{tabular}{cc}\n\t\t\t\\begin{tikzpicture}[spy using outlines={rectangle,yellow,magnification=3,size=4.0cm, connect spies, every spy on node\/.append style={very thick}}]\n\t\t\t\t\\node {\\includegraphics[width=9cm, height=9cm]{figures\/tsne_MCXapi.png}};\n\t\t\t\t\\spy[red] on (-0.3,0.2) in node [right] at (-5.5,6.3);\n\t\t\t\t\\spy[green,size=2.7cm] on (0.1,-1.2) in node [right] at (-6.9,-2.2);\n\t\t\t\t\\spy[blue,size=3.cm] on (0.5,2.1) in node [right] at (1.,6.);\n                    \\spy[orange,size=2.7cm] on (-2.5,1.0) in node [left] at (-4.2,1);\n\t\t\t\\end{tikzpicture} & \n\n\t\t\t\\begin{tikzpicture}[spy using outlines={rectangle,yellow,magnification=3,size=4cm, connect spies, every spy on node\/.append style={very thick}}]\n\t\t\t\t\\node {\\includegraphics[width=9cm, height=9cm]{figures\/tsne_api.png}};\n\t\t\t\t\\spy[red] on (-0.3,0.2) in node [right] at (-5.5,6.3);\n\t\t\t\t\\spy[green,size=2.7cm] on (0.1,-1.2) in node [right] at (-6.9,-2.2);\n\t\t\t\t\\spy[blue,size=3.cm] on (0.5,2.1) in node [right] at (1.,6.);\n                    \\spy[orange,size=2.7cm] on (-2.,1.0) in node [left] at (-4.2,1);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\large{MCX-API (RGB)} & \\large{API} \n\t\t\\end{tabular}\n\t}\n\t\\caption{Data visualizations in 2D by t-SNE for MCX-API(RGB) and API. The left plot is t-SNE for our proposed MCX-API. The right plot is t-SNE for base architecture API-Net. We blow up the intersection parts and outliers for a clear view.}\n\\label{fig:tsne}\n\\end{figure*}\n\n\\begin{figure*}[ht]\\centering\n\t\\resizebox{1.7\\columnwidth}{!}{\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_frame181.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n\t\t\t\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_mcxapi_layercam.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n   \n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_api_layercam.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\huge{DF} &\\huge{MCX-API (RGB)} & \\huge{API} \\\\\n\n   \t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_frame181.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n\t\t\t\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_mcxapi_layercam.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n   \n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_api_layercam.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\huge{F2F} &\\huge{MCX-API (RGB)} & \\huge{API} \n   \n\t\t\\end{tabular}\n\t}\n\t\\caption{Blow up in activation maps from LayerCAM analysis of MCX-API(RGB) and base architecture API-Net on DF and F2F faces.}\n\t\\label{fig:blowup-activation}\n\\end{figure*}\n\n\\begin{figure}[htp]\n    \\centering\n    \\subfigure[Visualization of the last block of the exit flow of MCX-API (RGB).]{\\label{fig:visualization-MCXapi}\\includegraphics[width=0.95\\linewidth]{figures\/visualization_MCXapi.pdf}}\n    \\subfigure[Visualization of the last block of the API-Net.]{\\label{fig:visualization-api}\\includegraphics[width=0.95\\linewidth]{figures\/visualization_api.pdf}}\n    \n    \\caption{Visualization of the last layer of MCX-API (RGB) and API networks. We utilize Grad-CAM~\\cite{selvaraju2017grad}, Grad-CAM++~\\cite{chattopadhay2018grad}, HiResCAM~\\cite{draelos2020use}, LayberCAM~\\cite{jiang2021layercam} and XGradCAM~\\cite{fu2020axiom} as our visualization tool. For larger figure, please refer to \\cref{fig:activation-map-large}.}\n    \\label{fig:activation-map}\n\\end{figure}\n\n\\subsection{Data Visualizations With t-SNE}\nThe results of a t-SNE 2D map for the feature vectors are illustrated in \\cref{fig:tsne}. We compare the t-SNE of our MCX-API and base architecture API-Net.\nWe notice that the five classes of Real, DF, F2F, FS, and NT for MCX-API are well separated with five different clusters as against the base architecture of API-Net. There is an unclear boundary between Real and NT, shown in the blue box for MCX-API. This overlapping can be the reason for the relatively lower accuracy obtained on NT. There are small areas overlapping between DF\/NT(yellow\/purple) and Real\/F2F(red\/blue). We further notice a few samples of Real (red dots) distributed in each fake class, leading to the errors of our proposed network.\n\n\\subsection{Visualizing Decisions With Attention Maps}\nWe apply different class activation visualization methods on the last layer of proposed network to analyze MCX-API shown in \\cref{fig:activation-map}. For comparison, we also show the visualization of the base API-Net. Precisely, we adopt Grad-CAM~\\cite{selvaraju2017grad}, Grad-CAM++~\\cite{chattopadhay2018grad}, HiResCAM~\\cite{draelos2020use}, LayberCAM~\\cite{jiang2021layercam} and XGradCAM~\\cite{fu2020axiom}. The visualization results are provided in \\cref{fig:visualization-MCXapi} for our proposed MCX-API and in \\cref{fig:visualization-api} for API-Net.\n\n\nThe activation map for Output Real is on the left part with a green background, and the activation map for Output Fake is on the right part with a pink background. The rows from top to bottom are the visualization for five classes of Real, DF, F2F, FS, and NT, respectively. We can observe that real images gains more attention within Output Real(left part) than Output Fake(right part). In contrast, fake images obtain more attention within Output Fake than Output Real. This explains the ability of our network to detect Deepfakes.\n\nWe further blow up the activation maps from LayerCAM for DF and F2F images in \\cref{fig:blowup-activation}. From visual analysis, it is evident that the MCX-API focuses more on the facial region, such as the eyes and the mouth. For instance, double eyebrows are found in the DF image (blue circle). MCX-API pays more attention than API around this region. \n\n\\section{Limitations of our work}\nWe notice in \\cref{tab:bosc-ffpp} that with the increase in color spaces, there are no apparent improvements in BOSC accuracy. We assume that there is redundant information among channels, and further work would be focused on finding the most helpful color information to extend our proposed approach. We also observe that no single configuration could perform reasonably well for all the unseen data, which is the biggest issue for Deepfake detection field. Introducing other information, such as temporal data and audio, would be a good idea as more inconsistency could be found by extending our work to video based approach.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThere is an imperative need for a generalized DeepFakes detection method to deal with the newer manipulation methods in visual media. In this paper, we proposed to apply the Multi-Channel Xception Attentive Pairwise Interaction (MCX-API) network to the Deepfakes detection field in a fine-grained manner. We conducted experiments on the publicly available FaceForensics++ dataset, and our approach obtained better performance than the SOTA approaches on both seen and unseen manipulation types. We obtain 98.48\\% BOSC accuracy on the FF++ dataset and 90.87\\% BOSC accuracy on the CelebDF dataset suggesting a promising direction for the generalization of DeepFake detection. Comprehensive ablation studies have been conducted to understand our algorithm better. We further explain the performance of our network by using t-SNE and attention maps. The results showed that Deepfake had been well separated from real videos. While our approach has indicated a promising solution to obtain a generalized detection mechanism, we have listed certain limitations that can pave the way for future work.  \n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\n\n\\subsection{The aim of this article}\n\n\\subsubsection{}\\label{history}\n\nLet $A$ be an abelian variety defined over a number field $k$.\n\nThen, by a celebrated theorem of Mordell and Weil, the abelian group that is formed by the set $A(k)$ of points of $A$ with coefficients in $k$ is finitely generated.\n\nIt is also conjectured that the Hasse-Weil $L$-series $L(A,z)$ for $A$ over $k$ has a meromorphic continuation to the entire complex plane and satisfies a functional equation with central point $z=1$ and, in addition, that the Tate-Shafarevich group $\\sha(A_k)$ of $A$ over $k$ is finite.\n\nAssuming these conjectures to be true, the Birch and Swinnerton-Dyer Conjecture concerns the leading coefficient $L^\\ast(A,1)$ in the Taylor expansion of $L(A,z)$ at $z=1$. This conjecture was originally formulated for elliptic curves in a series of articles of Birch and Swinnerton-Dyer in the 1960's (see, for example, \\cite{birch}) and then reinterpreted and extended to the setting of abelian varieties by Tate in \\cite{tate} to give the following prediction.\n\\medskip\n\n\\noindent{}{\\bf Conjecture} (Birch and Swinnerton-Dyer)\n\\begin{itemize}\n\\item[(i)] The order of vanishing of $L(A,z)$ at $z=1$ is equal to the rank of $A(k)$.\n\\item[(ii)] One has\n\\begin{equation}\\label{bsd equality} L^\\ast(A,1) = \\frac{\\Omega_A\\cdot R_A\\cdot {\\rm Tam}(A)}{\\sqrt{D_k}^{{\\rm dim}(A)}\\cdot |A(k)_{\\rm tor}|\\cdot |A^t(k)_{\\rm tor}|}\\cdot |\\sha(A_k)|.\\end{equation}\n\\end{itemize}\n\\medskip\n\nHere $\\Omega_A$ is the canonical period of $A$, $R_A$ the discriminant of the canonical N\\'eron-Tate height pairing on $A$, ${\\rm Tam}(A)$ the product over the (finitely many) places of $k$ at which $A$ has bad reduction of local `Tamagawa numbers', $D_k$ the absolute value of the discriminant of $k$,  $A(k)_{\\rm tor}$ the torsion subgroup of $A(k)$ and $A^t$ the dual abelian variety of $A$.\n\nIt should be noted at the outset that, even if one assumes $L(A,z)$ can be meromorphically continued to $z=1$ and $\\sha(A_k)$ is finite, so that both sides of (\\ref{bsd equality}) make sense, the predicted equality is itself quite remarkable.\n\nFor instance, the $L$-series is defined via an Euler product over places of $k$ so that its leading coefficient at $z=1$ is intrinsically local and analytic in nature whilst the most important terms on the right hand side of (\\ref{bsd equality}) are both global and algebraic in nature.\n\nIn addition, and more concretely, whilst isogenous abelian varieties give rise to the same $L$-series, the individual terms that occur in the `Euler characteristic' on the right hand side of  (\\ref{bsd equality}) are not themselves isogeny invariant and it  requires a difficult theorem of Tate to show that the validity of (\\ref{bsd equality}) is invariant under isogeny.\n\nFor these, and many other, reasons, the above conjecture, which in the remainder of the introduction we abbreviate to ${\\rm BSD}(A_k)$, has come to be regarded as one of the most important problems in arithmetic geometry today.\n\nNevertheless, there are various natural contexts in which it seems likely that ${\\rm BSD}(A_k)$ does not encompass the full extent of the interplay between the analytic and algebraic invariants of $A$. Moreover, a good understanding of the finer connections that can arise could lead to much greater insight into concrete questions such as, for example, the growth of ranks of Mordell-Weil groups in extensions of number fields.\n\nFor instance, if $A$ has a large endomorphism ring, then it seems reasonable to expect there to be a version of ${\\rm BSD}(A_k)$ that reflects the existence of such endomorphisms.\n\nThe earliest example of such a refinement appears to be Gross's formulation in \\cite{G-BSD} of an equivariant\nBirch and Swinnerton-Dyer conjecture for elliptic curves $A$ with complex multiplication by the maximal order $\\mathcal{O}$ of an imaginary quadratic field.\n\nThis conjecture incorporates natural refinements of both ${\\rm BSD}(A_k)$(i) and ${\\rm BSD}(A_k)$(ii) and is supported by numerical evidence obtained by Gross and Buhler in \\cite{GrossBuhler} and by theoretical evidence obtained by Rubin in \\cite{rubin2}.\n\nIn a different direction one can study the leading coefficients of the Hasse-Weil-Artin $L$-series $L(A,\\psi,z)$ that are obtained from $A$ and finite dimensional complex characters $\\psi$ of the absolute Galois group $G_k$ of $k$.\n\nIn this setting, general considerations led Deligne and Gross to the expectation that for any finite dimensional character $\\chi$ of $G_k$  over a number field $E$ the order of vanishing ${\\rm ord}_{z=1}L(A,\\sigma\\circ\\chi,z)$ at $z=1$ of $L(A,\\sigma\\circ\\chi,z)$ should be independent of the choice of an embedding $\\sigma: E \\to \\CC$. This prediction in turn led them naturally to the conjecture that for each complex character $\\psi$ one should have\n\\begin{equation}\\label{dg equality} {\\rm ord}_{z=1}L(A,\\psi,z) = {\\rm dim}_\\CC\\bigl( \\Hom_{\\CC[\\Gal(F\/k)]}(V_\\psi,\\CC\\otimes_\\ZZ A^t(F))\\bigr)\\end{equation}\n(cf. \\cite[p.127]{rohrlich}). Here $F$ is any finite Galois extension of $k$ such that $\\psi$ factors through the projection $G_k \\to \\Gal(F\/k)$ and $V_\\psi$ is any  $\\CC[\\Gal(F\/k)]$-module of character $\\psi$ (see also Conjecture \\ref{conj:ebsd}(ii) below).\n\nThis prediction is a natural generalization of ${\\rm BSD}(A_k)$(i) and is known to have important, and explicit, consequences for ${\\rm ord}_{z=1}L(A,\\psi,z)$ (see, for example, the recent article \\cite{bisatt} of Bisatt and Dokchitser).\n\nIn addition, for rational elliptic curves $A$ and characters $\\psi$ for which $L(A,\\psi,1)$ does not vanish, there is by now strong evidence for the conjecture of Deligne and Gross.\n\nSuch evidence has been obtained by Bertolini and Darmon \\cite{BD} in the setting of ring-class characters of imaginary quadratic fields, by Kato \\cite{kato} in the setting of linear characters of $\\QQ$ (in this regard see also Rubin \\cite[\\S8]{rubin}), by Bertolini, Darmon and Rotger \\cite{bdr} for odd, irreducible two-dimensional\nArtin representations of $\\QQ$ and by Darmon and Rotger \\cite{dr} for certain self-dual Artin representations of $\\QQ$ of dimension at most four.\n (We also recall in this context that, in the setting of \\cite{bdr}, recent work of Kings, Loeffler and Zerbes \\cite{klz} proves the finiteness of components of the $p$-primary part of the Tate-Shafarevich group of $A$ over $F$ for a large set of primes $p$.)\n\nWrite $\\mathcal{O}_\\psi$ for the ring of integers of the number field generated by the values of $\\psi$. Then, as a refinement of the conjectural equality (\\ref{dg equality}), and a natural analogue of ${\\rm BSD}(A_k)$(ii) relative to $\\psi$, it would be of interest to understand a precise conjectural formula in terms of suitable `$\\psi$-components' of the standard algebraic invariants of $A$ for the fractional $\\mathcal{O}_\\psi$-ideal that is generated by the product of the leading coefficient of $L(A,\\psi,z)$ at $z=1$ and an appropriate combination of `$\\psi$-isotypic' periods and regulators.\n\nSuch a formula might also reasonably be expected to lead to concrete predictions concerning the behaviour of natural arithmetic invariants attached to the abelian variety.\n\nFor example, in the recent article of Dokchitser, Evans and Wiersema \\cite{vdrehw}, inexplicit versions of such a formula have been shown to lead, under suitable hypotheses, to predictions concerning the non-triviality of Tate-Shafarevich groups and the existence of points of infinite order on $A$ over extension fields of $k$.\n\nHowever, the formulation of an explicit such conjecture has hitherto been straightforward only if one avoids the $p$-primary support of such fractional ideals for primes $p$ that divide the degree of the extension of $k$ that corresponds to the kernel of $\\psi$.\n\nIn addition, such a conjectural formula would not itself take account of any connections that might exist between the leading coefficients of $L(A,\\psi,z)$ for characters $\\psi$ that are not in the same orbit under the action of $G_\\QQ$.\n\nIn this direction, Mazur and Tate \\cite{mt} have in the special case that $k =\\QQ$, $A$ is an elliptic curve and $\\psi$ is linear predicted an explicit family of such congruence relations that refine ${\\rm BSD}(A_k)$(ii).\n\nThese congruences rely heavily on an explicit formula in terms of modular symbols for the values $L(A,\\psi,1)$ for certain classes of tamely ramified Dirichlet characters $\\psi$ that Mazur had obtained in \\cite{mazur79}. They are expressed in terms of the discriminants of integral group-ring valued pairings constructed by using the geometrical theory of bi-extensions and are closely linked to earlier work of Mazur, Tate and Teitelbaum in \\cite{mtt} regarding the formulation of $p$-adic analogues of ${\\rm BSD}(A_k)$(ii).\n\nThe conjecture of Mazur and Tate has in turn motivated much subsequent work and the formulation of several new conjectures involving the values $L(A,\\psi,1)$.\n\nSuch conjectures include the congruence relations that are formulated by Bertolini and Darmon in \\cite{bert} and \\cite{bert2} and involve a natural notion of `derived height pairings' and links to the Galois structure of Selmer modules that are predicted by Kurihara in \\cite{kuri}.\n\nIt has, however, proved to be much more difficult to formulate explicit refinements of ${\\rm BSD}(A_k)$(ii) that involve congruence relations between the values of derivatives of Hasse-Weil-Artin $L$-series.\n\nIn this direction, Darmon \\cite{darmon} has used the theory of Heegner points to formulate an analogue of the Mazur-Tate congruence conjecture for the first derivatives of Hasse-Weil-Artin $L$-series that arise from rational elliptic curves and ring class characters of imaginary quadratic fields.\n\nHowever, aside from this example, the only other such explicit study that we are aware of is due to Kisilevsky and Fearnley who in \\cite{kisilevsky} and \\cite{kisilevsky2} formulated, and studied numerically, conjectures for the `algebraic parts' of the leading coefficients of Hasse-Weil-Artin $L$-series that arise from rational elliptic curves and certain restricted families of Dirichlet characters.\n\n\\subsubsection{}In a more general setting, the formulation by Bloch and Kato \\cite{bk} of the `Tamagawa number conjecture' for the motive $h^1(A_k)(1)$ offers a different approach to the formulation of ${\\rm BSD}(A_k)$.\n\nIn particular, the subsequent re-working of this conjecture by Fontaine and Perrin-Riou in \\cite{fpr}, and its `equivariant' extension to motives with coefficients, as described by Flach and the first author in \\cite{bufl99}, in principle provides a systematic means of studying refined versions of ${\\rm BSD}(A_k)$.\n\nIn this setting it is known, for example, that the conjectures formulated by Gross in \\cite{G-BSD} are equivalent to the equivariant Tamagawa number conjecture for the motive $h^1(A_k)(1)$ with respect to the coefficient ring $\\mathcal{O}$ (cf. \\cite[\\S4.3, Rem. 10]{bufl99}).\n\nTo study Hasse-Weil-Artin $L$-series it is convenient to fix a finite Galois extension $F$ of $k$ of group $G$.\n\nThen the equivariant Tamagawa number conjecture for $h^1(A_{F})(1)$ with respect to the integral group ring $\\ZZ[G]$ is formulated as an equality of the form\n\\begin{equation}\\label{etnc eq} \\delta(L^\\ast(A_{F\/k},1)) = \\chi(h^1(A_{F})(1),\\ZZ[G]).\\end{equation}\n\\noindent{}Here $\\delta$ is a canonical homomorphism from the unit group $\\zeta(\\br [G])^\\times$ of the centre of $\\RR[G]$ to the relative algebraic $K$-group of the ring extension $\\bz [G] \\subseteq \\br\n[G]$ and $L^\\ast(A_{F\/k},1)$ is an element of $\\zeta(\\RR[G])^\\times$ that is defined using the leading coefficients $L^\\ast(A,\\psi,1)$ for each irreducible complex character $\\psi$ of $G$. Also, $\\chi(h^1(A_{F})(1),\\ZZ[G])$ is an adelic Euler characteristic that is constructed by\ncombining virtual objects (in the sense of Deligne) for each prime $p$ of the compactly supported \\'etale cohomology of the $p$-adic Tate modules of $A$ together with the Neron-Tate height pairing and period isomorphisms for $A$ over $F$, as well as an analysis of the finite support cohomology groups introduced by Bloch and Kato.\n\nThe equality (\\ref{etnc eq}) is known to constitute a strong and simultaneous refinement of the conjectures ${\\rm BSD}(A_L)$ as $L$ ranges over the  intermediate fields of $F\/k$.\n\nHowever, the rather technical, and inexplicit, nature of this equality means that it has proved to be very difficult to interpret in a concrete way, let alone to verify either theoretically or numerically.\n\nFor example, in the technically most demanding case in which $A$ has strictly positive rank over $F$ it has still only been verified numerically in a small number of cases by Navilarekallu in \\cite{tejaswi}, by Bley in \\cite{Bley1} and \\cite{Bley2}, by Bley and the second author in \\cite{bleymc} and by Wuthrich and the present authors in \\cite{bmw}.\n\nIn addition, the only theoretical evidence for the conjecture in this setting is its verification for certain restricted dihedral families of the form $F\/\\QQ$ where $F$ is an unramified abelian extension of an imaginary quadratic field (this is the main result of \\cite{bmw} and relies on the theorem of Gross and Zagier). In particular, the restriction on ramification that the latter result imposes on $F\/k$ means that many of the more subtle aspects of the conjecture are avoided.\n\nTo proceed we note that the conjectural equality (\\ref{etnc eq}) naturally decomposes into `components', one for each rational prime $p$, in a way that will be made precise in Appendix \\ref{consistency section}, and that each such $p$-component (which for convenience we refer to as `(\\ref{etnc eq})$_p$' in the remainder of this introduction) is itself of some interest.\n\nFor example, if $A$ has good ordinary reduction at $p$, then the compatibility result proved by Venjakob and the first named author in~\\cite[Th. 8.4]{BV2} shows, modulo the assumed non-degeneracy of classical $p$-adic height pairings, that the equality (\\ref{etnc eq})$_p$ is a consequence of the main conjecture of\nnon-commutative Iwasawa theory for $A$, as formulated by Coates et al in \\cite{cfksv} with respect to any compact $p$-adic Lie extension of $k$ that contains the cyclotomic $\\ZZ_p$-extension of $F$.\n\nThis means that the study of (\\ref{etnc eq}), and of its more explicit consequences, is directly relevant to attempts to properly understand the content of the main conjecture of non-commutative Iwasawa theory. It also shows that the $p$-adic congruence relations that are proved numerically by Dokchitser and Dokchitser in \\cite{dokchitsers} are related to the equality (\\ref{etnc eq}).\n\nTo study congruences in a more general setting we fix an embedding of $\\RR$ into the completion $\\CC_p$ of the algebraic closure of $\\QQ_p$. Then the long exact sequence of relative $K$-theory implies that the equality (\\ref{etnc eq})$_p$ determines the image of $L^\\ast(A_{F\/k},1)$ in $\\zeta(\\CC_p[G])^\\times$ modulo the image under the natural reduced norm map of the Whitehead group $K_1(\\ZZ_p[G])$ of $\\ZZ_p[G]$.\n\nIn view of the explicit description of the latter image that is obtained by Kakde in \\cite{kakde} or, equivalently, by the methods of Ritter and Weiss in \\cite{rw}, this means that (\\ref{etnc eq}) is essentially equivalent to an adelic family of (albeit inexplicit) congruence relations between the leading coefficients $L^\\ast(A,\\psi, 1)$, suitably normalised by a product of explicit equivariant regulators and periods, as $\\psi$ varies over the set of irreducible complex characters of $G$.\n\nThis is also the reason why the study of congruence relations between suitably normalised derivatives of Hasse-Weil-Artin $L$-series should be related to the construction of `$p$-adic $L$-functions' in the setting of non-commutative Iwasawa theory.\n\n\\subsubsection{}The main aim of the present article is then to develop general techniques that will allow one to understand the above congruence relations in a more explicit way, and in a much wider setting, than has previously been possible.\n\nIn this way we are led to the formulation (in Conjecture \\ref{conj:ebsd}) of a seemingly definitive refinement of the Birch and Swinnerton-Dyer formula (\\ref{bsd equality}) in the setting of Hasse-Weil-Artin $L$-series. We then derive a range of concrete consequences of this conjecture that are amenable to explicit investigation, either theoretically or numerically, in cases that (for the first time) involve a thoroughgoing mixture of difficult archimedean considerations that are related to refinements of the conjectural equality (\\ref{dg equality}) of Deligne and Gross, and of delicate $p$-adic congruence relations that are related to aspects of non-commutative Iwasawa theory.\n\nIn particular, we shall show that this family of predictions both refines and extends the explicit refinements of (\\ref{dg equality}) that were recalled in \\S\\ref{history}. It also gives insight into the more subtle aspects of the conjectural equality (\\ref{etnc eq}), and hence (via the results of \\cite{BV2}) of the main conjecture of non-commutative Iwasawa theory, that go well beyond the sort of concrete congruence conjectures that have been considered previously in connection to the central conjectures of either \\cite{bufl99} or \\cite{cfksv}.\n\nWe also believe that some of the general results obtained here can help contribute towards establishing a proper framework for the subsequent investigation of these important questions.\n\n\n\n\\subsection{The main contents}\n\n\\subsubsection{}\n\n\n\n\n\n\n\nAs a key part of our approach, we shall first associate two natural notions of Selmer complex to the $p$-adic Tate module of an abelian variety.\n\n The `classical Selmer complex' that we define in \\S\\ref{selmer section} is closely related to the `finite support cohomology' that was introduced by Bloch and Kato in \\cite{bk} and, as a result, its cohomology can be explicitly described in terms of Mordell-Weil groups and Selmer groups.\n\n Nevertheless, this complex is not well-suited to certain $K$-theoretical calculations since it is not always `perfect' over the relevant $p$-adic group ring.\n\n For this reason we shall in \\S\\ref{selmer section} also associate a notion of `Nekov\\'a\\v r-Selmer complex' to certain choices of $p$-adic submodules of the groups of semi-local points.\n\n This construction is motivated by the general approach of Nekov\\'a\\v r in \\cite{nek} and gives a complex that is always perfect and has cohomology that can be described in terms of the Selmer modules studied by Mazur and Rubin in \\cite{MRkoly}.\n\n Such Nekov\\'a\\v r-Selmer complexes will then play an important role in several subsequent $K$-theoretical computations.\n\nIn \\S\\ref{selmer section} we also explain how a suitably compatible family over all primes $p$ of $p$-adic modules of semi-local points, or a `perfect Selmer structure' for $A$ and $F\/k$ as we shall refer to it, gives rise to a canonical perfect complex of $G$-modules.\n\nWe shall then show that such structures naturally arise from a choice of global differentials and compute the cohomology groups of the associated Selmer complexes.\n\nIn \\S\\ref{ref bsd section} we formulate the Birch and Swinnerton-Dyer Conjecture for the variety $A$ and Galois extension $F$ of $k$, or `${\\rm BSD}(A_{F\/k})$' as we shall abbreviate it.\n\nUnder the assumed validity of an appropriate case of the Generalized Riemann Hypothesis, we shall first associate a $K_1$-valued leading coefficient element to the data $A$ and $F\/k$.\n\nAfter fixing a suitable choice of global differentials we can also associate a $K_1$-valued `period' element to $A$ and $F\/k$.\n\nThe central conjecture of this article then asserts that the image under the natural connecting homomorphism of the quotient of these $K_1$-valued invariants is equal to the Euler characteristic in a relative $K$-group of a pair comprising a Nekov\\'a\\v r-Selmer complex constructed from the given set of differentials and the classical N\\'eron-Tate height pairing for $A$ over $F$.\n\nThis conjectural equality also involves a small number of `Fontaine-Messing' correction terms that we use to\ncompensate for the choice of a finite set of places of $k$ that is necessary to state ${\\rm BSD}(A_{F\/k})$.\n\n\nIt can be directly shown that this conjecture recovers ${\\rm BSD}(A_{k})$ in the case that $F=k$, is consistent in several key respects and has good functorial properties under change of Galois extension $F\/k$.\n\nThe conjecture can also be interpreted as a natural analogue of the `refined Birch and Swinnerton-Dyer Conjecture' for abelian varieties over global function fields that was recently formulated, and in some important cases proved, by Kakde, Kim and the first author in \\cite{bkk}.\n\nIn \\S\\ref{k theory period sect} and \\S\\ref{local points section} we shall then concentrate on proving several technical results that will subsequently help us to derive explicit consequences from the assumed validity of ${\\rm BSD}(A_{F\/k})$.\n\nIn \\S\\ref{k theory period sect} these results include establishing the precise link between $K_1$-valued periods, classical periods, Galois resolvents and suitably modified Galois-Gauss sums.\n\n\nIn \\S\\ref{local points section} we shall prepare for subsequent $K$-theoretical computations with classical Selmer complexes by studying the cohomological-triviality of local points on ordinary abelian varieties.\n\nIn particular, we shall use these results to define a natural $K$-theoretical invariant of the twist matrix of such a variety.\n\n We shall then give a partial computation of these invariants and also explain how, in the case of elliptic curves, the (assumed) compatibility under unramified twist of suitable cases of the local epsilon constant conjecture (as formulated in its most general form by Fukaya and Kato in \\cite{fukaya-kato}) leads to an explicit description of the invariants.\n\nIn \\S\\ref{tmc} we shall impose several mild hypotheses on both the reduction types of $A$ and the ramification invariants of $F\/k$ that together ensure that the classical Selmer complex defined in \\S\\ref{selmer section} is perfect over the relevant $p$-adic group ring.\n\nWorking under these hypotheses, we shall then combine the results of \\S\\ref{k theory period sect} and \\S\\ref{local points section} together with a significant strengthening of the main computations that are made by Wuthrich and the present authors in \\cite{bmw} to derive a more explicit interpretation of ${\\rm BSD}(A_{F\/k})$.\n\nThese results are in many respects the technical heart of this article and rely heavily on the subtle, and still for the most part conjectural, arithmetic properties of wildly ramified Galois-Gauss sums.\n\nThe $K$-theoretical computations in \\S\\ref{tmc} also constitute a natural equivariant refinement and generalisation of several earlier computations in this area including those that are made by Venjakob in~\\cite[\\S3.1]{venjakob}, by the first author in \\cite{ltav}, by Bley in~\\cite{Bley1}, by Kings in~\\cite[Lecture 3]{kings} and by the second author in \\cite{dmc}.\n\nIn \\S\\ref{ecgs} we shall discuss concrete consequences of ${\\rm BSD}(A_{F\/k})$ concerning both the explicit Galois structure of Selmer complexes and modules and the formulation of precise refinements of the Deligne-Gross Conjecture.\n\nIn particular, in this section we shall address a problem explicitly raised by Dokchitser, Evans and Wiersema in \\cite{vdrehw} (see, in particular Remark \\ref{evans}).\n\nIn \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection} we shall then specialise to consider abelian extensions $F\/k$ and combine our approach with general techniques recently developed by Sano, Tsoi and the first author in \\cite{bst} in order to derive from ${\\rm BSD}(A_{F\/k})$ several explicit congruence relations between the suitably normalized derivatives of Hasse-Weil-Artin $L$-series.\n\nIn \\S\\ref{comparison section} we then prove that the pairing constructed by Mazur and Tate in \\cite{mt} using the theory of bi-extensions coincides with the inverse of a canonical `Nekov\\'a\\v r height pairing' that we define by using Bockstein homomorphisms arising naturally from Galois descent considerations.\n\nThis comparison result relies, in part, on earlier results of Bertolini and Darmon \\cite{bert2} and of Tan \\cite{kst} and is, we believe, of some independent interest.\n\nThe relations that are discussed in \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection} often take a very explicit form (see, for example, the discussion in \\S\\ref{explicit examples intro} below) and, when combined with the results of \\S\\ref{comparison section}, can be seen to extend and refine the earlier conjectures of Mazur and Tate \\cite{mt} and Darmon \\cite{darmon0} amongst others.\n\nThis approach also shows that for certain cyclic and dihedral extensions $F\/k$ the key formula that is predicted by ${\\rm BSD}(A_{F\/k})$  is equivalent to the validity of a family of explicit congruence relations that simultaneously involve both the N\\'eron-Tate and Mazur-Tate height pairings.\n\nIn this way we shall for the first time render refined versions of the Birch and Swinnerton-Dyer Conjecture accessible to numerical verification in cases in which they involve an intricate mixture of both archimedean phenomenon and delicate $p$-adic congruences.\n\nIn particular, in \\S\\ref{mrsconjecturesection} we give details of several such numerical verifications of the `$p$-component' of ${\\rm BSD}(A_{F\/k})$ for primes $p$ that divide the degree of $F\/k$ that Werner Bley has been able to perform by using this approach (see, in particular, Remark \\ref{bleyexamples rem} and Examples \\ref{bleyexamples}).\n\nIn \\S\\ref{mod sect} and \\S\\ref{HHP} we shall then specialise to consider applications of our general approach in two classical settings.\n\nFirstly, in \\S\\ref{mod sect} we consider rational elliptic curves over fields that are both abelian and tamely ramified over $\\QQ$. In this case\nwe can use the theory of modular symbols to give an explicit reinterpretation of ${\\rm BSD}(A_{F\/\\QQ})$ and thereby describe precise conditions under which the conjecture is valid.\n\nAs a concrete application of this result we then use it to deduce from Kato's theorem \\cite{kato} that for every natural number $n$ there are infinitely many primes $p$ and, for each such $p$, infinitely many abelian extensions $F\/\\QQ$ for which the $p$-component of ${\\rm BSD}(A_{F\/\\QQ})$ is valid whilst the degree and discriminant of $F\/\\QQ$ are each divisible by at least $n$ distinct primes and the Sylow $p$-subgroup of $\\Gal(F\/\\QQ)$ has exponent at least $p^n$ and rank at least $n$. This result is a considerable strengthening of the main result of Bley in \\cite{Bley3}.\n\nThen in \\S\\ref{HHP} we consider abelian extensions of imaginary quadratic fields and elliptic curves that satisfy the Heegner hypothesis.\n\nThe main result of this section is a significant extension of the main result of Wuthrich and the present authors in \\cite{bmw} and relies on Zhang's generalization of the theorem of Gross and Zagier relating first derivatives of Hasse-Weil-Artin $L$-series to the heights of Heegner points.\n\nIn this section we shall also point out an inconsistency in the formulation of a conjecture of Bradshaw and Stein in \\cite{BS} regarding Zhang's formula and offer a possible correction.\n\n\nThe article also contains two appendices. In Appendix \\ref{consistency section} we use techniques developed by Wuthrich and the present authors in \\cite{bmw} to explain the precise link between our central conjecture ${\\rm BSD}(A_{F\/k})$ and the conjectural equality (\\ref{etnc eq}).\n\nThis technical result may perhaps be of interest in its own right but also allows us to deduce from the general theory of equivariant Tamagawa numbers that our formulation of ${\\rm BSD}(A_{F\/k})$ is consistent in several key respects.\n\nFinally, in Appendix \\ref{exp rep section} we make explicit certain standard constructions in homological algebra relating to Poitou-Tate duality and also describe a general construction of algebraic height pairings from Bockstein homomorphisms.\n\nThe results of Appendix \\ref{exp rep section} are for the most part routine but nevertheless play an important role in the arguments that we use to compare height pairings in \\S\\ref{comparison section}.\n\n\\subsubsection{}\\label{explicit examples intro}To end the introduction we shall give some concrete examples of the sort of congruence predictions that result from our approach (all taken from the more general material given in \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection}).\n\nTo do this we fix a finite abelian extension of number fields $F\/k$ of group $G$ and an elliptic curve $A$ over $k$.\n\nWe also fix an odd prime $p$ that does not divide the order of the torsion subgroup of $A(F)$ and an isomorphism of fields $\\CC\\cong \\CC_p$ (that we do not explicitly indicate in the sequel). We write $\\widehat{G}$ for the set of irreducible complex characters of $G$.\n\nThe first prediction concerns the values at $z=1$ of Hasse-Weil-Artin $L$-series. To state it we set $F_p := \\QQ_p\\otimes_\\QQ F$ and write ${\\rm log}_{A,p}$ for the formal group logarithm of $A$ over $\\QQ_p\\otimes_\\QQ k$ and $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$.\n\nFor each subset $x_\\bullet = \\{x_{\\sigma}: \\sigma \\in \\Sigma(k)\\}$ of $A(F_p)$ and each character $\\psi$ in $\\widehat{G}$ we then define a `$p$-adic logarithmic resolvent' by setting\n\n\\begin{equation}\\label{log resol abelian} \\mathcal{LR}_\\psi(x_\\bullet) := {\\rm det}\\left(\\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{A,p}(x_{\\sigma'})))\\cdot \\psi(g)\\bigr)_{\\sigma,\\sigma' \\in \\Sigma(k)}\\right),\\end{equation}\nwhere we fix an ordering of $\\Sigma(k)$ and an extension $\\hat \\sigma$ to $F$ of each $\\sigma$ in $\\Sigma(k)$.\n\nNow if $S$ is any finite set of places of $k$ that contains all archimedean places, all that ramify in $F$, all at which $A$ has bad reduction and all above $p$, and $L_{S}(A,\\check\\psi,z)$ is the $S$-truncated $L$-series attached to $A$ and the contragredient $\\check\\psi$ of $\\psi$, then our methods predict that for any $x_\\bullet$ the sum\n\n\\begin{equation}\\label{first predict} \\sum_{\\psi \\in \\widehat{G}}\\frac{L_{S}(A,\\check\\psi,1)\\cdot \\mathcal{LR}_\\psi(x_\\bullet)}{\\Omega^\\psi_A\\cdot w_\\psi}\\cdot e_\\psi \\end{equation}\nbelongs to $\\ZZ_p[G]$ and annihilates the $p$-primary part $\\sha(A_{F})[p^\\infty]$ of the Tate-Shafarevich group of $A$ over $F$. Here $e_\\psi$ denotes the idempotent $|G|^{-1}\\sum_{g \\in G}\\check\\psi(g)g$ of $\\CC[G]$ and the periods $\\Omega^\\psi_A$ and Artin root numbers $w_\\psi$ are as explicitly defined in \\S\\ref{k theory period sect2}.\n\nIn particular, if one finds  for some choice of $x_\\bullet$ that the sum in (\\ref{first predict}) is a unit of $\\ZZ_p[G]$, then this implies that $\\sha(A_F)[p^\\infty]$ should be trivial.\n\nMore generally, the fact that each sum in (\\ref{first predict}) should belong to $\\ZZ_p[G]$ implies that for every $g$ in $G$, and every set of local points $x_\\bullet$, there should be a congruence\n\\[ \\sum_{\\psi \\in \\widehat{G}}\\psi(g)\\frac{L_{S}(A,\\check\\psi,1)\\cdot \\mathcal{LR}_\\psi(x_\\bullet)}{\\Omega_A^\\psi\\cdot w_\\psi}\n \\equiv 0 \\,\\,\\,({\\rm mod}\\,\\, |G|\\cdot \\ZZ_p).\\]\n\nIn concrete examples these congruences are strong restrictions on the values $L_{S}(A,\\check\\psi,1)$ that can be investigated  numerically but cannot be deduced by solely considering Birch and Swinnerton-Dyer type formulas for individual Hasse-Weil-Artin $L$-series\n\n\nIn general, our analysis leads to a range of congruence predictions that are both finer than the above and also involve the values at $z=1$ of higher derivatives of Hasse-Weil-Artin $L$-series, suitably normalised by a product of explicit regulators and periods.\n\nTo give an example of this sort of prediction, we shall focus on the simple case that $F\/k$ is cyclic of degree $p$ (although entirely similar predictions can be made in the setting of cyclic extensions of arbitrary $p$-power order and also for certain dihedral families of extensions).\n\nIn this case, under certain natural, and very mild, hypotheses on $A$ and $F\/k$ relative to $p$ there exist non-negative integers $m_0$ and $m_1$ with the property that the pro-$p$ completion $A(F)_p$ of $A(F)$ is isomorphic as a $\\ZZ_p[G]$-module to a direct sum of $m_0$ copies of $\\ZZ_p$ and $m_1$ copies of $\\ZZ_p[G]$.\n\nIn particular, if we further assume $m_0=2$ and $m_1= 1$, then we may fix points $P_{0}^1$ and $P_{0}^2$ in $A(k)$ and $P_1$ in $A(F)$ such that $A(F)_p$ is the direct sum of the $\\ZZ_p[G]$-modules generated by $P_{0}^1,$ $P_{0}^2$ and $P_1.$ (In Example \\ref{wuthrich example} the reader will find explicit examples of such pairs $A$ and $F\/k$ for the prime $p=3$.)\n\n\nThen, writing $L^{(1)}_{S}(A,\\check\\psi,1)$ for the value at $z=1$ of the first derivative of $L_{S}(A,\\check\\psi,z)$, our methods predict that, under mild additional hypotheses, there should exist an element $x$ of $\\ZZ_p[G]$ that annihilates $\\sha(A_{F})[p^\\infty]$ and is such that\n\\begin{equation}\\label{second predict} \\sum_{\\psi \\in \\widehat{G}}\\frac{L^{(1)}_{S}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{\\Omega^\\psi_A\\cdot i^{r_2}}\\cdot e_\\psi  = x\\cdot \\sum_{g\\in G}\\langle g(P_1),P_1\\rangle_{A_F}\\cdot g^{-1},\\end{equation}\nwhere $\\langle -,-\\rangle_{A_F}$ denotes the N\\'eron-Tate height pairing of $A$ relative to $F$, $\\tau^*(\\QQ,\\psi)$ is the (modified, global) Galois-Gauss sum of the character of $G_\\QQ$ that is obtained by inducing $\\psi$ and $r_2$ is the number of complex places of $k$.\n\nTo be more explicit, we write $R_A$ for the determinant of the N\\'eron-Tate regulator matrix of $A$ over $k$ with respect to the ordered $\\QQ$-basis $\\{P_{0}^1,P_{0}^2,\\sum_{g\\in G}g(P_1)\\}$ of $\\QQ\\cdot A(k)$ and for each non-trivial $\\psi$ in $\\widehat{G}$ we define a non-zero complex number by setting\n\\[ h^\\psi(P_1):=\\sum_{g\\in G}\\langle g(P_1),P_1\\rangle_{A_F}\\cdot\\psi(g)^{-1}.\\]\n\nWe finally write $S_{\\rm r}$ for the set of places of $k$ that ramify in $F$, $d_k$ for the discriminant of $k$ and $I_p(G)$ for the augmentation ideal of $\\ZZ_p[G]$. Then, under mild hypotheses, our methods predict that there should be containments\n\\[\n\\sum_{\\psi\\neq {\\bf 1}_G}\\frac{L_{S_{\\rm r}}^{(1)}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{\\Omega^\\psi_{A}\\cdot i^{r_2} \\cdot h^\\psi(P_1)}\\cdot e_\\psi  \\in I_p(G)^2\\,\\,\\text{ and }\\,\\, \\frac{L_{S_{\\rm r}}^{(3)}(A,1)\\sqrt{|d_k|}}{\\Omega_{A}\\cdot R_{A}}\\in \\ZZ_p,\\]\nand a congruence modulo $I_p(G)^3$ of the form\n\\begin{equation}\\label{examplecongruent}\n\\sum_{\\psi\\neq {\\bf 1}_G}\\frac{L_{S_{\\rm r}}^{(1)}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{ \\Omega^\\psi_{A}\\cdot i^{r_2} \\cdot h^\\psi(P_1)}\\cdot\\! e_\\psi\n \\equiv \\!\\frac{L_{S_{\\rm r}}^{(3)}(A,1)\\sqrt{|d_k|}}{\\Omega_{A}\\cdot R_{A}}\\cdot\\det\\left(\\begin{array}{cc}\n\\langle P_{0}^1,P_{0}^1\\rangle^{\\rm MT} & \\langle P_{0}^1,P_{0}^2\\rangle^{\\rm MT}\n\\\\\n\\langle P_{0}^2,P_{0}^1\\rangle^{\\rm MT} & \\langle P_{0}^2,P_{0}^2\\rangle^{\\rm MT}\n\\end{array}\\right)\\!.\n\\end{equation}\nHere $L_{S_{\\rm r}}^{(3)}(A,1)$ denotes the value at $z=1$ of the third derivative of $L_{S_{\\rm r}}(A,z)$ and\n\\[ \\langle\\,,\\rangle^{\\rm MT}:A(k)\\times A(k)\\to I_p(G)\/I_p(G)^2\\]\nis the canonical pairing that Mazur and Tate define in \\cite{mt0} by using the geometrical theory of bi-extensions.\n\nFurther, if $\\sha(A_F)[p^\\infty]$ is trivial, then the $p$-component of ${\\rm BSD}(A_{F\/k})$ is valid if and only if (\\ref{examplecongruent}) holds and, in addition, the $p$-component of the Birch and Swinnerton-Dyer Conjecture is true for $A$ over both $k$ and $F$.\n\n\n\nWe remark that even in the simplest possible case that $k = \\QQ$ and $p=3$, these predictions strongly refine those made by Kisilevsky and Fearnley in \\cite{kisilevsky} and cannot be deduced by simply considering leading term formulas for individual Hasse-Weil-Artin $L$-series.\n\n\n\n\n\n\n\\subsection{General notation} For the reader's convenience we give details here of some of the general notation and terminology that will be used throughout the article.\n\n\\subsubsection{}We write $|X|$ for the cardinality of a finite set $X$.\n\nFor an abelian group $M$ we write $M_{\\rm tor}$ for its torsion subgroup and $M_{\\rm tf}$ for the quotient of $M$ by $M_{\\rm tor}$.\n\nFor a prime $p$ and natural number $n$ we write $M[p^n]$ for the subgroup $\\{m \\in M: p^nm =0\\}$ of the Sylow $p$-subgroup $M[p^{\\infty}]$ of $M_{\\rm tor}$.\n\nWe set $M_p := \\ZZ_p\\otimes_\\ZZ M$, write $M^\\wedge_p$ for the  pro-$p$-completion $\\varprojlim_n M\/p^n M$ of $M$ and denote the Pontryagin dual $\\Hom(M,\\QQ\/\\ZZ)$ of $M$ by $M^\\vee$.\n\nIf $M$ is finite of exponent dividing $p^m$, then $M^\\vee$ identifies with $\\Hom_{\\ZZ_p}(M,\\QQ_p\/\\ZZ_p)$ and we shall (without explicit comment) use the canonical identification $\\QQ_p\/\\ZZ_p=\\varinjlim_n \\ZZ\/p^n\\ZZ$ to identify elements of $M^\\vee$ with their canonical image in the linear dual $\\Hom_{\\ZZ\/p^m\\ZZ}(M,\\ZZ\/p^m\\ZZ)$.\n\nIf $M$ is finitely generated, then for a field extension $E$ of $\\QQ$ we shall often abbreviate $E\\otimes_\\ZZ M$ to $E\\cdot M$.\n\n\nIf $M$ is a $\\Gamma$-module for some group $\\Gamma$, then we always endow $M^\\vee$ with the natural contragredient action of $\\Gamma$.\n\n\nWe recall that if $\\Gamma$ is finite, then a $\\Gamma$-module $M$ is said to be `cohomologically-trivial' if for all subgroups $\\Delta$ of $\\Gamma$ and all integers $i$ the Tate cohomology group $\\hat H^i(\\Delta,M)$ vanishes.\n\n\\subsubsection{}For any ring $R$ we write $R^\\times$ for its multiplicative group and $\\zeta(R)$ for its centre.\n\nUnless otherwise specified we regard all $R$-modules as left $R$-modules. We write ${\\rm Mod}(R)$ for the abelian category of $R$-modules and ${\\rm Mod}^{\\rm fin}(R)$ for the abelian subcategory of ${\\rm Mod}(R)$ comprising all $R$-modules that are finite.\n\nWe write $D(R)$ for the derived category of complexes of $R$-modules. If $R$ is noetherian, then we write $D^{\\rm perf}(R)$ for the full triangulated subcategory of $D(R)$ comprising complexes that are `perfect' (that is, isomorphic in $D(R)$ to a bounded complex of finitely generated projective $R$-modules).\n\nFor a natural number $n$ we write $\\tau_{\\le n}$ for the exact truncation functor on $D(R)$ with the property that for each object $C$ in $D(R)$ and each integer $i$ one has\n\\[ H^i(\\tau_{\\le n}(C)) = \\begin{cases} H^i(C), &\\text{if $i \\le n$}\\\\\n                                          0, &\\text{otherwise.}\\end{cases}\\]\n\n\\subsubsection{}For a Galois extension of number fields $L\/K$ we set $G_{L\/K} := \\Gal(L\/K)$. We also fix an algebraic closure $K^c$ of $K$ and set $G_K := G_{K^c\/K}$.\n\nFor each non-archimedean place $v$ of a number field we write $\\kappa_v$ for its residue field  and denote its  absolute norm $|\\kappa_v|$ by ${\\rm N}v$.\n\nWe write the dual of an abelian variety $A$ as $A^t$ and usually abbreviate its dimension ${\\rm dim}(A)$ to $d$.\n\nIf $A$ is defined over a number field $k$, then for each extension $F$ of $k$ we write the Tate-Shafarevich group of $A$ over $F$ as $\\sha(A_F)$.\n\nWe shall also use the following notation regarding sets of places of $k$.\n\n\\begin{itemize}\n\\item[-] $S_k^\\RR$ is the set of real archimedean places of $k$;\n\\item[-] $S_k^\\CC$ is the set of complex archimedean places of $k$;\n\\item[-] $S_k^\\infty (= S_k^\\RR \\cup S_k^\\CC$) is the set of archimedean places of $k$;\n\\item[-] $S_k^f$ is the set of non-archimedean places of $k$;\n\\item[-] $S_k^v$ is the set of places of $k$ that extend a place $v$ of a given subfield of $k$. In particular,\n\\item[-] $S_k^p$ is the set of $p$-adic places of $k$ for a given prime number $p$;\n\\item[-] $S_k^F$ is the set of places of $k$ that ramify in a given extension $F$ of $k$;\n\\item[-] $S_k^A$ is the set of places of $k$ at which a given abelian variety $A$ has bad reduction.\n\\end{itemize}\n\n For a fixed set of places $S$ of $k$ we also write $S(F)$ for the set of places of $F$ which lie above a place in $S$.\n\n\nFor a place $v$ of $k$ we set $G_{v} := G_{k_v^c\/k_v}$. We also write $k_v^{\\rm un}$ for the maximal unramified extension of $k$ in $k_v^c$ and set $I_{v} := G_{k_v^c\/k_v^{\\rm un}}$.\n\nWe fix a place $w$ of $F$ above $v$ and a corresponding embedding $F\\to k_v^c$. We write $G_w$ and $I_w$ for the images of $G_{v}$ and $I_{v}$ under the induced homomorphism $G_{v} \\to G$. We also fix a lift $\\Phi_v$ to $G$ of the Frobenius automorphism in $G_w\/I_w$.\n\nWe write $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$.\n\n\\subsection{Acknowledgements} We are very grateful to St\\'ephane Vigui\\'e for his help with aspects of the argument presented in \\S\\ref{ptduality} and to Werner Bley for providing us with the material in \\S\\ref{ell curve sect} and for pointing out a sign-error in an earlier version of the argument in \\S\\ref{comparison section}.\n\nWe are also grateful to both Werner Bley and Christian Wuthrich for their interest, helpful correspondence and tremendous generosity regarding numerical computations.\n\nIn addition, we would like to thank  Rob Evans, Masato Kurihara, Jan Nekov\\'a\\v r, Takamichi Sano, Kwok-Wing Tsoi and Stefano Vigni for helpful discussions and correspondence.\n\nFinally, it is a great pleasure to thank Dick Gross for his strong encouragement regarding this project and for several insightful remarks.\n\nThe second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the `Severo Ochoa Programme for Centres of Excellence in R\\&D' (SEV-2015-0554) as well as through project MTM2016-79400-P.\n\n\n\\section{Selmer complexes}\\label{selmer section}\n\nIn this section we define the Selmer complexes that play a key role in the conjecture we shall later formulate and establish some of their key properties.\n\nAt the outset we fix a finite set of places $S$ of $k$ with\n\\[ S_k^\\infty\\cup S_k^F \\cup S_k^A\\subseteq S.\\]\n\nWe also write $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$ and $\\Sigma_\\sigma(F)$ for each $\\sigma$ in $\\Sigma(k)$ for the set of embeddings $F \\to \\CC$ that extend $\\sigma$.\n\nFor each $v$ in $S_k^\\infty$ we fix a corresponding embedding $\\sigma_v$ in $\\Sigma(k)$ and an embedding $\\sigma'_v$ in $\\Sigma_{\\sigma_v}(F)$.\n\nWe then write $Y_{v,F}$ for the module $\\prod_{\\Sigma_{\\sigma_v}(F)}\\ZZ$ endowed with its natural action of $G\\times G_v$ (via which $G$ and $G_v$ respectively act via pre-composition and post-composition with the embeddings in $\\Sigma_{\\sigma_v}(F)$).\n\nFor each $v$ in $S_k^\\infty$ we set\n\\[ H_v(A_{F\/k}) := H^0(k_v,Y_{v,F}\\otimes_{\\ZZ}H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)),\\]\nregarded as a $G$-module via the given action on $Y_{v,F}$. We note that this $G$-module is free of rank $2d$ if $v$ is in $S_k^\\CC$ and spans a free $\\ZZ[1\/2][G]$-module of rank $d$ if $v$ is in $S_k^\\RR$.\n\nWe then define a $G$-module by setting\n\\[ H_\\infty(A_{F\/k}) := \\bigoplus_{v \\in S_k^\\infty}H_v(A_{F\/k}).\\]\n\nFinally we write $\\Sigma_k(F)$ for the set of $k$-embeddings $F \\to k^c$ and $Y_{F\/k}$ for the module $\\prod_{\\Sigma_k(F)}\\ZZ$, endowed with its natural action of $G\\times G_k$.\n\n\n\n\n\n\\subsection{Classical Selmer complexes}\\label{p-adiccomplexes} In this section we fix a prime number $p$.\n\n\n\\subsubsection{}We first record a straightforward (and well-known) result regarding pro-$p$ completions that will be useful in the sequel.\n\nWe let $B$ denote either $A$ or its dual variety $A^t$ and write $T_p(B)$ for the $p$-adic Tate module of $B$.\n\n\\begin{lemma}\\label{v not p} For each non-archimedean place $w'$ of $F$ the following claims are valid.\n\\end{lemma}\n\\begin{itemize}\n\\item[(i)] If $w'$ is not $p$-adic then the natural Kummer map $B(F_{w'})^\\wedge_p \\to H^1(F_{w'},T_p(B))$ is bijective.\n\\item[(ii)] There exists a canonical short exact sequence\n\\[ 0 \\to H^1(\\kappa_{w'}, T_p(B)^{I_{w'}}) \\to B(F_{w'})^\\wedge_p \\to H^0(F_{w'}, H^1(I_{w'}, T_{p}(B))_{\\rm tor})\\to 0.\\]\n\\end{itemize}\n\n\\begin{proof} We note first that if $w'$ does not divide $p$, then the module $H^1(F_{w'},T_p(B))$ is finite. This follows, for example, from Tate's local Euler characteristic formula, the vanishing of $H^0(F_{w'},T_p(B))$ and the fact that local duality identifies $H^2(F_{w'},T_p(B))$ with the finite module $H^0(F_{w'},T_p(B^t)\\otimes_{\\ZZ_p}\\QQ_p\/\\ZZ_p)^\\vee$.\n\nGiven this observation, claim (i) is obtained directly upon passing to the inverse limit over $m$ in the natural Kummer theory exact sequence\n\\begin{equation}\\label{kummerseq} 0 \\to B(F_{w'})\/p^m \\to H^1(F_{w'},T_p(B)\/p^m) \\to H^1(F_{w'},B)[p^m]\\to 0.\\end{equation}\n\n\nClaim (ii) is established by Flach and the first author in \\cite[(1.38)]{bufl95} (after recalling the fact that the group denoted $H^1_f(F_{w'},T_p(B))_{\\rm BK}$ in loc. cit. is equal to the image of $B(F_{w'})^\\wedge_p$ in $H^1(F_{w'},T_p(B))$ under the injective Kummer map).\n\\end{proof}\n\n\\begin{remark}\\label{Tamagawa remark}{\\em The cardinality of each module $H^0(F_{w'}, H^1(I_{w'}, T_{p}(B))_{\\rm tor})$ that occurs in Lemma \\ref{v not p}(ii) is the maximal power of $p$ that divides the Tamagawa number of $B$ at $w'$. }\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{}If $B$ denotes either $A$ or $A^t$, then for any subfield $E$ of $k$ and any place $v$ in $S_E^f$ we obtain a $G$-module by setting\n\\[ B(F_v) := \\prod_{w' \\in S_F^v}B(F_{w'}).\\]\n\nFor later purposes we note that if $E = k$, then this module is isomorphic to the module of $G_w$-coinvariants\n$Y_{F\/k}\\otimes_{\\ZZ[G_w]} B(F_w)$ of the tensor product $Y_{F\/k}\\otimes_{\\ZZ} B(F_w)$, upon which $G$ acts only the first factor but $G_k$ acts diagonally on both.\n\nIn a similar way,  the $p$-adic Tate module of the base change of $B$ through $F\/k$ is equal to\n\\[ T_{p,F}(B) := Y_{F\/k,p}\\otimes_{\\ZZ_p}T_p(B)\\]\n(where, again, $G$ acts only on the first factor of the tensor product whilst $G_k$ acts on both). We set $V_{p,F}(B) := \\QQ_p\\cdot T_{p,F}(B)$.\n\n\nWe can now introduce a notion of Selmer complex that will play an important role in the sequel.\n\n\\begin{definition}\\label{bkdefinition}{\\em For any finite subset $\\Sigma$ of $S_k^f$ that contains each of $S_k^p, S_k^F\\cap S_k^f$ and $S_k^A$, the `classical $p$-adic Selmer complex' ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ for the data $A, F\/k$ and $\\Sigma$ is the mapping fibre of the morphism\n\\begin{equation}\\label{bkfibre}\n\\tau_{\\le 3}(R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t))) \\oplus \\left(\\bigoplus_{v\\in\\Sigma} A^t(F_v)^\\wedge_p\\right)[-1]  \\xrightarrow{(\\lambda,\\kappa)} \\bigoplus_{v \\in \\Sigma} R\\Gamma (k_v, T_{p,F}(A^t))\n\\end{equation}\nin $D(\\ZZ_p[G])$. Here $\\lambda$ is the natural diagonal localisation morphism and $\\kappa$ is induced by the Kummer theory maps\n$A^t(F_v)^\\wedge_p\\to H^1(k_v,T_{p,F}(A^t))$ (and the fact that $H^0(k_v, T_{p,F}(A^t))$ vanishes for all $v$ in $\\Sigma$).\n\n}\n\\end{definition}\n\n\\begin{remark}{\\em If $p$ is odd, then $R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t)))$ is acyclic in degrees greater than two and so the natural morphism $\\tau_{\\le 3}(R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t))) \\to R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t)))$ in $D(\\ZZ_p[G])$ is an isomorphism. In this case the truncation functor $\\tau_{\\le 3}$ can therefore be omitted from the above definition.}\\end{remark}\n\nThe following result shows that the Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is independent, in a natural sense, of the choice of the set of places $\\Sigma$.\n\nFor this reason, in the sequel we shall usually write ${\\rm SC}_{p}(A_{F\/k})$ in place of ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$.\n\n\\begin{lemma}\\label{independenceofsigma} Let $\\Sigma$ and $\\Sigma'$ be any finite subsets of $S_k^f$ as in Definition \\ref{bkdefinition} with $\\Sigma\\subseteq\\Sigma'$. Then there is a canonical isomorphism ${\\rm SC}_{\\Sigma',p}(A_{F\/k})\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})$ in $D(\\ZZ_p[G])$.\n\\end{lemma}\n\n\\begin{proof} We recall that the compactly supported cohomology complex\n\\[ R\\Gamma_{c,\\Sigma}:=R\\Gamma_c(\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma},T_{p,F}(A^t))\\]\nis defined to be the mapping fibre of the diagonal localisation morphism\n\\begin{equation}\\label{compactloc} R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma},T_{p,F}(A^t)) \\to \\bigoplus_{v \\in S_k^\\infty\\cup\\Sigma}R\\Gamma(k_v,T_{p,F}(A^t))\\end{equation}\nin $D(\\ZZ_p[G])$.\n\nWe further recall that $R\\Gamma_{c,\\Sigma}$ is acyclic outside degrees $1, 2$ and $3$ (see, for example, \\cite[Prop. 1.6.5]{fukaya-kato}) and that\n $R\\Gamma(k_v,T_{p,F}(A^t))$ for each $v$ in $\\Sigma$ is acyclic outside degrees $1$ and $2$ and hence that the natural morphisms\n\\[ \\tau_{\\le 3}(R\\Gamma_{c,\\Sigma}) \\to R\\Gamma_{c,\\Sigma}\\,\\,\\text{ and } \\,\\,\\tau_{\\le 3}(R\\Gamma(k_v,T_{p,F}(A^t))) \\to R\\Gamma(k_v,T_{p,F}(A^t))\\]\nin $D(\\ZZ_p[G])$ are  isomorphisms.\n\nUpon comparing these facts with the definition of ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ one deduces the existence of a canonical exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{comparingtriangles}R\\Gamma_{c,\\Sigma}\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})\\to \\left(\\bigoplus_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\right)[-1]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty) \\to R\\Gamma_{c,\\Sigma}[1],\n\\end{equation}\nwhere we abbreviate $\\bigoplus_{ v\\in S_k^\\infty} R\\Gamma(k_v,T_{p,F}(A^t))$ to $R\\Gamma_\\infty$.\n\n\nIn addition, by the construction of \\cite[(30)]{bufl99}, there is a canonical exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{independencetriangle}\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})[-1]\\to R\\Gamma_{c,\\Sigma'}\\to R\\Gamma_{c,\\Sigma}\\to\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v}).\\end{equation}\n\nFinally we note that, since the choice of $\\Sigma$ implies that $A$ has good reduction at each place $v$ in $\\Sigma'\\setminus\\Sigma$, the module $H^0(F_{w'}, H^1(I_{w'}, T_{p}(A^t))_{\\rm tor})$ vanishes for every $w'$ in $S_F^v$.\n\nThus, since for each $v$ in $\\Sigma'\\setminus\\Sigma$ the complex $R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})$ is acyclic outside degree one, the exact sequences in Lemma \\ref{v not p}(ii) induce a canonical isomorphism\n\\begin{equation}\\label{firstrow}A^t(F_v)_p^\\wedge[-1]\\to R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})\\end{equation}\nin $D(\\ZZ_p[G])$.\n\nThese three facts combine to give a canonical commutative diagram in $D(\\ZZ_p[G])$ of the form\n\n\\begin{equation*}\\label{complexesdiag}\\xymatrix{\n\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}A^t(F_v)_p^\\wedge[-2] \\ar@{^{(}->}[d] \\ar[r]^{\\hskip-0.4truein\\sim}  &\n\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})[-1] \\ar[d] &\n\\\\\n\\bigoplus\\limits_{ v\\in \\Sigma'}A^t(F_v)_p^\\wedge[-2]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty)[-1] \\ar@{->>}[d] \\ar[r] &\nR\\Gamma_{c,\\Sigma'} \\ar[d] \\ar[r]  &\n{\\rm SC}_{\\Sigma',p}(A_{F\/k})\n\\\\\n\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge[-2]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty)[-1]  \\ar[r] &\nR\\Gamma_{c,\\Sigma}  \\ar[r]  &\n{\\rm SC}_{\\Sigma,p}(A_{F\/k}).\n}\\end{equation*}\nHere the first row is induced by the isomorphisms (\\ref{firstrow}), the second and third rows by the triangles (\\ref{comparingtriangles}) for $\\Sigma'$ and $\\Sigma$ respectively, the first column is the obvious short exact sequence and the second column is given by the triangle (\\ref{independencetriangle}).\n\nIn particular, since all rows and columns in this diagram are exact triangles, its commutativity implies the existence of a canonical isomorphism in $D(\\ZZ_p[G])$ from\n${\\rm SC}_{\\Sigma',p}(A_{F\/k})$ to ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$, as required.\n\\end{proof}\n\nTaken together, Lemma \\ref{v not p}(ii) and Remark \\ref{Tamagawa remark} imply that if $p$ is odd and the Tamagawa numbers of $A$ over $F$ are divisible by $p$, then the complex ${\\rm SC}_{p}(A_{F\/k})$ differs slightly from the `finite support cohomology' complex $R\\Gamma_f(k,T_{p,F}(A))$ that was defined (for odd $p$) and played a key role in the article \\cite{bmw} of Wuthrich and the present authors.\n\nFor such $p$ we have preferred to use ${\\rm SC}_{p}(A_{F\/k})$ rather than $R\\Gamma_f(k,T_{p,F}(A))$ in this article since it is more amenable to certain explicit constructions that we have to make in later sections.\n\nFor the moment, we record only the following facts about ${\\rm SC}_{p}(A_{F\/k})$ that will be established in Propositions \\ref{explicitbkprop} and \\ref{explicitbkprop2} below. We write $\\Sel_p(A_{F})$ for the classical $p$-primary Selmer group of $A$ over $F$. Then ${\\rm SC}_{p}(A_{F\/k})$ is acyclic outside degrees one, two and three and, assuming the Tate-Shafarevich group $\\sha(A_F)$ of $A$ over $F$ to be finite, there are canonical identifications for each odd $p$ of the form\n\\begin{equation}\\label{bksc cohom} H^i({\\rm SC}_{p}(A_{F\/k})) = \\begin{cases} A^t(F)_p, &\\text{if $i=1$,}\\\\\n\\Sel_p(A_F)^\\vee, &\\txt{if $i=2$,}\\\\\nA(F)[p^{\\infty}]^\\vee, &\\text{if $i=3$,}\\end{cases}\\end{equation}\nwhilst for $p=2$ there is a canonical identification $H^1({\\rm SC}_{2}(A_{F\/k})) =  A^t(F)_2$ and a canonical homomorphism $\\Sel_2(A_F)^\\vee \\to H^2({\\rm SC}_{2}(A_{F\/k}))$ with finite kernel and cokernel, and the module $H^3({\\rm SC}_{2}(A_{F\/k}))$ is finite.\n\n\\begin{remark}\\label{indeptremark}{\\em A closer analysis of the argument in Lemma \\ref{independenceofsigma} shows that, with respect to the identifications (\\ref{bksc cohom}) that are established (under the hypothesis that $\\sha(A_F)$ is finite)  in Proposition \\ref{explicitbkprop} below, the isomorphism ${\\rm SC}_{\\Sigma',p}(A_{F\/k})\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})$ constructed in Lemma \\ref{independenceofsigma} induces the identity map on all degrees of cohomology.}\\end{remark}\n\n\n\n\\subsection{Nekov\\'a\\v r-Selmer complexes} In this section we again fix a prime number $p$.\n\n\\subsubsection{}Whilst the modules that occur in (\\ref{bksc cohom}) are the primary objects of interest in the theory of abelian varieties, the complex ${\\rm SC}_{p}(A_{F\/k})$ is not always well-suited to our purposes since, except in certain special cases (that will be discussed in detail in \\S\\ref{tmc}), it does not belong to $D^{\\rm perf}(\\ZZ_p[G])$.\n\nFor this reason, we find it convenient to introduce the following alternative notion of Selmer complexes.\n\nThis construction is motivated by the general approach developed by Nekov\\'a\\v r in \\cite{nek}\n\n\\begin{definition}\\label{selmerdefinition}{\\em  Fix $\\ZZ_p[G]$-submodules $X$ of $A^t(F_p)^\\wedge_p$ and $X'$ of $H_{\\infty}(A_{F\/k})_p$. Then the `Nekov\\'a\\v r-Selmer complex' ${\\rm SC}_{S}(A_{F\/k};X,X')$ of the data $(A,F,S,X,X')$ is the mapping fibre of the  morphism\n\\begin{equation}\\label{fibre morphism}\nR\\Gamma(\\mathcal{O}_{k,S\\cup S_k^p},T_{p,F}(A^t)) \\oplus X[-1] \\oplus X'[0] \\xrightarrow{(\\lambda, \\kappa_1,\\kappa_2)} \\bigoplus_{v \\in S \\cup S_k^p} R\\Gamma (k_v, T_{p,F}(A^t))\n\\end{equation}\n in $D(\\ZZ_p[G])$. Here $\\lambda$ is again the natural diagonal localisation morphism, $\\kappa_1$ is the morphism\n\n \\[ X[-1]\\rightarrow \\bigoplus_{v \\in S_k^p}R\\Gamma (k_v, T_{p,F}(A^t))\\]\n\n induced by the sum over $v$ of the local Kummer maps (and the fact each group $H^0(k_v,T_{p,F}(A^t))$ vanishes) and $\\kappa_2$ is the morphism\n\n\\[ X'[0] \\to \\bigoplus_{v \\in S_k^\\infty}R\\Gamma (k_v, T_{p,F}(A^t))\\]\nthat is induced by the canonical comparison isomorphisms\n\\begin{equation}\\label{cancompisom} Y_{v,F,p}\\otimes_{\\ZZ} H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ) \\cong Y_{F\/k,p}\\otimes_{\\ZZ_p}T_{p}(A^t)= T_{p,F}(A^t) \\end{equation}\nfor each $v$ in $S_k^\\infty$\n}\\end{definition}\n\n\n\nIn the next result we establish the basic properties of these Nekov\\'a\\v r-Selmer complexes. In this result we shall write ${\\rm Mod}^\\ast(\\ZZ_p[G])$ for the category ${\\rm Mod}(\\ZZ_p[G])$ in the case that $p$ is odd and for the quotient of ${\\rm Mod}(\\ZZ_2[G])$ by its subcategory ${\\rm Mod}^{\\rm fin}(\\ZZ_2[G])$ in the case that $p = 2$.\n\n\n\n\\begin{proposition}\\label{prop:perfect} Let $X$ be a finite index $\\ZZ_p[G]$-submodule of $A^t(F_p)^\\wedge_p$ that is cohomologically-trivial as a $G$-module.\n\nLet $X'$ be a finite index projective $\\ZZ_p[G]$-submodule of $H_\\infty(A_{F\/k})_p$, with $X' = H_\\infty(A_{F\/k})_p$ if $p$ is odd.\n\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] ${\\rm SC}_{S}(A_{F\/k};X,X')$ is an object of $D^{\\rm perf}(\\ZZ_p[G])$ that is acyclic outside degrees one, two and three.\n\\item[(ii)] $H^3({\\rm SC}_{S}(A_{F\/k};X,X'))$ identifies with $A(F)[p^{\\infty}]^\\vee$.\n\\item[(iii)] If $\\sha(A_F)$ is finite, then in ${\\rm Mod}^\\ast(\\ZZ_p[G])$ there exists a canonical injective homomorphism\n\\[ H^1({\\rm SC}_{S}(A_{F\/k};X,X')) \\to A^t(F)_p \\]\nthat has finite cokernel and a canonical surjective homomorphism\n\\[  H^2({\\rm SC}_{S}(A_{F\/k};X,X')) \\to {\\rm Sel}_p(A_{F})^\\vee\\]\nthat has finite kernel.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Set $C_{S} := {\\rm SC}_{S}(A_{F\/k};X,X')$.\n\nThen, by comparing the definition of $C_{S}$ as the mapping fibre of (\\ref{fibre morphism}) with the definition of the compactly supported cohomology complex $R\\Gamma_c(A_{F\/k}) := R\\Gamma_c(\\mathcal{O}_{k,S\\cup S_k^p},T_{p,F}(A^t))$ as the mapping fibre of the morphism (\\ref{compactloc})\none finds that there is an exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{can tri}  R\\Gamma_c(A_{F\/k})\\to C_{S} \\to X[-1] \\oplus X'[0]\\to R\\Gamma_c(A_{F\/k})[1].\\end{equation}\n\nTo derive claim (i) from this triangle it is then enough to recall (from, for example, \\cite[Prop. 1.6.5]{fukaya-kato}) that $R\\Gamma_c(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one, two and three and note that both of the $\\ZZ_p[G]$-modules $X$ and $X'$ are finitely generated and cohomologically-trivial.\n\nThe above triangle also gives a canonical identification\n\\begin{equation}\\label{artinverdier} H^3(C_{S}) \\cong H^3(R\\Gamma_c(A_{F\/k})) \\cong H^0(k,T_{p,F}(A)\\otimes_{\\ZZ_p}\\QQ_p\/\\ZZ_p)^\\vee = A(F)[p^{\\infty}]^\\vee\\end{equation}\nwhere the second isomorphism is induced by the Artin-Verdier Duality Theorem.\n\nIn a similar way, if we set $\\Sigma:=(S\\cap S_k^f)\\cup S_k^p$ and abbreviate the classical $p$-adic Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ to $C'_\\Sigma$, then a direct comparison of the definitions of $C_{S}$ and $C_\\Sigma'$ shows that $C_{S}$ is isomorphic in $D(\\ZZ_p[G])$ to the mapping fibre of the morphism\n\\begin{equation}\\label{selmer-finite tri} C'_\\Sigma \\oplus X[-1] \\oplus X'[0]\n\\xrightarrow{(\\lambda', \\kappa'_1,\\kappa_2)}\n \\bigoplus_{v \\in \\Sigma} A^t(F_v)^\\wedge_p[-1] \\oplus \\bigoplus_{v \\in S_k^\\infty}\\tau_{\\le 3}(R\\Gamma(k_v,T_{p,F}(A^t)))\\end{equation}\nwhere $\\lambda'$ is the canonical morphism\n\\[ C'_\\Sigma \\to \\bigoplus_{v \\in \\Sigma} A^t(F_v)^\\wedge_p[-1]\\]\ndetermined by the definition of of $C'_\\Sigma$ as the mapping fibre of (\\ref{bkfibre}),\nand the morphism $$\\kappa'_1: X[-1]\\rightarrow \\bigoplus_{v \\in S_k^p} A^t(F_v)^\\wedge_p[-1]$$ is induced by the given inclusion\n $X \\subseteq A^t(F_p)^\\wedge_p$.\n\n\nThis description of $C_{S}$ gives rise to a canonical long exact sequence of $\\ZZ_p[G]$-modules\n\n\\begin{multline}\\label{useful1} 0 \\to {\\rm cok}(H^0(\\kappa_2)) \\to H^1(C_{S}) \\to H^1(C_\\Sigma') \\\\\n \\to (A^t(F_p)^\\wedge_p\/X) \\oplus\\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^p} A^t(F_v)^\\wedge_p \\oplus  \\bigoplus_{v \\in S_k^\\infty}H^1(k_v,T_{p,F}(A^t))\\\\ \\to H^2(C_{S})\\to H^2(C_\\Sigma') \\to \\bigoplus_{v \\in S_k^\\infty}H^2(k_v,T_{p,F}(A^t)). \\end{multline}\n\nIn addition, for each $v \\in S_k^\\infty$ the groups $H^1(k_v,T_{p,F}(A^t))$ and $H^2(k_v,T_{p,F}(A^t))$ vanish if $p$ is odd and are finite if $p=2$, whilst our choice of $X'$ ensures that ${\\rm cok}(H^0(\\kappa_2))$ is also a finite group of $2$-power order.\n\nClaim (iii) therefore follows upon combining the above sequence with the identifications of $H^1(C_\\Sigma')$ and $H^2(C_\\Sigma')$ given in (\\ref{bksc cohom}) for odd $p$, and in the subsequent remarks for $p=2$, that are valid whenever $\\sha(A_F)$ is finite.\n\n\\end{proof}\n\n\n\\begin{remark}\\label{mrselmer}{\\em If $p$ is odd, then the proof of Proposition \\ref{prop:perfect} shows that the cohomology group\n$H^1({\\rm SC}_{S}(A_{F\/k};X,H_\\infty(A_{F\/k})_p))$ coincides with the Selmer group $H^1_{\\mathcal{F}_X}(k,T_{p,F}(A^t))$ in the sense of Mazur and Rubin \\cite{MRkoly}, where $\\mathcal{F}_X$ is the Selmer structure with $\\mathcal{F}_{X,v}$ equal to the image of $X$ in $H^1(k_v,T_{p,F}(A^t))$ for $v\\in S_k^p$ and equal to $0$ for $v \\in S\\setminus S_k^p$.} \\end{remark}\n\n\n\\subsection{Perfect Selmer structures and integral complexes}\\label{perfect selmer integral} We write $\\ell(v)$ for the residue characteristic of a non-archimedean place $v$ of $k$.\n\n\n\\begin{definition}\\label{pgss def}{\\em A `perfect Selmer structure' for the pair $A$ and $F\/k$ is a collection\n\\[ \\mathcal{X} := \\{\\mathcal{X}(v): v \\}\\]\nover all places $v$ of $k$ of $G$-modules that satisfy the following conditions.\n\n\\begin{itemize}\n\\item[(i)] For each $v$ in $S_k^\\infty$ the module $\\mathcal{X}(v)$ is projective and a submodule of $H_v(A_{F\/k})$ of finite $2$-power index.\n\\item[(ii)] For each $v$ in $S_k^f$ the module $\\mathcal{X}(v)$ is cohomologically-trivial and a finite index $\\ZZ_{\\ell(v)}[G]$-submodule of $A^t(F_v)^\\wedge_{\\ell(v)}$.\n\\item[(iii)] For almost all (non-archimedean) places $v$ one has $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}.$\n\\end{itemize}\nWe thereby obtain a  projective $G$-submodule\n\\[ \\mathcal{X}(\\infty) := \\bigoplus_{v \\in S_k^\\infty}\\mathcal{X}(v)\\]\nof $H_\\infty(A_{F\/k})$ of finite $2$-power index and, for each rational prime $\\ell$, a finite index cohomologically-trivial $\\ZZ_\\ell[G]$-submodule\n\\[  \\mathcal{X}(\\ell) := \\bigoplus_{v\\in S_k^\\ell}\\mathcal{X}(v)\\]\nof $A^t(F_\\ell)^\\wedge_{\\ell}$.}\n\\end{definition}\n\n\n\\begin{remark}{\\em The conditions (ii) and (iii) in Definition \\ref{pgss def} are consistent since if $\\ell$ does not divide $|G|$, then any $\\ZZ_{\\ell}[G]$-module is automatically cohomologically-trivial for $G$.} \\end{remark}\n\nIn the following result we write $X_\\ZZ(A_F)$ for the `integral Selmer group' of $A$ over $F$ defined by Mazur and Tate in \\cite{mt}.\n\nWe recall that, if the Tate-Shafarevich group $\\sha(A_F)$ is finite, then $X_\\ZZ(A_F)$ is a finitely generated $G$-module and there exists an isomorphism of $\\hat \\ZZ[G]$-modules\n\\[ \\hat\\ZZ\\otimes_\\ZZ X_\\ZZ(A_F) \\cong {\\rm Sel}(A_F)^\\vee\\]\nthat is unique up to automorphisms that induce the identity map on both the submodule $X_\\ZZ(A_F)_{\\rm tor} = \\sha(A_F)^\\vee$ and quotient module $X_\\ZZ(A_F)_{\\rm tf} = \\Hom_\\ZZ(A(F), \\ZZ)$. (Here $\\hat\\ZZ$ denotes the profinite completion of $\\ZZ$).\n\nWe identify ${\\rm Mod}^{\\rm fin}(\\ZZ_2[G])$ as an abelian subcategory of ${\\rm Mod}(\\ZZ[G])$ in the obvious way and write ${\\rm Mod}^\\ast(\\ZZ[G])$ for the associated quotient category.\n\n\\begin{proposition}\\label{prop:perfect2} Assume that $\\sha(A_F)$ is finite. Then for any perfect Selmer structure $\\mathcal{X}$ for $A$ and $F\/k$\n there exists a complex $C_S(\\mathcal{X}) = {\\rm SC}_{S}(A_{F\/k};\\mathcal{X})$ in $D^{\\rm perf}(\\ZZ[G])$ that is unique up to isomorphisms in $D^{\\rm perf}(\\Z[G])$ that induce the identity map in all degrees of cohomology and has all of the following properties.\n\\begin{itemize}\n\\item[(i)] For each prime $\\ell$ there is a canonical isomorphism in $D^{\\rm perf}(\\ZZ_\\ell[G])$\n\\[ \\ZZ_\\ell\\otimes_\\ZZ C_S(\\mathcal{X}) \\cong {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}(\\ell),\\mathcal{X}(\\infty)_\\ell).\\]\n\\item[(ii)] $C_S(\\mathcal{X})$ is acyclic outside degrees one, two and three.\n\\item[(iii)] There is a canonical identification $H^3(C_S(\\mathcal{X})) = (A(F)_{\\rm tor})^\\vee$.\n\n\\item[(iv)] In ${\\rm Mod}^\\ast(\\ZZ[G])$ there exists a canonical injective homomorphism\n\\[ H^1(C_S(\\mathcal{X})) \\to A^t(F) \\]\nthat has finite cokernel and a canonical surjective homomorphism\n\\[ H^2(C_S(\\mathcal{X})) \\to X_\\ZZ(A_F)\\]\nthat has finite kernel.\n\\item[(v)] If $\\mathcal{X}(v)\\subseteq A^t(F_v)$ for all $v$ in $S\\cap S_k^f$ and $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}$ for all $v\\notin S$, then there exists an exact sequence in ${\\rm Mod}^\\ast(\\ZZ[G])$ of the form\n\\[ 0 \\to H^1(C_S(\\mathcal{X})) \\to A^t(F) \\xrightarrow{\\Delta_{S,\\mathcal{X}}} \\bigoplus_{v \\in S\\cap S_k^f}\\frac{A^t(F_v)}{\\mathcal{X}(v)} \\to\nH^2(C_S(\\mathcal{X})) \\to X_\\ZZ(A_F)\\to 0\\]\nin which $\\Delta_{S,\\mathcal{X}}$ is the natural diagonal map.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} We write $\\hat \\ZZ$ for the profinite completion of $\\ZZ$ and for each prime $\\ell$ set $C_S(\\ell) := {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}(\\ell),\\mathcal{X}(\\infty)_\\ell)$.\n\nTo construct a suitable complex $C_S(\\mathcal{X})$ we shall use the general result of \\cite[Lem. 3.8]{bkk} with the complex $\\widehat C$ in loc. cit. taken to be the object $\\prod_\\ell C_S(\\ell)$ of $D(\\hat \\ZZ[G])$.\n\nIn fact, since $\\mathcal{X}$ satisfies the conditions (i) and (ii) in Definition \\ref{pgss def}, Proposition \\ref{prop:perfect}(i) implies that each complex $C_S(\\ell)$ belongs to $D^{\\rm perf}(\\ZZ_\\ell[G])$ and is acyclic outside degrees one, two and three and so to apply \\cite[Lem. 3.8]{bkk} it is enough to specify for each $j \\in \\{1,2,3\\}$ a finitely generated $G$-module $M^j$ together with an isomorphism of $\\hat \\Z[G]$-modules of the form $\\iota_j: \\hat \\Z\\otimes_\\Z M^j \\cong \\prod_\\ell H^j(C_S(\\ell))$.\n\nBy Proposition \\ref{prop:perfect}(ii) it is clear that one can take $M^3 = A(F)_{\\rm tor}^\\vee$ and $\\iota_3$ the canonical identification induced by the decomposition $A(F)_{\\rm tor}^\\vee = \\prod_\\ell A(F)[\\ell^\\infty]^\\vee$.\n\nTo construct suitable modules $M^1$ and $M^2$,\n\n\nwe note first that the proof of Proposition \\ref{prop:perfect}(iii) combines with the fact that $\\mathcal{X}$ satisfies condition (iii) in Definition \\ref{pgss def} to give rise to a homomorphism of $\\hat\\ZZ[G]$-modules\n\\[ \\prod_\\ell H^1(C_S(\\ell)) \\xrightarrow{\\theta_1} \\hat\\ZZ\\otimes_\\ZZ A^t(F) \\]\nwith the property that $\\ker(\\theta_1)$ is finite of $2$-power order and ${\\rm cok}(\\theta_1)$ is finite, and to a diagram of homomorphisms of $\\hat\\ZZ[G]$-modules\n\\begin{equation}\\label{derived diag}\n\\prod_\\ell H^2(C_S(\\ell)) \\xrightarrow{\\theta_2} \\prod_\\ell H^2({\\rm SC}_{\\Sigma_\\ell,\\ell}(A_{F\/k})) \\xleftarrow{\\theta_3} \\hat\\ZZ\\otimes_\\ZZ X_{\\ZZ}(A_F)\\end{equation}\nin which $\\ker(\\theta_2)$ is finite whilst ${\\rm cok}(\\theta_2), \\ker(\\theta_3)$ and ${\\rm cok}(\\theta_3)$ are all finite of $2$-power order. Here for each prime number $\\ell$ we have also set $\\Sigma_\\ell:=(S\\cap S_k^f)\\cup S_k^\\ell$.\n\nIt is then straightforward to construct a commutative (pull-back) diagram of $G$-modules\n\\begin{equation}\\label{useful2 diagrams} \\begin{CD}\n M^1 @> >> A^t(F)\\\\\n @V \\iota_{11} VV @VV\\iota_{12} V\\\\\n  \\prod_\\ell H^1(C_S(\\ell)) @> \\theta_1>> \\hat\\ZZ\\otimes_\\ZZ A^t(F)\\end{CD}\\end{equation}\nin which $M^1$ is finitely generated, the upper horizontal arrow has finite kernel of $2$-power order and finite cokernel, the morphism $\\iota_{12}$ is the natural inclusion and the morphism $\\iota_{11}$ induces an isomorphism of $\\hat\\ZZ[G]$-modules $\\iota_1$ of the required sort.\n\nIn a similar way, there is a pull-back diagram of $G$-modules\n\\begin{equation*} \\begin{CD}\n M_2 @> \\theta_2' >> \\theta_3(X_{\\ZZ}(A_F))\\\\\n @V \\iota_{21} VV @VV\\iota_{22} V\\\\\n \\prod_\\ell H^2(C_S(\\ell)) @> \\theta_2 >> \\prod_\\ell H^2({\\rm SC}_{\\Sigma_\\ell,\\ell}(A_{F\/k}))\\end{CD}\\end{equation*}\nin which $M_2$ is finitely generated, $\\iota_{22}$ is the natural inclusion, $\\ker(\\theta_2')$ is finite, ${\\rm cok}(\\theta_2')$ is finite of $2$-power order and the morphism $\\iota_{21}$ induces a short exact sequence\n\\[0\\to \\hat \\ZZ\\otimes_\\ZZ M_2 \\to \\prod_\\ell H^2(C_S(\\ell)) \\to M_2' \\to 0 \\]\nin which $M_2'$ is finite of $2$-power order. Then, since $\\hat \\ZZ$ is a flat $\\ZZ$-module, one has\n\\[ {\\rm Ext}^1_{G}(M_2',M_2) = \\hat\\ZZ\\otimes_\\ZZ{\\rm Ext}^1_{G}(M_2',M_2) = {\\rm Ext}^1_{\\hat \\ZZ[G]}(M_2',\\hat\\ZZ\\otimes_\\ZZ M_2)\\]\nand so there exists an exact commutative diagram of $G$-modules\n\\[ \\begin{CD}\n0 @> >> \\hat \\ZZ\\otimes_\\ZZ M_2 @> \\iota_{21} >> \\prod_\\ell H^2(C_S(\\ell)) @> >>  M_2' @> >> 0\\\\\n& & @A AA @A\\iota_2 AA @\\vert\\\\\n0 @> >> M_2 @> >> M^2 @> >>  M_2' @> >> 0\\end{CD}\\]\nin which the left hand vertical arrow is the natural inclusion and $M^2$ is finitely generated.\n\nIt is then clear that $\\iota_2$ induces an isomorphism $\\hat\\ZZ\\otimes_\\ZZ M^2 \\cong \\prod_\\ell H^2(C_S(\\ell))$ and that the diagram\n\\[ M^2 \\xleftarrow{\\iota_{21}} M_2 \\xrightarrow{\\theta_2'} \\theta_3(X_{\\ZZ}(A_F)) \\xleftarrow{\\theta_3} X_{\\ZZ}(A_F)\\]\nconstitutes a morphism in ${\\rm Mod}^\\ast(\\ZZ[G])$. This morphism is surjective, has finite kernel and lies in a commutative diagram in ${\\rm Mod}^\\ast(\\ZZ[G])$\n\\begin{equation}\\label{derived diag2} \\begin{CD} M^2 @> >> X_\\ZZ(A_F)\\\\\n              @V \\iota_2 VV @VV V\\\\\n              \\prod_{\\ell}H^2(C_S(\\ell)) @> >> \\hat\\ZZ\\otimes_\\ZZ X_\\ZZ(A_F)\\end{CD}\\end{equation}\nin which the right hand vertical arrow is the inclusion map and the lower horizontal arrow corresponds to the diagram (\\ref{derived diag}).\n\nThese observations show that we can apply \\cite[Lem. 3.8]{bkk} in the desired way in order to obtain a complex $C_S(\\mathcal{X})$ in  $D^{\\rm perf}(\\ZZ[G])$ that has $H^j({\\rm SC}_{S}(A_{F\/k};\\mathcal{X})) = M^j$ for each $j$ in $\\{1,2,3\\}$ and satisfies all of the stated properties in claims (i)-(iv).\n\nTurning to claim (v) we note that the given conditions on the modules $\\mathcal{X}(v)$ imply that for each $v$ in $S\\cap S_k^f$ there is a direct sum decomposition of finite modules\n\\[ \\frac{A^t(F_v)}{\\mathcal{X}(v)} = \\frac{A^t(F_v)^\\wedge_{\\ell(v)}}{\\mathcal{X}(v)} \\oplus \\bigoplus_{\\ell \\not= \\ell(v)}A^t(F_v)^\\wedge_\\ell\\]\nand hence also a direct sum decomposition over all primes $\\ell$ of the form\n\\[ \\bigoplus_{v \\in S\\cap S_k^f} \\frac{A^t(F_v)}{\\mathcal{X}(v)} = \\bigoplus_\\ell \\left( \\bigoplus_{v \\in S_k^\\ell}\\frac{A^t(F_v)^\\wedge_{\\ell(v)}}{\\mathcal{X}(v)} \\oplus \\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^\\ell}A^t(F_v)^\\wedge_\\ell\\right).\\]\n\nThis shows that the kernel and cokernel of the map $\\Delta_{S,\\mathcal{X}}$ in claim (v) respectively coincide with the intersection over all primes $\\ell$ of the kernel and the direct sum over all primes $\\ell$ of the cokernel of the diagonal map\n\\[ A^t(F) \\to \\bigoplus_{v \\in S_k^\\ell}\\frac{A^t(F_v)^\\wedge_\\ell}{\\mathcal{X}(v)} \\oplus \\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^\\ell}A^t(F_v)^\\wedge_\\ell\\]\nthat occurs in the sequence (\\ref{useful1}) (with $p$ replaced by $\\ell$ and $X$ by $\\mathcal{X}(\\ell)$).\n\nGiven this fact, the exact sequence follows from the commutativity of the diagrams (\\ref{useful2 diagrams}) and (\\ref{derived diag2}) and the exactness of the sequence (\\ref{useful1}).\n\\end{proof}\n\n\\subsection{Global differentials and perfect Selmer structures}\\label{perf sel sect}\n\nWith a view to the subsequent formulation (in \\S\\ref{statement of conj section}) of our central conjecture we explain how a choice of global differentials gives rise to a natural perfect Selmer structure for $A$ and $F\/k$.\n\nIn the sequel we shall for a natural number $m$ write $[m]$ for the (ordered) set of integers $i$ that satisfy $1 \\le i\\le m$.\n\n\n\\subsubsection{}\\label{gamma section}For each $v$ in $S_k^\\RR$ we fix ordered $\\ZZ$-bases\n\\[ \\{\\gamma_{v,a}^+: a\\in [d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)^{c=1}$ and\n\\[ \\{\\gamma_{v,a}^-: a\\in [d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)^{c=-1}$, where $c$ denotes complex conjugation.\n\nFor each $v$ in $S_k^\\CC$ we fix an ordered $\\ZZ$-basis\n\\[ \\{\\gamma_{v,a}: a\\in [2d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$.\n\n\nFor each $v$ in $S_k^\\infty$ we then fix $\\tau_v\\in G$ with $\\tau_v(\\sigma_v')=c\\circ\\sigma_v'$ and write $H_v(\\gamma_\\bullet)$ for the free $G$-module with basis\n\\begin{equation}\\label{gamma basis}\\begin{cases} \\{ (1+\\tau_v)\\sigma_v'\\otimes \\gamma^+_{v,a} + (1-\\tau_v)\\sigma_v'\\otimes \\gamma^-_{v,a}: a \\in [d]\\}, &\\text{ if $v$ is real}\\\\\n                 \\{\\sigma_v'\\otimes\\gamma_{v,a}:a \\in [2d]\\}, &\\text{ if $v$ is complex.}\\end{cases}\\end{equation}\n\nThe direct sum\n\\[ H_\\infty(\\gamma_\\bullet) := \\bigoplus_{v \\in S_k^\\infty}H_v(\\gamma_\\bullet)\\]\nis then a free $G$-submodule of $H_\\infty(A_{F\/k})$ of finite $2$-power index.\n\nTo specify an ordered $\\ZZ[G]$-basis of $H_\\infty(\\gamma_\\bullet)$ we fix an ordering of $S_k^\\infty$ and then order the union of the sets\n (\\ref{gamma basis}) lexicographically.\n\n\\subsubsection{}\\label{perf sel construct}We next fix a N\\'eron model $\\mathcal{A}^t$ for $A^t$ over $\\mathcal{O}_k$ and, for each non-archimedean place $v$ of $k$, a N\\'eron model $\\mathcal{A}_v^t$ for $A^t_{\/k_v}$ over $\\mathcal{O}_{k_v}$.\n\nFor any subfield $E$ of $k$ and any non-archimedean place $v$ of $E$ we set $\\mathcal{O}_{F,v}:=\\prod_{w'\\in S_F^v}\\mathcal{O}_{F_{w'}}$.\n\nFor each non-archimedean place $v$ of $k$ we then set\n\\begin{equation}\\label{mathcalD} \\mathcal{D}_F(\\mathcal{A}^t_v) :=  \\mathcal{O}_{F,v}\\otimes_{\\mathcal{O}_{k_v}}\\Hom_{\\mathcal{O}_{k_v}}(H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t}), \\mathcal{O}_{k_v}).\\end{equation}\n\n\nWe finally fix an ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of the space of invariant differentials\n\\[ H^0(A^t_F,\\Omega^1_{A^t_F}) \\cong F\\otimes_k H^0(A^t,\\Omega^1_{A^t})\\]\nand write $\\mathcal{F}(\\omega_\\bullet)$ for the $G$-module generated by the elements of $\\omega_\\bullet$. In the sequel we often identify $\\omega_\\bullet$ with its dual ordered $\\QQ[G]$-basis in $\\Hom_{F}(H^0(A^t_F,\\Omega^1_{A^t_F}),F)$ and $\\mathcal{F}(\\omega_\\bullet)$ with the $G$-module generated by this dual basis.\n\nIn the sequel, for any subfield $E$ of $k$ and any place $v$ in $S_E^f$ we set $F_v:=\\prod_{w'\\in S_F^v}F_{w'}$.\n\nFor each non-archimedean place $v$ of $k$ we write $\\mathcal{F}(\\omega_\\bullet)_v$ for the $\\ZZ_{\\ell(v)}$-closure of the image of $\\mathcal{F}(\\omega_\\bullet)$ in $F_{v}\\otimes_k\\Hom_{k}(H^0(A^t,\\Omega^1_{A^t}), k)$ and\n\\begin{equation*}\\label{classical exp} {\\rm exp}_{A^t,F_v}: F_v\\otimes_k \\Hom_k(H^0(A^t,\\Omega^1_{A^t}),k) \\cong \\Hom_{F_v}(H^0(A^t_{F_v},\\Omega^1_{A^t_{F_v}}),F_v) \\cong \\QQ_{\\ell(v)}\\cdot A^t(F_v)^\\wedge_{\\ell(v)}\\end{equation*}\nfor the exponential map of $A^t_{F_v}$ relative to some fixed $\\mathcal{O}_{k_v}$-basis of $H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t})$.\n\nThen, if necessary after multiplying each element of $\\omega_\\bullet$ by a suitable natural number, we may, and will, assume that the following conditions are satisfied:\n\n\\begin{itemize}\n\\item[(i$_{\\omega_\\bullet}$)] for each $v$ in $S_k^f$ one has $\\mathcal{F}(\\omega_\\bullet)_{v}\\subseteq \\mathcal{D}_F(\\mathcal{A}_v^t)$;\n\\item[(ii$_{\\omega_\\bullet}$)] for each $v$ in $S\\cap S_k^f$, the map ${\\rm exp}_{A^t,F_v}$ induces an isomorphism of $\\mathcal{F}(\\omega_\\bullet)_{v}$ with a submodule of $A^t(F_v)$.\n\\end{itemize}\n\n\n\nWe then define $\\mathcal{X}=\\mathcal{X}_S(\\omega_\\bullet) = \\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ to be the perfect Selmer structure for $A$, $F\/k$ and $S$ that has the following properties:\n\\begin{itemize}\n\\item[(i$_\\mathcal{X}$)] If $v\\in S_k^\\infty$, then $\\mathcal{X}(v) = H_v(\\gamma_\\bullet)$.\n\\item[(ii$_\\mathcal{X}$)] If $v \\in S\\cap S_k^f$, then $\\mathcal{X}(v) = {\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)$.\n\\item[(iii$_\\mathcal{X}$)] If $v \\notin S$, then $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}$.\n\\end{itemize}\n\n\\begin{remark}{\\em This specification does define a perfect Selmer structure for $A$ and $F\/k$ since if $v$ does not belong to $S$, then the $G$-module $A^t(F_v)^\\wedge_{\\ell(v)}$ is cohomologically-trivial (by Lemma \\ref{useful prel}(ii) below).} \\end{remark}\n\n\\begin{remark}\\label{can structure groups}{\\em The perfect Selmer structure $\\mathcal{X}_S(\\omega_\\bullet) = \\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined above satisfies the conditions of Proposition \\ref{prop:perfect2}(v). As a consequence, if one ignores finite modules of $2$-power order, then the cohomology modules of the Selmer complex\n$C_S(\\mathcal{X}(\\omega_\\bullet)) = {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}_S(\\omega_\\bullet))$ can be described as follows: \\\n\n\\noindent{} - $H^1(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is the submodule of $A^t(F)$ comprising all elements $x$ with the property that, for each $v$ in $S$, the image of $x$ in $A^t(F_v)$ belongs to the subgroup ${\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)$.\n\n\\noindent{} - $H^2(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is an extension of the integral Selmer group $X_\\ZZ(A_F)$ by the (finite) cokernel of the diagonal map $A^t(F) \\to \\bigoplus_{v \\in S}\\bigl(A^t(F_v)\/{\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)\\bigr)$.\n\n\\noindent{} - $H^3(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is equal to $(A(F)_{\\rm tor})^\\vee$.}\\end{remark}\n\n\n\n\n\n\n\\section{The refined Birch and Swinnerton-Dyer Conjecture}\\label{ref bsd section}\n\nIn this section we formulate (as Conjecture \\ref{conj:ebsd}) a precise refinement of the Birch and Swinnerton-Dyer Conjecture.\n\n\\subsection{Relative $K$-theory} For the reader's convenience, we first quickly review some relevant facts of algebraic $K$-theory.\n\n\\subsubsection{}\\label{Relative $K$-theory}\n\nFor a Dedekind domain $R$ with field of fractions $F$, an $R$-order $\\mathfrak{A}$ in a finite dimensional separable $F$-algebra $A$ and a field extension $E$ of $F$ we set $A_E := E\\otimes_F A$.\n\nThe relative algebraic $K_0$-group $K_0(\\mathfrak{A},A_E)$ of the ring inclusion $\\mathfrak{A}\\subset A_E$ is described explicitly in terms of generators and relations by Swan in \\cite[p. 215]{swan}.\n\nFor any extension field $E'$ of $E$ there exists a canonical commutative diagram\n\\begin{equation} \\label{E:kcomm}\n\\begin{CD} K_1(\\mathfrak{A}) @> >> K_1(A_{E'}) @> \\partial_{\\mathfrak{A},A_{E'}} >> K_0(\\mathfrak{A},A_{E'}) @> \\partial'_{\\mathfrak{A},A_{E'}} >> K_0(\\mathfrak{A})\\\\\n@\\vert @A\\iota AA @A\\iota' AA @\\vert\\\\\nK_1(\\mathfrak{A}) @> >> K_1(A_E) @> \\partial_{\\mathfrak{A},A_E}  >> K_0(\\mathfrak{A},A_E) @> \\partial'_{\\mathfrak{A},A_E}  >> K_0(\\mathfrak{A})\n\\end{CD}\n\\end{equation}\nin which the upper and lower rows are the respective long exact sequences in relative $K$-theory of the inclusions $\\mathfrak{A}\\subset A_E$ and $\\mathfrak{A}\\subset A_{E'}$ and both of the vertical arrows are injective and induced by the inclusion $A_E \\subseteq A_{E'}$. (For more details see \\cite[Th. 15.5]{swan}.)\n\n\nIn particular, if $R = \\ZZ$ and for each prime $\\ell$ we set $\\mathfrak{A}_\\ell := \\ZZ_\\ell\\otimes_\\ZZ \\mathfrak{A}$ and $A_\\ell:=\n\\QQ_\\ell\\otimes _\\QQ A$, then we can regard each group $K_0(\\mathfrak{A}_\\ell,A_\\ell)$ as a subgroup of $K_0(\\mathfrak{A},A)$ by means of the canonical composite homomorphism\n\\begin{equation}\\label{decomp}\n\\bigoplus_\\ell K_0(\\mathfrak{A}_\\ell,A_\\ell) \\cong K_0(\\mathfrak{A},A)\\subset K_0(\\mathfrak{A},A_\\RR),\n\\end{equation}\nwhere $\\ell$ runs over all primes, the isomorphism is as described in the discussion following \\cite[(49.12)]{curtisr} and the inclusion is induced by the relevant case of $\\iota'$.\n\nFor an element $x$ of $K_0(\\mathfrak{A},A)$ we write $(x_\\ell)_\\ell$ for its image in $\\bigoplus_\\ell K_0(\\mathfrak{A}_\\ell,A_\\ell)$ under the isomorphism in (\\ref{decomp}).\n\nThen, if $G$ is a finite group and $E$ is a field of characteristic zero, taking reduced norms over the semisimple algebra $E[G]$ induces (as per the discussion in \\cite[\\S 45A]{curtisr}) an injective homomorphism\n\\[ {\\rm Nrd}_{E[G]}: K_1(E[G]) \\to \\zeta(E[G])^\\times. \\]\nThis homomorphism is bijective if $E$ is either algebraically closed or complete.\n\n\n\n\\subsubsection{}\\label{nad sec} We shall also use a description of $K_0(\\mathfrak{A},A_E)$ in terms of the formalism of `non-abelian determinants' that is given by Fukaya and Kato in \\cite[\\S1]{fukaya-kato}.\n\nWe recall, in particular, that any pair comprising an object $C$ of $D^{\\rm perf}(\\mathfrak{A})$ and a morphism of non-abelian determinants $\\theta: {\\rm Det}_{A_E}(E\\otimes_R C) \\to {\\rm Det}_{A_E}(0)$ gives rise to a canonical element of $K_0(\\mathfrak{A},A_E)$ that we shall denote by $\\chi_\\mathfrak{A}(C,\\theta)$.\n\nIf $E\\otimes_RC$ is acyclic, then one obtains in this way a canonical element $\\chi_\\mathfrak{A}(C,0)$ of $K_0(\\mathfrak{A},A_E)$.\n\nMore generally, if $E\\otimes_RC$ is acyclic outside of degrees $a$ and $a+1$ for any integer $a$, then a choice of isomorphism of $A_E$-modules $h: E\\otimes_RH^a(C) \\cong E\\otimes_RH^{a+1}(C)$ gives rise to a morphism $h^{\\rm det}: {\\rm Det}_{A_E}(E\\otimes_R C) \\to {\\rm Det}_{A_E}(0)$ of non-abelian determinants and we set\n\\[ \\chi_\\mathfrak{A}(C,h) := \\chi_\\mathfrak{A}(C,h^{\\rm det}).\\]\n\nWe recall the following general result concerning these elements (which follows directly from \\cite[Lem. 1.3.4]{fukaya-kato}) since it will be used often in the sequel.\n\n\\begin{lemma}\\label{fk lemma} Let $C_1 \\to C_2 \\to C_3 \\to C_1[1]$ be an exact triangle in $D^{\\rm perf}(\\mathfrak{A})$ that satisfies the following two conditions:\n\\begin{itemize}\n\\item[(i)] there exists an integer $a$ such that each $C_i$ is acyclic outside degrees $a$ and $a+1$;\n\\item[(ii)] there exists an exact commutative diagram of  $A_E$-modules\n\\[\\begin{CD}\n0 @> >> E\\otimes_R H^a(C_1) @> >> E\\otimes_R H^a(C_2) @> >> E\\otimes_R H^a(C_3) @> >> 0\\\\\n@. @V h_1VV @V h_2VV @V h_3VV \\\\\n0 @> >> E\\otimes_R H^{a+1}(C_1) @> >> E\\otimes_R H^{a+1}(C_2) @> >> E\\otimes_R H^{a+1}(C_3) @> >> 0\\end{CD}\\]\nin which each row is induced by the long exact cohomology sequence of the given exact triangle and each map $h_i$ is bijective.\n\\end{itemize}\n\nThen in $K_0(\\mathfrak{A},A_E)$ one has $\\chi_\\mathfrak{A}(C_2,h_2) = \\chi_\\mathfrak{A}(C_1,h_1) + \\chi_\\mathfrak{A}(C_3,h_3)$.\n\\end{lemma}\n\n\n\n\\begin{remark}\\label{comparingdets}{\\em If $\\mathfrak{A}$ is commutative, then $K_0(\\mathfrak{A},A_E)$ identifies with the multiplicative group of invertible $\\mathfrak{A}$-submodules of $A_E$. If, in this case, $C$ is acyclic outside degrees one and two, then for any isomorphism of $A_E$-modules $h: E\\otimes_RH^1(C)\\to E\\otimes_RH^2(C)$ one finds that the element $\\chi_{\\mathfrak{A}}(C,h)$ defined above corresponds under this identification to the inverse of the ideal $\\vartheta_{h}({\\rm Det}_{\\mathfrak{A}}(C))$ that is defined in \\cite[Def. 3.1]{bst}.}\\end{remark}\n\nFor convenience, we shall often abbreviate the notations $\\chi_{\\ZZ[G]}(C,h)$ and $\\chi_{\\ZZ_p[G]}(C,h)$ to $\\chi_G(C,h)$ and $\\chi_{G,p}(C,h)$ respectively.\n\nWhen the field $E$ is clear from context, we also write $\\partial_{G}$, $\\partial'_{G}$, $\\partial_{G,p}$ and $\\partial'_{G,p}$ in place of $\\partial_{\\ZZ[G],E[G]}$, $\\partial'_{\\ZZ[G],E[G]}$, $\\partial_{\\ZZ_p[G],E[G]}$ and $\\partial'_{\\ZZ_p[G],E[G]}$ respectively.\n\n\n\n\n\n\n\n\\subsection{Statement of the conjecture}\\label{statement of conj section}\n\nIn the sequel we fix a finite set of places $S$ of $k$ as in \\S\\ref{selmer section}. We also fix cycles $\\gamma_\\bullet$ and differentials $\\omega_\\bullet$ as in \\S\\ref{perf sel sect}.\n\n\n\\subsubsection{\n\nWe write $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ for the element of $K_1(\\RR[G])$ that is represented by the matrix of the canonical `period' isomorphism of $\\RR[G]$-modules\n\\begin{multline*} \\RR\\otimes_\\ZZ H_\\infty(\\gamma_\\bullet) = \\RR\\otimes_\\ZZ \\bigoplus_{v \\in S_k^\\infty}H^0(k_v,Y_{v,F}\\otimes_{\\ZZ}H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ))\\\\\n \\cong \\RR\\otimes_{\\QQ} \\Hom_F(H^0(A_F^t,\\Omega^1_{A_F^t}),F),\\end{multline*}\nwith respect to the ordered $\\ZZ[G]$-basis of $H_\\infty(\\gamma_\\bullet)$ specified in \\S\\ref{gamma section} and the ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of $ \\Hom_F(H^0(A_F^t,\\Omega^1_{A_F^t}),F)$.\n\nThis element $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ constitutes a natural `$K$-theoretical period' and can be explicitly computed in terms of the classical periods that are associated to $A$ (see Lemma \\ref{k-theory period} below).\n\n\nTo take account of the local behaviour of the differentials $\\omega_\\bullet$ we define a $G$-module\n\\[ \\mathcal{Q}(\\omega_\\bullet)_S := \\bigoplus_{v \\notin S} \\mathcal{D}_F(\\mathcal{A}_v^t)\/\\mathcal{F}(\\omega_\\bullet)_v,\\]\nwhere $v$ runs over all places of $k$ that do not belong to $S$.\n\nIt is easily seen that almost all terms in this direct sum vanish and hence that $\\mathcal{Q}(\\omega_\\bullet)_S$ is finite. This $G$-module is also cohomologically-trivial since $\\mathcal{D}_F(\\mathcal{A}_v^t)$ and $\\mathcal{F}(\\omega_\\bullet)_v$ are both free $\\ZZ_{\\ell(v)}[G]$-modules for each $v$ outside $S$.\n\nWe can therefore define an object of $D^{\\rm perf}(\\ZZ[G])$ by setting\n\\[ {\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}) := {\\rm SC}_S(A_{F\/k},\\mathcal{X}_S(\\omega_\\bullet)) \\oplus \\mathcal{Q}(\\omega_\\bullet)_S[0],\\]\nwhere we abbreviate the perfect Selmer structure $\\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined by the conditions (i$_\\mathcal{X}$), (ii$_\\mathcal{X}$) and (iii$_\\mathcal{X}$) in \\S\\ref{perf sel sect} to $\\mathcal{X}_S(\\omega_\\bullet)$.\n\nWe next write\n \\[\nh_{A,F}: A(F)\\times A^t(F) \\to \\RR\n\\]\nfor the classical N\\'eron-Tate height-pairing for $A$ over $F$.\n\nThis pairing is non-degenerate and hence, assuming $\\sha(A_{F})$ to be finite, it combines with the properties of the Selmer complex\n${\\rm SC}_{S}(A_{F\/k},\\mathcal{X}(\\omega_\\bullet))$ established in Proposition \\ref{prop:perfect}(ii) to induce a canonical isomorphism of $\\RR[G]$-modules\n\\begin{multline*} \\label{height triv}\nh_{A,F}': \\RR\\otimes_\\ZZ H^1({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k})) = \\RR\\otimes_\\ZZ A^t(F)\\\\ \\cong \\RR\\otimes_\\ZZ\\Hom_\\ZZ(A(F),\\ZZ) =   \\RR\\otimes_\\ZZ H^2({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k})).\\end{multline*}\nThis isomorphism then gives rise via the formalism recalled in \\S\\ref{nad sec} to a canonical element\n\\[ \\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F}) := \\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h'_{A,F})\\]\nof the relative algebraic $K$-group $K_0(\\ZZ[G],\\RR[G])$.\n\nOur conjecture will predict an explicit formula for this element in terms of Hasse-Weil-Artin $L$-series.\n\n\n\n\\subsubsection{}For every prime $\\ell$ the reduced norm maps ${\\rm Nrd}_{\\QQ_\\ell[G]}$ and ${\\rm Nrd}_{\\CC_\\ell[G]}$ discussed in \\S\\ref{Relative $K$-theory} are bijective and so there exists a composite homomorphism\n\\begin{equation}\\label{G,O hom} \\delta_{G,\\ell}: \\zeta(\\CC_\\ell[G])^\\times \\to K_1(\\CC_\\ell[G]) \\xrightarrow{\\partial_{\\ZZ_\\ell[G],\\CC_\\ell[G]}}\nK_0(\\ZZ_\\ell[G],\\CC_\\ell[G]) \\end{equation}\nin which the first map is the inverse of ${\\rm Nrd}_{\\CC_\\ell[G]}$. This homomorphism maps $\\zeta(\\QQ_\\ell[G])^\\times$ to the subgroup $K_0(\\ZZ_\\ell[G],\\QQ_\\ell[G])$ of $K_0(\\ZZ[G],\\QQ[G])$.\n\nIf now $v$ is any place of $k$ that does not belong to $S$, then $v$ is unramified in $F\/k$ and so the finite $G$-modules $$\\kappa_{F_v}:=\\prod_{w'\\in S_F^v}\\kappa_{F_{w'}}\\,\\,\\,\\, \\text{ and } \\,\\,\\,\\,\\tilde A^t_v(\\kappa_{F_v}):=\\prod_{w'\\in S_F^v}\\tilde A^t(\\kappa_{F_{w'}})$$ are both cohomologically-trivial by  Lemma \\ref{useful prel}(i) below. Here for any place $w'$ in $S_F^v$, $\\tilde A^t$ denotes the reduction of $A^t_{\/F_{w'}}$ to $\\kappa_{F_{w'}}$.\n\nFor any such $v$ we may therefore define an element of the subgroup $K_0(\\ZZ_{\\ell(v)}[G],\\QQ_{\\ell(v)}[G])$ of $K_0(\\ZZ[G],\\QQ[G])$ by setting\n\\begin{equation}\\label{localFM} \\mu_{v}(A_{F\/k}) := \\chi_{G,\\ell(v)}\\bigl(\\kappa_{F_v}^d[0]\\oplus\\tilde A^t_v(\\kappa_{F_v})_{\\ell(v)}[-1],0\\bigr)-\\delta_{G,\\ell(v)}(L_v(A,F\/k))\\end{equation}\nwhere $L_v(A,F\/k)$ is the element of $\\zeta(\\QQ[G])^\\times$ that is equal to the value at $z=1$ of the $\\zeta(\\CC[G])$-valued $L$-factor at $v$ of the motive $h^1(A_{F})(1)$, regarded as defined over $k$ and with coefficients $\\QQ[G]$, as discussed in \\cite[\\S4.1]{bufl99}.\n\nThe sum\n\\[ \\mu_{S}(A_{F\/k}) := \\sum_{v\\notin S}\\mu_{v}(A_{F\/k})\\]\nwill play an important role in our conjecture.\n\nWe shall refer to this sum as the `Fontaine-Messing correction term' for the data $A, F\/k$ and $S$ since, independently of any conjecture, the theory developed by Fontaine and Messing in \\cite{fm} implies that $\\mu_{v}(A_{F\/k})$ vanishes for all but finitely many $v$ and hence that $\\mu_S(A_{F\/k})$ is a well-defined element of $K_0(\\ZZ[G],\\QQ[G])$. (For details see Lemma \\ref{fm} below).\n\n\\subsubsection{}We write $\\widehat{G}$ for the set of irreducible complex characters of $G$. In the sequel, for each $\\psi$ in $\\widehat{G}$ we fix a $\\CC[G]$-module $V_\\psi$ of character $\\psi$.\n\nWe recall that a character $\\psi$ in $\\widehat{G}$ is said to be `symplectic' if the subfield of\n$\\bc$ that is generated by the values of $\\psi$ is totally real\nand $\\End_{\\br [G]}(V_\\psi)$ is isomorphic to the division ring\nof real Quaternions. We write $\\widehat{G}^{\\rm s}$ for the subset of\n$\\widehat{G}$ comprising such characters.\n\nFor each $\\psi$ in $\\widehat{G}$ we write $\\check\\psi$ for its contragredient character and\n\\[ e_\\psi:=\\frac{\\psi(1)}{|G|}\\sum_{g\\in G}\\psi(g^{-1})g\\]\nfor the associated central primitive idempotents of $\\CC[G]$.\n\nThese idempotents induce an identification of $\\zeta(\\CC[G])$ with $\\prod_{\\widehat{G}}\\CC$ and we write $x = (x_\\psi)_\\psi$ for the corresponding decomposition of each element $x$ of $\\zeta(\\CC[G])$.\n\nFor each $\\psi$ in $\\widehat{G}$ we write $L_{S}(A,\\psi,z)$ for the Hasse-Weil-Artin $L$-series of $A$ and $\\psi$, truncated by removing the Euler factors corresponding to places in $S$.\n\nWe can now state the central conjecture of this article.\n\n\\begin{conjecture}\\label{conj:ebsd}\nThe following claims are valid.\n\\begin{itemize}\n\\item[(i)] The group $\\sha(A_F)$ is finite.\n\\item[(ii)] For all $\\psi$ in $\\widehat{G}$ the function $L(A,\\psi,z)$ has an analytic continuation to $z=1$ where it has a zero of order\n $\\psi(1)^{-1}\\cdot {\\rm dim}_{\\CC}(e_\\psi(\\CC\\otimes_\\ZZ A^t(F)))$.\n\\item[(iii)] For all $\\psi$ in $\\widehat{G}^{\\rm s}$ the leading coefficient $L^*_S(A,\\psi,1)$ at $z=1$ of the function $L_S(A,\\psi,z)$ is a strictly positive real number. In particular, there exists a unique element $L^*_{S}(A_{F\/k},1)$ of $K_1(\\RR[G])$ with\n\\[ {\\rm Nrd}_{\\RR[G]}(L_S^*(A_{F\/k},1))_\\psi = L_S^*(A,\\check\\psi,1)\\]\nfor all $\\psi$ in $\\widehat{G}$.\n\\item[(iv)] In $K_0(\\ZZ[G],\\RR[G])$ one has\n\\[ \\partial_G\\left(\\frac{L_S^*(A_{F\/k},1)}{\\Omega_{\\omega_\\bullet}(A_{F\/k})}\\right) = \\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F}) + \\mu_{S}(A_{F\/k}).\\]\n\\end{itemize}\n\\end{conjecture}\n\nIn the sequel we shall refer to this conjecture as the `Birch and Swinnerton-Dyer Conjecture for the pair $(A,F\/k)$' and abbreviate it to ${\\rm BSD}(A_{F\/k})$.\n\n\n\n\\begin{remark}{\\em The assertion of ${\\rm BSD}(A_{F\/k})$(i) is the celebrated Shafarevich-Tate conjecture. The quantity $\\psi(1)^{-1}\\cdot {\\rm dim}_{\\CC}(e_\\psi(\\CC\\otimes_\\ZZ A^t(F)))$ is equal to the multiplicity with which the character $\\psi$ occurs in the rational representation $\\QQ\\otimes_\\ZZ A^t(F)$ of $G$ (and hence to the right hand side of the equality (\\ref{dg equality})) and so the assertion of ${\\rm BSD}(A_{F\/k})$(ii) coincides with a conjecture of Deligne and Gross (cf. \\cite[p. 127]{rohrlich}).}\\end{remark}\n\n\\begin{remark}{\\em Write $\\tau$ for complex conjugation. Then, by the Hasse-Schilling-Maass Norm Theorem (cf. \\cite[(7.48)]{curtisr}), the image of ${\\rm Nrd}_{\\RR[G]}$ is the subset of $\\prod_{\\widehat{G}}\\CC^\\times$ comprising $x$ with the property that $x_{\\psi^\\tau} = \\tau(x_\\psi)$ for all $\\psi$ in $\\widehat{G}$ and also that $x_\\psi$ is a strictly positive real number for all $\\psi$ in $\\widehat{G}^{\\rm s}$. This means that the second assertion of ${\\rm BSD}(A_{F\/k})$(iii) follows immediately from the first assertion, the injectivity of ${\\rm Nrd}_{\\RR[G]}$ and the fact that $L_S^*(A,\\psi^\\tau,1) = \\tau(L_S^*(A,\\psi,1))$ for all $\\psi$ in $\\widehat{G}$.\n\nThe first assertion of ${\\rm BSD}(A_{F\/k})$(iii) is itself motivated by the fact that if $\\psi$ belongs to $\\widehat{G}^{\\rm s}$, and $[\\psi]$ denotes the associated Artin motive over $k$, then one can show that $L^*_S(A,\\psi,1)$ is a strictly positive real number whenever the motive $h^1(A)\\otimes [\\psi]$ validates the `Generalized Riemann Hypothesis' discussed by Deninger in \\cite[(7.5)]{den}. However, since this fact does not itself provide any more evidence for ${\\rm BSD}(A_{F\/k})$(iii) we omit the details.} \\end{remark}\n\n\\begin{remark}\\label{weaker BSD}{\\em It is possible to formulate a version of ${\\rm BSD}(A_{F\/k})$ that omits claim (iii) and hence avoids any possible reliance on the validity of the Generalized Riemann Hypothesis. To do this we recall that the argument of \\cite[\\S4.2, Lem. 9]{bufl99} constructs a canonical `extended boundary homomorphism' of relative $K$-theory $\\delta_G: \\zeta(\\RR[G])^\\times \\to K_0(\\ZZ[G],\\RR[G])$ that lies in a commutative diagram\n\n\\[ \\xymatrix{\nK_1(\\RR[G]) \\ar@{^{(}->}[d]^{{\\rm Nrd}_{\\RR[G]}} \\ar[rr]^{\\hskip -0.2truein\\partial_G} & & K_0(\\ZZ[G],\\RR[G])\\\\\n\\zeta(\\RR[G])^\\times . \\ar[urr]^{\\delta_G}}\\]\n\nHence, to obtain a version of the conjecture that omits claim (iii) one need only replace the term on the left hand side of the equality in claim (iv) by the difference\n\\[ \\delta_G\\bigl(\\calL_S^*(A_{F\/k},1)\\bigr) - \\partial_G\\bigl(\\Omega_{\\omega_\\bullet}(A_{F\/k})\\bigr)\\]\nwhere $\\calL_S^*(A_{F\/k},1)$ denotes the element of $\\zeta(\\RR[G])^\\times$ with $\\calL_S^*(A_{F\/k},1)_\\psi = L_S^*(A,\\check\\psi,1)$ for all $\\psi$ in $\\widehat{G}$.}\\end{remark}\n\n\n\\begin{remark}\\label{rbsd etnc rem}{\\em The approach developed by Wuthrich and the present authors in \\cite[\\S4]{bmw} can be extended to show that\n the weaker version of ${\\rm BSD}(A_{F\/k})$ discussed in the last remark is equivalent to the validity of the equivariant Tamagawa number conjecture for the pair $(h^1(A_F)(1),\\ZZ[G])$, as formulated in \\cite[Conj. 4]{bufl99} (for details see Appendix A). Taken in conjunction with the results of Venjakob and the first author in \\cite{BV2}, this observation implies that the study of ${\\rm BSD}(A_{F\/k})$ and its consequences is relevant if one wishes to properly understand the content of the main conjecture of non-commutative Iwasawa theory, as formulated by Coates et al in \\cite{cfksv}.}\\end{remark}\n\n\\begin{remark}\\label{cons1}{\\em If, for each prime $\\ell$, we fix an isomorphism of fields $\\CC\\cong \\CC_\\ell$, then the exactness of the lower row in (\\ref{E:kcomm}) with $\\mathfrak{A} = \\ZZ_\\ell[G]$ and $A_E = \\CC_\\ell[G]$ implies that the equality in ${\\rm BSD}(A_{F\/k})$(iv) determines the image of $(L^*_S(A,\\psi, 1))_{\\psi\\in \\widehat{G}}$ in $\\zeta(\\CC_\\ell[G])^\\times$ modulo the image under the reduced norm map ${\\rm Nrd}_{\\QQ_\\ell[G]}$ of $K_1(\\ZZ_\\ell[G])$. In view of the explicit description of the latter image that is obtained by Kakde in \\cite{kakde} (or, equivalently, by the methods of Ritter and Weiss in \\cite{rw}), this means ${\\rm BSD}(A_{F\/k})$(iv) implicitly incorporates families of congruence relations between the leading coefficients $L^*_S(A,\\psi, 1)$ for varying $\\psi$ in $\\widehat{G}$.}\\end{remark}\n\n\n\n\n\\begin{remark}\\label{consistency remark}{\\em The formulation of ${\\rm BSD}(A_{F\/k})$ is consistent in the following respects.\n\\begin{itemize}\n\\item[(i)] Its validity is independent of the choices of set $S$ and ordered $\\QQ[G]$-basis $\\omega_\\bullet$.\n\n\\item[(ii)] Its validity for the pair $(A,F\/k)$ implies its validity for $(A_E,F\/E)$ for any intermediate field $E$ of $F\/k$ and for $(A,E\/k)$ for any such $E$ that is Galois over $k$.\n\n\\item[(iii)] Its validity for the pair $(A,k\/k)$ is equivalent, up to sign, to the Birch and Swinnerton-Dyer Conjecture for $A$ over $k$.\n\\end{itemize}\nEach of these statements can be proven directly but also follows from the observation in Remark \\ref{rbsd etnc rem} (see Remark \\ref{consistency} for more details).}\\end{remark}\n\n\n\\begin{remark}{\\em A natural analogue of ${\\rm BSD}(A_{F\/k})$ has been formulated, and in some important cases proved, in the setting of abelian varieties over global function fields by Kakde, Kim and the first author in \\cite{bkk}.} \\end{remark}\n\n\n\n\n\n\n\nMotivated at least in part by Remark \\ref{rbsd etnc rem}, our main aim in the rest of this article will be to describe, and in important special cases provide evidence for, a range of explicit consequences that would follow from the validity of ${\\rm BSD}(A_{F\/k})$.\n\n\\subsection{$p$-components}\\label{pro-p sect} To end this section we show that the equality in ${\\rm BSD}(A_{F\/k})$(iv) can be checked by considering separately `$p$-primary components' for each prime $p$.\n\nFor each prime $p$ and each isomorphism of fields $j: \\CC\\cong \\CC_p$, the inclusion $\\RR \\subset \\CC$ combines with the functoriality of $K$-theory to induce a homomorphism\n\\[ K_1(\\RR[G]) \\to K_1(\\CC_p[G])\\]\nand also pairs with the inclusion $\\ZZ \\to \\ZZ_p$ to induce a homomorphism\n\\[ K_0(\\ZZ[G],\\RR[G]) \\to K_0(\\ZZ_p[G],\\CC_p[G]).\\]\nIn the sequel we shall, for convenience, use $j_\\ast$ to denote both of these homomorphisms as well as the inclusion $\\zeta(\\RR[G])^\\times \\to \\zeta(\\CC_p[G])^\\times$ and isomorphism $\\zeta(\\CC[G])^\\times\\cong \\zeta(\\CC_p[G])^\\times$ that are induced by the action of $j$ on coefficients.\n\n\n\\begin{lemma}\\label{pro-p lemma} Fix $\\omega_\\bullet$ and $S$ as in ${\\rm BSD}(A_{F\/k})$. Then, to prove the equality in ${\\rm BSD}(A_{F\/k})$(iv) it suffices to prove, for every prime $p$ and every isomorphism of fields $j:\\CC\\cong \\CC_p$, that\n\\begin{multline}\\label{displayed pj} \\partial_{G,p}\\left(\\frac{j_*(L_S^*(A_{F\/k},1))}{j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}\\right) = \\chi_{G,p}({\\rm SC}_S(A_{F\/k},\\mathcal{X}(p),\\mathcal{X}(\\infty)_p),h^j_{A,F})\\\\ +\\chi_{G,p}( \\mathcal{Q}(\\omega_\\bullet)_{S,p} [0],0) + \\mu_{S}(A_{F\/k})_p,\\end{multline}\nwhere we write $\\mathcal{X}$ for $\\mathcal{X}_S(\\omega_\\bullet)$ and $h^{j}_{A,F}$ for $\\CC_p\\otimes_{\\RR,j}h^{{\\rm det}}_{A,F}$\n\\end{lemma}\n\n\\begin{proof} We consider the diagonal homomorphism of abelian groups\n\\begin{equation}\\label{local iso} K_0(\\ZZ[G],\\RR[G]) \\xrightarrow{(\\prod j_*)_p}  \\prod_p\\left(\\prod_{j: \\CC\\cong \\CC_p} K_0(\\ZZ_p[G],\\CC_p[G])\\right),\\end{equation}\nwhere the products run over all primes $p$ and all choices of isomorphism $j$.\n\nThe key fact that we shall use is that this map is injective. This fact is certainly well-known but, given its importance, we shall, for completeness, prove it.\n\nWe consider the exact sequences that are given by the lower row of (\\ref{E:kcomm}) with $\\mathfrak{A}= R[G]$ and $A_E = E[G]$ for the\npairs $(R,E)=(\\ZZ,\\QQ)$, $(\\ZZ,\\RR)$, $(\\ZZ_p,\\QQ_p)$ and\n$(\\ZZ_p,\\CC_p)$ and the maps between these sequences which are\ninduced by the obvious inclusions and by an embedding\n$j:\\RR\\to\\CC_p$.\n\nBy an easy diagram chase one obtains a\ncommutative diagram of short exact sequences\n\\begin{equation*}\n\\xymatrix{\n0 \\ar[r] & K_0(\\ZZ[G],\\QQ[G]) \\ar[r] \\ar[d] & K_0(\\ZZ[G],\\RR[G]) \\ar[r] \\ar[d] &\nK_1(\\RR[G])\/K_1(\\QQ[G]) \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & K_0(\\ZZ_p[G],\\QQ_p[G]) \\ar[r] & K_0(\\ZZ_p[G],\\CC_p[G]) \\ar[r] &\nK_1(\\CC_p[G])\/K_1(\\QQ_p[G]) \\ar[r] & 0.\n}\n\\end{equation*}\nTherefore it suffices to show that the maps\n\\begin{equation}\n\\label{equation_K_injectivity_left}\nK_0(\\ZZ[G],\\QQ[G])\\to\\prod_{p,j} K_0(\\ZZ_p[G],\\QQ_p[G])\n\\end{equation}\nand\n\\begin{equation}\n\\label{equation_K_injectivity_right}\nK_1(\\RR[G])\/K_1(\\QQ[G])\\to\\prod_{p,j}\nK_1(\\CC_p[G])\/K_1(\\QQ_p[G])\n\\end{equation}\nare injective. The injectivity of (\\ref{equation_K_injectivity_left})\nfollows immediately from the relevant case of the isomorphism in (\\ref{decomp}).\n\nLet $x\\in K_1(\\RR[G])$ be such that for all $p$ and all $j$ one has\n\\[ j_*(x)\\in K_1(\\QQ_p[G])\\subseteq K_1(\\CC_p[G]).\\]\n\nWe now use the (injective) maps ${\\rm Nrd}_{\\RR[G]}$ and ${\\rm Nrd}_{\\QQ[G]}$ to identify $K_1(\\RR[G])$ and $K_1(\\QQ[G])$ with $\\im({\\rm Nrd}_{\\RR[G]})$ and $\\im({\\rm Nrd}_{\\QQ[G]})$ respectively.\n\nThen, $x=\\sum_{g\\in G} c_gg$ is an element of $\\im({\\rm Nrd}_{\\RR[G]})$ such that\n\\begin{equation}\n\\label{equation_lemma_K_injectivity}\nj_*(x)=\\sum_{g\\in G} j(c_g)g\\in\\zeta(\\QQ_p[G])^\\times.\n\\end{equation}\n\nWe claim that $\\sum_{g\\in G} c_gg\\in\\QQ[G]$. Let $g\\in G$ and consider the\ncoefficient $c_g$.\n\nIf, firstly, $c_g$ was transcendental over $\\QQ$, then\nthere would be an embedding $j:\\RR\\to\\CC_p$ such that\n$j(c_g)\\not\\in\\QQ_p$, thereby contradicting\n(\\ref{equation_lemma_K_injectivity}).\n\nTherefore $c_g$ is algebraic\nover $\\QQ$. Now $j(c_g)\\in\\QQ_p$ for all $p$ and embeddings\n$j$ implies that all primes are completely split in the\nnumber field $\\QQ(c_g)$ and therefore $\\QQ(c_g)=\\QQ$.\n\nHence $x$ belongs to $\\im({\\rm Nrd}_{\\RR[G]})\\cap\\QQ[G]$ which, by the Hasse-Schilling-Maass Norm Theorem, is equal to $\\im({\\rm Nrd}_{\\QQ[G]})$.\n\nThis shows the injectivity of (\\ref{equation_K_injectivity_right}) and hence also of the map (\\ref{local iso}).\nThe injectivity of (\\ref{local iso}) in turn implies that the equality of ${\\rm BSD}(A_{F\/k})$(iv) is valid if and only if its image under each maps $j_*$ is valid.\n\nSet $\\mathcal{X} := \\mathcal{X}_S(\\omega_\\bullet)$. Then\n\\begin{multline*} j_*(\\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})) = \\chi_{G,p}(\\ZZ_p\\otimes_\\ZZ {\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}), \\CC_p\\otimes_{\\RR,j}h_{A,F}))\\\\\n=\\chi_{G,p}({\\rm SC}_S(A_{F\/k},\\mathcal{X}(p),\\mathcal{X}(\\infty)_p),h^j_{A,F}) +\\chi_{G,p}( \\mathcal{Q}(\\omega_\\bullet)_{S,p}[0],0),\\end{multline*}\nwhere the first equality is by definition of the map $j_*$ and the second by Proposition \\ref{prop:perfect2}(i).\n\nGiven this, the claim follows from the obvious equality $j_*(\\mu_{S}(A_{F\/k})) = \\mu_{S}(A_{F\/k})_p$ and the commutativity of the diagram\n\\begin{equation*}\\label{commute K thry} \\begin{CD} K_1(\\RR[G]) @> \\partial_G >> K_0(\\ZZ[G],\\RR[G])\\\\\n@VV j_{*} V @VV j_{*}V\\\\\nK_1(\\CC_p[G]) @> \\partial_{G,p} >> K_0(\\ZZ_p[G],\\CC_p[G]).\\end{CD}\\end{equation*}\n\\end{proof}\n\n\n\\begin{remark}{\\em In the sequel we shall say, for any given prime $p$, that the `$p$-primary component' {\\rm BSD}$_p(A_{F\/k})$(iv) of the equality in {\\rm BSD}$(A_{F\/k})$(iv) is valid if for every choice of isomorphism of fields $j:\\CC\\cong \\CC_p$ the equality (\\ref{displayed pj}) is valid.} \\end{remark}\n\n\\section{Periods and Galois-Gauss sums}\\label{k theory period sect}\n\nTo prepare for arguments in subsequent sections, we shall now explain the precise link between the $K$-theoretical period $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ that occurs in ${\\rm BSD}(A_{F\/k})$ and the classical periods that are associated to $A$ over $k$.\n\n\\subsection{Periods and Galois resolvents}\\label{k theory period sect2} At the outset we fix an ordered $k$-basis $\\{\\omega'_j: j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$.\n\nFor each $v$ in $S_\\RR^k$ we then set\n\\[ \\Omega_{A,v}^+ := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}^{+}}\\sigma_{v,*}(\\omega'_b)\\right)_{a,b}\\right)\\]\nand\n\n\\[ \\Omega_{A,v}^- := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}^{-}}\\sigma_{v,*}(\\omega'_b)\\right)_{a,b}\\right),\\]\nwhere the elements $\\gamma_{v,a}^+$ and $\\gamma_{v,a}^-$ of $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$ are as specified in \\S\\ref{gamma section} and in both matrices $(a,b)$ runs over $[d]\\times [d]$.\n\nFor each $v$ in $S_k^\\CC$ we also set\n\\[ \\Omega_{A,v} := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}}\\sigma_{v,*}(\\omega'_b),c\\!\\left(\\int_{\\gamma_{v,a}}\\sigma_{v,*}(\\omega'_b)\\right)\\right)_{a,b}\\right)\\]\nwhere the elements $\\gamma_{v,a}$ of $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$ are again as specified in \\S\\ref{gamma section} and $(a,b)$ runs over $[2d]\\times [d]$.\n\nWe note that, by explicitly computing integrals, the absolute values of these determinants can be explicitly related to the periods that occur in the classical formulation of the Birch and Swinnerton-Dyer conjecture (see, for example, Gross \\cite[p. 224]{G-BSD}).\n\nFor each archimedean place $v$ of $k$ and character $\\psi$ we then set\n\\[\\Omega^\\psi_{A,v} := \\begin{cases} \\Omega_{A,v}^{\\psi(1)}, &\\text{ if $v \\in S_k^\\CC$,}\\\\\n                                            (\\Omega^+_{A,v})^{1-\\psi_v^-(1)}(\\Omega^-_{A,v})^{\\psi_v^-(1)}, &\\text{ if $v \\in S_k^\\RR$}\\end{cases} \\]\nwith\n\\[\\psi_v^-(1) := \\psi(1) - {\\rm dim}_\\CC(H^0(G_w,V_\\psi)),\\]\nwhere again $V_\\psi$ is a fixed choice of $\\CC[G]$-module of character $\\psi$.\n\nFor each $\\psi$ we set\n\\[ \\Omega_A^\\psi := \\prod_{v \\in S_k^\\infty}\\Omega^\\psi_{A,v}\\]\nand we then finally define an element of $\\zeta(\\CC[G])^\\times$ by setting\n\\begin{equation}\\label{period def} \\Omega_A^{F\/k} := \\sum_{\\psi \\in \\widehat{G}}\\Omega^\\psi_A\\cdot e_\\psi.\\end{equation}\n\nFor each $v$ in $S_k^\\RR$, resp. in $S_k^\\CC$, we also set\n\\[ w_{v,\\psi} := \\begin{cases} i^{\\psi^-_v(1)}, &\\text{if $v\\in S_k^\\RR$,}\\\\\n                              i, &\\text{if $v\\in S_k^\\CC$.}\\end{cases}\\]\n\nFor each character $\\psi$ we then set\n\\[ w_{\\psi} :=  \\prod_{v \\in S_k^\\infty}w_{v,\\psi}\\]\nand then define an element of $\\zeta(\\CC[G])^\\times$ by setting\n\\begin{equation}\\label{root number def} w_{F\/k} := \\sum_{\\psi\\in \\widehat{G}} w_\\psi\\cdot e_\\psi .\\end{equation}\n\n\n\n\\begin{lemma}\\label{k-theory period} Set $n := [k:\\QQ]$. Fix an ordered $\\QQ[G]$-basis $\\{z_i: i \\in [n]\\}$ of $F$ and write $\\omega_\\bullet$ for the (lexicographically ordered) $\\QQ[G]$-basis  $\\{ z_i\\otimes \\omega'_j: i \\in [n], j \\in [d]\\}$ of $H^0(A_F^t,\\Omega^1_{A_F^t})$. Then in $\\zeta(\\RR[G])^\\times$ one has\n\\[ {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k})) = \\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot {\\rm Nrd}_{\\QQ[G]}\n\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{-d} \\]\nwhere we have fixed an extension $\\hat\\sigma$ to $\\Sigma(F)$ of each embedding $\\sigma$ in $\\Sigma(k)$.\\end{lemma}\n\n\\begin{proof} This follows from the argument of \\cite[Lem. 4.5]{bmw}.\\end{proof}\n\n\n\\subsection{Galois resolvents and Galois-Gauss sums}\n\nUnder suitable conditions, one can also choose the $\\QQ[G]$-basis $\\{z_i: i \\in [n]\\}$ of $F$ so that the reduced norm of the Galois resolvent matrix that occurs in Lemma \\ref{k-theory period} can be explicitly described in terms of Galois-Gauss sums.\n\nBefore explaining this we first recall the relevant notions of Galois-Gauss sums.\n\n\\subsubsection{}\\label{mod GGS section}\n\nThe `global Galois-Gauss sum of $F\/k$' is the element\n\\[ \\tau(F\/k) :=\\sum_{\\psi \\in \\widehat{G}}\\tau(\\QQ,\\psi)\\cdot e_\\psi\\]\nof $\\zeta(\\QQ^c[G])^\\times$.\n\nHere we regard each character $\\psi$ of $G$ as a character of $G_k$ via the projection $G_k \\to G$ and then write\n$\\tau(\\QQ,\\psi)$ for the global Galois-Gauss sum (as defined by Martinet in \\cite{martinet})\nof the induction of $\\psi$ to $G_\\QQ$.\n\n\nTo define suitable modifications of these sums\nwe then define the `unramified characteristic' of $v$ at each character $\\psi$ in $\\widehat{G}$ by setting\n\\[ u_{v,\\psi} := {\\rm det}(-\\Phi_v^{-1}\\mid V_\\psi^{I_w})\\in \\QQ^{c,\\times}.\\]\n\nFor each character $\\psi$ in $\\widehat{G}$ we set\n\\[ u_\\psi :=  \\prod_{v\\in S_k^F}u_{v,\\psi}.\\]\n\nWe then define elements of $\\zeta(\\QQ[G])^\\times$ by setting\n\\begin{equation}\\label{u def} u_v(F\/k) := \\sum_{\\psi\\in \\widehat{G}}u_\\psi\\cdot e_\\psi\\end{equation}\nand\n\\[ u_{F\/k} := \\prod_{v\\in S_k^F}u_v(F\/k) = \\sum_{\\psi\\in \\widehat{G}}u_\\psi\\cdot e_\\psi.\\]\n\nWe finally define the `modified global Galois-Gauss sum of $\\psi$' to be the element\n\\[ \\tau^\\ast(\\QQ,\\psi) := u_\\psi\\cdot \\tau(\\QQ,\\psi)\\]\nof $\\QQ^c$, and the `modified global Galois-Gauss sum of $F\/k$' to be the element\n\\[ \\tau^\\ast(F\/k) := u_{F\/k}\\cdot \\tau(F\/k) = \\sum_{\\psi\\in \\widehat{G}}\\tau^\\ast(\\QQ,\\psi)\\cdot e_\\psi \\]\nof $\\zeta(\\QQ^c[G])^\\times$.\n\n\n\n\\begin{remark}{\\em The modified Galois-Gauss sums $\\tau^\\ast(\\QQ,\\psi)$ defined above play a key role in the proof of the main results of classical Galois module theory, as discussed by Fr\\\"ohlich in  \\cite{frohlich}. In Lemma \\ref{imprimitive GGS} below, one can also find a more concrete reason as to why such terms should arise naturally in the setting of leading term conjectures.}\n\\end{remark}\n\n\\subsubsection{} The next result shows that under mild hypotheses the Galois-resolvent matrix that occurs in Lemma \\ref{k-theory period} can be explicitly interpreted in terms of the elements $\\tau^\\ast(F\/k)$ introduced above.\n\n\\begin{proposition}\\label{lms}The following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] For any ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of $H^0(A_F^t,\\Omega^1_{A_F^t})$ there exists an element $u(\\omega_\\bullet)$ of $\\zeta(\\QQ[G])^\\times$ such that\n\\[ {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k})) = u(\\omega_\\bullet)\\cdot \\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot \\tau^\\ast(F\/k)^{-d}.\\]\n\n\\item[(ii)] Fix a prime $p$ and set $\\mathcal{O}_{F,p} := \\ZZ_p\\otimes_\\ZZ\\mathcal{O}_F.$ Then if no $p$-adic place of $k$ is wildly ramified in $F$, there is an ordered $\\ZZ_p[G]$-basis $\\{z^p_{i}\\}_{i \\in [n]}$ of $\\mathcal{O}_{F,p}$ for which one has\n\\[ {\\rm Nrd}_{\\QQ_p[G]}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z^p_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right) = \\tau^\\ast(F\/k).\\]\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} It is enough to prove claim (i) for any choice of $\\QQ[G]$-basis $\\omega_\\bullet$. Then, choosing $\\omega_\\bullet$ as in Lemma\n\\ref{k-theory period}, the latter result implies that it is enough to prove that the product\n\\[ {\\rm Nrd}_{\\QQ_p[G]}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)\\cdot \\tau^\\ast(F\/k)^{-1}\\]\nbelongs to $\\zeta(\\QQ[G])^\\times$ and this follows from the argument used by Bley and the first author to prove \\cite[Prop. 3.4]{bleyburns}.\n\nTurning to claim (ii) we note that if no $p$-adic place of $k$ is wildly ramified in $F$, then the $\\ZZ_p[G]$-module $\\mathcal{O}_{F,p}$ is free of rank $n$ (by Noether's Theorem) and so we may fix an ordered $\\ZZ_p[G]$-basis $z^p_\\bullet := \\{z^p_{i}: i \\in [n]\\}$.\n\nThe matrix\n\n\\[ M(z^p_\\bullet) := (  (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z^p_b)))\\cdot g)_{\\sigma\\in \\Sigma(k),b\\in [n]})\\]\nin ${\\rm GL}_{n}(\\CC_p[G])$ then represents, with respect to the bases $z^p_\\bullet$ of $F_p$ and $\\{\\hat \\sigma: \\sigma \\in \\Sigma(k)\\}$ of $Y_{F\/k,p}$, the isomorphism of $\\CC_p[G]$-modules\n\\[ \\mu_{F,p}: \\CC_p\\otimes_{\\QQ_p} F_p \\cong \\CC_p\\otimes_{\\ZZ_p}Y_{F\/k,p}\\]\nthat sends each $z\\otimes f$ to $(z\\hat\\sigma(f))_{\\sigma\\in\\Sigma(k)}$.\n\nHence one has\n\\begin{align*} \\delta_{G,p}\\bigl({\\rm Nrd}_{\\CC_p[G]}\\bigl(M(z^p_\\bullet)\\bigr)\\bigr) = \\, &\\partial_{G,p}\\bigl([M(z^p_\\bullet)]\\bigr)\\\\\n                                                                     = \\, &[\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu_{F,p}]\\\\\n                                                                     = \\, &\\delta_{G,p}(\\tau^\\ast(F\/k)),\\end{align*}\n                                                                    \nwhere $[M(z^p_\\bullet)]$ denotes the class of $M(z^p_\\bullet)$ in $K_1(\\CC_p[G])$ and the last equality follows from the proof of \\cite[Th. 7.5]{bleyburns}.\n\nNow the exact sequence of relative $K$-theory implies that kernel of $\\delta_{G,p}$ is equal to the image of $K_1(\\ZZ_p[G])$ under the map ${\\rm Nrd}_{\\QQ_p[G]}$.\n\nIn addition, the ring  $\\ZZ_p[G]$ is semi-local and so the natural map ${\\rm GL}_{n}(\\ZZ_p[G]) \\to K_1(\\ZZ_p[G])$ is surjective.\n\nIt follows that there exists a matrix $U$ in ${\\rm GL}_n(\\ZZ_p[G])$ with\n\\[ {\\rm Nrd}_{\\CC_p[G]}(M(z^p_\\bullet))\\cdot {\\rm Nrd}_{\\CC_p[G]}(U) = \\tau^\\ast(F\/k)\\]\nand so it suffices to replace the basis $z^p_\\bullet$ by its image under the automorphism of $\\mathcal{O}_{F,p}$ that corresponds to the matrix $U$. \\end{proof}\n\nTaken together, Lemma \\ref{k-theory period} and Proposition \\ref{lms}(ii) give an explicit interpretation of the $K$-theoretical periods that occur in the formulation of {\\rm BSD}$(A_{F\/k})$.\n\nHowever, the existence of $p$-adic places that ramify wildly in $F$ makes the situation more complicated and this leads to technical difficulties in later sections.\n\n\n\n\n\n\\section{Local points on ordinary varieties}\\label{local points section}\n\nIn \\S\\ref{tmc} we will impose several mild hypotheses on the reduction types of $A$ and the ramification invariants of $F\/k$ which together ensure that the classical Selmer complex is perfect over $\\ZZ_p[G]$. Under these hypotheses, we will then give a more explicit interpretation of the equality in ${\\rm BSD}(A_{F\/k})$(iv).\n\nAs a necessary preparation for these results, in this section we shall establish several preliminary results concerning the properties of local points on varieties with good ordinary reduction.\n\n\n\\subsection{Cohomological-triviality}For this purpose we assume to be given a finite Galois extension $N\/M$ of $p$-adic fields and set $\\Gamma := G_{N\/M}$. We fix a Sylow $p$-subgroup $\\Delta$ of $\\Gamma$. We write $\\Gamma_0$ for the inertia subgroup of $\\Gamma$ and set $N_0 := N^{\\Gamma_0}$.\n\nWe also assume to be given an abelian variety $B$, of dimension $d$, over $M$ that has good reduction and write $\\tilde B$ for the corresponding reduced variety.\n\n\\begin{lemma}\\label{useful prel} The following claims are valid.\n\\begin{itemize}\n\\item[(i)] If $N\/M$ is unramified, then the $\\Gamma$-modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ are cohomologically-trivial.\n\\item[(ii)] If $N\/M$ is at most tamely ramified, then the $\\ZZ_p[\\Gamma]$-modules $B(N)^\\wedge_p$ and $\\tilde B(\\kappa_{N})[p^\\infty]$ are cohomologically-trivial.\n\\item[(iii)] If the variety $B$ is ordinary and $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then the $\\ZZ_p[\\Gamma]$-module $B(N)^\\wedge_p$ is cohomologically-trivial.\n\\item[(iv)] Assume that $B$ is ordinary and write $u$ for the twist matrix (in ${\\rm GL}_{d}(\\ZZ_p)$) of its formal group over the completion of $M^{\\rm un}$. If $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then $B(N)^\\wedge_p$ is torsion-free, and hence projective over $\\ZZ_p[\\Gamma]$, if and only if for any non-trivial $d$-fold vector $\\underline{\\zeta}$ of $p$-th roots of unity in $N^{\\rm un}$ one has\n\\[ \\Phi_N(\\underline{\\zeta}) \\not= \\underline{\\zeta}^u,\\]\nwhere $\\Phi_N$ is the Frobenius automorphism in $G_{N^{\\rm un}\/N}$. In particular, this is the case if any $p$-power root of unity in $N^{\\rm un}$ belongs to $N$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} A standard Hochschild-Serre spectral sequence argument combines with the criterion of \\cite[Thm. 9]{cf} to show that  claim (i) is valid provided that each of the modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ is cohomologically-trivial with respect to every  subgroup $C$ of $\\Gamma$ of prime order (see the proof of \\cite[Lem. 4.1]{bmw} for a similar argument).\n\nWe therefore fix a subgroup $C$ of $\\Gamma$ that has prime order. Now cohomology over $C$ is periodic of order 2 and each of the modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ span free $\\QQ[\\Gamma]$-modules.\nIt thus follows from \\cite[Cor. to Prop. 11]{cf} that the Herbrand Quotient with respect to $C$ of each of these modules is equal to 1.\nTo prove claim (i) it is enough to show that the natural norm maps $B(N)\\to B(N^C)$, $\\tilde B(\\kappa_{N}) \\to \\tilde B(\\kappa_{N^C})$ and $\\kappa_{N}\\to \\kappa_{N^C}$ are surjective.\n\nSince the extension $N\/N^C$ is unramified, this surjectivity is well-known for the module $\\kappa_N$ and for the modules $B(N)$ and $\\tilde B(\\kappa_{N})$ it follows directly from the result of Mazur in \\cite[Cor. 4.4]{m}.\n\nTo prove claim (ii) we assume that $N\/M$ is tamely ramified. In this case the order of $\\Gamma_0$ is prime to $p$ and so the same standard Hochschild-Serre spectral sequence argument as in claim (i) implies claim (ii) is true if the modules $B(N_0)^\\wedge_p = (B(N)^\\wedge_p)^{\\Gamma_0}$ and $\\tilde B(\\kappa_{N})[p^\\infty] = \\tilde B(\\kappa_{N})[p^\\infty]^{\\Gamma_0}$ are cohomologically-trivial with respect to every subgroup $C$ of $\\Gamma\/\\Gamma_0$ of order $p$. Since $N_0\/N_0^C$ is unramified, this follows from the argument in claim (i).\n\nIn a similar way, to prove claim (iii) one is reduced to showing that if $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then for each subgroup $C$ of $\\Gamma_0$ of order $p$, the norm map ${\\rm N}_C: B(N)^\\wedge_p \\to B(N^C)^\\wedge_p$ is surjective.\n\nNow the main result of Lubin and Rosen in \\cite{LR} implies that the cokernel of ${\\rm N}_C$ is isomorphic to the cokernel of the natural action of ${\\rm I}_d-u$ on the direct sum of $d$-copies of $C$ and from the proof of \\cite[Th. 2]{LR} one knows that ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic divisor of $|\\tilde B(\\kappa_N)|$.\n But if $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then $\\tilde B(\\kappa_{N})[p^\\infty]$ also vanishes (as $\\Delta$ is a $p$-group) and so\n ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic unit. It follows that ${\\rm cok}(N_C)$ vanishes, as required to prove claim (iii).\n\nTo prove claim (iv) we assume $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes. Then claim (ii) implies $B(N)^\\wedge_p$ is a projective  $\\ZZ_p[\\Gamma]$-module if and only if $B(N)^\\wedge_p[p^\\infty]$ vanishes. In addition, from the lemma in \\cite[\\S1]{LR} (with $L = K = N$), we know that the group $B(N)^\\wedge_p[p^\\infty]$ is isomorphic to the subgroup of $(N^{{\\rm un},\\times})^d$ comprising $p$-torsion elements $\\underline{\\eta}$ which satisfy\n$\\Phi_N(\\underline{\\eta}) = \\underline{\\eta}^u$.\n\nThis directly implies the first assertion of claim (iv) and the second assertion then follows  because ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic unit and so $u\\not\\equiv 1$ (mod $p$). \\end{proof}\n\n\\begin{remark}{\\em A more general analysis of the cohomological properties of formal groups was recently given by Ellerbrock and Nickel in \\cite{ellerbrocknickel}.}\\end{remark}\n\n\\subsection{Twist matrices and $K$-theory}\\label{twist inv prelim}\n\nIn this section we fix an abelian variety $B$ over $M$ of dimension $d$.\nWe assume that $B$ has good ordinary reduction and is such that $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes.\n\nWe shall then use Lemma \\ref{useful prel} to define a natural invariant in $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ of the twist matrix of $B$ that will play an important role in the explicit interpretation of the equality in ${\\rm BSD}(A_{F\/k})$(iv) that will be given in \\S \\ref{tmc} below.\n\nAt the outset we recall that the complex $R\\Gamma(N,\\ZZ_p(1))$ belongs to $D^{\\rm perf}(\\ZZ_p[\\Gamma])$. Hence, following Lemma \\ref{useful prel}(iii), we obtain a complex in $D^{\\rm perf}(\\ZZ_p[\\Gamma])$ by setting\n\\[ C_{B,N}^{\\bullet} := R\\Gamma(N,\\ZZ_p(1))^d[1] \\oplus B(N)^\\wedge_p[-1].\\]\n\nThis complex is acyclic outside degrees zero and one. In addition, Kummer theory gives an identification $H^1(N,\\ZZ_p(1)) = (N^\\times)^\\wedge_p$ and the invariant map ${\\rm inv}_N$ of $N$ an isomorphism $ H^2(N,\\ZZ_p(1)) \\cong \\ZZ_p$.\n\nWe next fix a choice of isomorphism of $\\QQ_p[\\Gamma]$-modules $\\lambda_{B,N}$  which lies in a commutative diagram\n\\begin{equation}\\label{lambda diag}\\begin{CD}\n0 @> >> \\QQ_p\\cdot (U^{(1)}_{N})^d @> \\subset >> \\QQ_p\\cdot H^0(C_{B,N}^{\\bullet}) @> ({\\rm val}_N)^d >> \\QQ_p^d @> >> 0\\\\\n@. @V {\\rm exp}_{B,N}VV @V \\lambda_{B,N} VV @V \\times f_{N\/M}VV\\\\\n0 @> >> \\QQ_p \\cdot B(N)^\\wedge_p @> \\subset >> \\QQ_p\\cdot H^1(C_{B,N}^{\\bullet}) @> {\\rm can} >> \\QQ_p^d @> >> 0.\\end{CD}\\end{equation}\nHere $U_N^{(1)}$ is the group of 1-units of $N$, ${\\rm val}_N: \\QQ_p\\cdot (N^\\times)^\\wedge_p\\to \\QQ_p$ is the canonical valuation map on $N$, $f_{N\/M}$ is the residue degree of $N\/M$, `{\\rm can}' is induced by ${\\rm inv}_N$ and ${\\rm exp}_{B,N}$ is the composite isomorphism\n\\[ \\QQ_p\\cdot (U^{(1)}_{N})^d \\cong N^d \\cong \\QQ_p\\cdot B(N)^\\wedge_p\\]\nwhere the first isomorphism is induced by the $p$-adic logarithm on $N$ and the second by the exponential map of the formal group of $B$ over $N$.\n\nWe now introduce a useful general convention: for each element $x$ of $\\zeta(\\CC_p[\\Gamma])$ we write $^\\dagger x$ for the unique element of $\\zeta(\\CC_p[\\Gamma])^\\times$ with the property that for each $\\mu$ in $\\widehat{\\Gamma}$ one has\n\\begin{equation}\\label{dagger eq} e_\\mu (^\\dagger x) = \\begin{cases} e_\\mu x, &\\text{ if $e_\\mu x\\not= 0$,}\\\\\n  e_\\mu, &\\text{ otherwise.}\\end{cases}\\end{equation}\n(This construction is written as $x \\mapsto ^*\\!\\! x$ in \\cite{bleyburns,breuning2}).\n\nWe then define an element\n\\[ c_{N\/M} := \\frac{^\\dagger((|\\kappa_M|-\\Phi_{N\/M})e_{\\Gamma_0})}{^\\dagger((1-\\Phi_{N\/M})e_{\\Gamma_0})}\\]\nof $\\zeta(\\QQ[\\Gamma])^\\times$. Here and in the sequel, $\\Phi_{N\/M}$ is a fixed lift to $\\Gamma$ of the Frobenius automorphism in $\\Gamma\/\\Gamma_0$ and, for any subgroup $J$ of $\\Gamma$, $e_{J}$ denotes the idempotent $(1\/|J|)\\sum_{\\gamma\\in J}\\gamma$.\n\nWe finally obtain our desired element of $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ by setting\n\\[ R_{N\/M}(\\tilde B) := \\chi_{\\Gamma,p}(C_{B,N}^{\\bullet},\\lambda_{B,N}) +d\\cdot\\delta_{\\Gamma,p}(c_{N\/M}). \\]\n\n\n\\begin{proposition}\\label{basic props} Assume $B$ is ordinary and $\\tilde B(\\kappa_{N^\\Delta})[p^{\\infty}]$ vanishes.\n\\begin{itemize}\n\\item[(i)] $R_{N\/M}(\\tilde B)$ depends only upon $N\/M$ and the reduced variety $\\tilde B$.\n\\item[(ii)] $R_{N\/M}(\\tilde B)$ has finite order.\n\\item[(iii)] If $N\/M$ is tamely ramified, then\n $R_{N\/M}(\\tilde B)$ vanishes.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}  We set $C^\\bullet := C_{B,N}^\\bullet$, $\\lambda := \\lambda_{B,N}$ and $f := f_{N\/M}$, and also write $\\wp = \\wp_N$ for the maximal ideal in the valuation ring of $N$.\n\nThen, whilst $\\lambda$ can be chosen in many different ways to ensure that (\\ref{lambda diag}) commutes, it is straightforward to check that $\\chi_{\\Gamma,p}(C^{\\bullet},\\lambda)$ is independent of this choice. The fact that this element depends only on the (twist matrix of the) reduced variety $\\tilde B$ follows from Lemma \\ref{twist dependence} below. This proves claim (i).\n\nIt is convenient to prove claim (iii) before claim (ii) and so we assume $N\/M$ is tamely ramified.\n\nWe write $\\hat B$ for the formal group of $B$. In this case Lemma \\ref{ullom}(i) below implies that for each natural number $n$ the $\\ZZ_p[\\Gamma]$-modules $U^{(n)}:= \\mathbb{G}_m(\\wp^n)$ and $V^{(n)}:= \\hat B(\\wp^n)$ are cohomologically-trivial and there exist exact triangles in $D^{\\rm perf}(\\ZZ_p[\\Gamma])$ of the form\n\\begin{equation*}\\label{use tri}\n\\begin{cases} (U^{(n)})^d[0]\\oplus V^{(n)}[-1]\\xrightarrow{\\alpha} C^{\\bullet}\\to C_{\\alpha}^\\bullet \\to (U^{(n)})^d[1]\\oplus V^{(n)}[0]\\\\\nU^{(1)}[0] \\xrightarrow{\\beta} R\\Gamma(N,\\ZZ_p(1))[1] \\to C^\\bullet_{N,1} \\to U_N^1[1]\\\\\n(U^{(1)}\/U^{(n)})^d[0] \\xrightarrow{\\gamma} C_{\\alpha}^\\bullet \\to (C^\\bullet_{N,1})^d\\oplus (V^{(1)}\/V^{(n)})[-1] \\to (U^{(1)}\/U^{(n)})^d[1].\\end{cases}\\end{equation*}\nHere $\\alpha$ is the unique morphism such that $H^0(\\alpha)$ and $H^1(\\alpha)$ are respectively induced by the inclusions $U^{(n)}\\subset (N^\\times)^\\wedge_p$ and $V^{(n)} \\subseteq B(N)^\\wedge_p$ and so that the cohomology sequence of the first triangle induces identifications of $H^0(C_{\\alpha}^\\bullet)$ and $H^1(C_{\\alpha}^\\bullet)$ with $((N^\\times)^\\wedge_p\/U^{(n)})^d$ and $\\ZZ_p^d \\oplus V^{(1)}\/V^{(n)}$; $\\beta$ is the unique morphism so that $H^0(\\beta)$ is induced by the inclusion $U^{(1)} \\subset (N^\\times)^\\wedge_p$ and so the cohomology sequence of the second  triangle induces identifications of $H^0(C_{N,1}^\\bullet)$ and $H^1(C_{N,1}^\\bullet)$ with $(N^\\times)^\\wedge_p\/U^{(1)}$ and $\\ZZ_p$ respectively; $\\gamma$ is the unique morphism so that $H^{0}(\\gamma)$ is the inclusion $(U^{(1)}\/U^{(n)})^d \\subset H^0(C_{\\alpha}^\\bullet)$.\n\nIn particular, if $n$ is sufficiently large, then we may apply Lemma \\ref{fk lemma} to the first and third of the above triangles to deduce that\n\n\\begin{align}\\label{interm} &\\chi_{\\Gamma,p}(C^{\\bullet},\\lambda)\\\\ = \\,&\\chi_{\\Gamma,p}((U^{(n)})^d[0]\\oplus V^{(n)}[-1],{\\rm exp}_{B,N}) +  \\chi_{\\Gamma,p}(C_{\\alpha}^{\\bullet},\\lambda_{\\alpha})\\notag\\\\\n= \\,&\\chi_{\\Gamma,p}(C_{\\alpha}^{\\bullet},\\lambda_{\\alpha})\\notag\\\\\n= \\,& \\chi_{\\Gamma,p}((U^{(1)}\/U^{(n)})^d[0],0) + d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N) + \\chi_{\\Gamma,p}((V^{(1)}\/V^{(n)})[-1],0)\\notag\\\\\n= \\,& \\chi_{\\Gamma,p}((U^{(1)}\/U^{(n)})^d[0],0) + d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N) - \\chi_{\\Gamma,p}((V^{(1)}\/V^{(n)})[0],0)\\notag\\\\\n= \\, &d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N),\\notag\n\n \\end{align}\nwhere we write $\\lambda_{\\alpha}$ for the isomorphism of $\\QQ_p[\\Gamma]$-modules\n\\[ \\QQ_p\\cdot H^0(C_{\\alpha}^\\bullet) = \\QQ_p\\cdot ((N^\\times)^\\wedge_p\/U^{(n)})^d \\cong \\QQ_p^d = \\QQ_p\\cdot H^1(C_{\\alpha}^\\bullet)\\]\nthat is induced by the map $f\\cdot {\\rm val}_N$ and the second and last equalities in (\\ref{interm}) follow from Lemma \\ref{ullom}.\n\nBut $$\\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot {\\rm val}_N) = \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},{\\rm val}_N) + \\delta_{\\Gamma,p}(^\\dagger(f\\cdot e_\\Gamma))$$ whilst from \\cite[Th. 4.3]{bleyburns} one has\n\n\\[ \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},{\\rm val}_N) = -\\delta_{\\Gamma,p}(c_{N\/M}\\cdot ^\\dagger\\!(f\\cdot e_\\Gamma))= - \\delta_{\\Gamma,p}(c_{N\/M}) - \\delta_{\\Gamma,p}(^\\dagger\\!(f\\cdot e_\\Gamma)).\\]\nClaim (iii) is thus obtained by substituting these facts into the equality (\\ref{interm}).\n\n\nTo deduce claim (ii) from claim (iii) we recall that an element $\\xi$ of $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ has finite order if and only if for cyclic subgroup $\\Upsilon$ of $\\Gamma$ and every quotient $\\Omega = \\Upsilon\/\\Upsilon'$ of\norder prime to $p$ one has $(q^\\Upsilon_{\\Omega}\\circ\\rho^{\\Gamma}_\\Upsilon)(\\xi) = 0$. Here\n$$\\rho^{\\Gamma}_\\Upsilon:K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])\\to K_0(\\ZZ_p[\\Upsilon],\\QQ_p[\\Upsilon])$$ is the natural restriction map,\n$$q^\\Upsilon_{\\Omega}:K_0(\\ZZ_p[\\Upsilon],\\QQ_p[\\Upsilon])\\to K_0(\\ZZ_p[\\Omega],\\QQ_p[\\Omega])$$\nmaps the class of a triple $(P,\\phi,Q)$ to the class of $(P^{\\Upsilon'},\\phi^{\\Upsilon'},Q^{\\Upsilon'})$, and the stated general fact is proved in \\cite[Thm. 4.1]{ewt}.\n\nSince the extension $N^{\\Upsilon'}\/N^\\Upsilon$ is tamely ramified, it is thus enough to show that\n\\[ (q^\\Upsilon_{\\Omega}\\circ\\rho^{\\Gamma}_\\Upsilon)(R_{N\/M}(\\tilde B))=R_{N^{\\Upsilon'}\/N^\\Upsilon}(\\tilde B).\\]\n\nThis is proved by a routine computation in relative $K$-theory that uses the same ideas as in \\cite[Rem. 2.9]{breuning2}.\n In fact, the only point worth mentioning explicitly in this regard is that if $\\Gamma'$ is normal in $\\Gamma$, and we set $N' := N^{\\Gamma'}$, then the natural projection isomorphism $\\iota:\\ZZ_p[\\Gamma\/\\Gamma']\\otimes^{\\mathbb{L}}_{\\ZZ_p[\\Gamma]}R\\Gamma(N,\\ZZ_p(1)) \\cong R\\Gamma(N',\\ZZ_p(1))$ in $D^{\\rm perf}(\\ZZ_p[\\Gamma\/\\Gamma'])$ gives a commutative diagram of (trivial) $\\QQ_p[\\Gamma\/\\Gamma']$-modules\n\n\\[ \\begin{CD} \\QQ_p\\cdot H^2(N,\\ZZ_p(1))^{\\Gamma'} @> {\\rm inv}_N >> \\QQ_p\\\\\n@V H^2(\\iota)VV @VV \\times f_{N\/N'}V\\\\\n\\QQ_p\\cdot H^2(N',\\ZZ_p(1)) @> {\\rm inv}_{N'}>> \\QQ_p.\\end{CD}\\]\n\\end{proof}\n\n\\begin{lemma}\\label{ullom} If $N\/M$ is tamely ramified, the following claims are valid for all natural numbers $a$.\n\\begin{itemize}\n\\item[(i)] The $\\ZZ_p[\\Gamma]$-modules $U^{(a)}$ and $V^{(a)}$ are cohomologically-trivial.\n\\item[(ii)] One has $d\\cdot \\chi_{\\Gamma,p}((U^{(1)}\/U^{(a)})[0],0) = \\chi_{\\Gamma,p}((V^{(1)}\/V^{(a)})[0],0)$.\n\\item[(iii)] For all sufficiently large $a$ one has $\\chi_{\\Gamma,p}((U^{(a)})^d[0]\\oplus V^{(a)}[-1],{\\rm exp}_{B,N})=0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} The key fact in this case is that for every integer $i$ the $\\ZZ_p[\\Gamma]$-module $\\wp^i$ is cohomologically-trivial (by Ullom \\cite{Ullom}).\n\nIn particular, if we write $i_0$ for the least integer with $i_0 \\ge e\/(p-1)$, where $e$ is the ramification degree of $N\/\\QQ_p$, then for any integer $a \\ge i_0$ the formal logarithm ${\\rm log}_{B}$ and $p$-adic exponential map restrict to give isomorphisms of $\\ZZ_p[\\Gamma]$-modules\n\\begin{equation}\\label{iso1} V^{(a)}\\cong (\\wp^{a})^d,\\,\\,\\,\\,\\,\\,\\,\\,\\wp^a \\cong U^{(a)}\\end{equation}\nand so the $\\ZZ_p[\\Gamma]$-modules $V^{(a)}$ and $U^{(a)}$ are cohomologically-trivial.\n\nIn addition, for all $a$ the natural isomorphisms\n\\begin{equation}\\label{iso2} U^{(a)}\/U^{(a+1)} \\cong\\wp^a\/\\wp^{a+1},\\,\\,\\,\\,\\,\\,\\,\\, \\bigl(\\wp^a\/\\wp^{a+1}\\bigr)^d \\cong V^{(a)}\/V^{(a+1)}\\end{equation}\nimply that these quotient modules are also cohomologically-trivial. By using the tautological exact sequences for each $a < i_0$\n\\begin{equation}\\label{filter1} \\begin{cases} &0 \\to U^{(a+1)}\/U^{(i_0)} \\to U^{(a)}\/U^{(i_0)} \\to U^{(a)}\/U^{(a+1)} \\to 0,\\\\\n &0 \\to V^{(a+1)}\/V^{(i_0)} \\to V^{(a)}\/V^{(i_0)} \\to V^{(a)}\/V^{(i+1)} \\to 0\\end{cases}\\end{equation}\none can therefore deduce (by a downward induction on $a$, starting at $i_0$) that all modules $U^{(a)}$ and $V^{(a)}$ are cohomologically-trivial. This proves claim (i).\n\nIn addition, by repeatedly using the exact sequences (\\ref{filter1}) and isomorphisms (\\ref{iso2}) one computes that $d\\cdot \\chi_{\\Gamma,p}((U^{(1)}\/U^{(a)})[0],0)$ is equal to\n\n\\begin{align*} d\\cdot\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}((U^{(b)}\/U^{(b+1)})[0],0)                                                            = \\, &\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}(\\bigl((U^{(b)}\/U^{(b+1)})\\bigr)^d[0],0)\\\\\n                                                            = \\, &\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}((V^{(b)}\/V^{(b+1)})[0],0)\\\\\n                                                            = \\, &\\chi_{\\Gamma,p}((V^{(1)}\/V^{(b)})[0],0),\\end{align*}\nas required go prove claim (ii).\n\nFinally, claim (iii) is a direct consequence of the isomorphisms (\\ref{iso1}).\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{twist dependence} Let $B$ and $B'$ be abelian varieties over $M$, of the same dimension $d$, that have good ordinary reduction and are such that $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ and $\\tilde B'(\\kappa_{N^\\Delta})[p^{\\infty}]$ both vanish. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The $\\ZZ_p[\\Gamma]$-modules $B(N)^\\wedge_p$ and $B'(N)^\\wedge_p$ are cohomologically-trivial and the formal group logarithms induce an isomorphism of $\\QQ_p[\\Gamma]$-modules\n\\[ \\QQ_p\\cdot B(N)^\\wedge_p \\xrightarrow{{\\rm log}_{B,N}} N^d \\xrightarrow{{\\rm exp}_{B',N}} \\QQ_p\\cdot B'(N)^\\wedge_p.\\]\n\n\\item[(ii)] If the reduced varieties $\\tilde B$ and $\\tilde B'$ are isomorphic, then in $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ one has\n\\[ \\chi_{\\Gamma,p}(B(N)^\\wedge_p[0] \\oplus B'(N)^\\wedge_p[-1], {\\rm exp}_{B',N}\\circ {\\rm log}_{B,N}) = 0.\\]\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} Claim (i) follows directly from Lemma \\ref{useful prel}(iii).\n\nTo prove claim (ii) we write $M^{\\rm un}$ for the  maximal unramified extension of $M$, $\\hat M^{\\rm un}$ for its completion and $\\mathcal{O}$ for the valuation ring of $\\hat M^{\\rm un}$. We write $\\varphi_M$ for the Frobenius automorphism in $G_{M^{\\rm un}\/M}$.\n\nThen the formal group $\\hat B$ of $B$ is toroidal and so there exists an isomorphism of formal groups $f_1: \\hat B \\cong \\mathbb{G}_m^d$ over $\\mathcal{O}$. If we let $\\varphi_M$ act on the coefficients of $f_1$, then one has $f_1^{\\varphi_M} = u\\circ f_1$, where $u$ is the `twist matrix' of $B$. Thus $u$ belongs to ${\\rm GL}_d(\\ZZ_p)$ and depends only on $\\tilde B$ (by the argument of Mazur in \\cite[p. 216]{m}).\n\nIn particular, if $\\tilde B'$ is isomorphic to $\\tilde B$, then there exists an isomorphism of formal groups $f_2: \\hat B' \\cong \\mathbb{G}_m^d$ over $\\mathcal{O}$ for which one also has $f_2^{\\varphi_M} = u\\circ f_2$.\n\nWe now consider the isomorphism $\\phi:= f_2^{-1}\\circ f_1$ from $\\hat B$ to $\\hat B'$ over $\\mathcal{O}$. We fix an element $x$ of $\\mathcal{O}_N\\mathcal{O}^{\\rm un}$ and an element $g$  of $G_{N^{\\rm un}\/M}$ whose image in $G_{M^{\\rm un}\/M}$ is an integral power $\\varphi_M^a$ of $\\varphi_M$.\n\n\nThen one has \n\\begin{multline*} g(\\phi(x)) = \\phi^g(g(x)) = ((f_2^{\\varphi_M^a})^{-1}\\circ f_1^{\\varphi_M^a})(g(x))\\\\ = ((u^a\\circ f_2)^{-1}\\circ (u^a\\circ f_1))(g(x)) =\n(f_2^{-1}\\circ f_1)(g(x)) = \\phi(g(x)).\\end{multline*}\n\nThis means that $\\phi$ is an isomorphism of $\\ZZ_p[[G_{N^{\\rm un}\/M}]]$-modules and so restricts to give an isomorphism of $\\Gamma$-modules\n\\[ \\hat B(N) = \\hat B(N^{\\rm un})^{G_{N^{\\rm un}\/N}} \\cong \\hat B'(N^{\\rm un})^{G_{N^{\\rm un}\/N}} = \\hat B'(N).\\]\n\nUpon passing to pro-$p$-completions, and noting that the groups $\\tilde B(\\kappa_{N})[p^\\infty]$ and $\\tilde B'(\\kappa_{N})[p^\\infty]$ vanish, we deduce that $\\phi$ induces an isomorphism of $\\ZZ_p[\\Gamma]$-modules\n\\[ \\phi_p: B(N)^\\wedge_p = \\hat B(N)^\\wedge_p\\cong \\hat B'(N)^\\wedge_p = B'(N)^\\wedge_p.\\]\n\nThere is also a commutative diagram of formal group isomorphisms\n\\[\n\\begin{CD} \\hat B @> f_1  >> \\mathbb{G}_m^d @> (f_2)^{-1} >> \\hat B' \\\\\n@V {\\rm log}_B VV @V {\\rm log}_{\\mathbb{G}_m} VV @VV {\\rm log}_{B'}V\\\\\n\\mathbb{G}_a^d @> \\times f_1'(0)  >> \\mathbb{G}_a^d @> \\times f_2'(0)^{-1} >> \\mathbb{G}_a^d\\end{CD}\\]\n\nTaken together with the isomorphism $\\phi_p$, this diagram implies that the element\n\\[ \\chi_{\\Gamma,p}(B(N)^\\wedge_p[0] \\oplus B'(N)^\\wedge_p[-1], {\\rm exp}_{B',N}\\circ {\\rm log}_{B,N}) \\]\nis equal to the image under $\\partial_{\\Gamma,p}$ of the automorphism of the $\\QQ_p[\\Gamma]$-module $N^d$ that corresponds to the matrix\n $f_2'(0)^{-1}f_1'(0)$.\n\nIt is thus enough to note that, since the latter matrix belongs to ${\\rm GL}_d(\\mathcal{O}_M)$ it is represented by a matrix in ${\\rm GL}_{d[M:\\QQ_p]}(\\ZZ_p[\\Gamma])$ and so belongs to the kernel of $\\partial_{\\Gamma,p}$, as required. \\end{proof}\n\n\n\n\n\n\\subsection{Elliptic curves}\\label{ell curve sect} If $B$ is an elliptic curve over $\\QQ_p$, then it is possible in certain cases to formulate a precise conjectural formula for the elements $R_{N\/M}(\\tilde B)$ defined above.\n\nThis aspect of the theory will be considered in detail elsewhere. However, to give a brief idea of the general approach we fix an isomorphism of formal groups $f: \\hat{B} \\lra \\mathbb{G}_m$ as in the proof of Lemma \\ref{twist dependence}.\n\nThen, with $\\varphi$ denoting the Frobenius automorphism in $G_{\\Qu_p^{\\rm un} \/ \\Qp}$, the twist matrix of $\\tilde B$ is the unique element $u$ of $\\ZZ_p^\\times$ for which the composite $f^\\varphi\\circ f^{-1}$ is equal to the endomorphism $[u]_{\\mathbb{G}_m}$ of $\\mathbb{G}_m$.\n\n\\begin{lemma} $\\hat{B}$ is a Lubin-Tate formal group with respect to the parameter $u^{-1}p$.\n\\end{lemma}\n\\begin{proof} By using the equalities\n\\[\nf^\\varphi \\circ [u^{-1}p]_{\\hat{B}} \\circ f^{-1} = [u^{-1}p]_{\\mathbb{G}_m} \\circ f^\\varphi \\circ f^{-1} = [p]_{\\mathbb{G}_m}\n\\]\none computes that\n\n\\begin{eqnarray*}\n  [u^{-1}p]_{\\hat{B}} &=& \\left( f^\\varphi \\right)^{-1} \\circ [p]_{\\mathbb{G}_m} \\circ f \\\\\n &\\equiv& \\left( f^\\varphi \\right)^{-1} \\circ X^p \\circ f \\pmod{p} \\\\\n&=& \\left( f^{-1} \\right)^{\\varphi} \\left( f(X)^p \\right)  \\\\\n&=& \\left( f^{-1}(f(X)) \\right)^p \\\\\n&=& X^p.\n\\end{eqnarray*}\n\nThus, since it is well known that $ [u^{-1}p]_{\\hat{B}}  \\equiv u^{-1}p X \\pmod{\\deg 2}$, it follows that $[u^{-1}p]_{\\hat{B}} $ is a Lubin-Tate power series with respect to $u^{-1}p$, as claimed.\n\\end{proof}\n\n\n\nWe write $\\chi^{\\rm ur}$ for the restriction to $G_M$ of the character\n\\[\n\\chi_\\Qp^{\\rm ur} \\colon G_\\Qp \\lra \\Ze_p^\\times, \\quad \\varphi \\mapsto u^{-1}.\n\\]\n\nWe assume that the restriction of $\\chi^{\\rm ur}$ to $G_N$ is non-trivial and write $T$ for the (unramified) twist $\\Zp(\\chi^{\\rm ur})(1)$ of the representation $\\Zp(1)$.\n\nThen, by \\cite[Prop.~2.5]{IV} or \\cite[Lem. 3.2.1]{BC2}, the complex $R\\Gamma(N, T)$ is acyclic outside\ndegrees one and two and there are canonical identifications\n\\[\n H^i(N, T) = \\begin{cases} \\hat{B}(\\frp_N), &\\text{ if $i=1$,}\\\\\n  \\bigl( \\Zp \/ p^{\\omega_N} \\Zp \\bigr) (\\chi^{ur}),&\\text{ if $i=2$,}\\end{cases}\\]\nwhere $\\omega_N$ denotes the $p$-adic valuation of the element $1 - \\chi^{ur}(\\varphi^{f_{N\/\\Qp}})$.\n\nThese explicit descriptions allow one to interpret $R_{N\/M}(\\tilde B)$ in terms of differences between elements that occur in the formulations of the local epsilon constant conjecture for the representations $\\ZZ_p(1)$ and $T$, as studied by Benois and Berger \\cite{benoisberger}, Bley and Cobbe \\cite{BC2} and Izychev and Venjakob \\cite{IV}.\n\nIn this way one finds that the (assumed) compatibility of these conjectures for the representations $\\ZZ_p(1)$  and $T$ implies the following equality\n\n\\begin{equation}\\label{curve local eps conj} R_{N\/M}(\\tilde B) = \\delta_{\\Gamma,p}\\bigl(\\bigl(\\sum_{\\chi}u^{f_{M\/\\Qp}(s_M\\chi(1) + m_\\chi)}e_\\chi\\bigr)\n\\frac{^\\dagger((1 - (u\\cdot\\varphi^{-1})^{f_{M\/\\Qp}})e_{\\Gamma_0})}\n{^\\dagger((|\\kappa_M| - (u\\cdot\\varphi^{-1})^{-f_{M\/\\Qp}})e_{\\Gamma_0})}\\bigr)\n\\end{equation}\nwhere the conductor of each character $\\chi$ is $\\pi_M^{m_\\chi}\\calO_M$ and the different of $M\/\\QQ_p$ is $\\pi_M^{s_M}\\calO_M$.\n\nIn particular, the results of \\cite{BC2} imply that the equality (\\ref{curve local eps conj}) is unconditionally valid for certain natural families of wildly ramified extensions $N\/M$.\n\n\\section{Classical Selmer complexes and refined BSD}\\label{tmc}\n\nIn this section we study ${\\rm BSD}(A_{F\/k})$ under the assumption that $A$ and $F\/k$ satisfy the following list of hypotheses.\n\nIn this list we fix an {\\em odd} prime number $p$ and an intermediate field $K$ of $F\/k$ such that $\\Gal(F\/K)$ is a Sylow $p$-subgroup of $G$\n\\begin{itemize}\n\\item[(H$_1$)] The Tamagawa number of $A_{K}$ at each place in $S_K^A$ is not divisible by $p$;\n\\item[(H$_2$)] $S_K^A \\cap S_K^p = \\emptyset$ (that is, no place of bad reduction for $A_{K}$ is $p$-adic);\n\\item[(H$_3$)] For all $v$ in $S_K^p$ above a place in $S_k^F$ the reduction is ordinary and $A(\\kappa_v)[p^\\infty]$ vanishes;\n\\item[(H$_4$)] For all $v$ in $S_K^f\\setminus S_K^p$ above a place in $S_k^F$ the group $A(\\kappa_v)[p^\\infty]$ vanishes;\n\\item[(H$_5$)] $S_k^A\\cap S_k^F = \\emptyset$ (that is, no place of bad reduction for $A$ is ramified in $F$);\n\\item[(H$_6$)] $\\sha(A_F)$ is finite.\n\\end{itemize}\n\n\\begin{remark}\\label{satisfying H} {\\em For a fixed abelian variety $A$ over $k$ and extension $F\/k$ the hypotheses (H$_1$) and (H$_2$) are clearly satisfied by all but finitely many odd primes $p\n, (H$_4$) and (H$_5$) constitute a mild restriction on the ramification of $F\/k$ and (H$_6$) coincides with the claim of ${\\rm BSD}(A_{F\/k})$(i). However, the hypothesis~ (H$_3$) excludes the case that is called `anomalous' by Mazur in~\\cite{m} and, for a given $A$, there may be infinitely many primes $p$ for which there are $p$-adic places $v$ at which $A$ has good ordinary reduction but $A(\\kappa_v)[p]$ does not vanish.\nNevertheless, it is straightforward to describe examples of abelian varieties $A$ for which there are only finitely many such anomalous places -- see, for example, the result of Mazur and Rubin in~\\cite[Lem. A.5]{mr}.}\n\\end{remark}\n\n\\begin{remark} {\\em The validity of each of the hypotheses listed above is equivalent\nto the validity of the corresponding hypothesis with $A$ replaced by $A^t$ and we will often use this fact without explicit comment.}\n\\end{remark}\n\nIn this section we first verify and render fully explicit the computation (\\ref{bksc cohom}) of the cohomology of the Selmer complex introduced in Definition \\ref{bkdefinition}, thereby extending the computations given by Wuthrich and the present authors in \\cite[Lem. 4.1]{bmw}.\n\nSuch an explicit computation will be useful in the proof of the main result of \\S\\ref{comparison section} below. We shall also use Lemma \\ref{useful prel} to ensure that, under the hypotheses listed above, this complex belongs to the category $D^{\\rm perf}(\\ZZ_p[G])$.\n\nIn the main result of this section we shall then re-interpret ${\\rm BSD}(A_{F\/k})$ in terms of invariants that can be associated to the classical Selmer complex under the above listed hypotheses.\n\n\\subsection{The classical Selmer complex}\\label{explicitbk} We fix an odd prime number $p$ and a finite set of non-archimedean places $\\Sigma$ of $k$ with\n\\[ S_k^p\\cup (S_k^F\\cap S_k^f) \\cup S_k^A \\subseteq \\Sigma.\\]\n\nFor any such set $\\Sigma$ the classical Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is defined as the mapping fibre of the morphism (\\ref{bkfibre}) in $D(\\ZZ_p[G])$.\n\nWe further recall from Lemma \\ref{independenceofsigma} that this complex is, in a natural sense, independent of the choice of $\\Sigma$ and so will be abbreviated to ${\\rm SC}_{p}(A_{F\/k})$.\n\nIn the next result we describe consequences of Lemma \\ref{useful prel} for this complex and also give a description of its cohomology that will be useful in the computations that are carried out in \\S \\ref{comparison section} below.\n\n\\begin{proposition}\\label{explicitbkprop}\n\nSet $C:= {\\rm SC}_{p}(A_{F\/k})$.\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] The complex $C$ is acyclic outside degrees one, two and three and there is a canonical identification $H^3(C)=A(F)[p^{\\infty}]^\\vee$ and a canonical inclusion of $H^1(C)$ into $H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{p}(A^t)\\bigr)$.\n\\item[(ii)] Assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then for every non-archimedean place $v$ of $k$ the $G$-modules $A^t(F_v)^\\wedge_p$ and $\\ZZ_p\\otimes_\\ZZ A^t(F_v)$ are cohomologically-trivial. In addition, the module $A^t(F_v)^\\wedge_p$ vanishes for every place $v$ in $S_k^F\\setminus S_k^p$.\n\nIn particular, the complex $C$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$.\n\\item[(iii)] Assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then for each normal subgroup $J$ of $G$ there is a natural isomorphism in $D^{\\rm perf}(\\ZZ_p[G\/J])$ of the form\n\\[ \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_{p}(A_{F\/k}) \\cong {\\rm SC}_{p}(A_{F^J\/k}).\\]\n\\item[(iv)] If $\\sha(A_F)$ is finite, then $H^1(C)$ identifies with the image of the injective Kummer map $A^t(F)_p\\to H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{p}(A^t)\\bigr)$ and there is a canonical isomorphism of $H^2(C)$ with $\\Sel_p(A_F)^\\vee$ (that is described in detail in the course of the proof below).\n    \\end{itemize}\\end{proposition}\n\n\n\n\\begin{proof}\nThroughout this argument we abbreviate the rings $\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma}$ and $\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)}$ to $U_{k}$ and $U_{F}$ respectively.\n\nSince the complexes $R\\Gamma (k_v, T_{p,F}(A^t))$ for $v$ in $\\Sigma$ are acyclic in degrees greater than two, the first assertion of claim (i) follows directly from the definition of $C$ as the mapping fibre of the morphism (\\ref{bkfibre}).\n\nIn addition, the description of the complex ${\\rm SC}_{S_k^\\infty\\cup\\Sigma}(A_{F\/k},X)$ (for any module $X$ as in Proposition \\ref{prop:perfect}) as the mapping fibre of the morphism (\\ref{selmer-finite tri}) in $D(\\ZZ_p[G])$ also implies that $H^3(C)$ is canonically isomorphic to $H^3({\\rm SC}_{S_k^\\infty\\cup\\Sigma}(A_{F\/k},X))$. Proposition \\ref{prop:perfect}(ii) thus implies that $H^3(C)$ identifies with $A(F)[p^\\infty]^\\vee$.\n\nFinally, the explicit definition of $C$ as a mapping fibre (combined with Lemma \\ref{v not p}(i)) also gives an associated canonical long exact sequence\n\\begin{multline}\\label{longexact}0 \\to H^1(C) \\to H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\to\n\\bigoplus\\limits_{w'\\in S_F^p}T_p\\bigl(H^1(F_{w'},A^t)\\bigr) \\stackrel{\\delta}{\\to} H^2(C)\\\\\n \\to H^2\\bigl(U_{F},T_{p}(A^t)\\bigr) \\to \\bigoplus\\limits_{ w'\\in \\Sigma(F)}H^2\\bigl(F_{w'},T_{p}(A^t)\\bigr) \\to\n H^3(C) \\to 0\n\\end{multline}\nin which the third and sixth arrows are the canonical maps induced by localisation. In this sequence each term $T_p\\bigl(H^1(F_{w'},A^t)\\bigr)$ denotes the $p$-adic Tate module of $H^1(F_{w'},A^t)$, which we have identified with the quotient of $H^1(F_{w'},T_{p}(A^t))$ by the image of $A^t(F_{w'})_p^\\wedge$ under the canonical Kummer map.\n\n\nIn particular, the sequence (\\ref{longexact}) gives a canonical inclusion $H^1(C) \\subseteq H^1\\bigl(U_{F},T_{p}(A^t)\\bigr)$ and this completes the proof of claim (i).\n\n\n\n\n\nTurning to claim (ii) we note first that if $v$ does not belong to $S_k^A\\cup S_k^F$ then the cohomological-triviality of $A^t(F_v)^\\wedge_p$ follows directly from Lemma \\ref{useful prel}(ii).\n\nIn addition, if $v$ is $p$-adic, then $A^t(F_v)^\\wedge_p$ is cohomologically-trivial as a consequence of Lemma \\ref{useful prel}(ii) and (iii) and the given hypotheses (H$_2$) and (H$_3$).\n\nIt suffices therefore to consider the $G$-modules $A^t(F_v)^\\wedge_p$ for places in $(S_k^A\\cup S_k^F)\\setminus S_k^p$. For each such $v$ we write $C_v(A_F)$ for the direct sum over $w'$ in $S_k^v$ of the modules $H^0(F_{w'}, H^1(I_{w'}, T_{p}(A^t))_{\\rm tor})$ that occur in the exact sequence of Lemma \\ref{v not p}(ii).\n\nNow if $v$ belongs to $S_k^A$, then (H$_5$) implies $v$ is unramified in $F\/k$ and so the $\\ZZ_p[G]$-module\n$$T_{p,F}(A^t)^{I_v}\\cong\\ZZ_p[G]\\otimes_{\\ZZ_p}T_p(A^t)^{I_v}$$\nis free. In this case therefore, the natural exact sequence\n$$0\\to T_{p,F}(A^t)^{I_v}\\stackrel{1-\\Phi_v^{-1}}{\\longrightarrow}T_{p,F}(A^t)^{I_v}\\to H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})\\to 0$$\nimplies that the $G$-module $H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})$ is cohomologically-trivial.\n\nSince the conditions (H$_1$) and (H$_5$) combine in this case to imply that $C_v(A_F)$ vanishes (as in the proof of \\cite[Lem. 4.1(ii)]{bmw}) the cohomological-triviality of $A^t(F_v)^\\wedge_p$ therefore follows from the exact sequence in Lemma \\ref{v not p}(ii).\n\nFinally, we claim that $A^t(F_v)^\\wedge_p$ vanishes for each $v$ that belongs to $S_k^F\\setminus S_k^p$. To see this we note that, in this case, (H$_5$) implies $v$ does not belong to $S_k^A$ so that $C_v(A_F)$ vanishes whilst the conditions (H$_4$) and (H$_5$) also combine (again as in the proof of \\cite[Lem. 4.1(i)]{bmw}) to imply $H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})$ vanishes. From the exact sequence of Lemma \\ref{v not p}(ii) we can therefore deduce that $A^t(F_v)^\\wedge_p$ vanishes, as claimed.\n\nAt this stage we have proved that for every non-archimedean place $v$ of $k$, the $G$-module $A^t(F_v)^\\wedge_p$ is cohomologically-trivial. Since each $\\ZZ_p[G]$-module $A^t(F_v)^\\wedge_p$ is finitely generated this implies that each complex $A^t(F_v)^\\wedge_p[-1]$ is an object of $D^{\\rm perf}(\\ZZ_p[G])$.\n\nGiven this fact, the final assertion of claim (ii) is a consequence of the definition of $C$ as the mapping fibre of (\\ref{bkfibre}) and the fact that, since $p$ is odd, the complexes $R\\Gamma(U_k,T_{p,F}(A^t))$ and $R\\Gamma (k_v, T_{p,F}(A^t))$ for each $v$ in $\\Sigma$ each belong to $D^{\\rm perf}(\\ZZ_p[G])$ (as a consequence, for example, of \\cite[Prop. 1.6.5(2)]{fukaya-kato}).\n\nTo complete the proof of claim (ii) we fix a non-archimedean place $v$ of $k$ and consider instead the $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$. We recall that  there exists a short exact sequence of $G$-modules of the form\n\\begin{equation*}\\label{finalassertion}0\\to\\mathcal{O}_{F,v}^d\\to A(F_v)\\to C\\to 0\\end{equation*}\nin which the group $C$ is finite. From this exact sequence one may in turn derive short exact sequences\n\\begin{equation}\\label{completions}0\\to((\\mathcal{O}_{F,v})^\\wedge_p)^d\\to A(F_v)^\\wedge_p\\to C^\\wedge_p\\to 0\\end{equation} and\n\\begin{equation}\\label{tensorproducts}0\\to(\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v})^d\\to\\ZZ_p\\otimes_\\ZZ A(F_v)\\to \\ZZ_p\\otimes_\\ZZ C\\to 0.\\end{equation}\n\nWe assume first that $v$ is $p$-adic. In this case the canonical maps $\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v}\\to(\\mathcal{O}_{F,v})^\\wedge_p$ and $\\ZZ_p\\otimes_\\ZZ C\\to C^\\wedge_p$ are bijective and hence the exactness of the above sequences implies that the canonical map $\\ZZ_p\\otimes_\\ZZ A(F_v)\\to A(F_v)^\\wedge_p$ is also an isomorphism. The $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$ is thus cohomologically-trivial, as required.\n\nWe finally assume that $v$ is not $p$-adic. In this case, the exact sequence (\\ref{completions}) gives an isomorphism $$A(F_v)^\\wedge_p\\cong C^\\wedge_p=\\ZZ_p\\otimes_\\ZZ C$$ and thus from the exact sequence (\\ref{tensorproducts}) we derive a short exact sequence\n$$0\\to(\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v})^d\\to\\ZZ_p\\otimes_\\ZZ A(F_v)\\to A(F_v)^\\wedge_p\\to 0.$$\nSince we have already established the cohomological-triviality of $A(F_v)^\\wedge_p$, we know that the $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$ is cohomologically-trivial if and only if the $G$-module $\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v}$ is cohomologically-trivial. But the latter module is naturally a $\\QQ$-vector-space, and therefore is indeed cohomologically-trivial. This completes the proof of claim (ii).\n\nTurning to claim (iii) we note that the cohomological-triviality of the $\\ZZ_p[G]$-module $A^t(F_v)^\\wedge_p$ for each $v$ in $\\Sigma$ (as is proved by claim (ii) under the given hypotheses) implies that there are natural isomorphisms in $D(\\ZZ_p[G\/J])$ of the form\n\\begin{align*} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}A^t(F_v)^\\wedge_p[-1] \\cong\\, &(\\ZZ_p[G\/J]\\otimes_{\\ZZ_p[G]}A^t(F_v)^\\wedge_p)[-1]\\\\\n\\cong\\, &H_0(J,A^t(F_v)^\\wedge_p)[-1]\\\\\n\\cong\\, &H^0(J,A^t(F_v)^\\wedge_p)[-1]\\\\\n= \\, & A^t(F^J_v)^\\wedge_p[-1],\\end{align*}\nwhere the third isomorphism is induced by the map sending each element $x$ of $A^t(F_v)^\\wedge_p$ to its image under the action of $\\sum_{g \\in J}g$.\n\nThe existence of the isomorphism in claim (iii) is then deduced by combining these isomorphisms together with the explicit definitions of the complexes ${\\rm SC}_{p}(A_{F\/k})$ and ${\\rm SC}_{p}(A_{F^J\/k})$ as mapping fibres and the fact (recalled, for example, from \\cite[Prop. 1.6.5(3)]{fukaya-kato}) that there are standard Galois descent isomorphisms in $D(\\ZZ_p[G\/J])$ of the form\n\\begin{equation}\\label{global descent} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}R\\Gamma(U_k,T_{p,F}(A^t)) \\cong R\\Gamma(U_k,T_{p,F^J}(A^t))\\end{equation}\nand\n\\begin{equation}\\label{local descent} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}R\\Gamma (k_v, T_{p,F}(A^t))\\cong R\\Gamma (k_v, T_{p,F^J}(A^t))\\end{equation}\nfor each $v$ in $\\Sigma$.\n\nTo prove claim (iv) we assume $\\sha(A_F)$ is finite and first prove that\nthe image of $H^1(C)$ in $H^1\\bigl(U_{F},T_{p}(A^t)\\bigr)$ coincides with the image of the injective Kummer map $$\\kappa:A^t(F)_p\\to H^1\\bigl(U_{F},T_{p}(A^t)\\bigr).$$\nIn order to do so,\nwe identify ${\\rm cok}(\\kappa)$ with the $p$-adic Tate module $T_p\\bigl(H^1\\bigl(U_F,A^t\\bigr)\\bigr)$ of $H^1\\bigl(U_F,A^t\\bigr)$.\n\nIt is clear that any element of ${\\rm im}(\\kappa)$ is mapped to the image of $A^t(F_{w'})_p^\\wedge$ in $H^1\\bigl(F_{w'},T_{p}(A^t)\\bigr)$ by localising at any place $w'$ in $S_F^p$.\n\nThe exactness of (\\ref{longexact}) therefore implies that ${\\rm im}(\\kappa)$ is contained in $H^1(C)$ and furthermore that we have a commutative diagram\nwith exact rows\n\\begin{equation}\\label{sha diag} \\xymatrix{\n0 \\ar[r] & A^t(F)_p \\ar[r] \\ar[d]^{\\kappa} & H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[r] \\ar@{=}[d] & T_p\\bigl(H^1\\bigl(U_F,A^t\\bigr)\\bigr)  \\ar[r] \\ar[d] & 0\\\\\n0 \\ar[r] & H^1(C) \\ar[r] & H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[r] & \\bigoplus\\limits_{w'\\in S_F^p}T_p\\bigl(H^1(F_{w'},A^t)\\bigr),\n}\\end{equation}\nwhere the right-most vertical arrow is induced by the localisation maps.\nBut the assumed finiteness of $\\sha(A_F)$ (combined with \\cite[Ch. I, Cor 6.6]{milne}) then implies that this arrow is injective, and therefore the Snake Lemma implies that $\\im(\\kappa)=H^1(C)$, as required.\n\n\n\nTo conclude the proof of claim (iv) we use the canonical exact triangle\n\\begin{multline}\\label{compacttriangle}R\\Gamma_c\\bigl(U_F,T_p(A^t)\\bigr)\\to R\\Gamma\\bigl(U_k,T_{p,F}(A^t)\\bigr)\\to \\bigoplus\\limits_{ v\\in S_k^\\infty\\cup\\Sigma} R\\Gamma(k_v,T_{p,F}(A^t))\\\\ \\to R\\Gamma_c\\bigl(U_F,T_p(A^t)\\bigr)[1]\\end{multline}\nin $D(\\ZZ_p[G])$. We also write $\\Delta$ for the canonical composite homomorphism\n$$\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\to\\bigoplus\\limits_{ w'\\in \\Sigma(F)}H^1(F_{w'},T_{p}(A^t))\\to H^2_c(U_F,T_p(A^t)),$$\nwith the first arrow given by the local Kummer maps and the second arrow given by the long exact cohomology sequence associated to the triangle (\\ref{compacttriangle}). We then claim that there is a canonical commutative diagram\n\n\\begin{equation}\\label{Selmerdiagram}\\xymatrix\nH^2_c\\bigl(U_F,T_p(A^t)\\bigr) \\ar[d] \\ar@{=}[r] &\n H^2_c\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[d] \\ar[r]^-{w\\circ s^{-1}} &\nH^1\\bigl(U_F,A[p^\\infty])^\\vee\n \\ar[d] \\\\\nH^2(C) \\ar^-{\\sim}[r] &\n\\cok(\\Delta) \\ar[r]^-{\\sim} &\n\\Sel_p(A_F)^\\vee\n.\n}\\end{equation}\nHere the second and third vertical arrows are the canonical projection maps and $w$ and $s$ are the isomorphisms defined in (\\ref{themapw}) and (\\ref{themaps}) in Appendix \\ref{exp rep section} below.\n\nThe composition of the horizontal arrows in the bottom row of the diagram (\\ref{Selmerdiagram}) will then define the desired canonical isomorphism of $H^2(C)$ with $\\Sel_p(A_F)^\\vee$.\n\nTo verify the existence of the diagram (\\ref{Selmerdiagram}) we use the canonical exact sequence\n\\begin{multline}\\label{comparingsequence}0\\to\\bigoplus\\limits_{ v\\in S_k^\\infty} H^0(k_v,T_{p,F}(A^t)) \\to H^1_c\\bigl(U_F,T_p(A^t)\\bigr)\\to H^1(C)\\\\ \\to\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\stackrel{\\Delta}{\\to}H^2_c\\bigl(U_F,T_p(A^t)\\bigr)\\to H^2(C)\\to 0\\end{multline} associated to the exact triangle (\\ref{comparingtriangles}).\n(Here we have used the fact that, as $p$ is odd, the group $H^i(k_v,T_{p,F}(A^t))$ vanishes for every $v$ in $S_k^\\infty$ and every $i > 0$.)\n\nThis exact sequence induces the desired canonical isomorphism of $H^2(C)$ with $\\cok(\\Delta)$ and, by construction, the last map occurring in the sequence gives a vertical map making the first square of the diagram (\\ref{Selmerdiagram}) commute.\n\nIt is finally straightforward, using the commutativity of the diagram in Corollary \\ref{Tatepoitouexplicit} below, to deduce that the isomorphism $$w\\circ s^{-1}:H^2_c\\bigl(U_{F},T_{p}(A^t)\\bigr) \\to\nH^1\\bigl(U_{F},A[p^\\infty])^\\vee$$ induces an isomorphism $$\\cok(\\Delta)\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee.$$ This induced isomorphism completes the construction of the diagram (\\ref{Selmerdiagram}) and thus also the proof of claim (iv).\\end{proof}\n\n\n\n\n\n\nIn the next result we shall (exceptionally for \\S\\ref{tmc}) consider the prime $2$ and describe an analogue of Proposition \\ref{explicitbkprop} in this case.\n\n\\begin{proposition}\\label{explicitbkprop2} The following claims are valid for the complex $C:= {\\rm SC}_{2}(A_{F\/k})$.\n\\begin{itemize}\n\\item[(i)] $C$ is acyclic outside degrees one, two and three.\n\\item[(ii)] If $\\sha(A_F)$ is finite, then $H^1(C)$ identifies with the image of the injective Kummer map $A^t(F)_2\\to H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{2}(A^t)\\bigr)$ and there exists a\n    canonical homomorphism $\\Sel_2(A_F)^\\vee \\to H^2(C)$, the kernel and cokernel of which are both finite.\n\\item[(iii)]  The module $H^3(C)$ is finite.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Claim (i) is established by the same argument that is used to prove the first assertion of Proposition \\ref{explicitbkprop}(i).\n\nIn a similar way, the analysis concerning the diagram (\\ref{sha diag}) is also valid in the case $p=2$ and proves the first assertion of claim (ii).\n\nTo prove the remaining claims we set $U_F := \\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)}$ and note that the long exact cohomology sequence of the exact triangle (\\ref{comparingtriangles}) gives rise in this case to an exact sequence\n\\begin{multline*} \\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_2^\\wedge \\oplus\\bigoplus\\limits_{ v\\in S_k^\\infty} H^1(k_v,T_{2,F}(A^t)) \\to H^2_c\\bigl(U_F,T_2(A^t)\\bigr) \\to H^2(C)\\\\\n\\to \\bigoplus\\limits_{ v\\in S_k^\\infty} H^2(k_v,T_{2,F}(A^t)) \\to H^3_c(U_F,T_{2}(A^t)) \\to H^3(C) \\to \\bigoplus_{ v\\in S_k^\\infty} H^3(k_v,T_{2,F}(A^t)).\\end{multline*}\nIn addition, for each $v$ in $S_k^\\infty$ and each $j \\in \\{1,2,3\\}$ the group $H^j(k_v,T_{2,F}(A^t))$ is finite.\n\nGiven these facts, the second assertion of claim (ii) is a consequence of Artin-Verdier Duality (just as with the analogous assertion in Proposition \\ref{explicitbkprop}(iv)) and claim (iii) follows directly from the isomorphism (\\ref{artinverdier}).\\end{proof}\n\n\n\n\\subsection{Statement of the main result}\\label{somr tmc sec} We continue to assume that the  hypotheses (H$_1$)-(H$_6$) are satisfied.\n\nIn this case, for any isomorphism of fields $j:\\CC\\cong\\CC_p$, the isomorphism\n\\[ h^{j}_{A,F}:=\\CC_p\\otimes_{\\RR,j}h_{A,F}^{{\\rm det}}\\]\ncombines with the explicit descriptions given in Proposition \\ref{explicitbkprop} to give a canonical element\n\\[ \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})\\]\nof $K_0(\\ZZ_p[G],\\CC_p[G])$. By Lemma \\ref{independenceofsigma} (and Remark \\ref{indeptremark}) this element is in particular independent of the choice of set $\\Sigma$ with respect to which ${\\rm SC}_p(A_{F\/k})={\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is defined. In the rest of \\S\\ref{tmc} we may and will thus set $$\\Sigma:=S_k^p\\cup (S_k^F\\cap S_k^f) \\cup S_k^A.$$\n\nOur aim in the rest of \\S\\ref{tmc} is to interpret ${\\rm BSD}_p(A_{F\/k})$(iv) in terms of an explicit description of\n this element.\n\n\n\n\\subsubsection{} At the outset we note that Hypotheses (H$_2$) and (H$_3$) imply that for each $v$ in $S_k^p$ the restriction $A^t_v$ of $A^t$ to $k_v$ satisfies the conditions that are imposed on $B$ in Proposition \\ref{basic props} and hence that the element $R_{F_w\/k_v}(\\tilde A^t_v)$ of $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])$ is well-defined.\n\nWe write $d$ for ${\\rm dim}(A)$ and then define an element of $K_0(\\ZZ_p[G],\\QQ_p[G])$ by setting\n\n\\[ R_{F\/k}(\\tilde A^t_v) := {\\rm ind}^G_{G_w}(d\\cdot R_{F_w\/k_v}+ R_{F_w\/k_v}(\\tilde A^t_v)).\\]\nHere ${\\rm ind}^G_{G_w}$ is the induction homomorphism $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])\\to K_0(\\ZZ_p[G],\\QQ_p[G])$ and $R_{F_w\/k_v}$ is the canonical element of $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])$ that is defined by Breuning in \\cite{breuning2} (and will be explicitly recalled in the course of the proof of Proposition \\ref{heavy part} below).\n\nIn the sequel we will fix a finite set of places $S$ of $k$ as in \\S\\ref{selmer section} (and hence as in the statement of Conjecture \\ref{conj:ebsd}).\n\nWe abbreviate $S_k^F\\cap S_k^f$ to $S_{\\rm r}$ and set $S_{p,{\\rm r}} := S_k^p\\cap S_k^F$.\nWe shall also write $S_{p,{\\rm w}}$ and $S_{p,{\\rm t}}$ for the (disjoint) subsets of $S_{p,{\\rm r}}$ comprising places that are respectively wildly and tamely ramified in $F$ and $S_{p,{\\rm u}}$ for the set $S_k^p\\setminus S_{p,{\\rm r}}$ of $p$-adic places in $k$ that do not ramify in $F$.\n\nFor each place $v$ in $S_k^p$ and each character $\\psi$ in $\\widehat{G}$ we define a non-zero element\n\\[ \\varrho_{v,\\psi} := {\\rm det}({\\rm N}v\\mid V_\\psi^{I_w})\\]\nof $\\QQ^c$.\n\nWe then define an invertible element of $\\zeta(\\CC[G])$ by setting\n\\begin{equation}\\label{bkcharelement}\n \\mathcal{L}^*_{A,F\/k} := \\sum_{\\psi \\in \\widehat{G}} \\frac{L^{*}_{S_{\\rm r}}(A,\\check{\\psi},1)\\cdot \\tau^{\\ast}(\\QQ,\\psi)^d\\cdot \\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v,\\psi}^d}{\\Omega_A^\\psi\\cdot w_\\psi^d}\\cdot e_\\psi\\end{equation}\nwhere, for each $\\psi$ in $\\widehat{G}$, the period $\\Omega_A^\\psi$ and root number $w_\\psi$ are as defined in \\S\\ref{k theory period sect2}, the modified global Galois-Gauss sum $\\tau^{\\ast}(\\QQ,\\psi)$ is as defined in \\S \\ref{mod GGS section} and the Hasse-Weil-Artin $L$-series $L_{S_{\\rm r}}(A,\\check{\\psi},z)$ is truncated by removing only the Euler factors corresponding to places in $S_{\\rm r}$.\n\n\n\n\\subsubsection{}\n\nWe can now state the main result of this section. In order to do so we use the homomorphism\n$$\\delta_{G,p}:\\zeta(\\CC_p[G])^\\times\\to K_0(\\ZZ_p[G],\\CC_p[G])$$ defined in (\\ref{G,O hom}) and also the local Fontaine-Messing correction $\\mu_v(A_{F\/k})$ terms defined in (\\ref{localFM}).\n\n\\begin{theorem}\\label{bk explicit} Assume $A$, $F\/k$ and $p$ satisfy all of the hypotheses (H$_1$)-(H$_6$).\n\n\nThen the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if for every isomorphism of fields $j:\\CC\\cong \\CC_p$ one has\n\\[ \\delta_{G,p}(j_\\ast(\\mathcal{L}^*_{A,F\/k}))=\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})  + \\sum_{v \\in S_{p,{\\rm w}}}R_{F\/k}(\\tilde A^t_v)+\n\\sum_{v\\in S_{p,{\\rm u}}^*} \\mu_v(A_{F\/k}). \\\nHere $S^*_{p,{\\rm u}}$ is the subset of $S_{p,{\\rm u}}$ comprising places that divide the different of $k\/\\QQ$.\n\\end{theorem}\n\n\\begin{remark}\\label{emptysets}{\\em If the sets $S_{p,{\\rm w}}$ and $S^*_{p,{\\rm u}}$ are empty, then the above result implies that the equality in ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if one has\n\\[ \\delta_{G,p}(\\mathcal{L}^*_{A,F\/k})=\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}).\\]\nThis is, in particular, the case if $p$ is unramified in $F\/\\QQ$ and, in this way, Theorem \\ref{bk explicit} recovers the results of Wuthrich and the present authors in \\cite[Prop. 4.2 and Th. 4.3]{bmw}. More generally, the above equality is predicted whenever no $p$-adic place of $k$ is wildly ramified in $F$ and, in addition, $p$ is unramified in $k\/\\QQ$ (as is obviously the case if $k = \\QQ$) and this case will play an important role in the special settings considered in \\S\\ref{mod sect} and \\S\\ref{HHP}. }\\end{remark}\n\n\\begin{remark}\\label{breuning remark}{\\em In \\cite[Conj. 3.2]{breuning2} Breuning has conjectured that the terms $R_{F_w\/k_v}$ should always vanish. In \\cite{breuning} and \\cite{breuning2} he has proved this conjecture for all tamely ramified extensions, for all abelian\nextensions of $\\QQ_p$ with $p$ odd, for all $S_3$-extensions of $\\QQ_p$ and for certain families of\ndihedral and quaternion extensions. If $p$ is odd, then Bley and Debeerst \\cite{bleydebeerst} have also given an algorithmic proof of the conjecture for all Galois extensions of $\\QQ_p$ of degree at most $15$. More recently, Bley and Cobbe \\cite{BC} have proved the conjecture for certain natural families of wildly ramified extensions. }\\end{remark}\n\n\n\\begin{remark}{\\em If $A$ is an elliptic curve, then Remark \\ref{breuning remark} combines with the equality in (\\ref{curve local eps conj}) to give a completely explicit description of the elements $R_{F\/k}(\\tilde A_v)$. However, whilst the results of \\cite{BC2} imply that this description is unconditionally valid for certain families of wildly ramified extensions, it is, in general, conjectural.}\\end{remark}\n\n\n\\begin{remark}\\label{bsdinvariants}{\\em If $F=k$ then it can be shown that the element (\\ref{bkcharelement}) is equal to the product $(-1)^d\\cdot(L^\\ast(A,1)\/\\Omega_A)\\cdot (\\sqrt{|d_k|)}^{d}$ with $$\\Omega_A=\\prod_{v\\in S_k^\\CC}\\Omega_{A,v}\\cdot\\prod_{v\\in S_k^\\RR}\\Omega_{A,v}^+,$$ where the classical periods $\\Omega_{A,v}$ and $\\Omega_{A,v}^+$ are as defined in \\S\\ref{k theory period sect2}.}\n\\end{remark}\n\n\n\\subsection{The proof of Theorem \\ref{bk explicit}}\n\n\n\\subsubsection{}\\label{clever peiods}\n\nIn view of Lemma \\ref{pro-p lemma} it is enough for us to fix a field isomorphism $j:\\CC\\cong \\CC_p$ and show that the displayed equality in Theorem \\ref{bk explicit} is equivalent to (\\ref{displayed pj}).\n\n\nTaking advantage of Remark \\ref{consistency remark}(i), we first specify the set $S$ to be equal to $S^\\infty_k\\cup S_k^F\\cup S_k^A$. We shall next use the approach of \\S\\ref{k theory period sect} to make a convenient choice of differentials $\\omega_\\bullet$.\n\nFor each $v$ in $S_k^p$ we set\n\\[ \\mathcal{D}_v := \\Hom_{\\mathcal{O}_{k_v}}(H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t}), \\mathcal{O}_{k_v}),\\]\nwhere the N\\'eron models $\\mathcal{A}^t_v$ are as fixed at the beginning of \\S\\ref{perf sel sect}.\n\nFor each such $v$ we also fix a free (rank one) $\\mathcal{O}_{k_v}[G]$-submodule $\\mathcal{F}_v$ of $F_v = k_v\\otimes_k F$ and we assume that for each $v \\in S_{p,{\\rm u}}$ one has\n\\[ \\mathcal{F}_v = \\mathcal{O}_{F,v} = \\mathcal{O}_{k_v}\\otimes_{\\mathcal{O}_{k}}\\mathcal{O}_{F}.\\]\n\nWe then set\n\\[ \\Delta(\\mathcal{F}_v) := \\mathcal{F}_v\\otimes_{\\mathcal{O}_{k_v}}\\mathcal{D}_v.\\]\n\nFor each place $w'$ in $S_F^p$ we write $\\Sigma(F_{w'})$ for the set of $\\QQ_p$-linear embeddings $F_{w'} \\to \\QQ_p^c$, we define a $\\ZZ_p[G_{w'}]$-module $Y_{F_{w'}} := \\prod_{\\sigma \\in \\Sigma(F_{w'})}\\ZZ_p$ (upon which $G_{w'}$ acts via precomposition with the embeddings) and write\n\\[ \\pi_{F_{w'}}: \\QQ_p^c\\otimes_{\\ZZ_p}F_{w'} \\to \\QQ_p^c\\otimes_{\\ZZ_p}Y_{F_{w'}}\\]\nfor the isomorphism of $\\QQ_p^c[G_{w'}]$-modules that sends each element $\\ell\\otimes f$ to $(\\ell\\otimes \\sigma(f))_\\sigma$.\n\nFor each $v$ in $S_k^p$ we then consider the isomorphism of $\\QQ_p[G]$-modules\n\\[ \\pi_{F_v}: \\QQ_p^c\\otimes_{\\ZZ_p}F_v = \\prod_{w'\\in S_F^v}(\\QQ_p^c\\otimes_{\\ZZ_p}F_{w'}) \\xrightarrow{(\\pi_{F_{w'}})_{w'}} \\QQ_p^c\\otimes_{\\ZZ_p}\\bigoplus_{w' \\in S_F^v}Y_{F_{w'}} = \\QQ_p^c\\otimes_{\\ZZ_p}Y_{F_v},  \\]\nwhere we set $Y_{F_v} := \\bigoplus_{w'}Y_{F_{w'}}$.\n\nAfter fixing an embedding of $\\QQ^c$ into $\\QQ_p^c$ we obtain an induced identification of $\\bigoplus_{v \\in S_k^p}Y_{F_v}$ with the module $Y_{F,p} := \\bigoplus_{\\Sigma(F)}\\ZZ_p$, upon which $G$ acts via pre-composition on the embeddings.\n\nWe next fix\n\\begin{itemize}\n\\item[$\\bullet$] an ordered $k$-basis $\\{\\omega'_j:j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$ that generates over $\\mathcal{O}_{k,p}$ the module\n $\\mathcal{D}_p := \\prod_{v \\in S_k^p}\\mathcal{D}_v$, and\n\\item[$\\bullet$] an ordered $\\ZZ_p[G]$-basis $\\{z_b:b \\in [n]\\}$ of $\\mathcal{F}_p := \\prod_{v\\in S_k^p}\\mathcal{F}_v$.\n\\end{itemize}\n\nThen the (lexicographically ordered) set\n\\[ \\omega_\\bullet:= \\{ z_b\\otimes \\omega'_j: b \\in [n], j \\in [d]\\}\\]\nis a $\\QQ_p[G]$-basis of $H^0(A_F^t,\\Omega^1_{A_F^t}) = F\\otimes_kH^0(A^t,\\Omega^1_{A^t})$ and the\n\narguments of Lemma \\ref{k-theory period} and Proposition \\ref{lms} combine to show that\n\\begin{equation}\\label{norm resolvents} \\partial_{G,p}\\left(j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))\\right)=\\delta_{G,p}\\left(j_*(\\Omega_A^{F\/k}\\cdot w_{F\/k}^d)\\right) + \\sum_{v\\in S_k^p}d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\end{equation}\nin $K_0(\\ZZ_p[G],\\CC_p[G])$.\n\n\\subsubsection{}Now, if necessary, we can multiply $\\mathcal{F}$ by a sufficiently large power of $p$ in order to ensure that for every $v$ in $S_{p,{\\rm r}}$ the following two conditions are satisfied.\n\n\\begin{itemize}\n\\item[$\\bullet$] the $p$-adic exponential map induces a (well-defined) injective homomorphism from $\\mathcal{F}_v$ to $(F_v^\\times)^\\wedge_p$;\n\\item[$\\bullet$] the formal group exponential ${\\rm exp}_{A^t,F_v}$ that arises from the differentials $\\{\\omega'_j:j \\in [d]\\}$ induces an isomorphism of $\\Delta(\\mathcal{F}_v)$ with a submodule of $A^t(F_v)^\\wedge_p$.\n\\end{itemize}\n\nFor each $v$ in $S_k^p$ we now set\n\\[ X(v) := \\begin{cases}{\\rm exp}_{A^t,F_v}(\\Delta(\\mathcal{F}_v)), &\\text{ if $v \\in S_{p,{\\rm r}}$}\\\\\nA^t(F_v)^\\wedge_p, &\\text{ if $v \\in S_{p,{\\rm u}}$.}\n\\end{cases}\\]\nThen it is clear that, for any choice of $\\gamma_\\bullet$ as in \\S \\ref{perf sel sect} and our specific choices of $S$ and $\\omega_\\bullet$, the module $X(v)$ coincides with $\\mathcal{X}(v)$ for $\\mathcal{X}=\\mathcal{X}_S(\\{\\mathcal{A}_v^t\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$.\n\nThe description of the complex\n\\[ C_{X(p)} := {\\rm SC}_{S}(A_{F\/k};X(p),H_\\infty(A_{F\/k})_p)\\]\nas the mapping fibre of the morphism (\\ref{selmer-finite tri}) gives rise to an exact triangle\n\\[ C_{X(p)} \\to {\\rm SC}_p(A_{F\/k}) \\oplus X(p)[-1] \\xrightarrow{(\\lambda', \\kappa'_1)}\n\\bigoplus_{v \\in S_k^p\\cup S_k^A} A^t(F_v)_p^\\wedge[-1]\\to C_{X(p)}[1].\\]\nHere we have used the fact that, for $v\\in S_{\\rm r}\\setminus S_k^p$, the module $A^t(F_v)_p^\\wedge$ vanishes by Proposition \\ref{explicitbkprop}(ii).\n\nFurther, since Proposition \\ref{explicitbkprop}(ii) implies that the $\\ZZ_p[G]$-modules\n\\[ X(p):= \\prod_{v\\in S_k^p}X(v) \\,\\,\\text{ and }\\,\\, \\bigoplus_{v \\in S_k^p\\cup S_k^A} A^t(F_v)_p^\\wedge\\]\nare cohomologically-trivial and that the complex ${\\rm SC}_p(A_{F\/k})$ is perfect, Proposition \\ref{prop:perfect}(i) implies that this is a triangle in $D^{\\rm perf}(\\ZZ_p[G])$.\n\nBy applying Lemma \\ref{fk lemma} to this exact triangle, we can therefore deduce that there is in $K_0(\\ZZ_p[G],\\CC_p[G])$ an equality\n\n\\begin{align}\\label{first comp} &\\chi_{G,p}(C_{X(p)},h^{j}_{A,F})\\\\\n =\\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\sum_{v \\in S_k^A}\\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)\\notag\\\\\n &\\hskip 2truein - \\sum_{v\\in S_k^p}\\chi_{G,p}(X(v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm id})\\notag\\\\\n= \\, & \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\sum_{v \\in S_k^A}\\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)\\notag\\\\\n&\\hskip 2truein - \\sum_{v\\in S_{p,{\\rm r}}}\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\notag\\\\\n= \\, & \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\delta_{G,p}\\Bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\Bigr)\\notag\\\\\n&\\hskip 2truein - \\sum_{v\\in S_{p,{\\rm r}}}\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\notag\\end{align}\nwhere the last equality holds because, for each $v\\in S_k^A$, one has\n\\[ \\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)=  \\delta_{G,p}\\Bigl( L_v(A,F\/k)\\Bigr).\\]\nThis equality in turn follows upon combining the argument that gives \\cite[(13)]{bmw} with the exactness of the sequence of Lemma \\ref{v not p}(ii) for each place $w'$ of $F$ above a place in $S_k^A$ and the fact that the third term occurring in each of these sequences vanishes, as verified in the course of the proof of Proposition \\ref{explicitbkprop}(ii).\n\n\n\\subsubsection{}The equalities (\\ref{norm resolvents}) and (\\ref{first comp}) lead us to consider for each place $v$ in $S_k^p$ the element\n\\[ c(F\/k,\\tilde A^t_v) := \\begin{cases} d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]-\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v}), &\\text{if $v \\in S_{p,{\\rm r}}$,}\\\\\nd\\cdot[\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}], &\\text{if $v \\in S_{p,{\\rm u}}$}.\\end{cases}\\]\n\nIt is straightforward to check that for each $v \\in S_{p,{\\rm r}}$ this element is independent of the choice of $\\mathcal{F}_v$ and that Lemma \\ref{twist dependence} implies its dependence on $A$ is restricted to (the twist matrix of) the reduction of $A^t$ at $v$.\n\nThe key step, however, in the proof of Theorem \\ref{bk explicit} is the computation of this element in term of local Galois-Gauss sums that is described in the next result.\n\nFor each place $v$ in $S_k^f$ we define an\n`equivariant local Galois-Gauss sum' by setting\n\\[ \\tau_v(F\/k) := \\sum_{\\psi \\in \\widehat{G}}\\tau(\\QQ_{\\ell(v)},\\psi_v)\\cdot e_\\psi\\in \\zeta(\\QQ^c[G])^\\times.\\]\nHere $\\psi_v$ denotes the restriction of $\\psi$ to $G_w$ and $\\tau(\\QQ_{\\ell(v)},\\psi_v)$ is the Galois-Gauss sum (as defined in \\cite{martinet}) of the induction to $G_{\\QQ_{\\ell(v)}}$ of the character of $G_{k_v}$ that is obtained by composing $\\psi_v$ with the natural projection $G_{k_v} \\to G_w$.\n\nWe also define a modified local Galois-Gauss sum by setting\n\\[ \\tau_v^p(F\/k) := \\begin{cases} \\varrho_v(F\/k)\\cdot u_v(F\/k)\\cdot \\tau_v(F\/k), &\\text{ if $v \\in S_{p,{\\rm r}}$,}\\\\\n                                              u_v(F\/k)\\cdot \\tau_v(F\/k), &\\text{ otherwise,}\\end{cases}\\]\nwhere we set\n\\begin{equation}\\label{varrho def} \\varrho_{v}(F\/k) := \\sum_{\\psi\\in \\widehat{G}}\\varrho_{v,\\psi}\\cdot e_\\psi\\end{equation}\nand the element $u_v(F\/k)$ of $\\zeta(\\QQ[G])^\\times$ is as defined in (\\ref{u def}).\n\nFinally, for each $p$-adic place $v$ of $k$ we set\n\n\\[ U_v(F\/k) := {\\rm ind}^{G}_{G_w}(U_{F_w\/k_v}),\\]\nwhere $U_{F_w\/k_v}$ is the `unramified term' in $K_0(\\ZZ_p[G_w],\\QQ_p^c[G_w])$ that is defined by Breuning in \\cite[\\S2.5]{breuning2}.\n\n\\begin{proposition}\\label{heavy part} For each place $v$ in $S_k^p$ and any choice of $j$ one has\n\\[c(F\/k,\\tilde A^t_v) =  d\\cdot \\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + d\\cdot U_v(F\/k) - R_{F\/k}(\\tilde A^t_v).\\]\nin $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\\end{proposition}\n\n\\begin{proof} The term $\\delta_{G,p}(j_*(\\tau^p_v(F\/k)))$ is independent of the choice of $j$ by \\cite[Lem. 2.2]{breuning2}. To prove the claimed equality we consider separately the cases $v \\in S_{p,{\\rm r}}$ and $v \\in S_{p,{\\rm u}}$.\n\nWe assume first that $v$ belongs to $S_{p,{\\rm r}}$. Then for every place $w'$ in $S_F^v$ we consider the corresponding perfect complex of $\\ZZ_p[G_{w'}]$-modules $R\\Gamma(F_{w'},\\ZZ_p(1))$ as described in \\S \\ref{twist inv prelim} and obtain a perfect complex of $\\ZZ_p[G]$-modules\n$$R\\Gamma(F_v,\\ZZ_p(1)):=\\prod_{w'\\in S_F^v}R\\Gamma(F_{w'},\\ZZ_p(1)).$$\n\nSince Kummer theory canonically identifies the cohomology in degree one of this complex with $(F_v^\\times)_p^\\wedge$ we may define an additional perfect complex of $\\ZZ_p[G]$-modules $C^\\bullet_{\\mathcal{F}_v}$ through the exact triangle\n\\[ \\mathcal{F}_v[0] \\xrightarrow{\\alpha_v} R\\Gamma(F_v,\\ZZ_p(1))[1] \\to C^\\bullet_{\\mathcal{F}_v} \\to \\mathcal{F}_v[1] \\]\nin $D^{\\rm perf}(\\ZZ_p[G])$, with $H^0(\\alpha_v)$ induced by the $p$-adic exponential map ${\\rm exp}_p$.\n\nWe write $f$ for the residue degree of our fixed place $w$ in $S_F^v$. Then the long exact cohomology sequence of the above triangle implies that the normalised valuation map ${\\rm val}_{F\/k,v} := (f\\cdot({\\rm val}_{F_{w'}}))_{w'\\in S_F^v}$\ninduces an isomorphism of $\\QQ_p[G]$-modules\n\\begin{multline*} \\QQ_p \\cdot H^0(C^\\bullet_{\\mathcal{F}_v})  \\cong \\QQ_p\\cdot ((F_v^\\times)^\\wedge_p\/{\\rm exp}_p(\\mathcal{F}_v)) \\xrightarrow{ {\\rm val}_{F\/k,v}}\\prod_{w'\\in S_F^v}\\QQ_p \\\\\n\\xleftarrow{ ({\\rm inv}_{F_{w'}})_{w'}} \\QQ_p\\cdot \\prod_{w'\\in S_F^v}H^2(F_{w'},\\ZZ_p(1)) \\cong \\QQ_p\\cdot H^1(C^\\bullet_{\\mathcal{F}_v})\\end{multline*}\nwhich, by abuse of notation, we also denote by ${\\rm val}_{F\/k,v}$.\n\nIn addition, the chosen differentials $\\{\\omega'_a: a \\in [d]\\}$ induce an isomorphism of\n $\\mathcal{O}_{k_v}$-modules $\\mathcal{D}_{v}\\cong \\mathcal{O}^d_{k_v}$\nand hence an isomorphism of $\\mathcal{O}_{k_v}[G]$-modules $\\omega_{v,*}: \\Delta(\\mathcal{F}_v) \\cong \\mathcal{F}_v^d$.\n\nIn particular, if we write $C_{A^t,F}^{v,\\bullet}$ for the complex $\\prod_{w'\\in S_F^v} C_{A^t_v,F_{w'}}^\\bullet$, where each complex $C_{A^t_v,F_{w'}}^\\bullet$ is as defined at the beginning of \\S\\ref{twist inv prelim}, then there exists a canonical exact triangle\n\\[ \\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1] \\xrightarrow{\\iota_v} C_{A^t,F}^{v,\\bullet} \\xrightarrow{\\iota'_v} C^{\\bullet,d}_{\\mathcal{F}_v} \\to \\Delta(\\mathcal{F}_v)[1]\\oplus A^t(F_v)^\\wedge_p[0]\\]\nin $D^{\\rm perf}(\\ZZ_p[G])$.\nHere $C^{\\bullet,d}_{\\mathcal{F}_v}$ denotes the product of $d$ copies of $C^{\\bullet}_{\\mathcal{F}_v}$, $H^0(\\iota_v)$ is the composite map $({\\rm exp}_p)^d\\circ \\omega_{v,*}$ and $H^1(\\iota_v)$ is the identity map between\n$A^t(F_v)^\\wedge_p = H^1(A^t(F_v)^\\wedge_p[-1])$ and the direct summand $A^t(F_v)^\\wedge_p$ of $H^1(C_{A^t,F}^{v,\\bullet})$.\n\nThe long exact cohomology sequence of this triangle also gives an exact commutative diagram of $\\QQ_p[G]$-modules\n\\[\\minCDarrowwidth1em\\begin{CD} 0 @> >>\\! \\QQ_p\\cdot \\Delta(\\mathcal{F}_v)\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^0(\\iota_v) >>\\! \\QQ_p\\cdot H^0(C_{A^t,F}^{v,\\bullet})\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^0(\\iota'_v) >> \\!\\QQ_p\\cdot H^0(C^{\\bullet,d}_{\\mathcal{F}_v})\\! @> >>\\! 0\\\\\n@. @V{\\rm exp}_{A^t,F_v} VV @V  \\lambda^v_{A^t,F} VV @V ({\\rm val}_{F\/k,v})^d VV \\\\\n0 @> >>\\! \\QQ_p\\cdot A^t(F_v)_p^\\wedge\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^1(\\iota_v) >>\\! \\QQ_p\\cdot H^1(C_{A^t,F}^{v,\\bullet})\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^1(\\iota'_v)>> \\!\\QQ_p\\cdot H^1(C^{\\bullet, d}_{\\mathcal{F}_v}) @> >>\\! 0,\\end{CD}\\]\nin which $\\lambda^v_{A^t,F} = (\\lambda_{A^t_v,F_{w'}})_{w'\\in S_F^v}$, where each map $\\lambda_{A^t_v,F_{w'}}$ is fixed as in diagram (\\ref{lambda diag}).\n\nAfter recalling the definition of $R_{F_w\/k_v}(\\tilde A^t_v)$ and applying Lemma \\ref{fk lemma} to this commutative diagram one can therefore derive an equality\n\\begin{align*}\\label{third}\n&\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\\\\n= \\, &\\chi_{G,p}(C_{A^t,F}^{v,\\bullet},\\lambda_{A^t,F}^v) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n= \\, &{\\rm ind}^G_{G_w}(\\chi_{G_w,p}(C_{A^t_v,F_{w}}^\\bullet,\\lambda_{A^t_v,F_{w}})) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n= \\, &{\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v))-{\\rm ind}^G_{G_w}(d\\cdot\\delta_{G_w,p}(c_{F_w\/k_v})) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n=\\, & {\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v)) - d(\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v})).\\notag\\end{align*}\n\nIt follows that\n\\[ c(F\/k,\\tilde A^t_v) = d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]-{\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v)) + d(\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v}))\\]\nand hence that the claimed result is true in this case if one has\n\\begin{multline}\\label{wanted at last}  [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}] + \\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v})\n\\\\ = \\delta_{G,p}(j_*(\\tau^p_v(F\/k))) + U_v(F\/k) - {\\rm ind}^G_{G_w}(R_{F_w\/k_v}).\\end{multline}\n\nTo prove this we note that the very definition of $R_{F_w\/K_v}$ in \\cite[\\S3.1]{breuning2} implies that\n\\begin{multline*} {\\rm ind}^G_{G_w}(R_{F_w\/k_v})\\\\ = \\delta_{G,p}(j_*(\\tau_v(F\/k))) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}] - \\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,v}) + U_v(F\/k) - \\delta_{G,p}(m_{F_w\/k_v}),\\end{multline*}\nwhere ${\\rm val}'_{F\/k,v}$ denotes the isomorphism of $\\QQ_p[G]$-modules\n\\[ \\QQ_p\\cdot H^1(C^\\bullet_{\\mathcal{F}_v})\\cong \\QQ_p\\cdot H^2(C^\\bullet_{\\mathcal{F}_v})\\]\nthat is induced by the maps ($({\\rm val}_{F_{w'}}))_{w'\\in S_F^v}$ and we use the element\n\\[ m_{F_w\/k_v}:= \\frac{^\\dagger(f\\cdot e_{G_w})\\cdot \\, ^\\dagger((1- \\Phi_v\\cdot {\\rm N}v^{-1})e_{I_w})}{^\\dagger((1-\\Phi_v^{-1})e_{I_w})}\\]\nof $\\zeta(\\QQ[G])^\\times$. (Here we use the notational convention introduced in (\\ref{dagger eq}). In addition, to derive the above formula for ${\\rm ind}^G_{G_w}(R_{F_w\/k_v})$ we have relied on \\cite[Prop. 2.6]{breuning2} and the fact that in loc. cit. Breuning uses the `opposite' normalization of Euler characteristics to that used here, so that the term $\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,v})$ appears in the corresponding formulas in loc. cit. with a negative sign.)\n\nTo deduce the required equality (\\ref{wanted at last}) from this formula it is then enough to note that\n\\[ \\chi_{G}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,p})\n= \\chi_{G}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,p}) - \\delta_{G,p}(^\\dagger(f\\cdot e_{G_w})),\\]\nand that an explicit comparison of definitions shows that\n\\[  j_*(\\tau^p_v(F\/k))\\cdot m_{F_w\/k_v} = j_*(\\tau_v(F\/k))\\cdot ^\\dagger(f\\cdot e_{G_w})\\cdot  c_{F_w\/k_v}.\\]\n\nTurning now to the case $v\\in S_{p,{\\rm u}}$ we only need to prove that for each such place $v$ one has\n\\[ d\\cdot[\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}] = d\\cdot \\delta_{G,p}(j_*(u_v(F\/k)\\cdot\\tau_v(F\/k))) + d\\cdot U_v(F\/k) - R_{F\/k}(\\tilde A^t_v).\\]\n\nNow, since each such $v$ is unramified in $F\/k$ the term $R_{F\/k}(\\tilde A^t_v)$ vanishes (as a consequence of Proposition \\ref{basic props}(iii) and Remark \\ref{breuning remark}) and so it is enough to note that\n\\[ [\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}] = \\delta_{G,p}((u_v(F\/k)\\cdot \\tau_v(F\/k))) + U_v(F\/k),\\]\nas is shown in the course of the proof of \\cite[Th. 3.6]{breuning2}.\n\\end{proof}\n\n\n\n\\subsubsection{}We next record a result concerning the decomposition of global Galois-Gauss sums as a product of local Galois-Gauss sums.\n\n\\begin{lemma}\\label{gauss} In $K_0(\\ZZ_p[G],\\QQ^c_p[G])$ one has\n\\[ \\delta_{G,p}(j_*(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k))) = \\sum_{v \\in S_{k}^p}(\\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k)).\\]\n\\end{lemma}\n\n\\begin{proof} We observe first that the difference $\\xi$ of the left and right hand sides of this claimed equality belongs to $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nThis follows from the fact that both the term\n\\[ \\delta_{G,p}(j_*(\\tau^\\ast(F\/k))) - \\sum_{v \\in S_k^p}[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}],\\]\nand for each $v \\in S_k^p$ the term\n\\[ \\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}],\\]\nbelong to $K_0(\\ZZ_p[G],\\QQ_p[G])$ (the former as a consequence of \\cite[Prop. 3.4 and (34)]{bleyburns} and the latter as a consequence of  \\cite[Prop. 3.4]{breuning2}).\n\nThus, by Taylor's Fixed Point Theorem for group determinants (as discussed in \\cite[Chap. 8]{Taylor}) it is enough for us to show that $\\xi$ belongs to the kernel of the natural homomorphism $\\iota: K_0(\\ZZ_p[G],\\QQ_p^c[G]) \\to K_0(\\mathcal{O}^t_p[G],\\QQ_p^c[G])$ where $\\mathcal{O}_p^t$ is the valuation ring of the maximal tamely ramified extension of $\\QQ_p$ in $\\QQ_p^c$.\n\nNow from \\cite[Prop. 2.12(i)]{breuning2} one has $\\iota(U_v(F\/k)) = 0$ for all $v$ in $S_k^p$. In addition, for each non-archimedean place $v$ of $k$ that is not $p$-adic the vanishing of $\\iota(\\delta_{G,p}(j_*(u_v(F\/k)\\cdot \\tau_v(F\/k))))$ is equivalent to the result proved by  Holland and Wilson in\n\\cite[Th. 3.3(b)]{HW3} (which itself relies crucially on the work of Deligne and Henniart in \\cite{deligne-henniart}).\n\nThe vanishing of $\\iota(\\xi)$ is thus a consequence of the fact that the classical decomposition of global Galois-Gauss sums as a product of local Galois-Gauss sums implies that\n\\[ \\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k) = \\prod_{v}\\tau^p_v(F\/k)\\]\nwhere $v$ runs over all places of $k$ that divide the discriminant of $F$, since for any place $v$ that does not ramify in $F$ one has $\\tau(\\QQ_{\\ell(v)},\\psi_v) = 1$ for all  $\\psi$ in $\\widehat{G}$.\n\\end{proof}\n\n\\subsubsection{}We can now complete the proof of Theorem \\ref{bk explicit}.\n\nTo this end we note first that the definition (\\ref{bkcharelement}) of $\\mathcal{L}^*_{A,F\/k}$ implies that\n\n\\begin{align*} &\\delta_{G,p}(j_\\ast(\\mathcal{L}^*_{A,F\/k})) - \\partial_{G,p}\\left(\\frac{j_*(L_S^*(A_{F\/k},1))}{j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}\\right)\\\\\n=\\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) + d\\cdot\\bigl(\\delta_{G,p}(j_*(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k))) - \\sum_{v\\in S_k^p}[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr)\\\\\n= \\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) + d\\cdot\\sum_{v \\in S_{k}^p}\\bigl(\\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr)\n\\\\\n= \\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) +\\sum_{v \\in S_{k}^p} \\bigl(c(F\/k,\\tilde A^t_v) - d\\cdot [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr) +\\sum_{v \\in S_{k}^p}R_{F\/k}(\\tilde A^t_v)\\\\\n= \\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})+\\sum_{v \\in S_{k}^p}R_{F\/k}(\\tilde A^t_v) - \\chi_{G,p}(C_{X(p)},h^{j}_{A,F})\\\\\n= \\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})+\\sum_{v \\in S_{p,{\\rm w}}}R_{F\/k}(\\tilde A^t_v) - \\chi_{G,p}(C_{X(p)},h^{j}_{A,F}).\\end{align*}\nHere the first equality follows from (\\ref{norm resolvents}), the second from Lemma \\ref{gauss}, the third from Proposition \\ref{heavy part}, the fourth from the definition of the terms $c(F\/k,\\tilde A^t_v)$ and the equality (\\ref{first comp}) and the last from the fact that $R_{F\/k}(\\tilde A^t_v)$ vanishes for each $v \\in S_k^p\\setminus S_{p,{\\rm w}}$ as  consequence of Proposition \\ref{basic props}(iii) and Remark \\ref{breuning remark}.\n\nIt is thus sufficient to show that the above displayed equality implies that the equality (\\ref{displayed pj}), with set of places $S$ and differentials $\\omega_\\bullet$ chosen as in \\S\\ref{clever peiods}, is equivalent to the equality stated in Theorem \\ref{bk explicit}.\n\nBut this is true since our choice of periods $\\omega_\\bullet$ and lattices $\\mathcal{F}_v$ as in \\S\\ref{clever peiods} implies that the module $Q(\\omega_\\bullet)_{S,p}$ vanishes and because\n\n\\begin{align*} \\mu_S(A_{F\/k})_p =\\, &\\sum_{v \\in S_k^p\\setminus S}\\mu_v(A_{F\/k})\\\\\n =\\, &\\sum_{v \\in S_{p,{\\rm u}}}\\mu_v(A_{F\/k})\\\\\n  =\\, &\\sum_{v \\in S^\\ast_{p,{\\rm u}}} \\mu_v(A_{F\/k})\\end{align*}\nwhere the last equality follows from Lemma \\ref{fm} below.\n\nThis completes the proof of Theorem \\ref{bk explicit}.\n\n\\begin{lemma}\\label{fm} For any place $v\\in S_k^A$ for which the residue characteristic $\\ell(v)$ is unramified in $F$, the term\n $\\mu_{v}(A_{F\/k})$ vanishes. \\end{lemma}\n\n\\begin{proof} We fix a place $v$ as in the statement of the lemma and set $p:= \\ell(v)$. We write $\\mathcal{O}_{F_v}$ for the integral closure of $\\ZZ_p$ in $F_v$ and set $\\wp_{F_v} := p\\cdot\\mathcal{O}_{F_v}$.\n\nThen, since $p$ does not ramify in $F\/\\QQ$, the $\\ZZ_p[G]$-modules $\\mathcal{O}_{F_v}$ and $\\wp_{F_v}$ are projective and $\\wp_{F_v}$ is the direct sum of the maximal ideals of the valuation rings in each field component of $F_v=\\prod_{w'\\in S_F^v}F_{w'}$.\n\nWe use the canonical comparison isomorphism of $\\QQ_p[G]$-modules\n\\[ \\nu_v: \\Hom_{F_v}(H^0(A^t_{F_v},\\Omega^1_{A^t_{F_v}}),F_v) \\cong {\\rm DR}_v(V_{p,F}(A^t))\/F^0\\]\nand the exponential map ${\\rm exp}_{\\rm BK}: {\\rm DR}_v(V_{p,F}(A^t))\/F^0\\to H^1_f(k,V_{p,F}(A^t))$ of Bloch and Kato.\n\nWe recall, in particular, that in this case there is a natural identification of spaces $H^1_f(k,V_{p,F}(A^t)) = \\QQ_p\\cdot A^t(F_v)^\\wedge_p$ under which the composite ${\\rm exp}_{\\rm BK}\\circ\\nu_v$ sends the free $\\ZZ_p[G]$-lattice $\\mathcal{D}_F(\\mathcal{A}_v^t)$ defined in (\\ref{mathcalD}) to ${\\rm exp}_{A^t,F_v}((\\mathcal{O}_{F_v})^d)$, where ${\\rm exp}_{A^t,F_v}$ is the classical exponential map of $A^t$ over $F_{v}$ (cf. the result of Bloch and Kato in \\cite[Exam. 3.11]{bk}).\n\nIn particular, since $A^t$ has good reduction over the absolutely unramified algebra $F_{v}$, the theory of Fontaine and Messing \\cite{fm} implies (via the proof of \\cite[Lem. 3.4]{bmw}) that there exists an exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to {\\rm exp}_{A^t,F_v}((\\mathcal{O}_{F_v})^d) \\xrightarrow{\\subseteq} A^t(F_v)^\\wedge_p \\to N \\to 0\\]\nwhere $N$ is a finite module that has finite projective dimension and is such that\n\\[ \\chi_{G,p}(N[-1],0) = \\delta_{G,p}(L_v(A,F\/k))\\]\nin $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nOn the other hand, since $F_{v}$ is absolutely unramified, the map ${\\rm exp}_{A^t,F_v}$ restricts to give a short exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to \\wp_{F_v}^d \\xrightarrow{{\\rm exp}_{A^t,F_v}} A^t(F_v)^\\wedge_p \\to \\tilde A^t_v(\\kappa_{F_v})_p \\to 0.\\]\n\nBy comparing these exact sequences, and noting that $\\mathcal{O}_{F_v}\/\\wp_{F_v}$ identifies with the ring $\\kappa_{F_v}$, one obtains a short exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to \\kappa_{F_v}^d \\to \\tilde A^t_v(\\kappa_{F_v})_p \\to N \\to 0\\]\nin which each term is both finite and of finite projective dimension.\n\nThus, upon taking Euler characteristics of this exact sequence, one finds that\n\\begin{align*} \\delta_{G,p}(L_v(A,F\/k)) =\\,&\\chi_{G,p}(N[-1],0)\\\\\n = \\, &\\chi_{G,p}(\\tilde A^t_v(\\kappa_{F_v})_p[-1],0) - \\chi_{G,p}(\\kappa_{F_v}^d[-1],0)\\\\\n = \\, &\\chi_{G,p}(\\tilde A^t_v(\\kappa_{F_v})_p[-1],0) + \\chi_{G,p}(\\kappa_{F_v}^d[0],0)\\\\\n  =\\, &\\chi_{G,p}(\\kappa_{F_v}^d[0]\\oplus\\tilde A^t_v(\\kappa_{F_v})_p[-1],0),\\end{align*}\n\n\\noindent{}and hence that the element $\\mu_{v}(A_{F\/k})$ vanishes, as required.\n\\end{proof}\n\n\n\\section{Euler characteristics and Galois structures}\\label{ecgs}\n\nIn this section we consider consequences of ${\\rm BSD}(A_{F\/k})$ concerning both the Galois structure of Selmer complexes and modules and the formulation of refinements of the Deligne-Gross Conjecture.\n\n\n\n\\subsection{Galois structures of Selmer complexes}\\label{Galoiscomplexes} In this section we fix a finite set $S$ of places of $k$ as described at the beginning of \\S\\ref{selmer section} as well as data $\\{\\mathcal{A}^t_v\\}_v$ and $\\omega_\\bullet$ as in \\S\\ref{perf sel construct}. We then write\n\\[ \\Upsilon = \\Upsilon(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,S)\\]\nfor the finite set of non-archimedean places $v$ of $k$ that are such that either $v$ belongs to $S$ or divides the different of $k\/\\QQ$ or the lattice $\\mathcal{F}(\\omega_\\bullet)_{v}$ differs from $\\mathcal{D}_F(\\mathcal{A}_v^t)$.\n\nWe then consider the perfect Selmer structure $\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)$ that is defined in \\S\\ref{perf sel construct}.\n\n\n\\begin{proposition}\\label{gec} If ${\\rm BSD}(A_{F\/k})$ is valid, then for any data $S$, $\\{\\mathcal{A}^t_v\\}_v$ and $\\omega_\\bullet$ as above the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The Selmer complex ${\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))$ is represented by a bounded complex of finitely generated free $G$-modules.\n\\item[(ii)] Set $\\ZZ' := \\ZZ[1\/2]$. If neither of the groups $A(F)$ and $A^t(F)$ has an element of odd order, then the $\\ZZ'[G]$-module $\\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)))$ has a presentation with the same number of generators and relations.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} We set $C_{\\omega_\\bullet} := {\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))$ and write $\\chi_G(C_{\\omega_\\bullet})$ for its Euler characteristic in $K_0(\\ZZ[G])$.\n\nThen, the definition of $\\Upsilon$ implies immediately that the module $\\mathcal{Q}(\\omega_\\bullet)_\\Upsilon$ vanishes and also combines with Lemma \\ref{fm} to imply that $\\mu_{\\Upsilon}(A_{F\/k})$ vanishes.\n\nThe vanishing of $\\mathcal{Q}(\\omega_\\bullet)_\\Upsilon[0]$ implies ${\\rm SC}_{\\Upsilon,\\omega_\\bullet}(A_{F\/k}) = C_{\\omega_\\bullet}$ and hence that\n\\[ \\partial'_G(\\chi_G({\\rm SC}_{\\Upsilon,\\omega_\\bullet}(A_{F\/k}),h_{A,F})) = \\chi_G(C_{\\omega_\\bullet}),\\]\nwhere $\\partial'_{G}$ denotes the canonical connecting homomorphism $K_0(\\ZZ[G],\\RR[G]) \\to K_0(\\ZZ[G])$ of relative $K$-theory.\n\nThen, given the vanishing of $\\mu_{\\Upsilon}(A_{F\/k})$, the equality in ${\\rm BSD}(A,F\/k)$(iv) implies that\n\\[ \\chi_G(C_{\\omega_\\bullet}) = \\partial'_G\\bigl(\\partial_G(L_\\Upsilon^*(A_{F\/k},1)\/\\Omega_{\\omega_\\bullet}(A_{F\/k}))).\\]\n\nHowever, the exactness of the lower row of diagram (\\ref{E:kcomm}), with $\\mathfrak{A} = \\ZZ[G]$ and $A_E = \\RR[G]$, implies that the composite homomorphism $\\partial'_G\\circ \\partial_G$ is zero and so it follows that the Euler characteristic $\\chi_G(C_{\\omega_\\bullet})$ must vanish.\n\nNow, by a standard resolution argument, we may fix a bounded complex of finitely generated $G$-modules $C^\\bullet$ that is isomorphic in $D(\\ZZ[G])$ to $C_{\\omega_\\bullet}$ and is such that, for some integer $a$, all of the following properties are satisfied: $C^i = 0$ for all $i < a$; $C^a$ is projective of rank (over $\\ZZ[G]$) at least two; $C^{i}$ is free for all $i \\not= a$.\n\nFrom the vanishing of $\\chi_G(C_{\\omega_\\bullet}) = \\chi_G(C^\\bullet)$ it then follows that the class of $C^a$ in $K_0(\\ZZ[G])$ coincides with that of a free $G$-module.\n\nThus, since the rank over $\\ZZ[G]$ of $C^a$ is at least two, we may conclude from the Bass Cancellation Theorem (cf. \\cite[(41.20)]{curtisr}) that $C^a$ is a free $G$-module, as required to prove claim (i).\n\nTurning to claim (ii), we note that if $\\ZZ'\\otimes_\\ZZ A(F)$ and $\\ZZ'\\otimes_\\ZZ A^t(F)$ are torsion-free, then Proposition \\ref{prop:perfect2} implies that the complex $C'_{\\omega_\\bullet} := \\ZZ'\\otimes_\\ZZ C_{\\omega_\\bullet}$ is acyclic outside degrees one and two and that $H^1(C'_{\\omega_\\bullet})$ is torsion-free.\n\nThis implies that $C'_{\\omega_\\bullet}$ is isomorphic in $D^{\\rm perf}(\\ZZ'[G])$ to a complex of finitely generated $\\ZZ'[G]$-modules of the form $(C')^1 \\xrightarrow{d'} (C')^2$ where $(C')^1$ is projective and $(C')^2$ is free.\n\nThe vanishing of the Euler characteristic of $C'_{\\omega_\\bullet}$ then implies, by the same argument as in claim (i), that the module $(C')^1$ is free.\n\nIn addition, the fact that the $\\RR[G]$-modules generated by $H^1(C'_{\\omega_\\bullet})$ and $H^2(C'_{\\omega_\\bullet})$ are isomorphic implies that the free modules $(C')^1$ and $(C')^2$ must have the same rank.\n\nGiven this, claim (ii) follows from the tautological exact sequence\n\\[ (C')^1 \\xrightarrow{d'} (C')^2 \\to \\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))) \\to 0.\\]\n\\end{proof}\n\n\\begin{remark}{\\em An explicit description of the module $\\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)))$ that occurs in Proposition \\ref{gec}(ii) can be found in Remark \\ref{can structure groups}.}\\end{remark}\n\n\n\n\n\\subsection{Refined Deligne-Gross-type conjectures}\n\nIn this section we address a problem raised by Dokchitser, Evans and Wiersema in \\cite[Rem. 14]{vdrehw} by explaining how ${\\rm BSD}(A_{F\/k})$ leads to an explicit formula for the fractional ideal that is generated by the product of the leading coefficients of Hasse-Weil-Artin $L$-series\n by a suitable combination of `isotypic' periods and regulators. (See, in particular, Remark \\ref{evans} below.)\n\n\\subsubsection{}We fix a character $\\psi$ in $\\widehat {G}$. We also then fix a subfield $E$ of $\\bc$ that is both Galois and of finite degree over\n$\\QQ$ and also large enough to ensure that, with\n$\\mathcal{O}$ denoting the ring of algebraic integers of $E$, there exists a finitely generated ${\\mathcal\nO}[G]$-lattice $T_\\psi$ that is free over $\\mathcal{O}$ and such that the $\\bc[G]$-module\n$V_\\psi:= \\bc \\otimes_{\\mathcal{O}}T_\\psi$ has character $\\psi$.\n\n\nWe then obtain a left, respectively right,\nexact functor from the category of $G$-modules to the category of $\\mathcal{O}$-modules by setting\n\n\\begin{align*} X^\\psi &:= \\Hom_{{\\mathcal O}}(T_\\psi,{\\mathcal O}\n\\otimes_{\\ZZ} X)^G,\n\\\\ X_\\psi &:= \\Hom_{{\\mathcal O}}(T_\\psi,{\\mathcal O}\\otimes_{\\ZZ} X)_G,\n\\end{align*}\nwhere the $\\Hom$-sets are endowed with the natural diagonal\n$G$-action.\n\nIt is easily seen that for any $G$-module $X$ there is a natural isomorphism of $\\mathcal{O}$-modules\n\\begin{equation}\\label{func iso} \\Hom_\\ZZ(X,\\ZZ)_\\psi \\cong \\Hom_\\mathcal{O}(X^{\\check{\\psi}},\\mathcal{O}).\\end{equation}\n\nFor a given odd prime number $p$, each maximal ideal $\\mathfrak p$ of $\\mathcal{O}$ that divides $p$ and each $\\mathcal{O}$-module $X$ we set $X_\\mathfrak{p} := \\mathcal{O}_\\mathfrak{p}\\otimes_{\\mathcal{O}}X$.\n\nWe also write $I(\\mathcal{O}_\\mathfrak{p})$ for the multiplicative group of invertible $\\mathcal{O}_\\mathfrak{p}$-submodules of $\\CC_p$ and we use the composite homomorphism of abelian groups\n\\[ \\rho_{\\mathfrak{p}}^{\\psi}: K_0(\\ZZ_p [G],\\CC_p[G]) \\to K_0(\\mathcal{O}_\\mathfrak{p} ,\\CC_p) \\xrightarrow{\\iota_\\mathfrak{p}} I(\\mathcal{O}_\\mathfrak{p}).\\]\nHere the first map is induced by the composite functor $X \\mapsto X^\\psi\\to (X^\\psi)_\\mathfrak{p}$ and $\\iota_\\mathfrak{p}$ is the canonical isomorphism induced by\n the upper row of (\\ref{E:kcomm}) with $\\A = {\\mathcal O}_\\mathfrak{p}$ and $E' = \\bc_p$ and the canonical\nisomorphisms $K_1(\\bc_p) \\xrightarrow{\\sim} \\bc_p^\\times$ and\n$K_1({\\mathcal O}_\\mathfrak{p}) \\xrightarrow{\\sim} {\\mathcal O}_\\mathfrak{p}^\\times$.\n\nFor any finite ${\\mathcal O}$-module $X$ we also set\n\\[ {\\rm char}_{\\mathfrak{p}}(X) := \\mathfrak{p}^{{\\rm length}_{{\\mathcal O}_{{\\mathfrak\np}}}(X_{\\mathfrak p})}.\\]\n\n\\subsubsection{}\\label{explicit ec section} Using the isomorphism (\\ref{func iso}), we define $R^\\psi_A$ to be the determinant, with respect to a choice of $\\mathcal{O}$-bases of $A^t(F)^{\\psi}$ and  $A(F)^{\\check\\psi}$ of the isomorphism of $\\CC$-spaces\n\\[ h^\\psi_{A,F}: \\CC\\cdot A^t(F)^{\\psi} \\cong \\CC\\cdot \\Hom_\\ZZ(A(F),\\ZZ)^\\psi \\cong \\CC\\cdot \\Hom_\\mathcal{O}(A(F)^{\\check\\psi},\\mathcal{O})\\]\nthat is induced by the N\\'eron-Tate height pairing of $A$ relative to $F$.\n\nMotivated by \\cite[Def. 12]{vdrehw}, we then define a non-zero complex number by setting\n\\[ \\mathcal{L}^\\ast(A,\\psi) := \\frac{L^\\ast(A,\\check\\psi,1)\\cdot \\tau^\\ast(\\QQ,\\psi)^d}{\\Omega_A^\\psi\\cdot w^d_\\psi\\cdot R_A^\\psi}.\\]\n\nFinally, after recalling the integral Selmer group $X_\\ZZ(A_F)$ of $A$ over $F$ that is defined by Mazur and Tate \\cite{mt} (and discussed in \\S\\ref{perfect selmer integral}), we note that if $\\sha(A_F)$ is finite then the kernel $\\sha_\\psi(A_F)$ of the natural surjective homomorphism of $\\mathcal{O}$-modules\n\\[ X_\\ZZ(A_F)_\\psi \\to \\Hom_\\ZZ(A(F),\\ZZ)_\\psi \\cong \\Hom_\\mathcal{O}(A(F)^{\\check\\psi},\\mathcal{O})\\]\nis finite.\n\nWe can now state the main result of this section.\n\n\\begin{proposition}\\label{ref deligne-gross} If ${\\rm BSD}(A_{F\/k})$ is valid, then so are the following claims.\n\\begin{itemize}\n\\item[(i)] For every $\\omega$ in $G_\\QQ$ one has $\\mathcal{L}^\\ast(A,\\omega\\circ \\psi) = \\omega(\\mathcal{L}^\\ast(A,\\psi))$. In particular, the complex number $\\mathcal{L}^\\ast(A,\\psi)$ belongs to $E$.\n\n\\item[(ii)] Assume that no place of $k$ at which $A$ has bad reduction is ramified in $F$. Then for every odd prime $p$ that satisfies the conditions (H$_1$)-(H$_4$) listed in \\S\\ref{tmc} and for which neither $A(F)$ nor $A^t(F)$ has a point of order $p$, and every maximal ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$, there is an equality of fractional $\\mathcal{O}_\\mathfrak{p}$-ideals\n\n\\[ \\mathcal{L}^\\ast(A,\\psi)\\cdot \\mathcal{O}_{\\mathfrak{p}} = \\frac{{\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F))\\cdot \\prod_{v\\in S^*_{p,{\\rm u}}}\\rho_\\mathfrak{p}^\\psi(\\mu_v(A_{F\/k}))}{|G|^{r_{A,\\psi}}\\cdot \\prod_{v \\in S_k^p\\cap S_k^F}\\varrho_\\psi^d\\cdot\\prod_{v\\in S_k^F\\cap S_k^f}P_v(A,\\check\\psi,1)}.\\]\nHere $$r_{A,\\psi}:={\\rm dim}_\\CC(\\CC\\cdot A^t(F)^\\psi)$$\nwhile $S^*_{p,{\\rm u}}$ is the set of $p$-adic places of $k$ that are unramified in $F$ but divide the different of $k\/\\QQ$ and, for every $v\\in S_k^F\\cap S_k^f$, $P_v(A,\\check\\psi,1)$ is the value at $z=1$ of the Euler factor at $v$ of the $\\check\\psi$-twist of $A$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} The first assertion of claim (i) is equivalent to asserting that the element\n\\[ \\mathcal{L}^\\ast := \\sum_{\\psi \\in \\widehat{G}}\\mathcal{L}^\\ast(A,\\psi)\\cdot e_\\psi\\]\nbelongs to the subgroup $\\zeta(\\QQ[G])^\\times$ of $\\zeta(\\RR[G])^\\times$.\n\nRecalling that $\\zeta(\\QQ[G])^\\times$ is the full pre-image under $\\delta_G$ of the subgroup $K_0(\\ZZ[G],\\QQ[G])$ of $K_0(\\ZZ[G],\\RR[G])$, it is therefore enough to prove that $\\delta_G(\\mathcal{L}^\\ast)$ belongs to $K_0(\\ZZ[G],\\QQ[G])$.\n\nTo do this we fix any basis of differentials $\\omega_\\bullet$ as in the statement of ${\\rm BSD}(A_{F\/k})$ and write $\\mathcal{L}^\\ast$ as a product $(\\mathcal{L}^\\ast_1)^{-1}\\cdot \\mathcal{L}^\\ast_2\\cdot (\\mathcal{L}^\\ast_3)^{-1}$ with\n\\[ \\mathcal{L}^\\ast_1 := {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))^{-1} \\cdot \\sum_{\\psi\\in \\widehat{G}} \\Omega_A^\\psi\\cdot w^d_\\psi\\cdot\\tau^\\ast(\\QQ,\\psi)^{-d}\\cdot e_\\psi,\\]\n\\[ \\mathcal{L}^\\ast_2 := {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))^{-1}\\cdot \\sum_{\\psi \\in \\widehat{G}}L^\\ast(A,\\check\\psi,1)\\cdot e_\\psi,\\]\nand\n\\[ \\mathcal{L}^\\ast_3 := \\sum_{\\psi\\in \\widehat{G}}R_A^\\psi\\cdot e_\\psi.\\]\n\nProposition \\ref{lms}(i) implies $\\mathcal{L}^\\ast_1$ belongs to $\\zeta(\\QQ[G])^\\times$. In addition, for any set of places $S$ as in the statement of ${\\rm BSD}(A_{F\/k})$, the element $\\mu_S(A_{F\/k})$ belongs to $K_0(\\ZZ[G],\\QQ[G])$. We next note that $\\delta_G(\\mathcal{L}^\\ast_2)$ differs from the left hand side of the equality in ${\\rm BSD}(A_{F\/k})$(iv) by\n$$\\sum_{v\\in S\\cap S_k^f}\\delta_G(L_v(A,F\/k)),$$ where $L_v(A,F\/k)\\in\\zeta(\\QQ[G])^\\times$ is the equivariant Euler factor of $(A,F\/k)$ at $v$ (see Appendix \\ref{consistency section} below).\n\nThis difference belongs to $K_0(\\ZZ[G],\\QQ[G])$ and therefore the validity of  ${\\rm BSD}(A_{F\/k})$ implies that $$\\delta_G(\\mathcal{L}^\\ast_2)- \\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})$$ also belongs to $K_0(\\ZZ[G],\\QQ[G])$.\n\nTo prove $\\mathcal{L}^\\ast \\in \\zeta(\\QQ[G])^\\times$ it is therefore enough to note that $\\delta_G(\\mathcal{L}^\\ast_3)$ also differs from $\\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})$ by an element of $K_0(\\ZZ[G],\\QQ[G])$ (as one verifies by straightforward computation).\n\nThe second assertion of claim (i) is true since if $\\omega$ is any element of $G_{\\QQ^c\/E}$, then $\\omega\\circ \\psi = \\psi$ and so $\\omega(\\mathcal{L}^\\ast(A,\\psi)) = \\mathcal{L}^\\ast(A,\\omega\\circ\\psi) = \\mathcal{L}^\\ast(A,\\psi)$.\n\nTurning to claim (ii) we note that the given hypotheses imply that the data $A, F\/k$ and $p$ satisfy the conditions of Theorem \\ref{bk explicit}. To prove claim (ii) it is therefore enough to show that, if the difference between the left and right hand sides of the equality in Theorem \\ref{bk explicit} belongs to $K_0(\\ZZ_p[G],\\QQ_p[G])$, then its image under $\\rho_{\\mathfrak{p}}^{\\psi}$ is the equality in claim (ii). Since this image is independent of the choice of isomorphism $j:\\CC\\cong\\CC_p$ we will omit it from all notations.\n\nThe group $I(\\mathcal{O}_\\mathfrak{p})$ is torsion-free and so Proposition \\ref{basic props}(ii) implies that each term $R_{F\/k}(\\tilde A^t_v)$ that occurs in Theorem \\ref{bk explicit} belongs to $\\ker(\\rho_{\\mathfrak{p}}^{\\psi})$.\n\nThus, since $\\mathcal{L}^\\ast(A,\\psi)$ differs from the element $\\mathcal{L}^\\ast_{A,\\psi}$ defined in (\\ref{bkcharelement}) by the equality\n\\[ \\mathcal{L}^\\ast(A,\\psi) = \\mathcal{L}^\\ast_{A,\\psi}\\cdot (R_A^\\psi)^{-1}\\prod_{v \\in S_k^p\\cap S_k^F}\\varrho_{\\psi}^{-d}\\cdot\\prod_{v\\in S_k^F\\cap S_k^f}P_v(A,\\check\\psi,1)^{-1}\\]\nthe claimed result will follow if we can show that $\\rho_{\\mathfrak{p}}^{\\psi}$ sends the element\n\\[ \\chi_{G,p}( {\\rm SC}_p(A_{F\/k}),h_{A,F})-\\delta_{G,p}(\\mathcal{L}^\\ast_3)\\]\nof $K_0(\\ZZ_p[G],\\QQ_p[G])$ to the ideal $|G|^{-r_{A,\\psi}}\\cdot {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F))$.\n\nNow, under the given hypotheses, Proposition \\ref{explicitbkprop} implies that the complex $C := {\\rm SC}_p(A_{F\/k})$ is acyclic outside degrees one and two and has cohomology $A^t(F)_p$ and $X_\\ZZ(A_F)_p=\\Sel_p(A_F)^\\vee$ in these respective degrees.\n\nIn particular, since $A^t(F)_p$ is torsion-free we can fix a representative of $C$ of the form $C^1 \\xrightarrow{d} C^2$, where $C^1$ and $C^2$ are free $\\ZZ_p[G]$-modules of the same rank.\n\nThen the tautological exact sequence\n\\begin{equation} 0 \\rightarrow H^1(C) \\xrightarrow{\\iota} C^1 \\xrightarrow{d}\nC^2 \\xrightarrow{\\pi} H^2(C) \\rightarrow\n0\\label{tatseq}\\end{equation}\ninduces a commutative diagram of ${\\mathcal O}_p$-modules with exact rows\n\\[\\begin{CD}\n@. @. C^1_{\\psi} @> d_\\psi >> C^2_{\\psi} @> \\pi_\\psi\n>> H^2(C)_\\psi @>  >> 0\\\\ @. @. @V {t^1_\\psi} VV @V\n{t^2_\\psi} VV  \\\\ 0 @> >> H^{1}(C)^\\psi @> \\iota^\\psi >>\n C^{1,\\psi} @> d^\\psi >> C^{2,\\psi}.\\end{CD}\\]\nEach vertical morphism $t^i_\\psi$ here is induced by sending each $x$ in $\\Hom_{{\\mathcal O}_p}(T_{\\psi,p}, {\\mathcal O}_p\\otimes_{\\ZZ_p}C^i)$ to $\\sum_{g \\in G}g(x)$ and is bijective since the $\\ZZ_p[G]$-module $C^i$ is free.\n\nThis diagram gives rise to an exact sequence of ${\\mathcal O}_\\mathfrak{p}$-modules\n\\begin{equation}\\label{scal}\n0\\rightarrow\nH^{1}(C)^\\psi_\\mathfrak{p}\n\\xrightarrow{\\iota^\\psi}C_\\mathfrak{p}^{1,\\psi}\\xrightarrow{d^\\psi}\nC_\\mathfrak{p}^{2,\\psi} \\xrightarrow{\\pi_\\psi\\circ (t^2_\\psi)^{-1}} H^2(C)_{\\psi,\\mathfrak{p}}\n\\rightarrow 0\\end{equation}\nwhich in turn implies that\n\\begin{equation}\\label{firstform} \\rho_{\\mathfrak{p}}^{\\psi}(\\chi_{G,p}( {\\rm SC}_p(A_{F\/k}),h_{A,F})) = \\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(C^{\\bullet,\\psi}_{\\mathfrak{p}}, \\tilde h^\\psi)\\bigr).\\end{equation}\nHere $C^{\\bullet,\\psi}_{\\mathfrak{p}}$ denotes the complex $C^{1,\\psi}_\\mathfrak{p} \\xrightarrow{d^\\psi} C^{2,\\psi}_\\mathfrak{p}$ and $\\tilde h^\\psi$ the composite isomorphism of $\\CC_p$-modules\n\\[ \\CC_p\\cdot H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}}) \\cong \\CC_p\\cdot (A^t(F)^\\psi)_\\mathfrak{p} \\xrightarrow{h^\\psi} \\Hom_{\\CC_p}(\\CC_p\\cdot (A(F)^{\\check\\psi})_\\mathfrak{p},\\CC_p) \\cong \\CC_p\\cdot H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\\]\nin which the first and third isomorphisms are induced by the maps in (\\ref{scal}) and $h^\\psi$ is induced by the isomorphism $h^\\psi_{A,F}$.\n\nGiven the definition of each term $R_A^\\psi$ it is, on the other hand, clear that\n\\begin{equation}\\label{secondform} \\rho^\\psi_\\mathfrak{p}(\\delta_{G,p}(\\mathcal{L}^\\ast_3)) = \\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], h^\\psi)\\bigr).\\end{equation}\n\nNow, since $\\mathcal{O}_\\mathfrak{p}$ is a discrete valuation ring it is straightforward to construct an exact triangle in $D(\\mathcal{O}_\\mathfrak{p})$ of the form\n\\[ H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2] \\to C^{\\bullet,\\psi}_{\\mathfrak{p}} \\to H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2] \\to H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-1]\\]\nand Lemma \\ref{fk lemma} applies to this triangle to imply that\n\\begin{align}\\label{thirdform}\n  &\\chi_{\\mathcal{O}_\\mathfrak{p}}(C^{\\bullet,\\psi}_{\\mathfrak{p}}, \\tilde h^\\psi) - \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^1(C^{\\bullet,\\psi}_{p})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], h^\\psi)\\notag\\\\\n=\\,&\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\tilde h^\\psi\\circ ( h^\\psi)^{-1}) +\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2], 0).\\end{align}\n\nNext we note the definition of $\\sha_\\psi(A_F)$ ensures $\\sha_\\psi(A_F)_\\mathfrak{p} = H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}$ and hence that\n\\begin{equation}\\label{fourthform}\\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2], 0)\\bigr) =  {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F)).\\end{equation}\n\nIn addition, after identifying both $\\CC_p\\cdot H^2(C)_{\\psi,\\mathfrak{p}}$ and $\\CC_p\\cdot H^2(C)^\\psi_{\\mathfrak{p}}$ with $e_\\psi(\\CC_p\\cdot H^2(C)_{\\mathfrak{p}})$ in the natural way, the map $t^2_\\psi$ that occurs in (\\ref{scal}) induces upon the latter space the map given by multiplication by $|G|$.\n\nTo derive claim (ii) from the displayed formulas (\\ref{firstform}), (\\ref{secondform}), (\\ref{thirdform}) and (\\ref{fourthform}) it is thus enough to note that\n\\[ \\dim_{\\CC_p}(\\CC_p\\cdot H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}}))=\\dim_{\\CC_p}(\\CC_p\\cdot (A^t(F)^\\psi)_\\mathfrak{p})=r_{A,\\psi},\\]\nand hence that\n\\begin{multline*} \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\tilde h^\\psi\\circ ( h^\\psi)^{-1})\\\\ = \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\cdot |G|^{-1})\\end{multline*}\nis sent by $\\iota_\\mathfrak{p}$ to the ideal generated by $|G|^{-r_{A,\\psi}}$.\n\\end{proof}\n\n\\begin{remark}\\label{evans}{\\em Fix a Galois extension $F$ of $k = \\QQ$ and an elliptic curve $A$ whose conductor $N_A$ is prime to the discriminant $d_F$ of $F$ and is such that $A(F)$ is finite. Then for each $\\psi$ in $\\widehat{G}$ one has $r_{A,\\psi}=0$, and hence $R_A^\\psi = 1$, so that the complex number $\\mathcal{L}^\\ast(A,\\psi)$ agrees up to a unit of $\\mathcal{O}$ with the element $\\mathcal{L}(A,\\psi)$ that is defined in \\cite[Def. 12]{vdrehw}. Now fix an odd prime $p$ that is prime to $d_F$, to $N_A$, to the order of $A(F)$, to the order of the group of points of the reduction of $A$ at any prime divisor of $d_F$ and to the Tamagawa number of $A_F$ at each place of bad reduction. Then the data $A, F\/k$ and $p$ satisfy the hypotheses of Proposition \\ref{ref deligne-gross}(ii) and the sets  $S^*_{p,{\\rm u}}$ and $S_k^p\\cap S_k^F$ are empty (the former since $k = \\QQ$). The explicit formula in the latter result therefore simplifies to give\n\\[ \\mathcal{L}(A,\\psi)\\cdot \\mathcal{O}_{\\mathfrak{p}} = {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F)) \\cdot\\prod_{\\ell\\mid d_F}P_\\ell(A,\\check\\psi,1)^{-1}\\]\nwhere in the product $\\ell$ runs over all prime divisors of $d_F$. This formula shows that, in any such case, the fractional $\\mathcal{O}$-ideal generated by $\\mathcal{L}(A,\\psi)$ should depend crucially on the structure of $\\sha(A_F)$ as a $G$-module, as already suggested in this context by Dokchitser, Evans and Wiersema in \\cite[Rem. 40]{vdrehw}. In particular, this observation is both consistent with, and helps to clarify, the result of [loc. cit., Th. 37(2)]. }\n\\end{remark}\n\n\n\\section{Abelian congruence relations and module structures}\\label{congruence sec}\n\nIn both this and the next section we apply the general results of Sano, Tsoi and the first author in \\cite{bst} to derive from the assumed validity of ${\\rm BSD}_p(A_{F\/k})$(iv) families of $p$-adic congruences that can be much more explicit than those discussed in Remark \\ref{cons1}.\n\nIn particular, in this section we will focus on congruence relations that express links to the Galois structures of Selmer and Tate-Shafarevich groups.\n\n\n\nIn contrast to \\S\\ref{tmc}, in this section we are not required to assume any of the hypotheses (H$_1$)-(H$_5$).\n\nHowever, we will now, unless explicitly stated otherwise, restrict to the case that $G$ is abelian and hence will not distinguish between the leading coefficient element $L_S^*(A_{F\/k},1)$ in $K_1(\\RR[G])$ and its reduced norm $\\sum_{\\psi \\in \\widehat G}L_S^*(A,\\check\\chi,1)\\cdot e_\\psi$ in $\\RR[G]^\\times$.\n\nAn extension of the results of this section to the general (non-abelian) setting will be given in the upcoming article \\cite{dmckwt}.\n\n\\subsection{Statement of the main result}\\label{8.1}\n\nThroughout this section we give ourselves a fixed odd prime $p$ and an isomorphism of fields $\\CC\\cong\\CC_p$ (that we will usually not mention).\n\nWe also fix a finite set $S$ of places of $k$ with\n\\[ S_k^\\infty\\cup S_k^p\\cup  S_k^F \\cup S_k^A\\subseteq S.\\]\n\nWe write $x\\mapsto x^\\#$ for the $\\CC$-linear involution of $\\CC[G]$ that inverts elements of $G$. We also set $n := [k:\\QQ]$ and write $d$ for the dimension of $A$.\n\n\\subsubsection{}In order to state the main result of this section we must first extend the definition of logarithmic resolvents given in (\\ref{log resol abelian}) to the setting of abelian varieties.\n\nTo do this we do not require $F\/k$ to be abelian but we do assume to be given an ordered $k$-basis $\\{\\omega'_j: j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$ and we use this basis to define a classical period $\\Omega_A^{F\/k}$ in $\\CC[G]^{\\times}$ as in (\\ref{period def}).\n\n\nFor each index $j$ we then write ${\\rm log}_{\\omega'_j}$ for the formal logarithm of $A^t$ over $F_p$ that is defined with respect to $\\omega_j'$.\n\nWe also fix an ordering of $\\Sigma(k)$. We write $\\CC_p[G]^{nd}$ for the direct sum of $nd$ copies of $\\CC_p[G]$ and fix a bijection between the standard basis of this module and the lexicographically-ordered direct product $[d]\\times \\Sigma(k)$.\n\nThen for any ordered subset\n\\begin{equation}\\label{setx} x_\\bullet:= \\{x_{(i,\\sigma)}: (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}\\end{equation}\nof $A^t(F_p)^\\wedge_p$ we define a logarithmic resolvent element of $\\zeta(\\CC_p[G])$ by setting\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) := {\\rm Nrd}_{\\QQ^c_p[G]}\\left(\\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{\\omega_j'}(x_{(j',\\sigma')})))\\cdot g \\bigr)_{(j,\\sigma),(j',\\sigma')}\\right) \\]\nwhere the indices $(j,\\sigma)$ and $(j',\\sigma')$ run over $[d]\\times \\Sigma(k)$ and ${\\rm Nrd}_{\\QQ^c_p[G]}(-)$ denotes the reduced norm of the given matrix in ${\\rm M}_{dn}(\\QQ_p^c[G])$.\n\nIt is clear that if $A$ is an elliptic curve (so $d=1$) and $F\/k$ is abelian, then the `$\\psi$-component' of this definition agrees with (\\ref{log resol abelian}).\n\n\\subsubsection{}\\label{statementstructure} For each non-archimedean place $v$ of $k$ that does not ramify in $F\/k$ and at which $A$ has good reduction we define an element of $\\QQ [G]$ by setting\n\\[ P_v(A_{F\/k},1) := 1-\\Phi_v\\cdot a_v +  \\Phi_v^2\\cdot {\\rm N}v^{-2}.\\]\nHere $a_v$ is the integer $1 + {\\rm N}v - | A(\\kappa_v)|$\n\nFor a non-negative integer $a$ we write $\\widehat{G}_{A,(a)}$ for the subset of $\\widehat{G}$ comprising characters $\\psi$ for which the  $L$-series $L(A,\\psi,z)$ vanishes at $z=1$ to order at least $a$. This definition ensures that the $\\CC[G]$-valued function\n\\[ L^{(a)}_{S}(A_{F\/k},z) := \\sum_{\\psi \\in \\widehat{G}_{A,(a)}}z^{-a}L_S(A,\\check\\psi,z)\\cdot e_\\psi\\]\nis holomorphic at $z=1$.\n\nFor each $a$ we also define idempotents of $\\QQ[G]$ by setting\n\\[ e_{(a)} = e_{F,(a)} := \\sum_{\\psi \\in \\widehat{G}_{A,(a)}}e_\\psi\\]\nand\n\\[ e_{a} = e_{F,a}:= \\sum_{\\psi \\in \\widehat{G}_{A,(a)}\\setminus \\widehat{G}_{A,(a+1)}}e_\\psi\\]\n(so that $e_{(a)} = \\sum_{b \\ge a}e_b$).\n\nThe N\\'eron-Tate height pairing for $A$ over $F$ induces a canonical isomorphism of $\\RR[G]$-modules\n\\[ h_{A_{F\/k}}:\\RR\\cdot A^t(F) \\cong \\Hom_\\RR(\\RR\\cdot A(F),\\RR) = \\RR\\cdot \\Hom_{\\ZZ[G]}(A(F),\\ZZ[G]).\\]\nFor each non-negative integer $a$ this pairing combines with our fixed isomorphism of fields $\\CC\\cong \\CC_p$ to induce an isomorphism of $\\CC_p[G]$-module\n\n\\begin{equation}\\label{athpower} {\\rm ht}^{a}_{A_{F\/k}}: \\CC_p\\cdot {\\bigwedge}^a_{\\ZZ_p[G]}\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G]) \\cong \\CC_p\\cdot{\\bigwedge}^a_{\\ZZ_p[G]} A^t(F)_p.\\end{equation}\n\n\nIn the following result we shall also use (the scalar extension of) the canonical `evaluation' pairing $${\\bigwedge}^a_{\\ZZ_p[G]} A^t(F)_p\\times{\\bigwedge}^a_{\\ZZ_p[G]} \\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])\\to \\ZZ_p[G]$$\nand write ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ for the $a$-th Fitting ideal of the $\\ZZ_p[G]$-module $\\Sel_p(A_F)^\\vee$.\n\n\nThe proof of this result will be given in \\S\\ref{proof of big conj}.\n\n\n\\begin{theorem}\\label{big conj} Fix an ordered maximal subset $x_\\bullet:= \\{x_{(i,\\sigma)}: (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$ and a finite non-empty set $T$ of places of $k$ that is disjoint from $S$\n\nIf ${\\rm BSD}(A_{F\/k})$ is valid, then for any non-negative integer $a$, any subsets $\\{\\theta_j: j \\in [a]\\}$ and $\\{\\phi_i: i\\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])$ and $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ respectively, and any element $\\alpha$ of $\\ZZ_p[G]\\cap \\ZZ_p[G]e_{(a)}$ the product\n\\begin{equation}\\label{key product} \\alpha^{1+2a}\\cdot (\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#) \\cdot  \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\end{equation}\nbelongs to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$  and annihilates $\\sha(A^t_{F})[p^\\infty]$.\n\\end{theorem}\n\n\nWe remark on several ways in which this result either simplifies or becomes more explicit.\n\n\\begin{remarks}\\label{more explicit rem}{\\em \\\n\n\\noindent{}(i) If $A(F)$ does not contain an element of order $p$, then our methods will show that the prediction in Theorem \\ref{big conj} should remain true if the term $\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#$ is omitted. For more details see Remark \\ref{omit T} below.\n\n\\noindent{}(ii) If one fixes a subset $\\{\\phi_i: i \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ of cardinality $a$ that generates a free direct summand of rank $a$, then our approach combines with \\cite[Th. 3.10(ii)]{bst} to suggest that, as the subset $\\{\\theta_j: j \\in [a]\\}$ varies, elements of the form (\\ref{key product}) can be used to give an explicit description of the $a$-th Fitting ideal ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$.}\\end{remarks}\n\n\\begin{remark}\\label{e=1 case}{\\em In special cases one can either show, or is led to predict, that the idempotent $e_{(a)}$ is equal to $1$ and hence that the element $\\alpha$ in (\\ref{key product}) can be taken to be $1$.\n This is, for example, the case if $a = 0$, since each function $L(A, \\psi,z)$ is holomorphic at $z=1$ and, in the setting of abelian extensions of $\\QQ$, this case will be considered in detail in \\S\\ref{mod sect}. This situation can also arises naturally in cases with $a > 0$, such as the following.\n\n\\noindent{}(i) If $F$ is a ring class field of an imaginary quadratic field $k$ and suitable hypotheses are satisfied by an elliptic curve $A\/\\QQ$ and the extension $F\/\\QQ$, then the existence of a Heegner point in $A(F)$ with non-zero trace to $A(k)$ combines with the theorem of Gross and Zagier to imply that $e_{(1)} = 1$. This case will be considered in detail in \\S\\ref{HHP}.\n\n\\noindent{}(ii) As a generalization of (i), if $F$ is a generalized dihedral extension of a field $F'$, $k$ is the unique quadratic extension of $F'$ in $F$, all $p$-adic places split completely in $k\/F'$ and the rank of $A(k)$ is odd, then the result of Mazur and Rubin in \\cite[Th. B]{mr2} combines with the prediction of ${\\rm BSD}(A_{F\/k})$(ii) to imply that $e_{(1)} = 1$.\n\n\\noindent{}(iii) Let $F$ be a finite extension of $k$ inside a $\\ZZ_p$-extension $k_\\infty$, set $\\Gamma:= G_{k_\\infty\/k}$ and write $r_\\infty$ for the corank of ${\\rm Sel}_p(A_{k_\\infty})$ as a module over the Iwasawa algebra $\\ZZ_p[[\\Gamma]]$. Then, if $A$, $F\/k$  and $p$ satisfy all of the hypotheses listed at the beginning of \\S\\ref{tmc}, one can show that the inverse limit $\\varprojlim_{F'}A^t(F')_p$, where $F'$ runs over all finite extensions of $F$ in $k_\\infty$, is a free $\\ZZ_p[[\\Gamma]]$-module of rank $r_\\infty$ and this in turn combines with the prediction of ${\\rm BSD}(A_{F\/k})$(ii) to imply that $e_{(r_\\infty)}=1$.\n}\n\\end{remark}\n\n\\begin{remark}\\label{new remark SC}{\\em Under suitable additional hypotheses it is also possible to obtain more explicit versions of the containments predicted by Theorem \\ref{big conj}. To describe an example, assume that neither $A(F)$ nor $A^t(F)$ has a point of order $p$, that $p$ is unramified in $k$, that all $p$-adic places of $k$ are at most tamely ramified in $F$ and that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1)$-(H$_5$) that are listed in \\S\\ref{tmc}. Then, after taking account of the equality in Remark \\ref{emptysets}, the argument that is used to prove Theorem \\ref{big conj} can be directly applied to the Selmer complex ${\\rm SC}_p(A_{F\/k})$ rather than to the complex $C_{S,X}$ that occurs in \\S\\ref{proof of big conj}. In this way one finds that ${\\rm BSD}(A_{F\/k})$ predicts under the above hypotheses that for any given non-negative integer $a$ and any data\n $\\alpha$, $\\{\\theta_j:j\\in[a]\\}$ and $\\{\\phi_i:i\\in[a]\\}$ as in Theorem \\ref{big conj}, the product\n\\begin{equation}\\label{key product2} \\alpha^{1+2a}\\cdot(e_{F,a}\\cdot\\mathcal{L}_{A,F\/k}^*)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\end{equation}\nshould belong to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ and annihilate $\\sha(A_F^t)[p^\\infty]$. Here $\\mathcal{L}_{A,F\/k}^*$ is the element defined in (\\ref{bkcharelement}) and hence is related to the values\nof $L$-functions that are truncated only at the places in $S_k^f\\cap S_k^F$ rather than at all places in $S$ (as in the expression (\\ref{key product})).\n}\\end{remark}\n\n\\begin{remark}\\label{integrality rk}{\\em To obtain concrete congruences from the result of Theorem \\ref{big conj} or its variant in Remark \\ref{new remark SC} one can, for example, proceed as follows. The stated results imply that the product expression $\\mathcal{L}$ in either (\\ref{key product}) or (\\ref{key product2}) belongs to $\\ZZ_p[G]$ and hence that for every $g$ in $G$ one has\n\\[ \\sum_{\\psi \\in \\widehat{G}_{A,a}}\\psi(g)\\mathcal{L}_\\psi \\equiv 0 \\,\\,\\,({\\rm mod}\\,\\, |G|\\cdot \\ZZ_p).\\]\nHere each element $\\mathcal{L}_\\psi$ of $\\CC_p$ is defined by the equality $\\mathcal{L} = \\sum_{\\psi\\in \\widehat{G}_{A,a}}\\mathcal{L}_\\psi\\cdot e_\\psi$ and so can be explicitly related to the value at $z=1$ of the function $z^{-a}L(A,\\check\\psi,z)$.}\\end{remark}\n\n\\subsection{Explicit regulator matrices}Motivated by Remark \\ref{e=1 case}, we consider in more detail the case that $e_{(a)} = 1$.\n\nIn this case, there exist $a$-tuples in $A^t(F)$ and $A(F)$ that are each linearly independent over $\\QQ[G]$ and this fact implies the expressions $(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))$ in Theorem \\ref{big conj} can be interpreted in terms of classical Neron-Tate heights.\n\nTo state the result we use the following notation: for ordered $a$-tuples $P_\\bullet = \\{P_i: i \\in [a]\\}$ of $A^t(F)$ and $Q_\\bullet = \\{Q_j: j \\in [a]\\}$ of $A(F)$  we define a matrix in ${\\rm M}_a(\\RR[G])$ by setting\n\\begin{equation}\\label{regulatormatrix} h_{F\/k}(P_\\bullet, Q_\\bullet) := (\\sum_{g \\in G}\\langle g(P_i),Q_j\\rangle_{A_F}\\cdot g^{-1})_{1\\le i,j\\le a},\\end{equation}\nwhere $\\langle -,-\\rangle_{A_F}$ denotes the Neron-Tate height pairing for $A$ over $F$\n\n\\begin{lemma}\\label{height pairing interp} Fix a natural number $a$ such that $e_{(a)} =1$ and then choose ordered $a$-tuples $P_\\bullet$ of $A^t(F)$ and $Q_\\bullet$ of $A(F)$ that are each linearly independent over $\\QQ[G]$. Then the matrix $e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet)$ belongs to ${\\rm GL}_a(\\RR[G]e_a)$ and one has\n\\begin{multline*}  e_a\\cdot\\left\\{(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\mid \\theta_j\\in \\Hom_{\\ZZ[G]}(A^t(F),\\ZZ[G]),\\phi_i\\in \\Hom_{\\ZZ[G]}(A(F),\\ZZ[G])\\right\\}\\\\\n= {\\rm det}(e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1}\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A^t(F)_{\\rm tf}\/\\langle P_\\bullet\\rangle)^\\vee_{\\rm tor})\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A(F)_{\\rm tf}\/\\langle Q_\\bullet\\rangle)^\\vee_{\\rm tor}) \\end{multline*}\nwhere $\\langle P_\\bullet\\rangle$ and $\\langle Q_\\bullet\\rangle$ denote the $G$-modules that are generated by $P_\\bullet$ and $Q_\\bullet$.\n  \\end{lemma}\n\n\\begin{proof} We write $N(P_\\bullet)$ and $N(Q_\\bullet)$ for the quotients of $A^t(F)_{\\rm tf}$ and $A(F)_{\\rm tf}$ by $\\langle P_\\bullet\\rangle$ and $\\langle Q_\\bullet\\rangle$. Then, by taking $\\ZZ$-linear duals of the tautological short exact sequence\n\\[ 0 \\to \\langle P_\\bullet\\rangle \\xrightarrow{\\iota_{P_\\bullet}} A^t(F)_{\\rm tf} \\to N(P_\\bullet) \\to 0\\]\none obtains an exact sequence $$A^t(F)^\\ast \\xrightarrow{\\iota_{P_\\bullet}^\\ast} \\langle P_\\bullet\\rangle^\\ast \\to N(P_\\bullet)_{\\rm tor}^\\vee \\to 0$$ and hence an equality\n\\[ \\im({\\bigwedge}^a_{\\ZZ[G]}\\iota_{P_\\bullet}^\\ast) = {\\rm Fit}^0_{\\ZZ[G]}(N(P_\\bullet)_{\\rm tor}^\\vee).\\]\n\nIn the same way one derives an equality\n\\[ \\im({\\bigwedge}^a_{\\ZZ[G]}\\iota_{Q_\\bullet}^\\ast) = {\\rm Fit}^0_{\\ZZ[G]}(N(Q_\\bullet)_{\\rm tor}^\\vee).\\]\n\nSince the maps $e_{a}(\\QQ\\otimes_\\ZZ\\iota_{P_\\bullet}^\\ast)$ and $e_{a}(\\QQ\\otimes_\\ZZ\\iota_{Q_\\bullet}^\\ast)$ are bijective, these equalities imply that the lattice\n\\[ e_a\\cdot\\left\\{(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\mid \\theta_j\\in A^t(F)^\\ast,\\phi_i\\in A(F)^\\ast\\right\\}\\]\nis equal to the product\n\\begin{multline*} \\left\\{e_a(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(e_a(\\wedge_{i=1}^{i=a}\\phi_i)))\\mid \\theta_j\\in \\langle P_\\bullet\\rangle^\\ast,\\phi_i\\in \\langle Q_\\bullet\\rangle^\\ast\\right\\}\\\\\n\\times {\\rm Fit}^0_{\\ZZ[G]}(N(P_\\bullet)_{\\rm tor}^\\vee)\\cdot{\\rm Fit}^0_{\\ZZ[G]}(N(Q_\\bullet)_{\\rm tor}^\\vee). \\end{multline*}\n\nThis implies the claimed result since, writing $P_i^\\ast$ and $Q_j^\\ast$ for the elements of $\\langle P_\\bullet\\rangle^\\ast$ and $\\langle Q_\\bullet\\rangle^\\ast$ that are respectively dual to $P_i$ and $Q_j$, one has\n\\begin{multline*} \\left\\{e_a(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(e_a(\\wedge_{i=1}^{i=a}\\phi_i)))\\mid \\theta_j\\in \\langle P_\\bullet\\rangle^\\ast,\\phi_i\\in \\langle Q_\\bullet\\rangle^\\ast\\right\\}\\\\\n=  \\ZZ[G]\\cdot (e_a\\wedge_{i=1}^{i=a}P_i^\\ast)({\\rm ht}^a_{A_{F\/k}}(e_a\\wedge_{i=1}^{i=a}Q^\\ast_i))\\end{multline*}\nand\n\\begin{align*} e_a(\\wedge_{i=1}^{i=a}P_i^\\ast)({\\rm ht}^a_{A_{F\/k}}(e_a\\wedge_{i=1}^{i=a}Q^\\ast_i)) =\\, &{\\rm det}\\bigl( (e_aP_i^\\ast(h^{-1}_{A_{F\/k}}(e_a Q^\\ast_j)))_{1\\le i, j\\le a}\\bigr)\\\\\n =\\, &{\\rm det}(e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1}.\\end{align*}\nWe note that the last equality is true because the definition of ${\\rm ht}^a_{A_{F\/k}}$ implies that for every $j$ one has\n\\[ h^{-1}_{A_{F\/k}}(e_a Q^\\ast_j) = \\sum_{b=1}^{b=a} ((e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1})_{bj}P_b.\\]\n\\end{proof}\n\n\\subsection{Bounds on logarithmic resolvents}\\label{p-adic sec} The result of Lemma \\ref{height pairing interp} means that in many cases the only term in the product expression (\\ref{key product}) that is not explicitly understood is the logarithmic resolvent of the chosen semi-local points.\n\nWe shall now partly address this issue. For subsequent purposes (related to the upcoming article \\cite{dmckwt}) we do not here require $G$ to be abelian.\n\n\n\n\\subsubsection{} We start by deriving an easy consequence of the arguments in Proposition \\ref{lms} and Lemma \\ref{ullom}. For each natural number $i$ we set $\\wp_{F_p}^i:=\\prod_{w'\\in S_F^p}\\wp_{F_{w'}}^i$, where $\\wp_{F_{w'}}$ denotes the maximal ideal in the valuation ring of $F_{w'}$. We also set $\\hat A^t(\\wp_{F_p}^i):=\\prod_{w'\\in S_F^p}\\hat A^t_{w'}(\\wp_{F_{w'}}^i)$, where $\\hat A^t_{w'}$ denotes the formal group of $A^t_{\/F_{w'}}$.\n\n\\begin{proposition}\\label{explicit log resolve} If all $p$-adic places are tamely ramified in $F\/k$ and $\\hat A^t(\\wp_{F_p})$ is torsion-free, then there exists an ordered $\\ZZ_p[G]$-basis $x_\\bullet$ of $\\hat A^t(\\wp_{F_p})$ for which one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) = \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d,\\]\nwhere the elements $\\varrho_v(F\/k)$ of $\\zeta(\\QQ[G])^\\times$ are as defined in (\\ref{varrho def}).\n\\end{proposition}\n\n\\begin{proof} Since all $p$-adic places of $k$ are tamely ramified in $F$ Lemma \\ref{ullom} implies that the $\\ZZ_p[G]$-module $\\hat A^t(\\wp^{i}_{F_p})$ is cohomologically-trivial for all $i$. Hence, if $\\hat A^t(\\wp_{F_p})$ is torsion-free, then it is a projective $\\ZZ_p[G]$-module (by \\cite[Th. 8]{cf}) and therefore free of rank $nd$ (since $\\QQ_p\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p})$ is isomorphic to $F^d_p$).\n\nIn this case we fix an ordered basis $x_\\bullet$ of $\\hat A^t(\\wp_{F_p})$ and regard it as a $\\QQ_p^c[G]$-basis of $\\QQ_p^c\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p})$.\n\nWe also regard $\\{(i,\\hat\\sigma): (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}$ as a $\\QQ_p^c[G]$-basis of the direct sum $(\\QQ_p^c\\cdot Y_{F\/k,p})^d$ of $d$ copies of $\\QQ_p^c\\cdot Y_{F\/k,p}$.\n\nThen, with respect to these bases, the matrix\n\\begin{equation}\\label{log resolve matrix} \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{\\omega_j'}(x_{(j',\\sigma')})))\\cdot g \\bigr)_{(j,\\sigma),(j',\\sigma')}\\end{equation}\nrepresents the composite isomorphism of $\\QQ^c_p[G]$-modules\n\\[ \\mu': \\QQ_p^c\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p}) \\xrightarrow{({\\rm log}_{\\omega_j'})_{j\\in [d]}}\n(\\QQ_p^c\\otimes_\\QQ F)^d \\xrightarrow{(\\mu)_{i \\in [d]}} (\\QQ_p^c\\cdot Y_{F\/k,p})^d,\\]\nwhere $\\mu$ is the isomorphism $\\QQ_p^c\\otimes_\\QQ F \\cong \\QQ_p^c\\cdot Y_{F\/k,p}$ that sends each element $\\lambda\\otimes f$ to $(\\lambda\\cdot \\hat\\sigma(f))_{\\sigma\\in \\Sigma(k)}$. This fact implies that\n\\begin{equation}\\label{eq 1}\\delta_{G,p}(\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)) = [\\hat A^t(\\wp_{F_p}) , Y^d_{F\/k,p}; \\mu'] \\end{equation}\nin $K_0(\\ZZ_p[G],\\QQ^c_p[G])$.\n\nIn addition, since for any large enough integer $i$ the image of $\\hat A^t(\\wp^i_{F_p})$ under each map ${\\rm log}_{\\omega_j'}$ is equal to $\\wp^i_{F,p}$, the `telescoping' argument of Lemma \\ref{ullom} implies that\n\\begin{align}\\label{eq 2} [\\hat A^t(\\wp_{F_p}) , Y^d_{F\/k,p}; \\mu'] = \\, &d\\cdot [\\wp_{F_p} , Y_{F\/k,p}; \\mu]\\\\\n                                                                 = \\, &d\\cdot [\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu] + d\\cdot \\chi_{G,p}\n                                                                 \\bigl((\\mathcal{O}_{F,p}\/\\wp_{F_p})[0],0\\bigr).\\notag\n                                                                 \\end{align}\n\nNext we note that if $f_v$ is the absolute residue degree of a $p$-adic place $v$, then the normal basis theorem for\n$\\mathcal{O}_{F_w}\/\\wp_{F_w}$ over the field with $p$ elements implies that there exists a short\nexact sequence of $G_w\/I_w$-modules\n\n\\[ 0 \\to \\ZZ [G_w\/I_w] ^{f_{v}} \\xrightarrow{\\times p} \\ZZ[G_w\/I_w]^{f_{v}} \\to \\mathcal{O}_{F_w}\/\\wp_{F_w} \\to 0.\\]\nBy using these sequences (for each such $v$) one computes that\n\n\\begin{equation*}\\label{eq 3} \\chi_{G,p}\\bigl((\\mathcal{O}_{F,p}\/\\wp_{F_p})[0],0\\bigr) = \\delta_{G,p}\\bigl(\\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr).\\end{equation*}\n\n\nUpon combining this equality with (\\ref{eq 1}) and (\\ref{eq 2}) we deduce that\n\\[ \\delta_{G,p}(\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\prod_{v \\in S_k^p}\\varrho_v(F\/k))^{-d}) = d\\cdot [\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu]\\]\nand from here one can deduce the claimed result by using the argument of Proposition \\ref{lms}.\\end{proof}\n\n\n\\begin{remark}\\label{new add}{\\em Assume that all $p$-adic places are at most tamely ramified in $F\/k$, that neither $\\widehat{A^t}(\\wp_{F_p})$ nor $A(F)$ has an element of order $p$ and that $e_{(a)} = 1$ so there exist ordered $a$-tuples $P_\\bullet$ and $Q_\\bullet$  as in Lemma \\ref{height pairing interp}. Then Proposition \\ref{explicit log resolve}, Lemma \\ref{height pairing interp} and Remark \\ref{more explicit rem}(i) combine with Theorem \\ref{big conj} to imply ${\\rm BSD}_p(A_{F\/k})$(iv) predicts that any element in the set\n\\begin{equation*}\\label{explicit ann}\n\\frac{L^{(a)}_{S}(A_{F\/k},1)\\cdot \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot {\\rm det}(h_{F\/k}(P_\\bullet, Q_\\bullet))} \\cdot {\\rm Fit}^0_{\\ZZ[G]}((A^t(F)\/\\langle P_\\bullet\\rangle)^\\vee_{\\rm tor})\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A(F)\/\\langle Q_\\bullet\\rangle)^\\vee_{\\rm tor})\\end{equation*}\nbelongs to ${\\rm Fit}^a_{\\ZZ_{p}[G]}({\\rm Sel}_{p}(A_{F})^\\vee)$ and annihilates $\\sha(A^t_{F})[p^\\infty]$}.\\end{remark}\n\n\\begin{remark}\\label{new add2}{\\em To obtain a variant of the prediction in Remark \\ref{new add} that may in some cases be more amenable to numerical investigation assume that $p$ is unramified in $k$, that all $p$-adic places of $k$ are at most tamely ramified in $F$, that neither $A(F)$ nor $A^t(F)$ has an element of order $p$, that $e_{(a)} = 1$ and that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then by using Remark \\ref{new remark SC} in place of Theorem \\ref{big conj} the same approach as in Remark \\ref{new add} allows one to show that under these hypotheses the prediction in Remark \\ref{new add} should be true after replacing the terms $L^{(a)}_{S}(A_{F\/k},1)$ and $S_k^p$ that occur in the given displayed expression by $L^{(a)}_{S_{\\rm r}}(A_{F\/k},1)$ and $S_{p,{\\rm r}}$ respectively, where, as in (\\ref{bkcharelement}), we set $S_{\\rm r} := S_{k}^f\\cap S_k^F$ and $S_{p,{\\rm r}}:= S_k^p\\cap S_k^F$.\n}\\end{remark}\n\n\\begin{example}\\label{wuthrich example}{\\em Christian Wuthrich kindly supplied us with the following concrete applications of Remark \\ref{new add}.  Set $k = \\QQ$ and $K = \\QQ(\\sqrt{229})$ and write $F$ for the Galois closure of the field $L = \\QQ(\\alpha)$ with $\\alpha^3-4\\alpha+1 = 0$. Then $K \\subset F$ and the group $G := G_{F\/k}$ is dihedral of order six. Let $A$ denote either of the curves 3928b1 (with equation $y^2 = x^3-x^2 + x + 4$) or 5864a1 (with equation $y^2 = x^3-x^2 -24x + 28$). Then ${\\rm rk}(A_\\QQ)= 2$, ${\\rm rk}(A_K) = {\\rm rk}(A_L) = 3$ and ${\\rm rk}(A_F) = 5$ and, since $\\sha_3(A_K)$ vanishes (as can be shown via a computation with Heegner points on the quadratic twist of $A$), these facts combine with \\cite[Cor. 2.10(i)]{bmw0} to imply the $\\ZZ_{3}[G]$-module $A(F)_{3}$ is isomorphic to $\\ZZ_{3}[G](1+\\tau) \\oplus \\ZZ_{3}\\oplus \\ZZ_{3}$, with $\\tau$ the unique non-trivial element in $G_{F\/L}$. In particular, if we set $\\Gamma := G_{F\/K}$, then we can choose a point $P$ that generates over $\\ZZ_{3}[\\Gamma]$ a free direct summand of $A(F)_{3}$. In addition, ${\\rm Fit}^1_{\\ZZ_{3}[\\Gamma]}({\\rm Sel}_{3}(A_{F})^\\vee)$ is contained in\n\\[ {\\rm Fit}^1_{\\ZZ_{3}[\\Gamma]}(\\ZZ_{3}[G](1+\\tau) \\oplus \\ZZ_{3}\\oplus \\ZZ_{3}) = {\\rm Fit}^0_{\\ZZ_{3}[\\Gamma]}(\\ZZ_{3}\\oplus \\ZZ_{3}) = I_{3}(\\Gamma)^2,\\]\nwhere $I_{3}(\\Gamma)$ is the augmentation ideal of $\\ZZ_{3}[\\Gamma]$. Finally, we note that $3$ splits in $K$ and is unramified in $F$ so that for each $3$-adic place $v$ of $K$ the element $\\varrho_v(F\/K)$ is equal to $3$. After taking account of these facts, Remark \\ref{new add} (with $F\/k$ taken to be $F\/K$, $a$ to be $1$ and $P_1 = Q_1$ to be $P$) shows ${\\rm BSD}_3(A_{F\/k})$(iv) predicts that\n\\[ 9\\cdot \\tau^\\ast(F\/K)\\cdot\\frac{L^{(1)}_{S}(A_{F\/K},1)}{\\Omega_A^{F\/k}} = x\\cdot \\sum_{g \\in G}\\langle g(P),P\\rangle_{A_F}\\cdot g^{-1}\\]\nfor an element $x$ of $I_{3}(\\Gamma)^2$ that annihilates the $3$-primary component of $\\sha(A_{F})$. Here we have also used the fact that $w_{F\/K}=1$ because each of the real places $v$ of $K$ has trivial decomposition group in $\\Gamma$ so $\\psi^-_v(1)=1-1=0$ and thus $w_\\psi=1$ for each $\\psi\\in\\widehat\\Gamma$.}\n\\end{example}\n\n\\subsubsection{}It is possible to prove less precise versions of Proposition \\ref{explicit log resolve} without making any hypotheses on ramification and to thereby obtain more explicit versions of the prediction that is made in Theorem \\ref{big conj}.\n\nFor example, if $F_p^\\times$ has no element of order $p$, $\\mathfrak{M}$ is any choice of maximal $\\ZZ_p$-order in $\\QQ_p[G]$ that contains $\\ZZ_p[G]$ and $x$ is any element of $\\zeta(\\ZZ_p[G])$ such that $x\\cdot \\mathfrak{M} \\subseteq \\ZZ_p[G]$, then one can deduce from the result of \\cite[Cor. 7.8]{bleyburns} that for every element $y$ of the ideal\n\\[ \\zeta(\\ZZ_p[G])\\cdot p^d\\cdot x^{1+2d}((1-e_G)+ |G|\\cdot e_G)\\]\nthere exists an ordered $nd$-tuple $x(y)_\\bullet$ of points in $A^t(F_p)$ for which  one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x(y)_\\bullet) = y\\cdot \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d\\]\nin $\\zeta(\\QQ^c[G])$.\n\nHowever, we shall not prove this result here both because it is unlikely to be the best possible `bound' that one can give on logarithmic resolvents in terms of Galois-Gauss sums and also because, from the perspective of numerical investigations, the following much easier interpretation of logarithmic resolvents in terms of Galois resolvents is likely to be more helpful.\n\n\\begin{lemma} Let $i_0$ be the least integer with $i_0 > e_{F,p}\/(p-1)$, where $e_{F,p}$ is the maximal absolute ramification degree of any $p$-adic place of $F$. Let $z$ be an element of $F$ that belongs to the $i_0$-th power of every prime ideal of $\\mathcal{O}_F$ above $p$.\n\nThen, for any integral basis $\\{a_i:  i \\in [n]\\}$ of $\\mathcal{O}_k$, there exists an ordered $nd$-tuple $x_\\bullet$ of points in $A^t(F_p)$ for which  one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) = {\\rm Nrd}_{\\QQ_p^c[G]}\\left( \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z\\cdot a_i))\\cdot g\\bigr)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^d. \\]\n\\end{lemma}\n\n\\begin{proof} The definition of $i_0$ ensures that the formal group logarithm of $A^t$ over $F_p$ gives an isomorphism of $\\hat A^t(\\wp_{F_p}^{i_0})$ with a direct sum $(\\wp_{F_p}^{i_0})^d$ of $d$ copies of $\\wp_{F_p}^{i_0}$ (cf. \\cite[Th. 6.4(b)]{silverman}). Here the individual copies of $\\wp_{F_p}^{i_0}$ in the sum are parametrised by the differentials $\\{\\omega_j': j \\in [n]\\}$ that are used to define ${\\rm log}_{A^t}$.\n\nThe choice of $z$ also implies that $Z := \\{z\\cdot a_i:  i \\in [n]\\}$ is a subset of $\\wp_{F_p}^{i_0}$. We may therefore choose a pre-image $x_\\bullet$ in $\\hat A^t(\\wp_{F_p}^{i_0})$ of the ordered $nd$-tuple in $(\\wp_{F_p}^{i_0})^d$ that is obtained by placing a copy of $Z$ in each of the $d$ direct summands.\n\nFor these points $x_\\bullet$ the interpretation of the matrix (\\ref{log resolve matrix}) that is given in the proof of  Proposition \\ref{explicit log resolve} shows that it is equal to a $d\\times d$ diagonal block matrix with each diagonal block equal to the matrix\n\\[  \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z\\cdot a_i))\\cdot g\\bigr)_{\\sigma\\in \\Sigma(k),i\\in [n]}.\\]\n\nSince $\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)$ is defined to be the reduced norm of the matrix (\\ref{log resolve matrix}) the claimed equality is therefore clear.\\end{proof}\n\n\\subsection{The proof of Theorem \\ref{big conj}}\\label{proof of big conj}\n\n\\subsubsection{}We start by proving a technical result that is both necessary for the proof of Theorem \\ref{big conj} and will also be of further use in the sequel.\n\nIn this result we use the terminology of `characteristic elements' from \\cite[Def. 3.1]{bst}.\n\n\\begin{lemma}\\label{modifiedlemma} Fix an ordered subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ as in Theorem \\ref{big conj}. Write $X$ for the $\\ZZ_p[G]$-module generated by $x_\\bullet$ and $C_{S,X}$ for the Selmer complex ${\\rm SC}_S(A_{F\/k};X,H_\\infty(A_{F\/k})_p)$ from Definition \\ref{selmerdefinition}. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The module $H^1(C_{S,X})$ is torsion-free.\n\\item[(ii)] For any finite non-empty set of places $T$ of $k$ that is disjoint from $S$, there exists an exact triangle in $D^{\\rm perf}(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{modifiedtriangle}\\bigoplus_{v\\in T}R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-2]\\to C_{S,X}\\stackrel{\\theta}{\\to} C_{S,X,T}\\to\\bigoplus_{v\\in T}R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-1]\\end{equation}\nin which $C_{S,X,T}$ is acyclic outside degrees one and two and there are canonical identifications of $H^1(C_{S,X,T})$ with $H^1(C_{S,X})$ and of  $\\Sel_p(A_F)^\\vee$ with a subquotient of $H^2(C_{S,X,T})$ in such a way that $\\QQ_p\\cdot\\Sel_p(A_F)^\\vee=\\QQ_p\\cdot H^2(C_{S,X,T})$.\n \\item[(iii)] Following claim (ii) we write\n\\[ h^{T}_{A,F}: \\CC_p\\cdot H^1(C_{S,X,T}) \\to \\CC_p\\cdot H^2(C_{S,X,T})\\]\nfor the isomorphism $(\\CC_p\\otimes_{\\ZZ_p}H^2(\\theta))\\circ(\\CC_p\\otimes_{\\RR,j}h_{A,F})$ of $\\CC_p[G]$-modules. Then, if ${\\rm BSD}_p(A_{F\/k})$(iv) is valid, there exists a characteristic element $\\mathcal{L}$ for  $(C_{S,X},h_{A,F})$ in $\\CC_p[G]^\\times$ with the property that for any non-negative integer $a$ one has\n\\[ e_a\\cdot\\mathcal{L}^{-1}=\\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet).\\]\n\nIn addition, in this case the element\n\\[ \\mathcal{L}_T := (\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)^{-1}\\cdot\\mathcal{L}\\]\nis a characteristic element for  $(C_{S,X,T},h^{T}_{A,F})$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n\nSince $p$ is odd there exists a homomorphism of $\\ZZ_p[G]$-modules $\\phi$ from the module $H_\\infty(A_{F\/k})_p = \\bigoplus_{v\\in S_k^\\infty}H^0(k_v,T_{p,F}(A^t))$ to $A^t(F_p)^\\wedge_p$ that sends the ordered $\\ZZ_p[G]$-basis of $H_\\infty(A_{F\/k})_p$ specified at the end of \\S\\ref{gamma section} to $x_\\bullet$.\n\nThen, comparing the triangle (\\ref{can tri}) with the construction of \\cite[Prop. 2.8 (ii)]{bst} immediately implies that $C_{S,X}$ is isomorphic in $D^{\\rm perf}(\\ZZ_p[G])$ to the complex that is denoted in loc. cit. by $C_\\phi(T_{p,F}(A^t))$.\n\nGiven this, claim (i) follows directly from \\cite[Prop. 2.8 (ii)]{bst} and claim (ii) from \\cite[Prop. 2.8 (iii)]{bst} with $C_{S,X,T}:=C_{\\phi,T}(T_{p,F}(A^t))$.\n\nTurning to claim (iii) we note that each place $v$ in $T$ is not $p$-adic, does not ramify in $F\/k$ and is of good reduction for $A$.\n\nEach complex $R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))$ is therefore well-defined. Since these complexes are acyclic outside degree one, where they have finite cohomology, we may therefore apply Lemma \\ref{fk lemma} to the triangle (\\ref{modifiedtriangle}) to deduce that\n\\begin{align*}\\chi_{G,p}(C_{S,X},h_{A,F})-\\chi_{G,p}(C_{S,X,T},h^{T}_{A,F})= \\, &\\sum_{v\\in T}\\chi_{G,p}(R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-2],0)\\\\\n=\\, &\\delta_{G,p}({\\det}_{\\QQ_p[G]}(1-\\Phi_v|\\QQ_p\\cdot T_{p,F}(A^t)(-1)))\\\\\n=\\, &\\delta_{G,p}(P_v(A_{F\/k},1)^\\#)\n\\end{align*}\nin $K_0(\\ZZ_p[G],\\CC_p[G])$.\n\nThus if $\\mathcal{L}$ is a characteristic element of $(C_{S,X},h_{A,F})$, then $(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)^{-1}\\cdot\\mathcal{L}$ is a characteristic element for  $(C_{S,X,T},h^{T}_{A,F})$, as claimed in the final assertion of claim (iii).\n\nIt is thus enough to deduce from ${\\rm BSD}_p(A_{F\/k})$(iv) the existence of a characteristic element $\\mathcal{L}$ of $(C_{S,X},\\CC_p\\otimes_{\\RR}h_{A,F})$ with the required interpolation property.\n\nNow, since $S$ contains all $p$-adic places of $k$, the module $\\mathcal{Q}(\\omega_\\bullet)_{S,p}$ vanishes and the $p$-primary component of the term $\\mu_{S}(A_{F\/k})$ is also trivial.\n\nIn addition, as the validity of {\\rm BSD}$_p(A_{F\/k})$(iv) is independent of the choice of global periods and we can assume firstly that $\\omega_\\bullet$ is the set $\\{ z_i\\otimes \\omega'_j: i \\in [n], j \\in [d]\\}$ fixed in Lemma \\ref{k-theory period} and secondly that the image of $\\mathcal{F}(\\omega_\\bullet)_p$ under the formal group exponential ${\\rm exp}_{A^t,F_p}$ (defined with respect to the differentials $\\{\\omega_j': j \\in [d]\\})$ is contained in $X$.\n\nThen the assumed validity of the equality (\\ref{displayed pj}) in this case combines with the equality in Lemma \\ref{k-theory period} to imply that the element\n\n\\begin{equation}\\label{char el 1} \\frac{L_S^*(A_{F\/k},1)}{{\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}= \\frac{L_S^*(A_{F\/k},1)}{ \\Omega_A^{F\/k}\\cdot w_{F\/k}^{d}}\\cdot {\\rm det}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{d}\\end{equation}\nof $\\zeta(\\CC_p[G])^\\times$ is the inverse of a characteristic element of $(C_{S,\\omega_\\bullet},h_{A,F})$.\n\nHere we write $C_{S,\\omega_\\bullet}$ for the Selmer complex ${\\rm SC}_S(A_{F\/k};\\mathcal{X}(p),H_\\infty(A_{F\/k})_p)$ where $\\mathcal{X}$ is the perfect Selmer structure $\\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined in \\S\\ref{perf sel sect}. (In addition, the fact that it is the inverse of a characteristic element results from a comparison of our chosen normalisation of non-abelian determinants compared with that of \\cite[(10)]{bst}, as described in Remark \\ref{comparingdets}).\n\nIn particular, since $\\mathcal{X}(p)$ is by definition equal to ${\\rm exp}_{A^t,F_p}(\\mathcal{F}(\\omega_\\bullet)_p)$ and hence, by assumption, contained in $X$, a comparison of the definitions of $C_{S,\\omega_\\bullet}$ and $C_{S,X}$ shows that there is an exact triangle\n\\[  C_{S,\\omega_\\bullet} \\to C_{S,X} \\to \\bigl(X\/\\mathcal{X}(p))[-1] \\to C_{S,\\omega_\\bullet}[1]\\]\nin $D^{\\rm perf}(\\ZZ_p[G])$.\n\nSince the product\n\n\\[ \\xi := \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot {\\rm det}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{-d}\\]\nis equal to the determinant of a matrix that expresses a basis of the free $\\ZZ_p[G]$-module $\\mathcal{X}(p)$ in terms of the basis $x_\\bullet$ of $X$, the above triangle implies that the product\n\\[ \\frac{L_S^*(A_{F\/k},1)}{ \\Omega_A^{F\/k}\\cdot w_{F\/k}^{d}}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\]\nof $\\xi$ and the element (\\ref{char el 1}) is the inverse of a characteristic element for  $(C_{S,X},h_{A,F})$.\n\nThe claimed interpolation formula is thus a consequence of the fact that $L_S^{(a)}(A_{F\/k},1)$ is equal to $e_a\\cdot L_S^*(A_{F\/k},1)$. \\end{proof}\n\n\n\\subsubsection{}We are now ready to prove Theorem \\ref{big conj}.\n\nTo do this we will apply the general result of \\cite[Th. 3.10(i)]{bst} to the complex $C_{S,X,T}$, isomorphism $h^{T}_{A,F}$ and characteristic element $\\mathcal{L}_T$ constructed in Lemma \\ref{modifiedlemma}.\n\n In order to do so, we fix an ordered subset $\\Phi:=\\{\\phi_i: i \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ of cardinality $a$. We fix a pre-image $\\phi_i'$ of each $\\phi_i$ under the surjective composite homomorphism\n\\[ H^2(C_{S,X})\\to\\Sel_p(A_F)^\\vee\\to \\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G]),\\]\nwhere the first arrow is the canonical map from Proposition \\ref{prop:perfect}(iii) and the second is induced by the canonical short exact sequence\n\\begin{equation}\\label{sha-selmer}\n \\xymatrix{0 \\ar[r] & \\sha(A_F)[p^\\infty]^\\vee \\ar[r] & \\Sel_p(A_F)^\\vee \\ar[r]&  \\Hom_{\\ZZ_p}(A(F)_p,\\ZZ_p)\\ar[r] & 0.}\n\\end{equation}\n\nWe set $\\Phi':=\\{\\phi'_i:i \\in [a]\\}$ and consider the image $H^2(\\theta)(\\Phi')$ of $\\Phi'$ in $H^2(C_{S,X,T})$, where $\\theta$ is the morphism that occurs in the triangle (\\ref{modifiedtriangle}) (so that $H^2(\\theta)$ is injective).\n\n\nWe next write $\\iota:H^1(C_{S,X,T})=H^1(C_{S,X})\\to A^t(F)_p$ for the canonical homomorphism in Proposition \\ref{prop:perfect}(iii).\n\nThen, with $\\mathcal{L}_T$ the element specified in Lemma \\ref{modifiedlemma}(iii), a direct comparison of the definitions of $h_{A,F}^{T}$ and ${\\rm ht}_{A_{F\/k}}^{a}$ shows that the `higher special element' that is associated via \\cite[Def. 3.3]{bst} to the data $(C_{S,X,T},h^{T}_{A,F},\\mathcal{L}_T,H^2(\\theta)(\\Phi'))$ coincides with the pre-image under the bijective map $\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G]}^a\\iota$ of the element\n\\begin{equation}\\label{hse interpret}(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)\\cdot \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot {\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i).\\end{equation}\n(Here we have also used the fact that, since ${\\rm BSD}(A_{F\/k})$(ii) is assumed to be valid, the idempotent $e_a$ defined here coincides with the idempotent denoted $e_a$ in \\cite[\\S3.1]{bst} for the complex $C_{S,X,T}$.)\n\nTo proceed we fix an ordered subset $\\{\\theta_j: j \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])$ and identify it with its image under the injective map\n\\[ \\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G]) \\to \\Hom_{\\ZZ_p[G]}(H^1(C_{S,X,T}),\\ZZ_p[G])\\]\ninduced by $\\iota$. We also set $\\mathfrak{A} := \\ZZ_p[G]e_{(a)}$ and $M := \\mathfrak{A}\\otimes_{\\ZZ_p[G]}H^2(C_{S,X,T})$.\n\nThen the above interpretation of the higher special element in terms of the product (\\ref{hse interpret}) combines with the general result of \\cite[Th. 3.10(i)]{bst} to imply that for any element $\\alpha$ of $\\ZZ_p[G]\\cap\\mathfrak{A}$ and any element $y$ of $\\ZZ_p[G]$ that annihilates ${\\rm Ext}^2_{\\mathfrak{A}}(M,\\mathfrak{A})$\nthe product\n\\[\\alpha \\cdot y^a \\cdot(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)\\cdot \\frac{L^{(a)}_{S_F}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\]\nboth belongs to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ and annihilates $(\\Sel_p(A_F)^\\vee)_{\\rm tor}$.\n\nIn addition, the exact sequence (\\ref{sha-selmer}) identifies $(\\Sel_p(A_F)^\\vee)_{\\rm tor}$ with $\\sha(A_F)[p^\\infty]^\\vee$ and the Cassels-Tate pairing identifies $\\sha(A_F)[p^\\infty]^\\vee$ with $\\sha(A^t_F)[p^\\infty]$.\n\nTo deduce the result of Theorem \\ref{big conj} from here, it is therefore enough to show that $\\alpha^2$ annihilates ${\\rm Ext}^2_{\\mathfrak{A}}(M,\\mathfrak{A})$.\n\nTo do this we use the existence of a convergent first quadrant cohomological spectral sequence\n\\[ E_2^{pq} = {\\rm Ext}_{\\mathfrak{A}}^p(M,{\\rm Ext}^q_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})) \\Rightarrow {\\rm Ext}^{p+q}_{\\ZZ_p[G]}(M,\\mathfrak{A})\\]\n(cf. \\cite[Exer. 5.6.3]{weibel}).\n\nIn particular, since the long exact sequence of low degree terms of this spectral sequence gives an exact sequence of $\\ZZ_p[G]$-modules\n\\[ \\Hom_{\\ZZ_p[G]}(M,{\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})) \\to {\\rm Ext}_{\\mathfrak{A}}^2(M,\\mathfrak{A}) \\to {\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A}),\\]\nwe find that it is enough to show that the element $\\alpha$ annihilates both ${\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})$ and ${\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A})$.\n\nTo verify this we write $\\mathfrak{A}^\\dagger$ for the ideal $\\{x \\in \\ZZ_p[G]: x\\cdot e_{(a)} = 0\\}$ so that there is a natural short exact sequence of $\\ZZ_p[G]$-modules $0 \\to \\mathfrak{A}^\\dagger \\to \\ZZ_p[G] \\to \\mathfrak{A} \\to 0$.\n\nThen by applying the exact functor ${\\rm Ext}^\\bullet_{\\ZZ_p[G]}(-,\\mathfrak{A})$ to this sequence one obtains a surjective homomorphism\n\\[ \\Hom_{\\ZZ_p[G]}(\\mathfrak{A}^\\dagger,\\mathfrak{A}) \\twoheadrightarrow {\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A}).\\]\n\nIn addition, since $\\ZZ_p[G]$ is Gorenstein, by applying the exact functor ${\\rm Ext}^{\\bullet}_{\\ZZ_p[G]}(M,-)$ to the above sequence one finds that  there is a natural isomorphism\n\\[ {\\rm Ext}^{3}_{\\ZZ_p[G]}(M,\\mathfrak{A}^\\dagger) \\cong {\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A}).\\]\n\nTo complete the proof of Theorem \\ref{big conj} it is thus enough to note that the left hand modules in both of the last two displays are annihilated by $\\alpha$ since the definition of $\\mathfrak{A}^\\dagger$ implies immediately that $\\alpha\\cdot \\mathfrak{A}^\\dagger = 0$.\n\n\\begin{remark}\\label{omit T}{\\em If $A(F)$ does not contain an element of order $p$, then \\cite[Th. 3.10(i)]{bst} can be directly applied to the complex $C_{S,X}$ rather than to the auxiliary complex $C_{S,X,T}$. This shows that, in any such case, the prediction in Theorem \\ref{big conj} should remain true if the term $\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#$ is omitted.}\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Abelian congruence relations and height pairings}\\label{mrsconjecturesection}\n\n\n\n\n\n\n\nIn this section we continue to investigate the $p$-adic congruence relations between the leading coefficients of Hasse-Weil-Artin $L$-series that are encoded by ${\\rm BSD}_p(A_{F\/k})$(iv) in the case that $F\/k$ is abelian.\n\nMore concretely in \\S\\ref{mtchd} we will show that, beyond the integrality properties that are discussed in Theorem \\ref{big conj} and Remark \\ref{integrality rk}, elements of the form (\\ref{key product}) can be expected to satisfy additional congruence relations in the integral augmentation filtration that  involve Mazur-Tate regulators.\n\n\nThen in \\S \\ref{cycliccongssection} we specialise to the case of cyclic extensions in order to make these additional congruence relations fully explicit.\n\nIn particular, in this way we render the equality ${\\rm BSD}_p(A_{F\/k})$(iv) amenable to (numerical) verification even in cases in which it incorporates a thorough-going mixture of both archimedean phenomena and delicate $p$-adic congruences.\n\nFinally, in \\S\\ref{dihedral} we explain how these results extend to certain families of non-abelian extensions.\n\nThroughout this section, just as in \\S\\ref{8.1}, we give ourselves a fixed odd prime $p$ and isomorphism of fields $j:\\CC\\cong\\CC_p$ (explicit mention of which we usually omit), a finite set $S$ of places of $k$ with\n\\[ S_k^\\infty\\cup S_k^p\\cup  S_k^F \\cup S_k^A\\subseteq S\\]\nand a fixed ordered $k$-basis $\\{\\omega_j'\\}_{j\\in[d]}$ of $H^0(A^t,\\Omega^1_{A^t})$ with associated classical period $\\Omega_A^{F\/k}$.\n\nExcept in \\S\\ref{dihedral} we shall always assume in this section that $F\/k$ is abelian. In addition, we shall always assume that $p$ is chosen so that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\n\n\n\n\\subsection{A Mazur-Tate conjecture for higher derivatives}\\label{mtchd}\n\nIn this section we formulate a Mazur-Tate type conjecture for higher derivatives of Hasse-Weil-Artin $L$-series. We then show that, under the hypotheses listed in \\S \\ref{tmc}, this conjecture would follow from the validity of ${\\rm BSD}(A_{F\/k})$.\n\n\\subsubsection{\n\nWe first quickly recall the construction of canonical height pairings of Mazur and Tate \\cite{mt0}.\n\nTo do this we fix a subgroup $J$ of $G$ and set $E := F^J$.\nWe recall that the subgroups of `locally-normed elements' of $A(E)$ and $A^t(E)$ respectively are defined by setting \\begin{equation}\\label{localnorms}U_{F\/E}:=\\bigcap_v \\bigl(A(E)\\cap N_{F_w\/E_v}(A(F_w))\\bigr),\\,\\,\\,\\,U^t_{F\/E}:=\\bigcap_v \\bigl(A^t(E)\\cap N_{F_w\/E_v}(A^t(F_w))\\bigr).\\end{equation}\nHere each intersection runs over all (finite) primes $v$ of $E$ and $w$ is a fixed prime of $F$ above $v$. In addition, $N_{F_w\/E_v}$ denotes the norm map of $F_w\/E_v$ and each intersection of the form $A(E)\\cap N_{F_w\/E_v}(A(F_w))$, resp. $A^t(E)\\cap N_{F_w\/E_v}(A^t(F_w))$, takes place inside $A(E_v)$, resp. $A^t(E_v)$.\n\nWe recall from Lemma \\ref{useful prel}(i) that each of the expressions displayed in (\\ref{localnorms}) is in general a finite intersection of subgroups of $A(E)$, resp. of $A^t(E)$, and that the subgroups $U_{F\/E}$ and $U^t_{F\/E}$ have finite index in $A(E)$ and $A^t(E)$ respectively.\n\nWe note for later use that, whenever $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc}, then Proposition \\ref{explicitbkprop}(ii) (together with the duality of these hypotheses) implies that $$U_{F\/E,p}:=\\ZZ_p\\otimes_{\\ZZ}U_{F\/E}\\,\\,\\text{ and }\\,\\,U^t_{F\/E,p}:=\\ZZ_p\\otimes_{\\ZZ}U^t_{F\/E}$$ are equal to $A(E)_p$ and to $A^t(E)_p$ respectively (for every given subgroup $J$ of $G$).\n\nIn general, Mazur and Tate \\cite{mt0} construct, by using the theory of biextensions, a canonical height pairing \\begin{equation}\\label{tanpairing}\\langle\\,,\\rangle^{\\rm MT}_{F\/E}:U^t_{F\/E}\\otimes_\\ZZ U_{F\/E}\\to J.\\end{equation}\nThis pairing will be a key ingredient of our conjectural congruence relations. To formulate our conjecture we must first describe how to make reasonable choices of points on which to evaluate the Mazur-Tate pairing.\n\n\\begin{definition}\\label{separablepair}{\\em Fix a subgroup $J$ of $G$ and set $E:=F^J$. We define a `$p$-separable choice of points of $A$ for $F\/E$ of rank $(a,a')$' (with $a'\\geq a\\geq 0$) to be a pair $(\\mathcal{Y},\\mathcal{Y}')$ chosen as follows.\n\nLet $\\mathcal{Y}= \\{y_i: i \\in [a]\\}$ be any ordered finite subset of $A(F)_p$ that generates a free $\\ZZ_p[G]$-direct-summand $Y$ of $A(F)_p$ of rank equal to $|\\mathcal{Y}|=a$. Then $\\Tr_J(Y)=Y^J$ is a $\\ZZ_p[G\/J]$-direct-summand of $A(E)_p$\n\nWe then let $$\\mathcal{Y}'=\\Tr_J(\\mathcal{Y})\\cup\\{w_i:a<i\\leq a'\\}$$ be any ordered finite subset of $U_{F\/E,p}$, containing $\\Tr_J(\\mathcal{Y}):=\\{\\Tr_J(y_i):i \\in [a]\\}$, that generates a free $\\ZZ_p[G\/J]$-direct-summand $Y'$ of $A(E)_p$ of rank equal to $|\\mathcal{Y}'|=a'\\geq a$.}\n\\end{definition}\n\n\n\n\n\n\n\n\\begin{remarks}\\label{p-separable}{\\em\nTo identify $p$-separable choices of points as above, one can proceed as follows. Write $G=P\\times H$ where $P$ is the Sylow $p$-subgroup of $G$. Then the argument of \\cite[Prop. 2.1]{burns nagoya} shows that a finite subset $\\mathcal{Y}$ of $A(F)_p$ generates a free $\\ZZ_p[G]$-direct-summand $Y$ of $A(F)_p$ of rank equal to $|\\mathcal{Y}|=a$ if and only if the images in $A(F^P)_p\/p\\cdot A(F^P)_p$ of the elements $$h\\cdot\\sum_{g\\in P}g(y),$$ for each $h\\in H$ and each $y\\in\\mathcal{Y}$, are linearly independent over $\\mathbb{F}_p$. An analogous observation applies to the choice of the set $\\mathcal{Y}'$ in a $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$.}\n\\end{remarks}\n\nIn the rest of \\S \\ref{mrsconjecturesection}, for any $\\ZZ_p[G]$-module $M$ we write $M^*$ for the linear dual $\\Hom_{\\ZZ_p}(M,\\ZZ_p)$, endowed with the natural contragredient action of $G$.\n\n\nFor given integers $a'\\geq a\\geq 0$ and a given $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$, we always use the following notation.\nWe fix a decomposition $A(F)_p=Y\\oplus Z$. Since $A(F)_p^*=Y^*\\oplus Z^*$ and\n$Y^*$ is $\\ZZ_p[G]$-free of rank $a$ we may and will denote by $y_i^*$ the element of $Y^*\\subseteq A(F)_p^*$ which maps $y_i$ to 1 and $y_j$ to 0 for $j\\neq i$.\n\nIn a similar way, to each element $y'\\in\\mathcal{Y}'$ we associate a canonical dual element $(y')^*$ of $A(E)_p^*$.\n\nFix now any maximal subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$ and also set $x_\\bullet^J:= \\{\\Tr_J(x):x\\in x_\\bullet\\}\\subseteq A^t(E_p)^\\wedge_p$.\nBy using the isomorphism ${\\rm ht}_{A_{F\/k}}^{a}$ considered in (\\ref{athpower}) as well as the analogous isomorphism ${\\rm ht}_{A_{E\/k}}^{a'}$, one then obtains elements\n\\[ \\eta_{\\mathcal{Y},x_\\bullet} := \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) \\cdot {\\rm ht}_{A_{F\/k}}^{a}(\\wedge_{y\\in\\mathcal{Y}}y^*)\\in\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G]}^aA^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J} := \\frac{L^{(a')}_{S}(A_{E\/k},1)}{\\Omega_A^{E\/k}\\cdot w_{E\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{E\/k}}(x_\\bullet^J)\\cdot {\\rm ht}_{A_{E\/k}}^{a'}(\\wedge_{y'\\in\\mathcal{Y}'}(y')^*)\\in\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p.\\]\n\n\nFor each $a<i\\leq a'$ we may also define composite homomorphisms of $\\ZZ_p[G\/J]$-modules\n\\begin{equation}\\label{htan}{\\rm ht}^{{\\rm MT}}_{w_i}:U^t_{F\/E,p}\\to \\ZZ_p\\otimes_\\ZZ J\\cong I_p(J)\/I_p(J)^2\\to\\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2\\end{equation} by setting\n\\begin{equation}\\label{minussignMT}{\\rm ht}^{{\\rm MT}}_{w_i}(P):=-\\langle P,w_i\\rangle^{\\rm MT}_{F\/E}.\\end{equation} Here $\\mathcal{I}_p(J)$ denotes the ideal of $\\ZZ_p[G]$ generated by the augmentation ideal $I_p(J)$ of $\\ZZ_p[J]$, the isomorphism maps each $g$ in $J$ to the class of $g-1$ and the last arrow is induced by the inclusion $I_p(J)\\subseteq\\mathcal{I}_p(J)$. (We also note that the occurrence of the minus sign in (\\ref{minussignMT}) will be clarified by the result of Theorem \\ref{thecomptheorem} below).\n\n\n\n\\subsubsection{\n\nFor any finitely generated $\\ZZ_p[G]$-module $M$ and any non-negative integer $r$ we write $\\bigcap_{\\ZZ_p[G]}^rM$ for the $r$-th `Rubin lattice' of $M$ inside $\\QQ_p\\cdot\\bigwedge_{\\ZZ_p[G]}^rM$, as defined originally by Rubin \\cite{rub} (see also \\cite[\\S 4.2]{bks}).\n\nThen the homomorphisms ${\\rm ht}^{{\\rm MT}}_{w_i}$ can be combined as in \\cite[\\S3.4.1]{bst} to give a canonical homomorphism\n\\begin{equation}\\label{canbecombined}\n{\\rm ht}_{\\mathcal{Y}'}^{\\rm MT}: {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} U^t_{F\/E,p} \\longrightarrow ({\\bigcap}_{\\ZZ_p[G\/J]}^{a} U^t_{F\/E,p})\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\n\\end{equation}\n\nFollowing an original idea of Darmon, we define the `norm operator'\n$$\\mathcal{N}_J : {\\bigcap}_{\\ZZ_p[G]}^a A^t(F)_p \\longrightarrow ({\\bigcap}_{\\ZZ_p[G]}^a A^t(F)_p) \\otimes_{\\ZZ_p} \\ZZ_p[J]\/I_p(J)^{1+a'-a}$$\nto be the homomorphism induced by sending each $x$ to $\\sum_{\\sigma \\in J} \\sigma (x) \\otimes \\sigma^{-1}$.\n\nWe also use \\cite[Prop. 4.12]{bks} to obtain a canonical injective homomorphism of $\\ZZ_p[G]$-modules\n\\begin{multline*} \\nu_J:({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(E)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}\\\\ \\rightarrow ({\\bigcap}_{\\ZZ_p[G]}^{a} A^t(F)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\\end{multline*}\n\n\n\n\nWe can now state the central conjecture of this section.\n\n\\begin{conjecture}\\label{mrsconjecture} Assume that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\nFix any subgroup $J$ of $G$ (with $E := F^J$) and any $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$. Then one has\n\\[ \\eta_{\\mathcal{Y},x_\\bullet}\\in {\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'}U^t_{F\/E,p}\\]\nand in $({\\bigcap}_{\\ZZ_p[G]}^{a} A^t(F)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$ there is an equality\n\\[ \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{\\rm MT}(\\eta_{\\mathcal{Y}',x_\\bullet^J})).\\]\n\\end{conjecture}\n\n\\begin{remark}\\label{explicit pairings}{\\em\nLet $k = \\QQ$, $J = G$, $a=0$, $a' = {\\rm dim}_\\QQ(A(k))$ and $\\mathcal{Y}=\\{y_i: i \\in [a']\\}$ be any $\\ZZ$-basis of $\\Hom_\\ZZ(A(k),\\ZZ)$. \nThen, after taking account of Remark \\ref{new add}, one finds that the congruence in Conjecture \\ref{mrsconjecture} specialises in this case to refine and extend the conjectures formulated (for certain elliptic curves $A$ and abelian fields $F$) by Mazur and Tate in \\cite{mt} and also incorporates the main conjecture of Darmon \\cite{darmon0}. }\\end{remark}\n\n\\begin{remark}\\label{indeptpoints\n{\\em Fix any set $x_\\bullet$, subgroup $J$ of $G$ and integers $a'\\geq a\\geq 0$ as in Conjecture \\ref{mrsconjecture}.\nThen the general result of \\cite[Prop. 4.17]{bks} can be used to show that the validity of Conjecture \\ref{mrsconjecture} is independent of the $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$.}\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\nThe following result establishes conditions under which Conjecture \\ref{mrsconjecture} is a consequence of ${\\rm BSD}(A_{F\/k})$.\n\nThe proof of this result will be given in \\S \\ref{mrsproof} below.\n\n\\begin{theorem}\\label{rbsdimpliesmt} Fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc}. Assume also that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and that Hypothesis \\ref{hypbd} below is satisfied.\n\nThen if ${\\rm BSD}(A_{F\/k})$ is valid, so is Conjecture \\ref{mrsconjecture}.\n\\end{theorem}\n\n\n\n\\begin{remarks}\\label{mrstheorem}{\\em Each of the hypotheses (H$_1$)-(H$_5$) as well as Hypothesis \\ref{hypbd} is satisfied widely  (see Remarks \\ref{satisfying H} and \\ref{rmkbd} respectively). However, without assuming any of these hypotheses it is also possible to deduce from ${\\rm BSD}(A_{F\/k})$ less explicit versions of the congruence relations that are predicted by Conjecture \\ref{mrsconjecture}.\n\nTo be more precise we fix an odd prime number $p$ for which neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and set $$C_{S,x_\\bullet}:={\\rm SC}_S(A_{F\/k};\\langle x_\\bullet\\rangle_{\\ZZ_p[G]},H_\\infty(A_{F\/k})_p).$$ Then, without any additional hypotheses on $A$, $F\/k$ or $p$, the same argument as used to prove Theorem \\ref{rbsdimpliesmt} shows ${\\rm BSD}(A_{F\/k})$ implies that\n\\[ \\eta_{\\mathcal{Y},x_\\bullet}\\in {\\bigcap}_{\\ZZ_p[G]}^{a}H^1(C_{S,x_\\bullet})\\subseteq{\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'}H^1(C_{S,x_\\bullet})^J\\subseteq{\\bigcap}_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p\\]\nand that in $\\left({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(F)_p\\right)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$ one has an equality\n\\[ \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{C_{S,x_\\bullet}}(\\eta_{\\mathcal{Y}',x_\\bullet^J})).\\]\nHere ${\\rm ht}_{\\mathcal{Y}'}^{C_{S,x_\\bullet}}$ is the `Bockstein homomorphism' (\\ref{htcy}) associated to the Nekov\\'a\\v r-Selmer complex $C_{S,x_\\bullet}$  in the course of the proof of Theorem \\ref{rbsdimpliesmt} below. In this context we recall also from Remark \\ref{mrselmer} that the cohomology group $H^1(C_{S,x_\\bullet})$ may be interpreted as a Selmer group in the sense of Mazur and Rubin \\cite{MRkoly}. }\n\\end{remarks}\n\n\n\n\n\\subsection{Cyclic congruence relations}\\label{cycliccongssection}\n\nIn this section we make the congruences in Conjecture \\ref{mrsconjecture} more explicit in the case of certain cyclic extensions.\n\nWe always assume the hypotheses (H$_1$)-(H$_6$) listed in \\S\\ref{tmc} and, motivated by\nRemarks \\ref{emptysets} and \\ref{new remark SC}, we further strengthen the hypothesis (H$_3$) by assuming that $p$ is unramified in $F$.\n\nIn this situation, the argument that is used in the proof of Theorem \\ref{rbsdimpliesmt} below may be applied to the classical Selmer complex.\n\nIn this way, the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) implies (via Theorem \\ref{bk explicit}) congruence relations in the spirit of Conjecture \\ref{mrsconjecture} that circumvent the choice of semi-local points and apply directly to the elements $\\calL^*_{A,F\/k}$ and $\\calL^*_{A,E\/k}$ as defined in (\\ref{bkcharelement}).\n\nWe shall also now assume that the following additional conditions on $A$, $F\/k$ and  $p$  are satisfied:\n\n\\begin{itemize}\n\\item[(H$_7$)] $F\/k$ is cyclic of order $p^m$ (for some natural number $m$);\n\\item[(H$_8$)] neither $A(k)$ nor $A^t(k)$ has a point of order $p$;\n\\item[(H$_{9}$)] for every field $L$ with $k\\subseteq L\\subset F$ and $L \\not=F$ the group $\\sha(A_L)[p^\\infty]$ vanishes;\n\\item[(H$_{10}$)] the group $H^1({\\rm Gal}(k(A[p^{m}])\/k),A[p^{m}])$ vanishes.\n\\end{itemize}\n\nWe impose the hypothesis (H$_{10}$) since it recovers Hypothesis \\ref{hypbd} below in this specific setting and hence will allow us to apply the result of Theorem \\ref{rbsdimpliesmt}. We also note that this hypothesis is satisfied widely (see Remark \\ref{rmkbd}).\n\n\n\n\nOn the other hand, the hypotheses (H$_7$), (H$_8$) and (H$_9$) are introduced since they allow us to apply the main result of Yakovlev \\cite{yakovlev} in order to deduce that $A(F)_p$ and $A^t(F)_p$ are (isomorphic) permutation $\\ZZ_p[G]$-modules and hence that it is possible to fix an optimal collection of points as in Definition \\ref{separablepair}.\n\n\nThen, by combining such a choice of points with the methods of \\cite{bleymc,dmc}, one can make fully explicit the congruence relations that are implied by ${\\rm BSD}_p(A_{F\/k})$(iv).\n\nIn order to stress the key ideas we shall focus on the simple cases $m=1$ and $m=2$. In particular, in the case $m=1$ we will show that the validity of  ${\\rm BSD}_p(A_{F\/k})$(iv) is under certain natural hypotheses equivalent to the validity of a single explicit congruence relation.\n\n\\subsubsection{\nWe first assume that $m=2$. In this case we denote by $J$ the subgroup of $G$ of order $p$ and set $E := F^J$ and $\\Gamma := G\/J$. For each $i\\in \\{0,1,2\\}$ we also write $F_i$ for the intermediate field of $F\/k$ with $[F_i:k] = p^i$ and set $\\Gamma_i:= G_{F_i\/k}$ (so that $\\Gamma_2 = G$ and $\\Gamma_1 = \\Gamma$).\n\nThen, under the given hypotheses, Yakovlev's result implies (via the general approach of \\cite[\\S7.2.5]{ltav}) that for each $i \\in \\{0,1,2\\}$ there exists a non-negative integers $m_i$ and subsets $\\calP_{(i)}=\\{P_{i,j}\\mid j \\in [m_i]\\}$ of $A(F_i)_p$ and $\\calP^t_{(i)}=\\{P^t_{i,j}:j\\in[m_i]\\}$ of $A^t(F_i)_p$ such that the $\\ZZ_p[\\Gamma_i]$-modules generated by each of the points $P_{i,j}$ and $P^t_{i,j}$ are free of rank one and there are direct sum decompositions of $\\ZZ_p[G]$-modules\n\\begin{equation}\\label{global points}\n A(F)_p = \\bigoplus_{i=0}^{i=2}\\bigoplus_{j=1}^{j=m_i}\\ZZ_p[\\Gamma_i]\\cdot P_{i,j}\\end{equation}\n\n and\n\\begin{equation}\\label{global points2} A^t(F)_p = \\bigoplus_{i=0}^{i=2}\\bigoplus_{j=1}^{j=m_i}\\ZZ_p[\\Gamma_i]\\cdot P^t_{i,j},\n\\end{equation}\n\nso that the integers $m_i$ are uniquely determined by the equalities\n$${\\rm rk}(A_k)=m_0+m_1+m_2,\\,\\,\\,\\,\\,\\,{\\rm rk}(A_E)=m_0+p(m_1+m_2),\\,\\,\\,\\,\\,\\,{\\rm rk}(A_F)=m_0+pm_1+p^2m_2.$$\nFor a full proof of these facts see \\cite[Prop. 2.2]{bleymc}.\n\nWe then obtain `regulator matrices' \\begin{align*}R_{F}(\\calP_{\\bullet}):=\\, &h_{F\/k}(\\calP^t_{(2)};\\,\\calP_{(2)})\\in {\\rm M}_{m_2}(\\RR[G]),\\\\\\ R_{E}(\\calP_{\\bullet}):=\\, &h_{E\/k}(\\calP^t_{(1)}\\cup\\calP_{(2)}^{t,J};\\,\\calP_{(1)}\\cup \\calP_{(2)}^J)\\in {\\rm M}_{m_1+m_2}(\\RR[\\Gamma])\\\\\nR_{k}(\\calP_{\\bullet}) := \\, &h_{k\/k}(\\calP^t_{(0)}\\cup \\calP^{t,\\Gamma}_{(1)}\\cup \\calP_{(2)}^{t,G};\\, \\calP_{(0)}\\cup \\calP^\\Gamma_{(1)}\\cup \\calP^G_{(2)}) \\in {\\rm M}_{{\\rm rk}(A_k)}(\\RR)\\end{align*}\nas in (\\ref{regulatormatrix}), where we write $\\calP_{(2)}^{t,J}$ for the set $\\{\\sum_{g \\in J}g(P_{2,j}^t)\\mid j \\in [m_i]\\}$ and define $\\calP_{(2)}^J$, $\\calP^{t,\\Gamma}_{(1)}$, $\\calP_{(2)}^{t,G}$, $\\calP^\\Gamma_{(1)}$ and $\\calP^G_{(2)}$ similarly, and in each union we order the points  lexicographically\n\nWe define an idempotent\n\\[\ne_{\\langle 2\\rangle} := \\begin{cases}1,& \\text{ if } m_0=m_1=0,\\\\ 1-e_G,& \\text{ if } m_0\\neq 0,m_1=0,\\\\ 1-e_J,& \\text{ if } m_1\\neq 0, \\end{cases}\\]\nin $\\QQ[G]$ and\n\\[ e_{\\langle 1\\rangle} := \\begin{cases}1,& \\text{ if } m_0=0,\\\\ 1-e_\\Gamma,& \\text{ if } m_0\\neq 0, \\end{cases}\n\\]\nin $\\QQ[\\Gamma]$.\n\nThen the matrix $e_{\\langle i\\rangle}\\cdot R_{F_i}(\\calP_{\\bullet})$ belongs to ${\\rm GL}_{m_2}(\\CC[G]e_{\\langle 2\\rangle})$ if $i=2$ and to  ${\\rm GL}_{m_1+ m_2}(\\CC[\\Gamma]e_{\\langle 1\\rangle})$ if $i = 1$ and so we may set\n\\begin{align}\\label{lmin}\n\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}} := \\, &({\\rm det}(e_{\\langle 2\\rangle}\\cdot R_{F}(\\calP_{\\bullet})))^{-1}\\cdot e_{\\langle 2\\rangle}\\calL^*_{A,F\/k}\\in e_{\\langle 2\\rangle}\\cdot\\CC[G]^\\times\\\\ \\notag\n\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}} := \\, &({\\rm det}(e_{\\langle 1\\rangle}\\cdot R_{E}(\\calP_{\\bullet})))^{-1}\\cdot e_{\\langle 1\\rangle}\\calL^*_{A,E\/k}\\in e_{\\langle 1\\rangle}\\cdot\\CC[\\Gamma_1]^\\times\\\\ \\notag\n\\calL^\\ast_{k,\\calP_{\\bullet}} :=\\, &{\\rm det}(R_k(\\calP_{\\bullet}))^{-1}\\cdot\\calL^*_{A,k}\\in\\RR^\\times.\\end{align\n\n\\begin{remark} {\\em This notation is motivated by fact that the decompositions in (\\ref{global points}) and (\\ref{global points2}) imply $e_{\\langle 2\\rangle}$ and $e_{\\langle 1\\rangle}$ are equal to the idempotents $e_{F,m_2}$ and $e_{E,m_1+m_2}$ that occur in \\S \\ref{statementstructure} with respect to $A$ and the extensions $F\/k$ and $E\/k$ respectively and so the terms $e_{\\langle 2\\rangle}\\calL^*_{A,F\/k}$ and $e_{\\langle 1\\rangle}\\calL^*_{A,E\/k}$ involve the values at $z=1$ of the $m_1$-th and $(m_1+m_2)$-th derivatives of Hasse-Weil-Artin $L$-series. We further recall from Remark \\ref{bsdinvariants} that the term $\\calL^*_{A,k}$ can be explicitly described in terms of invariants occurring in the classical Birch and Swinnerton-Dyer conjecture for $A$ over $k$.}\\end{remark}\n\n\nBy using the approach of the second author in \\cite{dmc}, one finds that Theorem \\ref{rbsdimpliesmt} implies the explicit congruences in the following result. We do not give the proof of this result but rather refer the reader to loc. cit.\n\nIn this result we use the canonical identification\n$$\\mathcal{I}_p(J)^{m_1}\/\\mathcal{I}_p(J)^{m_1+1}\\cong\\ZZ_p[\\Gamma]\\otimes_{\\ZZ_p}I_p(J)^{m_1}\/I_p(J)^{m_1+1}.$$\n\n\n\n\\begin{theorem}\\label{cyclicthm}\nFix an odd prime number $p$ that is unramified in $F$ and such that the data $A, F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_{10}$) with $m=2$. Then if ${\\rm BSD}_p(A_{F\/k})$(iv) is valid one has\n\\begin{equation*}\\label{totallyexplicit}\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}\\equiv  (-1)^{m_1}\\cdot\\bigl(\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}}\\otimes {\\rm det}\\bigl(\\langle \\calP_{(1)}^t,\\calP_{(1)}\\rangle^{\\rm MT}_{F\/E}\\bigr)\\bigr)\\,\\,\\,\\,\\,({\\rm mod}\\,\\,\\mathcal{I}_p(J)^{m_1+1})\\end{equation*}\nand\n\\begin{equation*}\\label{totallyexplicit}\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}\\equiv (-1)^{m_0+m_1}\\cdot\\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot {\\rm det}\\bigl(\\langle\\calP_{(0)}^t\\cup \\calP_{(1)}^{t,\\Gamma},\\calP_{(0)}\\cup \\calP^\\Gamma_{(1)}\\rangle^{\\rm MT}_{F\/k}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G)^{m_0+m_1+1}).\\end{equation*}\n\n\\end{theorem}\n\n\n\\begin{remark}\\label{changeofptsremark}{\\em\nThe Wedderburn components of the elements $\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}$ and $\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}}$  recover the products that are studied by Fearnley and Kisilevsky \\cite{kisilevsky,kisilevsky2}. In particular, the numerical computations in loc. cit., as well as the numerical computations performed by Bley and the second author in \\cite{bleymc}, can be interpreted as supporting evidence for ${\\rm BSD}(A_{F\/k})$.\n}\n\\end{remark}\n\n\\subsubsection{\n\n\nWe now assume that $m=1$. In this case Yakovlev's result implies, under the given hypotheses, that for each $i \\in \\{0,1\\}$ there exists a non-negative $m_i$ and subsets $\\calP_{(i)}$ of $A(F_i)_p$ and $\\calP^t_{(i)}$ of $A^t(F_i)_p$ for which there are direct sum decompositions of $\\ZZ_p[G]$-modules\n\\[ A(F)_p = \\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P_{0,j}\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P_{1,j}\\]\nand\n\\[ A^t(F)_p = \\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P^t_{0,j}\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P^t_{1,j}.\\]\n\nBy using such a choice of points as well as the idempotent $$e_{F,m_1}:= \\begin{cases}1,& \\text{ if } m_0=0,\\\\ 1-e_G,& \\text{ if } m_0\\neq 0, \\end{cases}$$ one obtains elements $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ of $e_{F,m_1}\\cdot\\CC[G]^\\times$ and $\\calL^*_{k,\\calP_\\bullet}$ of $\\RR^\\times$ exactly as in (\\ref{lmin}).\n\n\nIn this special case we obtain the following refinement of Theorem \\ref{cyclicthm}. This result will be proved in \\S\\ref{proofmequals1} below.\n\n\\begin{theorem}\\label{mequals1} Fix an odd prime number $p$ that is unramified in $F$ and such that the data $A, F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_{10}$) with $m=1$. Then the following claims are valid.\n\n\\begin{itemize}\\item[(i)] If ${\\rm BSD}_p(A_{F\/k})$(iv) is valid one has\n\\begin{equation*}\\calL^{(m_1)}_{F,\\calP_\\bullet}\\equiv (-1)^{m_0}\\cdot\\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot {\\rm det}\\bigl(\\langle\\calP_{(0)}^t,\\calP_{(0)}\\rangle^{\\rm MT}_{F\/k}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G)^{m_0+1}).\\end{equation*}\n\n\\item[(ii)] Assume that the $p$-part of the Birch and Swinnerton-Dyer Conjecture is valid for $A$ over both $k$ and $F$ and that the group $\\sha(A_F)[p^\\infty]$ vanishes.\n\nThen ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if the element $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ belongs to $I_p(G)^{m_0}$ and satisfies the displayed congruence in claim (i).\\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\\label{equivalenceeasy}{\\em The argument that we use to prove Theorem \\ref{mequals1} can be extended, under suitable hypotheses, to the setting of cyclic extensions of arbitrary $p$-power degree. In this way one obtains an explicit family of congruence relations (of the same general form as in Theorem \\ref{mequals1}(i)) whose validity is equivalent to that of ${\\rm BSD}_p(A_{F\/k})$(iv) whenever the $p$-part of the Birch and Swinnerton-Dyer Conjecture holds for $A$ over each intermediate field of $F\/k$. This will be explained in forthcoming work of Bley and the second author.\n}\\end{remark}\n\n\\begin{remark}\\label{bleyexamples rem}{\\em Theorems \\ref{cyclicthm} and \\ref{mequals1} both illustrate the fact that, unless the relevant Mordell-Weil groups are projective as Galois modules, the equality ${\\rm BSD}_p(A_{F\/k})$(iv) encodes a family of congruence relations between the values at $z=1$ of higher derivatives of Hasse-Weil-Artin $L$-series that also involve (non-trivial) Mazur-Tate regulators. As far as we are aware, no such congruence relations have previously been identified, let alone verified, in the literature.}\\end{remark}\n\n\\begin{examples}\\label{bleyexamples} {\\em In view of the last remark, we are very grateful to Werner Bley for providing us with the following list of examples for which he has been able to numerically verify the conjectural equality ${\\rm BSD}_p(A_{F\/k})$(iv) by means of the congruence relation in Theorem \\ref{mequals1}. The full details of these computations will appear in future work.\n\nThroughout this list, $A$ is the elliptic curve with indicated Cremona reference, $k=\\QQ$, $p=3$ and $F$ is the unique degree three Galois extension of $\\QQ$ that is contained in $\\QQ(\\zeta_\\ell)$ for any of the indicated primes $\\ell$. In addition, one always has ${\\rm rk}(A_F)={\\rm rk}(A_\\QQ)=1$.\n\n\\begin{itemize}\\item[-] $A$ is the curve $37a1$ (with Weierstrass equation $y^2+y=x^3-x$) and $\\ell\\in\\{13,19\\}.$\n\\item[-] $A$ is  $43a1$ ($y^2+y=x^3+x^2)$ and $\\ell\\in\\{7,13,37\\}.$\n\\item[-] $A$ is $53a1$ ($y^2+xy+y=x^3-x^2$) and $\\ell\\in\\{13,19,31,43\\}.$\n\\item[-] $A$ is $58a1$ ($y^2+xy=x^3-x^2-x+1$) and $\\ell\\in\\{7,13,19,31,43\\}.$\n\\item[-] $A$ is $61a1$ ($y^2+xy=x^3-2x+1$) and $\\ell\\in\\{7,13,43\\}.$\n\\item[-] $A$ is $65a1$ ($y^2+xy=x^3-x$) and $\\ell\\in\\{19,37,43\\}.$\n\\item[-] $A$ is $65a2$ ($y^2+xy=x^3+4x+1$) and $\\ell\\in\\{19,37,43\\}.$\n\\item[-] $A$ is $77a1$ ($y^2+y=x^3+2x$) and $\\ell\\in\\{19,37\\}.$\n\\end{itemize}\n}\n\\end{examples}\n\n\\subsubsection{}\\label{proofmequals1}\nIn this section we prove Theorem \\ref{mequals1}.\n\nAt the outset we set $C:={\\rm SC}_p(A_{F\/k})$ and note that, under the hypotheses of Theorem \\ref{mequals1}, Theorem \\ref{bk explicit} (and Remark \\ref{emptysets}) implies that ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if the element\n$$\\xi:=\\delta_{G,p}(\\calL^*_{A,F\/k})-\\chi_{G,p}(C,h_{A,F})$$\nof $K_0(\\ZZ_p[G],\\CC_p[G])$ vanishes.\n\nIt is a routine exercise to deduce claim (i) of Theorem \\ref{mequals1} from this fact by combining Theorem \\ref{thecomptheorem} below with the approach used by the second author to prove \\cite[Cor. 3.5]{dmc} (and so we leave the details to the reader).\n\nTo prove claim (ii) we assume that the $p$-part of the Birch and Swinnerton-Dyer Conjecture holds for $A$ over both $k$ and $F$ and that $\\sha(A_F)[p^\\infty]$ vanishes. We also assume that $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ belongs to $I_p(G)^{m_0}$ and satisfies the congruence displayed in claim (i), and then proceed to prove that $\\xi$ vanishes.\n\nThe first key step is to explicitly compute the second term that occurs in the definition of $\\xi$ and to do this we  use the canonical identifications\n\\[ r_1:A^t(F)_p\\to H^1(C)\\,\\,\\text{ and }\\,\\, r_2:H^2(C)\\to \\Sel_p(A_F)^\\vee\\stackrel{\\sim}{\\to}A(F)_p^*\\]\nthat are described in Proposition \\ref{explicitbkprop}(iv). Here the second arrow in the definition of $r_2$ is the canonical map and so is bijective since $\\sha(A_F)[p^\\infty]$ is assumed to vanish.\n\nThese identifications imply that the complex $C$ corresponds to a unique element $\\varepsilon = \\varepsilon_{C,r_1,r_2}$ of the Yoneda Ext-group ${\\rm Ext}^2_{\\ZZ_p[G]}(A(F)_p^*,A^t(F)_p)$.\n\nFor each pair $(i,j)$ with $i\\in\\{0,1\\}$ and $j\\in[m_i]$ we define a dual point $P_{i,j}^*$ of $A(F)_p^*$ in the natural way and obtain a direct sum decomposition of $\\ZZ_p[G]$-modules\n$$A(F)_p^*=\\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P_{0,j}^*\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P_{1,j}^*$$\nas in \\cite[Lem. 4.1]{bleymc}.\n\nWe also fix a generator $\\sigma$ of $G$ and a free $\\ZZ_p[G]$-module $P:=\\bigoplus_{i,j}\\ZZ_p[G]b_{i,j}$ of rank $m_0+m_1$ and consider the exact sequence\n\\begin{equation}\\label{syzygy}0\\to A^t(F)_p\\stackrel{\\iota}{\\to}P\\stackrel{\\Theta}{\\to}P\\stackrel{\\pi}{\\to}A(F)_p^*\\to 0,\\end{equation} where we set $\\iota(P^t_{0,j})=\\Tr_G(b_{0,j})$, $\\iota(P^t_{1,j})=b_{1,j}$, $\\Theta(b_{i,j})=(\\sigma^{p^i}-1)b_{i,j}$ and $\\pi(b_{i,j})=P^*_{i,j}$.\n\nThis sequence defines a canonical isomorphism\n\\begin{equation}\\label{computeext}{\\rm Ext}^2_{\\ZZ_p[G]}(A(F)_p^*,A^t(F)_p)\\cong{\\rm End}_{\\ZZ_p[G]}(A^t(F)_p)\/\\iota_*\\bigl(\\Hom_{\\ZZ_p[G]}(P,A^t(F)_p)\\bigl),\\end{equation}\nwhere $\\iota_*$ denotes composition with $\\iota$.\n\nThen, since Proposition \\ref{explicitbkprop}(ii) implies $C$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ the results of \\cite[Lem. 4.2 and Lem. 4.3]{bleymc} imply that there exists an automorphism $\\phi$ of the $\\ZZ_p[G]$-module $A^t(F)_p$ that represents the image of $\\varepsilon$ under (\\ref{computeext}) and also fixes the element $P^t_{1,j}$ for every $j$ in $[m_1]$.\n\nIt follows that the exact sequence\n\\begin{equation}\\label{yonedarep}0\\to A^t(F)_p\\stackrel{\\iota\\circ\\phi^{-1}}{\\to}P\\stackrel{\\Theta}{\\to}P\\stackrel{\\pi}{\\to}A(F)_p^*\\to 0\\end{equation}\nis a representative of the extension class $\\varepsilon$.\n\nIn particular, for any choice of $\\CC_p[G]$-equivariant splittings\n$$s_1:\\CC_p\\cdot P\\to \\CC_p\\cdot A^t(F)_p\\oplus\\CC_p\\cdot\\im(\\Theta)$$\nand\n$$s_2:\\CC_p\\cdot P\\to \\CC_p\\cdot A(F)^*_p\\oplus\\CC_p\\cdot\\im(\\Theta)$$\nof the scalar extensions of the canonical exact sequences\n$$0\\to A^t(F)_p\\stackrel{\\iota}{\\to}P\\stackrel{\\Theta}{\\to}\\im(\\Theta)\\to 0$$\nand\n$$0\\to\\im(\\Theta)\\to P\\stackrel{\\pi}{\\to}A(F)_p^*,$$\nthe definition of non-abelian determinants implies that\n\\begin{align}\\label{separatingstuff}&-\\chi_{G,p}(C,h_{A,F})\\\\\n=&-\\delta_{G,p}({\\det}_{\\CC_p[G]}(s_2^{-1}\\circ(h_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ((\\CC_p\\cdot\\phi)\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_1))\\notag\\\\\n=&\\,\\delta_{G,p}({\\det}_{\\CC_p[G]}(s_1^{-1}\\circ(h^{-1}_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_2))\\notag\\\\\n&\\hskip 1.5truein +\\delta_{G,p}({\\det}_{\\QQ_p[G]}((\\QQ_p\\cdot\\phi^{-1})\\oplus{\\rm id}_{\\QQ_p\\cdot\\im(\\Theta)}))\\notag.\\end{align}\n\nIn addition, by an explicit computation as in \\cite[Prop. 4.4]{bleymc} one shows that\n\\begin{align}\\label{theregulatorterm}&{\\det}_{\\CC_p[G]}(s_1^{-1}\\circ(h^{-1}_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_2)\\\\\n=\\, &\\det(e_{F,m_1}\\cdot h_{F\/k}(\\calP^t_{(1)};\\calP_{(1)}))^{-1}\\cdot(\\sigma-1)^{-m_0}\\notag\\\\\n&\\hskip 1truein +(1-e_{F,m_1})\\cdot\\det(h_{k\/k}(\\calP^t_{(0)}\\cup\\calP^{t,G}_{(1)};\\calP_{(0)}\\cup\\calP^{G}_{(1)}))^{-1}.\\notag\\end{align}\n\nTo compute the second term that occurs in the final equality of (\\ref{separatingstuff}) we fix, as we may, any elements $\\Psi_{(0,j),(0,k)}$ of $\\ZZ_p[G]$ with the property that\n\\begin{equation}\\label{matrixPsi}\\phi^{-1}(P^t_{0,k})=\\sum_{j\\in[m_0]}\\Psi_{(0,j),(0,k)}(P^t_{0,j}).\\end{equation}\nThen the chosen properties of the representative $\\phi$ of $\\varepsilon$ fixed above imply that\n\\begin{equation}\\label{leftovertermMT}{\\det}_{\\QQ_p[G]}((\\QQ_p\\cdot\\phi^{-1})\\oplus{\\rm id}_{\\QQ_p\\cdot\\im(\\Theta)})=\ne_{F,m_1}+(1-e_{F,m_1})\\cdot\\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr).\\end{equation}\n\nThe equalities (\\ref{separatingstuff}), (\\ref{theregulatorterm}) and (\\ref{leftovertermMT}) now combine with the definition of $\\xi$ to imply that\n $\\xi=\\delta_{G,p}(\\mathcal{L})$ with\n\n\\[ \\mathcal{L} := \\calL_{F,\\calP_\\bullet}^{(m_1)}\\cdot(\\sigma-1)^{-m_0}+(1-e_{F,m_1})\\cdot\\calL^*_{k,\\calP_\\bullet}\n\\cdot\\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr).\\\n\nWe shall use this description to show $\\xi$ belongs to the finite group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$, or equivalently that $\\mathcal{L}$ belongs to the unit group of the maximal $\\ZZ_p$-order $\\mathcal{M}_p(G)$ in $\\QQ_p[G]$.\n\nTo do this we note first that the assumed containment $\\calL_{F,\\calP_\\bullet}^{(m_1)}\\in I_{G,p}^{m_0}$ and validity of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $k$ combine to imply $\\mathcal{L}$ belongs to $\\QQ_p[G]^\\times$ and hence that $\\xi$ belongs to\n$K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nNext we recall from the general result of \\cite[Thm. 4.1]{ewt} that, since $G$ has order $p$, the group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$ is the kernel of the homomorphism\n\\[ K_0(\\ZZ_p[G],\\QQ_p[G]) \\xrightarrow{(\\rho,q)}  K_0(\\ZZ_p,\\QQ_p)\\oplus K_0(\\ZZ_p,\\QQ_p)\\]\nwhere $\\rho$ is the canonical restriction of scalars map and $q$ is the homomorphism induced by mapping the class of a triple $(P,\\phi,Q)$ to the class of $(P^G,\\phi^G,Q^G)$.\n\nGiven this, the claimed containment $\\xi \\in K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$ follows from the fact that Remark \\ref{consistency remark}(ii) and (iii) combine to imply that the validities of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F$ and $k$ imply that  $\\rho(\\xi)$ and $q(\\xi)$ both vanish.\n\nAt this stage we know that $\\mathcal{L}$ belongs to $\\mathcal{M}_p(G)^\\times$ and hence that it belongs to $\\ZZ_p[G]^\\times$ (so that $\\xi$ vanishes) if and only if it belongs to $\\ZZ_p[G]$.\n\nNow if $m_0=0$ then $\\mathcal{L} =\\calL_{F,\\calP_\\bullet}^{(m_1)}$ and $\\ZZ_p[G]=I_p(G)^{m_0}$ and so our hypotheses imply directly that  $\\mathcal{L} \\in \\ZZ_p[G]$. In the sequel we will therefore assume that $m_0\\neq 0$, and hence that $e_{F,m_1}=1-e_G$.\n\nIn this case $\\mathcal{L}$ belongs to $\\ZZ_p[G]$ if and only if one has\n\\begin{equation*}\\label{totallyexplicit1}\\calL^{(m_1)}_{F,\\calP_\\bullet}\\cdot(\\sigma-1)^{-m_0}\\equiv \\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot \\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G))\\end{equation*}\nand so the assumed validity of the congruence in Theorem \\ref{mequals1}(i) reduces us to verifying that, for any $j$ and $k$ in $[m_0]$, one has\n\\begin{equation}\\label{PsicomputesMT}\\Psi_{(0,j),(0,k)}\\cdot(\\sigma-1)+I_p(G)^2=-\\langle P^t_{0,k},P_{0,j}\\rangle^{\\rm MT}_{F\/k}\\end{equation} in $I_p(G)\/I_p(G)^2$.\n\nTo verify these equalities we use the fact, proved in Theorem \\ref{thecomptheorem} below, that $\\langle \\,,\\rangle^{\\rm MT}_{F\/k}$\ncoincides with the inverse of the pairing induced by\n\\begin{align*}\\beta:A^t(k)_p\\to I_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p}(A(F)_p^*)_{G} \\to  I_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p}A(k)_p^*,\\end{align*}\n\nwhere the first arrow is the Bockstein homomorphism associated to the complex $C$ and the maps $r_1$ and $r_2$ (see \\S\\ref{bocksection} below), while the second arrow is induced by restriction to $A(k)_p$.\n\nNow $\\beta$ may be computed, through the representative (\\ref{yonedarep}) of the extension class $\\varepsilon$, as the connecting homomorphism that  arises when applying the snake lemma to the following commutative diagram (in which both rows and the third column are\nexact and the first column is a complex)\n\\[\\begin{CD}\n@. @. @. A^t(k)_p\\\\ @. @. @. @VV (\\iota\\circ\\phi^{-1})^{G} V  \\\\ 0 @>\n>> I_p(G)\\otimes_{\\ZZ_p[G]}P @> \\subseteq >> P @> \\Tr_{G}  >>\n P^G  @> >> 0\\\\\n@. @VV {\\rm id}\\otimes_{\\ZZ_p[G]}\\Theta V @VV \\Theta V @VV\\Theta^{G} V\\\\\n0 @>\n>> I_p(G)\\otimes_{\\ZZ_p[G]}P @> \\subseteq >> P @> \\Tr_{G} >>\n P^G  @> >> 0\\\\\n@. @VV ({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}  V \\\\ @.\nI_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p} (A(F)_p^*)_{G}.\n\\end{CD}\\]\n\nIn addition, the equality (\\ref{matrixPsi}) implies that\n$$\\iota(\\phi^{-1}(P^t_{0,k}))=\\iota(\\sum_{l\\in[m_0]} \\Psi_{(0,l),(0,k)} P^t_{0,l})=\\sum_{l\\in[m_0]} \\Psi_{(0,l),(0,k)}\\cdot\\Tr_G(b_{0,l})$$\nand so, since $$P^*_{0,l}(P_{0,j})=\\begin{cases}1,\\,\\,\\,\\,\\,\\,\\,\\,l=j,\\\\\n0,\\,\\,\\,\\,\\,\\,\\,\\,l\\neq j,\\end{cases}$$\nwe can finally use the above diagram to compute that\n\\begin{align*}-\\langle P^t_{0,k},P_{0,j}\\rangle_{F\/k}^{\\rm MT}=&\\left(({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}\\left(\\Theta(\\Psi_{(0,j),(0,k)}\\cdot b_{0,j})\\right)\\right)(P_{0,j})\\\\\n=&\\left(({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}\\left((\\sigma-1)\\Psi_{(0,j),(0,k)}\\cdot b_{0,j}\\right)\\right)(P_{0,j})\\\\\n=&\\left(\\left((\\sigma-1)+I_p(G)^2\\right)\\otimes\\Psi_{(0,j),(0,k)}\\cdot P^*_{0,j}\\right)(P_{0,j})\\\\\n=&\\Psi_{(0,j),(0,k)}\\cdot(\\sigma-1)+I_p(G)^2.\n\\end{align*}\nHere the first equality is given by Theorem \\ref{thecomptheorem}. This verifies the equalities (\\ref{PsicomputesMT}) and thus completes the proof of Theorem \\ref{mequals1}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Dihedral congruence relations}\\label{dihedral} With a view to extending the classes of extensions $F\/k$ for which the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) can be made fully explicit we consider the case that $F\/k$ is generalized dihedral of order $2p^n$.\n\nWe recall that this means the Sylow $p$-subgroup $P$ of $G$ is abelian and of index two and that the conjugation action of any lift to $G$ of the generator of $G\/P$ inverts elements of $P$. We write $K$ for the unique quadratic extension of $k$ in $F$.\n\nIn this setting we shall show that, in certain situations, the validity of ${\\rm BSD}_p(A_{F\/k})$(iv) can be checked by verifying congruences relative to the abelian extension $F\/K$.\n\nIn order to state the precise result we fix a finite Galois extension $E$ of $\\QQ$ in $\\bc$ that is large enough to ensure that, with $\\mathcal{O}$ denoting the ring of algebraic integers of $E$, there exists for each character $\\psi$ of $\\widehat{G}$ a finitely generated $\\mathcal{O}[G]$-lattice that is free over $\\mathcal{O}$ and spans a $\\bc[G]$-module of character $\\psi$.\n\nFor each $\\psi$ in $\\widehat{G}$ we recall the non-zero complex number $\\mathcal{L}^\\ast(A,\\psi)$ defined in \\S\\ref{explicit ec section}.\n\n\\begin{proposition}\\label{dihedral prop} Let $F\/k$ be generalized dihedral of degree $2p^n$ as above. Assume that $\\sha(A_F)$ is finite and that no place of $k$ at which $A$ has bad reduction is ramified in $F$. Assume also that $p$ satisfies the conditions (H$_1$)-(H$_4$) listed in \\S\\ref{tmc} and that neither $A(K)$ nor $A^t(K)$ has a point of order $p$.\n\nThen the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if the following three conditions are satisfied.\n\n\\begin{itemize}\n\\item[(i)] For every $\\psi$ in $\\widehat{G}$ and $\\omega$ in $G_\\QQ$, one has $\\mathcal{L}^\\ast(A,\\omega\\circ \\psi) = \\omega(\\mathcal{L}^\\ast(A,\\psi))$.\n\\item[(ii)] For every $\\psi$ in $\\widehat{G}$ and every prime ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$, the explicit formula for $\\mathcal{L}^\\ast(A, \\psi)\\cdot \\mathcal{O}_\\mathfrak{p}$ that is given in Proposition \\ref{ref deligne-gross}(ii) is valid.\n\\item[(iii)] The equality of ${\\rm BSD}_p(A_{F\/K})$(iv) is valid.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Since $F\/K$ is an extension of $p$-power degree, the assumption that neither $A(K)$ nor $A^t(K)$ has a point of order $p$ implies that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\n\nHence, in this case, the given assumptions imply that the data $A$, $F\/k$ and $p$ satisfy all of the hypotheses of Proposition \\ref{ref deligne-gross}.\n\nIn particular, if we write $\\xi$ for the difference between the left and right hand sides of the equality in Theorem \\ref{bk explicit}, then the argument of Proposition \\ref{ref deligne-gross} shows that the assumed validity of the given conditions (i) and (ii) implies that $\\xi$ belongs to\n $K_0(\\ZZ_p[G],\\QQ_p[G])$ and also to the kernel of the homomorphism $\\rho_\\mathfrak{p}^\\psi$ for every $\\psi$ in $\\widehat{G}$ and every prime ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$.\n\nThese facts combine with the general result of \\cite[Th. 4.1]{ewt} to imply that $\\xi$ belongs to the finite group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$.\n\nWe next recall from \\cite[Lem. 5.12(ii)]{bmw} that, since $G$ is assumed to be dihedral, the natural restriction map ${\\rm res}^G_P:K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor} \\to K_0(\\ZZ_p[P],\\QQ_p[P])$ is injective.\n\nIt follows that $\\xi$ vanishes, and hence by Theorem \\ref{bk explicit} that ${\\rm BSD}_p(A_{F\/k})$(iv) is valid, if the element ${\\rm res}^G_P(\\xi)$ vanishes.\n\nTo complete the proof, it is therefore enough to note that the functorial behaviour of the conjecture ${\\rm BSD}(A_{F\/k})$ under change of extension, as described in Remark \\ref{consistency remark}(ii) (and justified via Remark \\ref{consistency}(ii)), implies that ${\\rm res}^G_P(\\xi)$ vanishes if and only if the equality of ${\\rm BSD}_p(A_{F\/K})$(iv) is valid. \\end{proof}\n\n\\begin{remark}{\\em If $P$ is cyclic, then Proposition \\ref{dihedral prop} shows that in certain situations the validity of ${\\rm BSD}_p(A_{F\/k})$(iv) for the non-abelian extension $F\/k$ can be checked by verifying the relevant cases of the refined Deligne-Gross Conjecture formula in Proposition \\ref{ref deligne-gross}(ii) together with explicit congruences for the cyclic extension $F\/K$ of the form that are discussed in \\S\\ref{cycliccongssection}. In addition, if $P$ is cyclic, then the main result of Yakovlev in \\cite{yakovlev2} can be used to show that if the groups $\\sha(A_{F'})[p^\\infty]$ vanish for all proper subfields of $F$ that contain $K$, then the $\\ZZ_p[G]$-module $A(F)_p$ is a `trivial source module' and so has a very explicit structure. }\\end{remark}\n\n\\begin{example}{\\em For the examples described in Example \\ref{wuthrich example} the field $F$ is a dihedral extension of $k = \\QQ$ of degree $6$ and both of the given elliptic curves $A$ satisfy all of the hypotheses that are necessary to apply Proposition \\ref{dihedral prop} (in the case $p=3$ and $n=1$). In this way one finds that the validity of ${\\rm BSD}_3(A_{F\/\\QQ})$(iv) implies, and if $\\sha(A_F)[3^\\infty]$ vanishes is equivalent to, the validity of the relevant cases of the refined Deligne-Gross Conjecture together with the validity of an explicit congruence of the form described in Theorem \\ref{mequals1} for the cyclic extension $F\/K$ (and with $m_1= 1$ and $m_0 = 2$). Unfortunately, however, since $F\/\\QQ$ is of degree $6$ it seems that for the given curves $A$ the latter congruences are at present beyond the range of numerical investigation via the methods that have been used to verify the cases discussed in Example \\ref{bleyexamples}. }\\end{example}\n\n\n\\subsection{The proof of Theorem \\ref{rbsdimpliesmt}}\\label{mrsproof}\n\nWe fix a subgroup $J$ of $G$ and set $E := F^J$ and a maximal subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$.\n\nTo study the relationship between ${\\rm BSD}(A_{F\/k})$ and Conjecture \\ref{mrsconjecture} we set $X:=\\langle x_\\bullet\\rangle_{\\ZZ_p[G]}$ and consider both of the complexes $C_{S,X}:={\\rm SC}_S(A_{F\/k};X,H_\\infty(A_{F\/k})_p)$ and $C_{S,X,J}:={\\rm SC}_S(A_{E\/k};X^J,H_\\infty(A_{E\/k})_p)$.\n\nThen the definition of $C_{S,X}$ as the mapping fibre of the morphism (\\ref{fibre morphism}), the well-known properties of \\'etale cohomology under Galois descent and the fact that $X$ is a free $\\ZZ_p[G]$-module (see also (\\ref{global descent}), (\\ref{local descent}) and the argument that precedes them) imply that the object $$(C_{S,X})_J:=\\ZZ_p[G\/J]\\otimes_{\\ZZ_p[G]}^{\\mathbb{L}}C_{S,X}$$ of $D^{\\rm perf}(\\ZZ_p[G\/J])$ is isomorphic to $C_{S,X,J}$. This fact combines with \\cite[Lem. 3.33]{bst} to give canonical identifications\n$$H^1(C_{S,X})^J=H^1((C_{S.X})_J)=H^1(C_{S,X,J})$$ and\n$$H^2(C_{S,X})_J=H^2((C_{S,X})_J)=H^2(C_{S,X,J}).$$\n\nIn addition, our assumption that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ combines with Proposition \\ref{prop:perfect} to imply that,\nin the terminology of \\cite{bst}, the complex $C_{S,X}$ is a `strictly admissible' complex of $\\ZZ_p[G]$-modules. The approach of \\S 3.4.1 in loc. cit. therefore gives a `Bockstein homomorphism' of $\\ZZ_p[G\/J]$-modules \\begin{equation}\\label{htc}{\\rm ht}^{C_{S,X}}:H^1(C_{S,X})^J\\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2\\otimes_{\\ZZ_p[G\/J]}H^2(C_{S,X,J}).\\end{equation}\n\nFor any $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$\nand each index $i$ with $a< i \\le a'$ we then use the dual point $w_i^*$ to construct a composite homomorphism of $\\ZZ_p[G\/J]$-modules \\begin{equation}\\label{projection}H^2(C_{S,X,J})\\to\\Sel_p(A_E)^\\vee\\to A(E)_p^*\\to (Y')^*\\to\\ZZ_p[G\/J].\\end{equation} Here the first arrow is the canonical homomorphism of Proposition \\ref{prop:perfect}(iii), the second arrow is the canonical homomorpism occurring in (\\ref{sha-selmer}), the third arrow is the natural restriction maps and the fourth arrow maps an element of $(Y')^*$ to its coefficient at the basis element $w_i^*$.\n\nUpon composing ${\\rm ht}^{C_{S,X}}$ with each map (\\ref{projection}) we thereby obtain, for each index $i$ with $a<i\\leq a'$, a homomorphism of $\\ZZ_p[G\/J]$-modules\n\\begin{equation}\\label{htcwi} {\\rm ht}^{C_{S,X}}_{w_i}: H^1(C_{S,X})^J \\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2 \\end{equation}\nthat corresponds to the map denoted by ${\\rm Boc}^C_x$ in \\cite[\\S 3.4.1]{bst}.\nJust as in (\\ref{canbecombined}) these homomorphisms can then be combined to give a canonical homomorphism \\begin{equation}\\label{htcy}{\\rm ht}_{\\mathcal{Y}'}^{C_{S,X}}: {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} H^1(C_{S,X})^J \\longrightarrow ({\\bigcap}_{\\ZZ_p[G\/J]}^{a} H^1(C_{S,X})^J)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\\end{equation}\n\n\n\n\n\n\n\n\nWe will apply the general result of \\cite[Th. 3.34(ii)]{bst} to the complex $C_{S,X}$, the isomorphism $h_{A,F}$ and the characteristic element $\\mathcal{L}_{F\/k}$ for the pair $(C_{S,X},h_{A,F})$ constructed in Lemma \\ref{modifiedlemma}(iii) under the assumption that ${\\rm BSD}(A_{F\/k})$ is valid. We recall that \\begin{equation}\\label{charelementFk}e_{F,a}\\cdot\\mathcal{L}_{F\/k}^{-1}=\\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet).\\end{equation}\n\nSimilarly, if ${\\rm BSD}(A_{F\/k})$ is valid then the observations made in Remark \\ref{consistency remark}(ii) (see also Remark \\ref{consistency} below for more details) imply that ${\\rm BSD}(A_{E\/k})$ is also valid. Applying Lemma \\ref{modifiedlemma}(iii) to the complex $C_{S,X,J}$ and the isomorphim $h_{A,E}$ therefore implies the existence of a characteristic element $\\mathcal{L}_{E\/k}$ for this pair with the property that \\begin{equation}\\label{charelementEk}e_{E,a'}\\cdot\\mathcal{L}_{E\/k}^{-1}=\\frac{L^{(a')}_{S}(A_{E\/k},1)}{\\Omega_A^{E\/k}\\cdot w_{E\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{E\/k}}(x_\\bullet^J).\\end{equation}\n\nWe next fix any (injective) section $s$ to the canonical surjective composite homomorphism $$H^2(C_{S,X})\\to\\Sel_p(A_F)^\\vee\\to A(F)_p^*\\to Y^*,$$ where the last arrow is the natural restriction map. The set $s(\\mathcal{Y}^*):=\\{s(y^*):y\\in\\mathcal{Y}\\}$ is then (in the sense of \\cite[Def. 3.6]{bst}) a `separable subset' of $H^2(C_{S,X})$ (of cardinality $a$).\nThe `higher special element' associated via \\cite[Def. 3.3]{bst} to the data $$(C_{S,X},h_{A,F},\\mathcal{L}_{F\/k},s(\\mathcal{Y}^*))$$ is then, by (\\ref{charelementFk}) and a direct comparison of the definitions of $h_{A,F}$ and ${\\rm ht}_{A_{F\/k}}^{a}$, equal to $\\eta_{\\mathcal{Y},x_\\bullet}$.\n\nWe finally fix any (injective) section $s'$ to the canonical surjective composite homomorphism $$H^2(C_{S,X,J})\\to\\Sel_p(A_E)^\\vee\\to A(E)_p^*\\to (Y')^*,$$ where the last arrow is the natural restriction map. The set $s'((\\mathcal{Y}')^*):=\\{s'(y^*):y\\in\\mathcal{Y}'\\}$ is then a separable subset of $H^2(C_{S,X,J})$ (of cardinality $a'$). The higher special element associated via \\cite[Def. 3.3]{bst} to the data $$(C_{S,X,J},h_{A,E},\\mathcal{L}_{E\/k},s'((\\mathcal{Y}')^*))$$ is then, by (\\ref{charelementEk}) and a direct comparison of the definitions of $h_{A,E}$ and ${\\rm ht}_{A_{E\/k}}^{a'}$, equal to $\\eta_{\\mathcal{Y}',x_\\bullet^J}$.\n\nThe general result \\cite[Th. 3.34(ii)]{bst} therefore implies that\n\\[ \\eta_{\\mathcal{Y},x_\\bullet} \\in {\\bigcap}_{\\ZZ_p[G]}^{a} H^1(C_{S,X})\\subseteq{\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p,\\]\n\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} H^1(C_{S,X})^J\\subseteq{\\bigcap}_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p,\\]\nand that\n\\begin{equation}\\label{forreferenceinremark} \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{C_{S,X}}(\\eta_{\\mathcal{Y}',x_\\bullet^J}))\\end{equation}\nin $\\left({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(F)_p\\right)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$.\n\n\nTo complete the proof of Theorem \\ref{rbsdimpliesmt} we will combine the equality (\\ref{forreferenceinremark}) with the comparison of height pairings proved in Theorem \\ref{thecomptheorem} below.\n\n\n\n\nWe first note that $A$, $F\/E$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc} (see \\cite[Lem. 3.4]{bmw0}). We also note that the validity of these hypotheses (and their duality) combines with Proposition \\ref{explicitbkprop}(ii) to imply that $U_{F\/E,p}=A(E)_p$ and $U^t_{F\/E,p}=A^t(E)_p$.\n\nIt will thus be enough to prove that, for any index $a<i\\leq a'$, the restriction to $H^1(C_{S,X})^J$ of the homomorphism\n$${\\rm ht}_{w_i}^{{\\rm MT}}:A^t(E)_p\\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2$$ is equal to the homomorphism ${\\rm ht}_{w_i}^{C_{S,X}}$ of (\\ref{htcwi}).\n\nBy Theorem \\ref{thecomptheorem} below applied to $F\/E$, for any $P$ in $A(E)_p$ one has $${\\rm ht}_{w_i}^{{\\rm MT}}(P):=-\\langle P,w_i\\rangle^{\\rm MT}_{F\/E}=\\langle P,w_i\\rangle_{p}^{\\rm N}$$ in $\\ZZ_p\\otimes_\\ZZ J$, where $\\langle \\,,\\rangle_{p}^{\\rm N}$ is the Bockstein pairing associated to the classical Selmer complex ${\\rm SC}_p(A_{F\/E})$ as in (\\ref{thepairing}) below.\n\n\nIt is therefore be enough to prove that, for any index $a<i\\leq a'$ and for any $P\\in H^1(C_{S,X})^J$ one has \\begin{equation*}{\\rm ht}_{w_i}^{C_{S,X}}(P)=\\langle P,w_i\\rangle_{p}^{\\rm N}.\\end{equation*} But this equality follows easily from the compatibility of the maps $$H^1(C_{S,X})^J=H^1(C_{S,X,J})\\to H^1({\\rm SC}_p(A_{F\/E}))$$ and $$H^2(C_{S,X,J})\\to H^2({\\rm SC}_p(A_{F\/E}))$$ occurring in the exact sequence (\\ref{useful1}) with all the other maps involved in the construction of both Bockstein homomorphisms (and the naturality of such homomorphisms). This completes the proof of Theorem \\ref{rbsdimpliesmt}.\n\n\n\n\n\n\\section{Height pairing comparisons}\\label{comparison section}\n\n\n\n\n\n\n\n\n\n\nIn this section we continue to assume that $F\/k$ is abelian.\n\nWe fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S \\ref{tmc}. We also assume that neither $A(F)$ not $A^t(F)$ has a point of order $p$.\n\nIn this situation, Proposition \\ref{explicitbkprop} implies that the classical Selmer complex ${\\rm SC}_p(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one and two, where its cohomology modules canonically identify with $A^t(F)_p$ and $\\Sel_p(A_F)^\\vee$ respectively.\n\nVia the general formalism outlined in \\S \\ref{bocksection} one therefore obtains an associated canonical algebraic height pairing\n$$\\langle\\,,\\rangle^{\\rm N}_{p}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to \\ZZ_p\\otimes_{\\ZZ}G$$\nthat we refer to as the `Nekov\\'a\\v r height pairing'.\n\nProposition \\ref{explicitbkprop}(ii) also implies that the $p$-completion of the height pairing (\\ref{tanpairing}) (for $J=G$) that is defined using the theory of bi-extensions by Mazur and Tate in \\cite{mt0} gives a well-defined pairing\n$$\\langle\\,,\\rangle_p^{\\rm MT}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to \\ZZ_p\\otimes_{\\ZZ}G.$$\n\nIn this section we shall prove that the Nekov\\'a\\v r height pairing coincides with the inverse of the Mazur-Tate height pairing. This comparison result was already used in the proof of Theorem \\ref{rbsdimpliesmt} and is also, we believe, of some independent interest.\n\n\\subsection{Statement of the result}\n\nTo make the construction of the Nekov\\'a\\v r height pairing more precise we fix a finite subset $\\Sigma$ of $S_k^f$ that contains $S_k^p$, $S_k^F\\cap S_k^f$ and $S_k^A$.\n\nProposition \\ref{explicitbkprop} implies that the Selmer complex $C_{\\Sigma}={\\rm SC}_{\\Sigma,p}(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one and two.\n\nIn addition,\nwe can use Proposition \\ref{explicitbkprop}(iv) to interpret the global Kummer map as an isomorphism \\begin{equation}\\label{r1}r_1:A^t(F)_p\\to H^1(C_{\\Sigma}),\\end{equation} and also to define a canonical composite homomorphism \\begin{equation}\\label{r2}r_2:H^2(C_{\\Sigma})\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee{\\to} A(F)_p^*\\to A(k)_p^*.\\end{equation} Here the first arrow is the canonical isomorphism occurring in the diagram (\\ref{Selmerdiagram}) and the last arrow is simply the restriction to $A(k)_p$ of any element of the linear dual $A(F)_p^*$.\n\nBy using the general approach (and notation) of \\S \\ref{bocksection}, we can therefore define the Nekov\\'a\\v r height pairing by setting\n\n\\begin{equation}\\label{thepairing}\\langle\\,,\\rangle^{\\rm N}_{p}:=\\langle\\,,\\rangle_{C_\\Sigma,r_1,r_2}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to I_{p}(G)\/I_{p}(G)^2\\cong \\ZZ_p\\otimes_{\\ZZ}G,\\end{equation}\nwhere $I_p(G)$ denotes the augmentation ideal in $\\ZZ_p[G]$.\n\nThen Lemma \\ref{independenceofsigma} and Remark \\ref{indeptremark} combine with the naturality of Bockstein pairings to imply that this pairing is  independent of the choice of set $\\Sigma$.\n\nWe write $m_p(G)$ for the non-negative integer for which $p^{m_p(G)}$ is the exponent of the group $\\ZZ_p\\otimes_{\\ZZ}G$.\n\nThen the proof of our comparison of height pairings is greatly simplified by imposing the following additional technical hypothesis:\n\n\\begin{hyp}\\label{hypbd\nThe group $H^1({\\rm Gal}(F(A[p^{m_p(G)}])\/F),A[p^{m_p(G)}])$ vanishes.\n\\end{hyp}\n\n\\begin{remarks}\\label{rmkbd}{\\em\nHypothesis \\ref{hypbd} is motivated by the result \\cite[Lem. 2.13]{bert} of Bertolini and Darmon and \n is satisfied whenever multiplication-by-`$-1$' belongs to the image of the canonical Galois representation $G_F\\to\\Aut_{\\mathbb{F}_p}(A[p])$. In particular, if $A$ is an elliptic curve, then the hypothesis only excludes finitely many primes $p$ by a famous result of Serre \\cite{serre}.}\n\\end{remarks}\n\n\n\n\n\n\n\\begin{theorem}\\label{thecomptheorem} Fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S \\ref{tmc}. Assume also that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and that Hypothesis \\ref{hypbd} is satisfied.\n\nThen the Nekov\\'a\\v r pairing $\\langle\\,,\\rangle^{\\rm N}_{p}$ is equal to the inverse of the Mazur-Tate pairing $\\langle\\,,\\rangle^{\\rm MT}_p$\n\\end{theorem}\n\nThe proof of Theorem \\ref{thecomptheorem} will occupy the rest of this section. We thus assume throughout that all the hypotheses listed in Theorem \\ref{thecomptheorem} are valid.\n\n\\subsection{An auxiliary height pairing}\n\nIn this section we define a $p$-adic height pairing $\\langle\\,,\\rangle^{\\rm BD}_p$  whose construction is motivated by the work of Bertolini and Darmon \\cite{bert,bert2}.\n\nIn \\S\\ref{compproof} below we will then show that $\\langle\\,,\\rangle_p^{\\rm BD}$ coincides with the inverse of $\\langle\\,,\\rangle_p^{\\rm N}$. By then appealing to comparison results due to Bertolini and Darmon and to Tan, we will finally obtain the desired comparison of pairings.\n\nWe will now follow closely some of the constructions from \\cite{bert,bert2} and for this reason we will be rather brief in some of our arguments.\n\n\\subsubsection{} We again write $m:=m_p(G)$ for the non-negative integer such that $p^m$ is the exponent of the finite abelian $p$-group $\\ZZ_p\\otimes_{\\ZZ}G$.\n\nWe set $Z:=\\ZZ\/p^m\\ZZ$ and $R:=Z[G]$ and write $I_{R}$ for the augmentation ideal in $R$. We will also require all of the following notation.\n\nFor any finite set $S$ of non-archimedean places of $k$, we define $\\Sel^S_{p^m}(A^t\/F)$ to be the kernel of the localisation map $$H^1\\bigl(F,A^t[p^m]\\bigr)\\to\\bigoplus_{w'\\notin S(F)}H^1(F_{w'},A^t)$$ (with the sum running over all non-archimedean places of $F$ that do not belong to $S(F)$). When $S$ is taken to be empty one recovers the usual Selmer group associated to multiplication by $p^m$, which we denote by $\\Sel^{(p^m)}(A^t_F)$ instead.\n\nFor such a set $S$ and $B\\in\\{A,A^t\\}$ we also define $R$-modules\n$$B_S(\\mathbb{A}_F)\/p^m:=\\prod_{w'\\in S(F)}B(F_{w'})\/p^mB(F_{w'}),$$ $$H^1_{S}\\bigl(\\mathbb{A}_F,B[p^m]\\bigr):=\\prod_{w'\\in S(F)}H^1(F_{w'},B[p^m])$$ and $$H^1_{S}\\bigl(\\mathbb{A}_F,B\\bigr)[p^m]:=\\prod_{w'\\in S(F)}H^1(F_{w'},B)[p^m].$$\n\nBy abuse of notation, we denote by $\\lambda_{S}$ both the localisation map $$\\Sel^S_{p^m}(A^t\/F){\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)$$ and the induced map $$\\Sel^S_{p^m}(A^t\/F){\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m].$$\n\nWe also write $$\\kappa:A^t_S(\\mathbb{A}_F)\/p^m\\to H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)$$ and $$\\kappa':H^1_{S}\\bigl(\\mathbb{A}_F,A\\bigr)[p^m]^\\vee\\to H^1_{S}\\bigl(\\mathbb{A}_F,A[p^m]\\bigr)^\\vee$$ for the canonical maps induced by the local Kummer sequences.\n\nWe recall (see Corollary \\ref{Tatepoitouexplicit} below) that the local Tate duality isomorphism $$u_S:H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)\\to H^1_{S}\\bigl(\\mathbb{A}_F,A[p^m]\\bigr)^\\vee$$ induces an isomorphism\n\\begin{equation}\\label{inducedbyum}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\cong\\cok(\\kappa)\\stackrel{\\sim}{\\to}\\cok(\\kappa')\\cong\\bigl(A_S(\\mathbb{A}_F)\/p^m\\bigr)^\\vee.\\end{equation}\nBy composing this isomorphism with the dual of the restriction map $$\\Sel^{(p^m)}(A_F)\\to A_S(\\mathbb{A}_F)\/p^m$$ one finally obtains a map\n$$v_S:H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]{\\to}\n{\\rm Sel}^{(p^m)}(A_F)^\\vee.$$\n\n\n\n\n\\subsubsection{\nIn the sequel we denote by $P$ the Sylow $p$-subgroup of $G$ and set $K:=F^P$.\n\n\\begin{lemma}\\label{214} The restriction map $H^1(K,A[p^m])\\to H^1(F(A[p^m]),A[p^m])$ is injective.\\end{lemma}\n\\begin{proof} As in \\cite[Cor. 2.14]{bert}, the result follows immediately upon combining the validity of Hypothesis \\ref{hypbd} with the fact that $A(F)[p^m]$ vanishes (because $A(F)$ does not have a point of order $p$).\n\\end{proof}\n\nLemma \\ref{214} now implies the existence of `admissible' auxiliary sets of primes, as in \\cite[Def. 1.5]{bert2} or \\cite[Def. 2.20, Def. 2.22]{bert}:\n\n\\begin{lemma}\\label{tm} There is a finite set $T$ of non-archimedean places of $K$ such that:\n\\begin{itemize\n\\item[(i)] Every place in $T$ splits completely in $F\/K$.\n\\item[(ii)] If $v\\in T$ then $A(K_v)\/p^mA(K_v)$ is a free $Z$-module of rank $2\\dim(A)$.\n\\item[(iii)] The restriction map \\begin{equation}\\label{Kres}\\Sel^{(p^m)}(A_K)\\to\\bigoplus_{v\\in T}A(K_v)\/p^mA(K_v)\\end{equation} is injective.\n\\end{itemize}\n\\end{lemma}\n\nThe existence of such a set $T$ (of infinitely many such sets $T$, in fact) is proved in \\cite[Lem. 2.21]{bert} using a Tcheboratev density argument (and Lemma \\ref{214}).\n\nIn the sequel, we fix such an admissible set $T$ and, by abuse of notation, also denote by $T$ the set of places of $k$ below places in $T$.\n\nThe existence of the admissible set $T$ now leads us to consider the canonical complex of $R$-modules $$C_{\\rm BD}:=\\bigl[\\Sel^T_{p^m}(A^t\/F)\\stackrel{\\lambda_T}{\\to}H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\bigr],$$ with first term placed in degree 1. The desired auxiliary pairing $\\langle\\,,\\rangle_p^{\\rm BD}$ will then be obtained as a Bockstein pairing associated to $C_{\\rm BD}$.\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{bdlemma} The following claims are valid.\n\\begin{itemize}\\item[(i)] Let $S$ be any finite set of non-archimedean places of $k$ which contains $T$. Then the sequence\n\\begin{equation}\\label{notexact}0\\to{\\rm Sel}^{(p^m)}(A^t_F)\\to \\Sel^S_{p^m}(A^t\/F)\\stackrel{\\lambda_S}{\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\stackrel{v_S}{\\to}\n{\\rm Sel}^{(p^m)}(A_F)^\\vee\\to 0\\end{equation} is exact.\n\\item[(ii)] The $R$-modules $\\Sel^T_{p^m}(A^t\/F)$, $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ and $A_T(\\mathbb{A}_F)\/p^m$ are finitely generated and $G$-cohomologically-trivial.\n\\item[(iii)] Let $S\\supseteq T$ be any set as in claim (i). Then there is a canonical quasi-isomorphism from the complex $C_{\\rm BD}$ to the complex $$C_{{\\rm BD},S}:=\\bigl[\\Sel^S_{p^m}(A^t\/F)\\stackrel{\\lambda_S}{\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\bigr],$$ with first term placed in degree 1, which induces the identity map both on ${\\rm Sel}^{(p^m)}(A^t_F)$ and on ${\\rm Sel}^{(p^m)}(A_F)^\\vee$ (viewed as the cohomology modules of $C_{\\rm BD}$ and of $C_{{\\rm BD},S}$ via (\\ref{notexact})).\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} The exactness of (\\ref{notexact}) at the first three modules is proved in \\cite[Lem. 6.15]{milne}, and is independent of the fact that the set $S$ contains $T$. See also \\cite[Prop. 1.2]{bert2} and the discussion that follows it.\n\nIn order to prove claim (i), it is hence enough to show that $v_{T}$ is surjective (since then so must be $v_S$ for any set $S$ containing $T$), or equivalently that the restriction map \\begin{equation}\\label{tmrestriction}\\Sel^{(p^m)}(A_F)\\to A_{T}(\\mathbb{A}_F)\/p^m\\end{equation} is injective. This may be achieved by following the approach of \\cite[Lem. 2.25, Lem. 2.26, Lem 3.1]{bert} or \\cite[Prop. 1.6]{bert2}. We limit ourselves to giving a sketch of this argument.\n\nProposition \\ref{explicitbkprop}(ii) implies that, for any non-archimedean place $v$ of $k$, the module $A(F_v)$ is $P$-cohomologically-trivial.\n\n\n\n\n\nThis fact then combines with our assumption that $A(F)$ has no point of order $p$ to imply the restriction map $\\Sel^{(p^m)}(A_K)\\to\\Sel^{(p^m)}(A_F)^P$ is bijective, by the same argument used to prove \\cite[Lem 2.19]{bert} or \\cite[Lem. 1.4]{bert2}.\n\nSince the restriction map (\\ref{Kres}) is injective, one finally deduces that the restriction map (\\ref{tmrestriction}) is injective, just as in \\cite[Lem 2.25]{bert} or \\cite[Prop. 1.6]{bert2}.\n\nTo next consider claim (ii), it is enough to show that each of the modules $\\Sel^T_{p^m}(A^t\/F)$, $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ and $A_T(\\mathbb{A}_F)\/p^m$ is finitely generated and free over $Z[P]$. These facts in turn follow from the arguments of \\cite[\\S 3.1]{bert}.\n\nIndeed, one first uses properties (i) and (ii) of the set $T$, as given in Lemma \\ref{tm}, to see that $A_T(\\mathbb{A}_F)\/p^m$ is (finitely generated and) $Z[P]$-free.\nThe duality (\\ref{inducedbyum}) between $A_{T}(\\mathbb{A}_F)\/p^m$ and $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ then implies that the same property is true for the latter module.\n\nSince neither $A(F)$ nor $A^t(F)$ has a point of order $p$, the canonical exact sequences\n$$0\\to A(F)\/p^mA(F)\\to\\Sel^{(p^m)}(A_F)\\to\\sha(A_F)[p^m]\\to 0$$ and\n$$0\\to A^t(F)\/p^mA^t(F)\\to\\Sel^{(p^m)}(A^t_F)\\to\\sha(A^t_F)[p^m]\\to 0$$ combine with the Cassels-Tate pairing to imply that the finite groups ${\\rm Sel}^{(p^m)}(A^t_F)$ and ${\\rm Sel}^{(p^m)}(A_F)^\\vee$ have the same order. One may therefore combine the fact that $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ is $Z[P]$-free with the exactness of (\\ref{notexact}) (for $S=T$) to deduce, as in \\cite[Thm. 3.2]{bert}, that $\\Sel^{T}_{p^m}(A^t\/F)$ is also $Z[P]$-free.\n\n\nWe finally consider claim (iii). The canonical inclusions give vertical arrows in the commutative diagram\n\\begin{equation*}\\xymatrix{\n\\Sel^{T}_{p^m}(A^t\/F) \\ar[d] \\ar[r]^(0.45){\\lambda_T} & H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m] \\ar[d] \\\\\n\\Sel^{S}_{p^m}(A^t\/F)  \\ar[r]^(0.45){\\lambda_S} & H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\n}\\end{equation*}\nand hence define a morphism of complexes $C_{\\rm BD}\\to C_{{\\rm BD},S}$.\n\nIt is clear that this morphism  induces the identity map both on ${\\rm Sel}^{(p^m)}(A^t_F)$ and on ${\\rm Sel}^{(p^m)}(A_F)^\\vee$, so by the exactness of the respective sequences of the form (\\ref{notexact}) it is in particular a quasi-isomorphism, as required.\n\\end{proof}\n\n\\subsubsection{\nTo define the useful auxiliary pairing we let $S$ be any finite set of non-archimedean places of $k$ containing $T$.\n\nWe may then combine Lemma \\ref{bdlemma} with the general formalism outlined in \\S \\ref{bocksection} below to define a pairing\n$\\langle\\,,\\rangle_p^{\\rm BD}$ as follows.\n\nWe let $r_{1,m}$ denote the canonical composite map\n$$A^t(k)_p\/p^m\\to A^t(F)_p\/p^m\\to \\Sel^{(p^m)}(A^t_F)\\to H^1(C_{{\\rm BD},S}),$$  and $r_{2,m}$ denote the canonical composite map $$H^2(C_{{\\rm BD},S})\\stackrel{v_{S}}{\\to}{\\rm Sel}^{(p^m)}(A_F)^\\vee\\to(A(F)_p\/p^m)^\\vee\\to(A(k)_p\/p^m)^\\vee.$$\n\nAs in \\S \\ref{bocksection}, these canonical choices induce a canonical Bockstein homomorphism \\begin{equation*}\\label{thebdbock}\\beta_{C_{{\\rm BD},S}}:A^t(k)_p\/p^m{\\to}I_{R}\/I_{R}^2\\otimes_{Z} (A(k)_p\/p^m)^\\vee\\end{equation*} and hence a canonical pairing\n$$\\langle\\,,\\rangle_{C_{{\\rm BD},S}}:A^t(k)_p\/p^m\\otimes_{Z} A(k)_p\/p^m\\to I_{R}\/I_{R}^2\n\\cong Z\\otimes_{\\ZZ}G= \\ZZ_p\\otimes_{\\ZZ}G,$$\nor equivalently a canonical pairing\n$$\\langle\\,,\\rangle_p^{{\\rm BD}}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to  \\ZZ_p\\otimes_{\\ZZ}G.$$\n\nIn particular, Lemma \\ref{bdlemma}(iii) combines with the naturality of Bockstein homomorphisms to ensure that this pairing is independent of the choice of set $S$ containing $T$.\n\n\\subsubsection{\n In order to compare $\\langle\\,,\\rangle_p^{\\rm BD}$ to the Nekov\\'a\\v r height pairing we shall need to clarify the relationship between complexes of the form $C_{{\\rm BD},S}$ and classical Selmer complexes.\n\nTo do this we fix a finite subset $\\Sigma$ of $S_k^f$ with\n$$S_k^p\\cup (S_k^F \\cap S_k^f)\\cup S_k^A\\cup T\\subseteq\\Sigma.$$\n\nWe then use the classical Selmer complex $C_\\Sigma$ to define a complex of $R$-modules $$C_{\\Sigma}\/p^m:=Z\\otimes_{\\ZZ_p}^{\\mathbb{L}}C_{\\Sigma}.$$ We refer the reader to Lemma \\ref{reducedbock} in Appendix \\ref{exp rep section} below for the definition of the maps $q_{1}'$ and $q_{2}'$ that are associated, by using the fixed homomorphisms $r_1$ and $r_2$ displayed in (\\ref{r1}) and (\\ref{r2}) respectively, to the construction of this complex.\n\n\\begin{lemma}\\label{morphismphi} There exists a canonical morphism of complexes\n\\[ \\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m\\]\nwith the following properties:\n\\begin{itemize}\\item[(i)] The restriction of $q'_{1}$ to $A^t(k)_p\/p^m$ is equal to the composition $$A^t(k)_p\/p^m\\stackrel{r_{1,m}}{\\to}H^1(C_{{\\rm BD},\\Sigma})\\stackrel{H^1(\\phi)}{\\to}H^1(C_{\\Sigma}\/p^m).$$\n\\item[(ii)] $r_{2,m}$ is equal to $-1$ times the composition $$H^2(C_{{\\rm BD},\\Sigma})\\stackrel{H^2(\\phi)}{\\to}H^2(C_{\\Sigma}\/p^m)\\stackrel{q'_{2}}{\\to}A(k)_p^*\/p^m\\to(A(k)_p\/p^m)^\\vee,$$ where the last arrow is the canonical map.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} In order to explicitly define the morphism $\\phi$ we first describe an explicit representative of the classical Selmer complex $C_\\Sigma$. The claimed result is clearly independent of such a choice.\n\nTo do so we use the notation given in \\S \\ref{expcohcomp} below. In particular, for any $w'\\in\\Sigma(F)$ and any $i\\geq 0$ we let $R\\Gamma^i(F_{w'},T_p(A^t))$ denote the group of inhomogeneous $i$-cochains of $G_{w'}$ with coefficients in $T_p(A^t)$, and also write $d_{w'}^i$ for the usual differential.\n\nFor any $v\\in\\Sigma$ we then set\n$$C_v^2:=\\frac{\\prod_{w'\\in S_F^v}\\bigl(R\\Gamma^{1}(F_{w'},T_p(A^t))\/\\im(d^0_{w'})\\bigr)}{\\kappa\\bigl(A^t(F_v)_p^\\wedge\\bigr)}$$\nand also, for any $i\\geq 3$,\n$$C_v^i:=\\prod_{w'\\in S_F^v}R\\Gamma^{i-1}(F_{w'},T_p(A^t)).$$ We note that it is straightforward, by using the relevant exact sequences of the form (\\ref{kummerseq}), to verify that the module $C_v^2$ is $\\ZZ_p$-torsion-free.\n\nIf, for each $v\\in\\Sigma$, we set $D_v^i=C_v^{i-1}$, we let $\\partial_v^{i}$ be induced by $\\prod_{w'\\in S_F^v}d_{w'}^{i-1}$ and consider the complex $D^\\bullet_v=(D_v^{i},\\partial_v^{i})$, then there is a canonical exact triangle\n$$A^t(F_v)_p^\\wedge[-1]\\stackrel{\\kappa}{\\to}R\\Gamma(k_v,T_{p,F}(A^t))\\to D^\\bullet_v\\to A^t(F_v)_p^\\wedge[0]$$ in $D(\\ZZ_p[G])$.\n\nSimilarly, we write $G_{F,\\Sigma}$ for the Galois group of the maximal Galois extension of $F$ wich is unramified outside $\\Sigma(F)\\cup S_F^\\infty$, set $U_{F,\\Sigma}:=\\mathcal{O}_{F,\\Sigma\\cup S_F^\\infty}$, write $R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))$ for the group of inhomogeneous $i$-cochains of $G_{F,\\Sigma}$ with coefficients in $T_p(A^t)$, and also write $d^i$ for the usual differential.\n\n\nThen, since $R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))$ is acyclic in degrees less that one, we may obtain an explicit representative of $C_\\Sigma$ as the $-1$-shift of the explicit mapping cone of the diagonal localisation morphism\n$$\\tau_{\\geq 1}\\bigl(R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))\\bigr)\\stackrel{\\lambda}{\\to}\\bigoplus_{v\\in\\Sigma}D_v^\\bullet.$$\n(Here $\\tau_{\\geq 1}$ denotes the truncation in degree greater than or equal to 1 preserving cohomology).\n\nTo be explicit about this construction of a representative of $C_\\Sigma$, we set\n\\[\nC_{\\Sigma}^i :=\n\\begin{cases}\n  0,    &\\text{ if $i\\leq 0$}\\\\\n  R\\Gamma^1\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)\/{\\rm im}(d^0),    &\\text{ if $i=1$}\\\\\n\\bigl(\\bigoplus_{v\\in \\Sigma}C^i_v\\bigr)\\oplus R\\Gamma^i\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr),    &\\text{ if $i\\geq 2$.}\n \n\n\\end{cases}\n\\]\nWe let the differential $d_{\\Sigma}^1$ in degree $1$ be given by\n\\[d_{\\Sigma}^1(\\overline{c}):=\\bigl(-(\\overline{c_{w'}})_{w'\\in \\Sigma(F)},d^1(c)\\bigr)\\]\nand the differential $d_{\\Sigma}^i$ in degree $i\\geq 2$ be given by\n\\[d_{\\Sigma}^i(\\bigl((a_{w'})_{w'\\in \\Sigma(F)},b\\bigr)):=\\bigl(-(d^{i-1}_{w'}(a_{w'})+b_{w'})_{w'\\in \\Sigma(F)},d^i(b)\\bigr).\\]\nHere, in any degree $i$ and for any place $w'\\in \\Sigma(F)$, we have written $x_{w'}$ for the localisation at $w'$\nof an inhomogeneous $i$-cochain $x$ in $R\\Gamma^i\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)$.\n\nThe exactness of the sequence (\\ref{longexact}) combines with our assumption that $A^t(F)$ contains no point of order $p$ to imply that the module $H^1\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)$ is $\\ZZ_p$-torsion-free. One may deduce from this fact that the module $C_\\Sigma^1$ is also $\\ZZ_p$-torsion-free.\n\nNow, since each module $C_{\\Sigma}^i$ is $\\ZZ_p$-torsion-free, the complex $C_{\\Sigma}\/p^m$ may be represented by the sequence of modules $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^i$ together with the differentials $Z\\otimes_{\\ZZ_p}d_{\\Sigma}^i$.\n\nIn order to define the required morphism $\\phi$, we first define $$\\phi^1:\\Sel^{\\Sigma}_{p^m}(A^t\/F)\\to Z\\otimes_{\\ZZ_p}C_{\\Sigma}^1$$ as follows.\n\nWe first note that, since the set $\\Sigma$ is assumed to contain $S_k^p\\cup S_k^A$, the argument of \\cite[Prop. 6.5]{milne} proves that the canonical inflation map defines an isomorphism $$\\iota:H^1(U_{F,\\Sigma},A^t[p^m])\\stackrel{\\sim}{\\to}\\Sel^{\\Sigma}_{p^m}(A^t\/F).$$\n\nFor any $x\\in\\Sel^{\\Sigma}_{p^m}(A^t\/F)$ we may hence fix a 1-cocycle $y\\in R\\Gamma^1(U_{F,\\Sigma},A^t[p^m])$ representing $\\iota^{-1}(x)$ and use the map $l_\\Sigma$ defined in Lemma \\ref{lifts}(i) below to define $\\phi^1(x)$ as the class in $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^1$ of the lift $$l_\\Sigma(y)\\in Z\\otimes_{\\ZZ_p}R\\Gamma^1(U_{F,\\Sigma},T_p(A^t)).$$\n\nBefore proceeding we note that Lemma \\ref{lifts}(ii) ensures both that $\\phi^1$ is well-defined and also that\n\\begin{equation}\\label{itisacocycle}(Z\\otimes_{\\ZZ_p}d^1)(l_\\Sigma(y))=0.\n\\end{equation}\n\nWe next define $$\\phi^2:H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\to Z\\otimes_{\\ZZ_p}C_{\\Sigma}^2$$ as follows.\n\nFor any $(x_{w'})_{w'\\in\\Sigma(F)}$ in\n\\[ H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]=\\frac{H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)}\n{\\kappa\\bigl(A^t_\\Sigma(\\mathbb{A}_F)\/p^m\\bigr)}\\]\nwe fix a family of 1-cocycles\n$$(y_{w'})_{w'\\in\\Sigma(F)}\\in\\prod_{w'\\in\\Sigma(F)}R\\Gamma^1(F_{w'},A^t[p^m])$$ representing $(x_{w'})_{w'\\in\\Sigma(F)}$.\nWe then use the maps $l_{w'}$, as $w'$ ranges through $\\Sigma(F)$, defined in Lemma \\ref{lifts}(i) below to obtain a lift\n$$l_{w'}(y_{w'})\\in Z\\otimes_{\\ZZ_p}R\\Gamma^1(F_{w'},T_p(A^t)).$$\n\nWe finally define $\\phi^2((x_{w'})_{w'\\in\\Sigma(F)})$ as the class in $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^2$ of $(-(l_{w'}(y_{w'}))_{w'\\in\\Sigma(F)},0)$. Again one can use Lemma \\ref{lifts}(ii) to verify that $\\phi^2$ is well-defined.\n\n\nOur explicit description of the differential $d_{\\Sigma}^1$ then implies that, for any $x\\in\\Sel^{\\Sigma}_{p^m}(A^t\/F)$, one has\n\\begin{align*} \\bigl((Z\\otimes_{\\ZZ_p}d_{\\Sigma}^1)\\circ\\phi^1\\bigr)(x)=&\\bigl(-(\\overline{l_\\Sigma(y)_{w'}})_{w'},(Z\\otimes_{\\ZZ_p}d^1)(l_\\Sigma(y))\\bigr)\\\\\n=&\\bigl(-(\\overline{l_\\Sigma(y)_{w'}})_{w'},0\\bigr)\\\\\n=&\\bigl(-(\\overline{l_{w'}(y_{w'})})_{w'},0\\bigr)\\\\\n=&\\phi^2((\\overline{y_{w'}})_{w'})\\\\\n=&\\phi^2(\\lambda_\\Sigma(\\iota(\\overline{y})))\\\\\n=&(\\phi^2\\circ\\lambda_\\Sigma)(x).\\end{align*}\nHere again $y$ is a 1-cocycle representing $\\iota^{-1}(x)$, the second equality follows from (\\ref{itisacocycle}) and the third equality holds since, by Lemma \\ref{lifts}(ii), the maps $l_\\Sigma$ and $l_{w'}$ commute with localisation at each $w'$.\n\nPutting then $\\phi^i=0$ for $i\\neq 1,2$ therefore defines a morphism of complexes of $R$-modules $$\\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m.$$ Indeed, using Lemma \\ref{lifts}(ii) one verifies that\n\\[ \\bigl((Z\\otimes_{\\ZZ_p}d_{\\Sigma}^2)\\circ\\phi^2\\bigr)((x_{w'})_{w'})=0\\] for any $(x_{w'})_{w'}$ in $H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$.\n\nWe next prove that $\\phi$ satisfies the stated properties. Now, $q'_{1}$ is by definition equal to the composition\n$$A^t(F)_p\/p^m\\stackrel{r_1\/p^m}{\\to}H^1(C_\\Sigma)\/p^m\\stackrel{q_1}{\\to}H^1(C_\\Sigma\/p^m)$$\nwith $q_{1}$ induced by the canonical projection $$R\\Gamma^1\\bigl(U_{F,\\Sigma},T_p(A^t)\\bigr)\\to Z\\otimes_{\\ZZ_p}R\\Gamma^1\\bigl(U_{F,\\Sigma},T_p(A^t)\\bigr).$$ The defining property of the lifting map $l_\\Sigma$ given in Lemma \\ref{lifts}, combined with our definition of $\\phi^1$, therefore implies that $q'_1$ coincides with the canonical composition\n$$A^t(F)_p\/p^m\\to\\Sel^{(p^m)}(A^t_F)\\to H^1(C_{{\\rm BD},\\Sigma})\\stackrel{H^1(\\phi)}{\\to}H^1(C_\\Sigma\/p^m).$$\nThis fact directly gives property (i) of the morphism $\\phi$.\n\n\n\nTurning now to property (ii), we write $q_{2}$ for the canonical isomorphism $$H^2(C_\\Sigma)\/p^m\\to H^2(C_\\Sigma\/p^m)$$ induced by the canonical projection morphism $C_\\Sigma\\to C_\\Sigma\/p^m$ as in Lemma \\ref{reducedbock} below.\n\nThen an explicit computation using our definition of $\\phi^2$ shows that the composition\n\\begin{multline*}H^1_\\Sigma(\\mathbb{A}_F,A^t[p^m])\\to H^1_\\Sigma(\\mathbb{A}_F,A^t)[p^m]\\to H^2(C_{{\\rm BD},\\Sigma}) \\stackrel{H^2(\\phi)}{\\to}H^2(C_\\Sigma\/p^m)\\stackrel{q_{2}^{-1}}{\\to}H^2(C_{\\Sigma})\/p^m\\\\ \\stackrel{\\sim}{\\to}\\cok(\\Delta)\/p^m\\to\\frac{H^2_c\\bigl(U_{F,\\Sigma},A^t[p^m]\\bigr)}{\\im\\bigl(\\kappa(A^t_\\Sigma(\\mathbb{A}_F)\/p^m)\\bigr)}\\end{multline*}\ncoincides with $-1$ times the canonical map. In this display, the fifth arrow is the canonical isomorphism occurring in the diagram (\\ref{Selmerdiagram}) and the sixth arrow is the canonical projection.\n\n\n\n\nThe commutativity of the diagram (\\ref{Selmerdiagram}), as well as of the diagram in Corollary \\ref{Tatepoitouexplicit} below, can now be combined with the above given fact to show that the composition\n\\begin{equation*} H^2(C_{{\\rm BD},\\Sigma}) \\stackrel{H^2(\\phi)}{\\to}H^2(C_{\\Sigma}\/p^m)\\stackrel{q_{2}^{-1}}{\\to}H^2(C_{\\Sigma})\/p^m\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee\/p^m\\to\\Sel^{(p^m)}(A_F)^\\vee\\end{equation*} is equal to $-v_{\\Sigma}$.\n\nGiven this fact, and since $q'_{2}$ is by definition equal to $r_2\/p^m\\circ q_{2}^{-1}$, the required equality of composite maps is now clear from their definitions.\n\\end{proof}\n\n\n\n\\subsection{The proof of Theorem \\ref{thecomptheorem}}\\label{compproof}\n\nAs a first step we use Lemma \\ref{morphismphi} to show that the Nekov\\'a\\v r height pairing $\\langle\\,,\\rangle^{\\rm N}_{p}$ is equal to the inverse of the pairing $\\langle\\,,\\rangle_p^{\\rm BD}$.\n\nTo do this we recall that we have set $m:=m_p(G)$ and also fixed a finite set $T$ of places of $k$ as in Lemma \\ref{tm}.\n\nThen for any finite subset $\\Sigma$ of $S_k^f$ that contains $$S_k^p\\cup (S_k^F\\cap S_k^f)\\cup S_k^A\\cup T,$$ any $x\\in A^t(k)_p$ and $y\\in A(k)_p$ one computes that\n\\begin{align}\\label{explicit comp} \\langle x,y\\rangle^{\\rm N}_{p}=\\, &\\langle x,y\\rangle_{C_{\\Sigma},r_1,r_2}\\\\ =\\, &\\langle x+p^m,y+p^m\\rangle_{C_{\\Sigma}\/p^m,q'_{1},q'_{2}}\\notag\\\\\n= \\, &\\langle x+p^m,y+p^m\\rangle_{C_{{\\rm BD},\\Sigma},r_{1,m},-r_{2,m}}\\notag\\\\\n=  &-\\langle x+p^m,y+p^m\\rangle_{C_{{\\rm BD},\\Sigma},r_{1,m},r_{2,m}}\\notag\\\\\n=  &-\\langle x,y\\rangle_p^{\\rm BD}.\\notag\\end{align}\n\nHere the second equality is given by Lemma \\ref{reducedbock} below and the third equality, after fixing a morphism $\\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m$ as in Lemma \\ref{morphismphi}, is immediate from the naturality of Bockstein homomorphisms and the description of $H^1(\\phi)$ and $H^2(\\phi)$ given in that result.\n\nBertolini and Darmon have defined in \\cite[\\S 3.4.1]{bert} and \\cite[\\S 2.2]{bert2}\na pairing \\[\\langle\\,,\\rangle_1:A^t(k)\\times A(k)\\to  G.\\] (Although the definition of this pairing is only given in the case that $A$ is an elliptic curve and $F\/k$ is an abelian $p$-extension, it extends naturally to this more general setting).\n\nIn addition, the results of Bertolini and Darmon in \\cite[Thm. 2.8, Rem. 2.10]{bert2} and of Tan in \\cite[Prop. 3.1]{kst} combine to imply directly that the pairing obtained by tensoring $\\langle\\,,\\rangle_1$ with $\\ZZ_p$ coincides with the Mazur-Tate pairing $\\langle\\,,\\rangle_p^{\\rm MT}$.\n\nGiven the latter fact, and the equality of (\\ref{explicit comp}), the proof of Theorem \\ref{thecomptheorem} is therefore completed by means of the following result.\n\n\\begin{lemma}\\label{bdcomp}The pairing $\\langle\\,,\\rangle^{\\rm BD}_p$ coincides with the pairing obtained by tensoring $\\langle\\,,\\rangle_{1}$ with $\\ZZ_p$.\n\\end{lemma}\n\\begin{proof}\nAs necessary preparation for the proof of this result, we fix a minimal set of generators $\\gamma_1,\\ldots,\\gamma_r,\\gamma_{r+1},\\ldots,\\gamma_s$ of $G$, ordered so that for $1\\leq i\\leq r$ the order $o(\\gamma_i)$ of $\\gamma_i$ is a power of $p$ and for $r<i\\leq s$ the order $o(\\gamma_i)$ of $\\gamma_i$ is prime to $p$.\n\nWe also denote by $J$ the cartesian product of the sets $\\{0,1,\\ldots,o(\\gamma_i)-1\\}$ as $i$ ranges through all indices between 1 and $s$.\n\nFor each index $1\\leq i\\leq s$ we write $F_i$ for the fixed field in $F$ of the subgroup of $G$ generated by $\\{\\gamma_j:j\\neq i\\}$.\n\nFor each index $1\\leq i\\leq r$ we let $$D_i^{(1)}:=-\\sum_{j=0}^{o(\\gamma_i)-1}j\\gamma_i^j\\in R$$ be the `derivative operator' of \\cite[Lem. 2.1]{bert2}.\n\n\nTo proceed with the proof, it is enough to verify that, for any points $P\\in A^t(k)$ and $Q\\in A(k)$, the image of $\\langle P,Q\\rangle_1$ in $\\ZZ_p\\otimes_{\\ZZ}G\\cong I_R\/I_R^2$ is equal to $\\langle P+p^m,Q+p^m\\rangle_{C_{{\\rm BD},T}}$, since the latter element is by definition equal to $\\langle P,Q\\rangle_p^{{\\rm BD}}$.\n\nTo compute the term $\\langle P,Q\\rangle_1$, we use the fact that the $G$-modules $\\Sel^T_{p^m}(A^t\/F)$ and $A_T(\\mathbb{A}_F)\/p^m$ are cohomologically-trivial established in Lemma \\ref{bdlemma}(ii).\n\nLet $a\\in \\Sel^{T}_{p^m}(A^t\/F)$ be chosen so that $\\Tr_G(a)$ is equal to the image of $P+p^m$ under the composite map\n$$A^t(k)_p\/p^m\\to\\bigl(A^t(F)_p\/p^m\\bigr)^G\\to\\Sel^{T}_{p^m}(A^t\/F)^G.$$\n\nLet $b\\in A_{T}(\\mathbb{A}_F)\/p^m$ be chosen so that $\\Tr_G(b)$ is equal to the image $Q_{\\rm loc}$ of $Q+p^m$ under the composite map\n$$A(k)_p\/p^m\\to\\bigl(A(F)_p\/p^m\\bigr)^G\\to\\bigl(A_{T}(\\mathbb{A}_F)\/p^m\\bigr)^G.$$\n\n\n\nAfter interpreting the local duality pairing $\\langle\\,,\\rangle_{T}$ of (\\ref{localpairing}) below as taking values in $Z$ in the canonical way as is done throughout \\cite{bert2}, the image of $\\langle P,Q\\rangle_1$ in $I_{R}\/I_{R}^2$ is by definition equal to\n\\begin{align}\\label{bdcomput}\\notag &\\bigl(\\sum_{g\\in G}\\langle g^{-1}(\\lambda_{T}(a)),b\\rangle_{T}\\cdot g\\bigr)+I_{R}^2\\\\\n\\notag =&\\bigl(\\sum_{j_\\bullet\\in J}\\langle(\\gamma_1^{-j_1}\\cdot\\ldots\\cdot\\gamma_s^{-j_s})(\\lambda_T(a)),b\\rangle_T\\cdot(\\gamma_1^{j_1}\\cdot\\ldots\\cdot\\gamma_s^{j_s})\\bigr)+I_{R}^2\\\\\n\\notag =&\\bigl(\\sum_{i=1}^{i=r}\\sum_{j_\\bullet\\in J}\\langle(\\gamma_1^{-j_1}\\cdot\\ldots\\cdot\\gamma_s^{-j_s})(\\lambda_T(a)),b\\rangle_T\\cdot j_i(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}})(\\lambda_T(a)),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2.\n\\end{align}\n\n\nWe now let, as we may, $a'_1,\\ldots,a'_s\\in H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ be chosen so that\n\\begin{equation}\\label{augmentationdivision}\\sum_{l=1}^{l=s}(\\gamma_l-1)\\cdot a_l'=\\lambda_{T}(a).\\end{equation}\n\nThen (\\ref{bdcomput}) implies that the image of $\\langle P,Q\\rangle_1$ in $I_{R}\/I_{R}^2$ is equal to\n\n\\begin{align*}&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}})\\bigl(\\sum_{l=1}^{l=s}(\\gamma_l-1)\\cdot a_l'\\bigr),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}}\\cdot(\\gamma_i-1)) (a_i'),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle \\Tr_G(a_i'),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle a_i',\\Tr_G(b)\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle a_i',Q_{\\rm loc}\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}Q_{\\rm loc}(u_T(a_i'))\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2.\n\\end{align*}\nIn this last expression we have identified $Q_{\\rm loc}$ with its double-dual.\n\nThe claimed result is thus equivalent to the equality $$\\langle P+p^m,Q+p^m\\rangle_{C_{{\\rm BD},T}}=\\bigl(\\sum_{i=1}^{i=r}Q_{\\rm loc}(u_T(a_i'))\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2$$ in $I_{R}\/I_{R}^2$.\n\nBut, taking into account the equality (\\ref{augmentationdivision}), and given that the $R$-modules $\\Sel^T_{p^m}(A^t\/F)$ and $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ are finitely generated and $G$-cohomologically-trivial by Lemma \\ref{bdlemma}(ii),\nthis equality follows directly from a diagram chase to compute the Bockstein homomorphism $\\beta_{C_{{\\rm BD},T}}$.\n\\end{proof}\n\n\n\n\n\n\\section{Modular symbols}\\label{mod sect} In this section we shall use the theory of modular symbols to study ${\\rm BSD}(A_{F\/k})$ in the case $A$ is an elliptic curve and $F$ is a totally real tamely ramified abelian extension of $k = \\QQ$. The results in this section will, in particular, extend the main results of Bley in \\cite{Bley3}.\n\nFor each natural number $d$ we set $\\zeta_d := {\\rm exp}(2\\pi i\/d)$. We write $\\QQ(d)$ for $\\QQ(\\zeta_d)$ and $\\QQ(d)^+$ for the\nmaximal real subfield of $\\QQ(d)$ and set $G_d := G_{\\QQ(d)\/\\QQ}$ and $G^+_d := G_{\\QQ(d)^+\/\\QQ}$.\n\nFor any element $a$ of $(\\ZZ\/d\\ZZ)^\\times$ we write $\\sigma_a$ for the element of $G_d$ that sends $\\zeta_d$ to $\\zeta_d^a$.\n\n\\subsection{Regularized Mazur-Tate elements} We first recall the definition and relevant properties of modular symbols as introduced by Mazur and Tate in \\cite{mt} and normalized by Darmon in \\cite{darmon} and explain their connection to the leading term element that occurs in Theorem \\ref{bk explicit}.\n\n\n\\subsubsection{}\\label{ell cirve section}We fix an elliptic curve $A$ that is defined over $\\QQ$ and has conductor $N$. We also fix a modular parametrisation $\\varphi_A: X_0(N) \\to A$ and write $c(\\varphi_A)$ for the associated Manin constant.\n\nWe write $\\Lambda$ for a N\\'eron lattice for $A$ in $\\CC$ and then identify $A(\\CC)$ with $\\CC\/\\Lambda$ and the N\\'eron differential $\\omega_0$ of $A$ with ${\\rm d}z$.\n\nWe define strictly positive real numbers $\\Omega^+$ and $\\Omega^-$ by the condition that $\\Lambda$ is a sublattice of $\\ZZ\\cdot\\Omega^+ + \\ZZ\\cdot i\\Omega^-$ with minimal index.\n\nThen the pullback under $\\varphi_A$ of $\\omega_0$ defines a cusp form of weight $2$ on $X_0(N)$ so\n\\[ \\varphi_A^*(\\omega_0) = c(\\varphi_A)f(q){\\rm d}q\/q = c(\\varphi_A)2\\pi if (z){\\rm d}z\\]\nwhere $f(q) = \\sum_{n \\ge 1}a_nq^n$, with $q = {\\rm exp}(2\\pi iz)$, is normalized such that $a_1 = 1$.\n\nWe now fix a natural number $c$ that is square-free and prime to $N$. For each integer $a$ we follow Mazur and Tate \\cite{mt} in defining the modular symbols $[ \\frac{a}{c}]^+$ and $[ \\frac{a}{c}]^{-}$ via the equality\n\\[ 2\\pi i\\int_{a\/c}^{a\/c + i\\infty}f(z){\\rm d}z = \\left[ \\frac{a}{c}\\right]^+\\Omega^+ + i\\left[ \\frac{a}{c}\\right]^{-}\\Omega^-.\\]\nRecalling that $[\\frac{a}{c}]^+$ depends only on the residue class of $a$ modulo $t$, we then follow Darmon \\cite{darmon} in defining the regularized modular\nsymbols\n\n\\[ \\left[ \\frac{a}{c}\\right]^* := \\sum_{t\\mid c}\\mu(c\/t)\\left[\\frac{a(c\/t)^{-1}}{t}\\right]^+.\\]\nHere $t$ runs over all natural numbers that divide $c$, $\\mu(-)$ is the M\\\"obius function and $(c\/t)^{-1}$ denotes the inverse of $c\/t$ modulo $t$ (which exists since $c$ is assumed to be squarefree).\n\nWe finally define the `regularized Mazur-Tate element of $A$ at level $c$' to be the element\n\\[ \\Theta^{\\rm MT}_{A,c} := \\frac{1}{2}\\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\left[\\frac{a}{c}\\right]^*\\sigma_a\\]\nof $\\QQ[G_c]$. In the sequel we identify $\\Theta^{\\rm MT}_{A,c}$ with its image under the canonical projection $\\QQ[G_c]\\to\\QQ[G_c^+]$.\n\n\\subsubsection{} In the next result we describe the precise link between the Mazur-Tate element $\\Theta^{\\rm MT}_{A,c}$ and the leading term element $\\calL^*_{A,\\QQ(c)^+\/\\QQ}$ that occurs in Theorem \\ref{bk explicit} (as defined in (\\ref{bkcharelement}) with respect to a fixed odd prime $p$).\n\nTo state the result we write $\\Upsilon_{c,0}$ for the subset of $\\widehat{G_c^+}$ comprising all characters $\\psi$\nfor which $L(A,\\check\\psi,1)$ does not vanish and define an idempotent of $\\CC[G_c^+]$ by setting\n\\[ e_{c,0} := \\sum_{\\psi\\in \\Upsilon_{c,0}} e_\\psi.\\]\n(It is then in fact straightforward to check that $e_{c,0}$ belongs to $\\QQ[G_c^+]$.)\n\n\n\nIn the sequel, we denote by $c_\\psi$ the conductor of a Dirichlet character $\\psi$.\n\n\\begin{lemma}\\label{gammatheta} For any odd prime number $p$, in $\\QQ[G_c^+]$ one has\n\\[ \\Theta^{\\rm MT}_{A,c}=  (c(\\varphi_A)\/c_\\infty)\\cdot \\bigl(\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi\\big)\\cdot e_{c,0}\\calL^*_{A,\\QQ(c)^+\/\\QQ}\\]\nwhere $c_\\infty$ is the number of connected components of $A(\\RR)$ and we set\n\\[ m_p^\\psi := \\begin{cases} p^{-1}, &\\text{ if $p$ divides $c$ but not $c_\\psi$},\\\\\n                             1, &\\text{otherwise.}\\end{cases}\\]\n\\end{lemma}\n\n\\begin{proof} The key point is that, by Darmon \\cite[Prop. 2.3]{darmon}, for every $\\psi\\in\\widehat{G_c^+}$ one has \\begin{equation}\\label{chiMT}(\\Theta^{\\rm MT}_{A,c})_\\psi=c(\\varphi_A)\\cdot\\frac{c}{c_\\psi}\\cdot\\frac{L_c(A,\\check{\\psi},1)\\tau_c(\\psi)}{2\\Omega^+}.\\end{equation}\nHere the $L$-function $L_c(A,\\check{\\psi},z)$ is truncated at all rational primes dividing $c$ and we use the (in general, imprimitive) Gauss sum\n\\[ \\tau_c(\\psi):= \\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\psi(a)\\cdot \\zeta^a_c.\\]\n\nSince the definition of $e_{c,0}$ implies that $L_c(A,\\check{\\psi},1)\\not= 0$ if and only if $e_\\psi e_{c,0}\\not= 0$, the equalities (\\ref{chiMT}) combine to imply that $\\Theta^{\\rm MT}_{A,c} = e_{c,0}\\cdot\\Theta^{\\rm MT}_{A,c}$.\n\nIn addition, since each character $\\psi$ in $\\widehat{G_c^+}$ is even one has both $w_{\\infty,\\psi}=1$ and\n\\[ 2\\Omega^+ = c_\\infty\\cdot\\Omega_{A,\\infty}^+ = c_\\infty\\cdot \\Omega_{A,\\infty}^\\psi\\]\nwhilst, if $p$ divides $c$, an explicit computation of each factor $\\rho_{p,\\psi}$ occurring in (\\ref{bkcharelement}) shows that it is equal to $(m_p^\\psi)^{-1}$.\n\nThe claimed equality now follows directly upon substituting these facts and the result of Lemma \\ref{imprimitive GGS} below into the equality of (\\ref{chiMT}), recalling the explicit definition (\\ref{bkcharelement}) of the element $\\mathcal{L}^*_{A,\\QQ(c)^+\/\\QQ}$ and noting that, in terms of the notation of \\S\\ref{somr tmc sec}, one has $L_c(A,\\check{\\psi},1) = L_{S_{\\rm r}}(A,\\check{\\psi},1)$.\n\\end{proof}\n\n\\begin{lemma}\\label{imprimitive GGS} For each $\\psi$ in $\\widehat{G_c^+}$ one has $\\tau_c(\\psi) = \\tau^\\ast(\\QQ,\\psi)$.\\end{lemma}\n\n\\begin{proof} Since $\\psi$ is linear, the definition of the Galois-Gauss sum $\\tau(\\QQ,\\psi)$ implies that it is equal to the primitive Gauss sum $\\tau_{c_\\psi}(\\psi)$.\n\nUpon recalling the explicit definition of $\\tau^\\ast(\\QQ,\\psi)$ from \\S\\ref{mod GGS section}, it is thus enough to show that\n\\[ \\tau_c(\\psi) = \\tau_{c_\\psi}(\\psi)\\cdot \\prod_{\\ell \\mid (c\/c_\\psi)}(-\\psi(\\sigma_\\ell)^{-1})\\]\nwhere $\\ell$ runs over all prime divisors of $c\/c_\\psi$.\n\nTo prove this it is in turn enough to show that for each prime divisor $\\ell$ of $c\/c_\\psi$ one has an equality $\\tau_c(\\psi) = -\\psi(\\sigma_\\ell)^{-1}\\cdot \\tau_{c'}(\\psi)$ with $c' := c\/\\ell$.\n\nTo show this we  fix integers $s$ and $t$ such that $s\\ell + tc' = 1$. Then\n$\\zeta_c = \\zeta_\\ell^t\\cdot \\zeta_{c'}^s$ and hence\n\n\\begin{align*} \\tau_c(\\psi) = \\, &\\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\psi(a)\\cdot \\zeta_c^a\\\\\n= \\, &\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_1a_2)\\cdot \\zeta_\\ell^{ta_1}\\zeta_{c'}^{sa_2}\\\\\n= \\, &\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\left(\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\zeta_\\ell^{ta_1}\\right)\\psi(a_2)\\cdot\\zeta_{c'}^{sa_2}\\\\\n= \\, &-\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_2)\\cdot\\zeta_{c'}^{sa_2}\\\\\n= \\, &-\\psi(s)^{-1}\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_2)\\cdot\\zeta_{c'}^{a_2}\\\\\n= \\, &-\\psi(\\sigma_\\ell)^{-1}\\cdot\\tau_{c'}(\\psi).\n\\end{align*}\nThe third equality here follows from the fact that $\\psi(a_1) = 1$ for all $a_1$ in $(\\ZZ\/\\ell\\ZZ)^\\times$, the fourth from the equality $\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\zeta_\\ell^{ta_1} = -1$ and the last from the fact that $\\sigma_\\ell$ acts as multiplication by $s$ on $(\\ZZ\/c'\\ZZ)^\\times$.\n\\end{proof}\n\n\n\\begin{remark}\\label{euler factor rem}{\\em For any abelian extension $F$ of $\\QQ$ and any subfield $F'$ of $F$ we write $\\pi_{F\/F'}$ for the natural projection map $\\QQ[G_{F\/\\QQ}] \\to \\QQ[G_{F'\/\\QQ}]$.\n\nThen, if $p$ is any prime that does not divide $c$, the interpolation formula (\\ref{chiMT}) implies that\n\\[\\pi_{\\QQ(pc)\/\\QQ(c)}(\\Theta^{\\rm MT}_{A,pc}) =- \\sigma_p(p -\\sigma_p^{-1}a_p + \\sigma_p^{-2})\\cdot \\Theta_{A,c}^{\\rm MT}\\]\nwith $a_p=1+p-|A(\\mathbb{F}_p)|$.\n}\\end{remark}\n\n\n\n\\subsection{Mazur-Tate elements and refined BSD}\n\n\\subsubsection{}We now fix a tamely ramified real abelian extension $F$ of $\\QQ$ and set $G := G_{F\/\\QQ}$.\n\nWe also fix an odd prime $p$, write $F'$ for the maximal subfield of $F$ whose degree over $\\QQ$ is prime to $p$ and set $G' := G_{F'\/\\QQ}$.\n\nWe fix an elliptic curve $A$ that is defined over $\\QQ$ and whose conductor $N$ is prime to the conductor $c$ of $F$. We always assume that the Tate-Shafarevich group $\\sha(A_{F'})$ of $A$ over $F'$ is finite.\n\nThen, since the conductor $c$ of $F$ is square-free, we can define an element of $\\QQ[G]$ by setting\n\\[ \\Theta^{\\rm MT}_{A,F} := \\pi_{\\QQ(c)\/F}(\\Theta^{\\rm MT}_{A,c}).\\]\nWe also write $e_{F,0}$ for the idempotent $\\pi_{\\QQ(c)^+\/F}(e_{c,0})$ of $\\QQ[G]$.\n\n\nIn this section we shall prove the following results.\n\n\\begin{theorem}\\label{MT result} We assume that $A$, $F$ and $p$ are such that $(c,N)=1$ and also satisfy all of the following conditions.\n\n\\begin{itemize}\n\\item[(a)]   $p$ is prime to $|A(F')_{\\rm tor}|$;\n\\item[(b)]  $p$ is prime to $N$ and to the Tamagawa number of $A$ at any prime divisor of $N$;\n\\item[(c)] If $p$ divides $c$, then $p$ is prime to $|A(\\kappa_v)|$ for any $p$-adic place $v$ of $F'$ and, in addition, $A$ has ordinary reduction at $p$;\n\\item[(d)] $p$ is prime to $|A(\\kappa_v)|$ for any place $v$ of $F'$ of residue characteristic dividing $c$.\n \\item[(e)] $p$ is prime to the Manin constant $c(\\varphi_A)$.\n\n\\end{itemize}\n\nThen the following assertions are valid.\n\\begin{itemize}\n\\item[(i)] The projection to $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$ of the equality in ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if one has\n\\begin{equation*}\\label{refMT}\\ZZ_p[G]\\cdot\\Theta^{\\rm MT}_{A,F} = e_{F,0}\\cdot{\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee).\\end{equation*}\n\n\n\\item[(ii)] If $L(A_{F},1)$ is non-zero, then ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if $\\Theta^{\\rm MT}_{A,F}$ belongs to ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ and, in addition, the $p$-part of the Birch and Swinnerton-Dyer conjecture is valid for $A$ over $F'$.\n\\end{itemize}\n\\end{theorem}\n\nThe following explicit corollary of Theorem \\ref{MT result} extends the main results of Bley in \\cite{Bley3} (for more details see Remark \\ref{mod bley rem} below).\n\n\\begin{corollary}\\label{bleycor} Fix an elliptic curve $A$ over $\\QQ$ with $L(A_{\\QQ},1)\\not=0$. Then for each natural number $n$ there are infinitely many primes $p$, and for each such prime $p$ infinitely many abelian extensions $F\/\\QQ$ that satisfy all of the following properties.\n\n\\begin{itemize}\n\\item[$\\bullet$] ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid.\n\\item[$\\bullet$] The exponent of $G_{F\/\\QQ}$ is divisible by $p^n$.\n\\item[$\\bullet$] The $p$-rank of $G_{F\/\\QQ}$ is at least $n$.\n\\item[$\\bullet$] The order of $G_{F\/\\QQ}$ is divisible by at least $n$ distinct primes.\n\\item[$\\bullet$] The discriminant of $F$ is divisible by at least $n$ distinct primes.\n\\end{itemize}\n\\end{corollary}\n\n\n\n\\subsubsection{} We shall now prove Theorem \\ref{MT result}.\n\nAt the outset we note that the given assumptions (a)-(e) imply that the hypotheses (H$_1$)-(H$_5$) listed in \\S\\ref{tmc} are satisfied and so Proposition \\ref{explicitbkprop} implies that the complex\n\\[ C_{F,0}:=\\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}^{\\mathbb{L}}{\\rm SC}_p(A_{F\/\\QQ})\\]\nbelongs to $D^{\\rm perf}(\\ZZ_p[G]e_{F,0})$.\n\nIn addition, by the main result of Kato in \\cite{kato}, one knows that $e_\\psi(\\CC_p\\otimes_{\\ZZ_p} A(F)_p)$ vanishes for every character $\\psi$\n in $\\Upsilon_{c,0}$ and hence that the complex $\\QQ_p\\otimes_{\\ZZ_p}C_{F,0}$ is acyclic.\n\nIt follows that for every field isomorphism $j:\\CC\\cong\\CC_p$, one has\n\\begin{equation*}\\label{zeroEuler}\\pi_{0,*}(\\chi_{G,p}({\\rm SC}_p(A_{F\/\\QQ}),h^j_{A,F}))=\\chi_{\\ZZ_p[G]e_{F,0}}(C_{F,0},0)\\end{equation*}\nin $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$, where $$\\pi_{0,\\ast}:K_0(\\ZZ_p[G],\\CC_p[G])\\to K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$$ denotes the canonical projection homomorphism.\n\nNow, in Theorem \\ref{bk explicit}, the sets $S_{p,{\\rm w}}$ and $S_{p,{\\rm u}}^\\ast$ are empty (since $F$ is a tamely ramified extension of $k = \\QQ$) and so the above equality implies that the image under $\\pi_{0,\\ast}$ of the equality in ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if one has\n\\begin{equation}\\label{first approx} \\pi_{0,\\ast}(\\delta_{G,p}(\\calL^*_{A,F\/\\QQ})) = \\chi_{\\ZZ_p[G]e_{F,0}}(C_{F,0},0).\\end{equation}\n\nTo interpret this equality we identify $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$ with the multiplicative group of invertible $\\ZZ_p[G]e_{F,0}$-lattices in $\\CC_p[G]e_{F,0}$ (as in Remark \\ref{comparingdets}).\n\nIn particular, since the description of the cohomology of ${\\rm SC}_p(A_{F\/\\QQ})$ given in (\\ref{bksc cohom}) combines with hypothesis (a) and the acyclicity of $\\QQ_p\\otimes_{\\ZZ_p}C_{F,0}$ to imply\n\\[ H^i(C_{F,0}) = \\begin{cases} \\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}\\Sel_p(A_{F})^\\vee, &\\text{if $i=2$,}\\\\\n0, &\\text{otherwise,}\\end{cases}\\]\nit follows that the right hand side of (\\ref{first approx}) identifies with the lattice\n\\[{\\rm Fit}^0_{\\ZZ_p[G]e_{F,0}}(\\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}\\Sel_p(A_{F})^\\vee) = e_{F,0}\\cdot {\\rm Fit}^0_{\\ZZ_p[G]}(\\Sel_p(A_{F})^\\vee).\\]\n\nIn addition, the left hand side of (\\ref{first approx}) identifies with the $\\ZZ_p[G]$-module generated by $e_{F,0}\\calL^*_{A,F\/\\QQ}$.\n Thus, since both of the terms $c_\\infty$ and $c(\\varphi_A)$ belong to $\\ZZ_p^\\times$ (the former because $p$ is odd and the latter by hypothesis (e)), to derive claim (i) from the formula of Lemma \\ref{gammatheta} it is enough to show that $$\\pi_{\\QQ(c)^+\/F}\\bigl(\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi\\bigr)$$ belongs to $\\ZZ_p[G]^\\times$.\n\nTo verify this assertion it is enough to show that $\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi$ belongs to $\\ZZ_p[G_c^+]^\\times$ and this fact follows directly from Lemma \\ref{bley lemma} below with $A=G_c^+$, $i=1$ and, for each positive divisor $d$ of $c$, the subgroup $H_d$ of $G_c^+$ taken to be $G_{\\QQ(c)^+\/\\QQ(d)^+}$.\n\nIn order to prove claim (ii), we assume that $L(A_{F},1)\\not= 0$. This implies, by Kato \\cite{kato}, that the groups $\\sha(A_F)$ and $A(F)$ are both finite and also that the idempotent $e_{F,0}$ is equal to $1$.\n\nIn particular,\nthe module $\\Sel_p(A_F)^\\vee$ is canonically isomorphic to $\\sha(A_F)[p^\\infty]^\\vee$ (see the exact sequence (\\ref{sha-selmer})).\n\nThe module $\\Sel_p(A_F)^\\vee$  is therefore also canonically isomorphic to $\\sha(A_F)[p^\\infty]$ via the Cassels-Tate pairing.\n\nClaim (i) therefore implies that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if ${\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee) = {\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ is generated over $\\ZZ_p[G]$ by $\\Theta^{\\rm MT}_{A,F}$.\n\nNow, by Nakayama's Lemma, the latter assertion is valid if and only if ${\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F}$\n and, in addition, the module of coinvariants\n\n \\[ \\ZZ_p[G']\\otimes_{\\ZZ_p[G]}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\/(\\ZZ_p[G]\\cdot \\Theta^{\\rm MT}_{A,F})\\bigr)\\]\n\nvanishes, or equivalently\n\\begin{equation}\\label{conv reinter} \\pi_{F\/F'}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\\bigr) = \\ZZ_p[G']\\cdot \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}).\\end{equation}\n\nWe next note that, since ${\\rm SC}_p(A_{F\/\\QQ})$ is acyclic outside degree two, the isomorphism\n\n\\[ \\ZZ_p[G']\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_p(A_{F\/\\QQ})\\cong {\\rm SC}_p(A_{F'\/\\QQ})\\]\nin $D(\\ZZ_p[G'])$ coming from Proposition \\ref{explicitbkprop}(iii) induces an isomorphism of $\\ZZ_p[G']$-modules\n\\begin{multline}\\label{comp iso} \\ZZ_p[G']\\otimes_{\\ZZ_p[G]}{\\rm Sel}_p(A_{F})^\\vee \\cong H^2(\\ZZ_p[G']\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_p(A_{F\/\\QQ})) \\\\ \\cong H^2({\\rm SC}_p(A_{F'\/\\QQ})) \\cong {\\rm Sel}_p(A_{F'})^\\vee , \\end{multline}\nand hence an equality\n\\begin{equation}\\label{proj1} \\pi_{F\/F'}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\\bigr) = {\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee).\\end{equation}\n\nIn addition, if we write $c'$ for the conductor of $F'$, then the distribution relation in Remark \\ref{euler factor rem} implies that\n\\begin{equation}\\label{dist rel 2} \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}) = \\Theta^{\\rm MT}_{A,F'}\\cdot \\prod_{\\ell \\mid (c\/c')}-\\sigma_\\ell(\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}).\\end{equation}\n\nWe claim next that for each prime factor $\\ell$ of $c\/c'$ the element $\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}$ is a unit in $\\ZZ_p[G']$. This is because any such $\\ell$ cannot be equal to $p$ (since $F\/F'$ is a $p$-extension and $F\/\\QQ$ is tamely ramified) so that $\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}$ is a generator of the Fitting ideal of the $\\ZZ_p[G']$-module $\\bigoplus_{v\\in S_{F'}^\\ell} A(\\kappa_v)$, whilst the given condition (d) implies that the latter direct sum contains no element of order $p$.\n\nGiven this, the equality (\\ref{dist rel 2}) implies that $\\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F})$ differs from $\\Theta^{\\rm MT}_{A,F'}$ by a unit in $\\ZZ_p[G']$ and thus combines with (\\ref{proj1}) to imply that (\\ref{conv reinter}) is valid if and only if the invertible $\\ZZ_p[G']$-sublattice\n\\begin{equation}\\label{last ideal} {\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee)\\cdot (\\ZZ_p[G']\\cdot (\\Theta^{\\rm MT}_{A,F'}))^{-1}\\end{equation}\nof $\\QQ_p[G']$ is trivial.\n\nIn addition, if ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_F)[p^\\infty])={\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F}$ then the above arguments would also imply that ${\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F'}$. Since $p\\nmid |G|'$, the lattice (\\ref{last ideal}) would then be trivial if and only if its image under the natural restriction map $\\varrho_{G'}$ to the group of invertible $\\ZZ_p$-sublattices of $\\QQ_p$ is trivial.\n\nIn particular, since the argument of claim (i) shows that the triviality of (\\ref{last ideal}) is equivalent to the validity of ${\\rm BSD}_p(A_{F'\/\\QQ})$(iv), the observations in Remark \\ref{consistency remark}(ii) and (iii) imply that the triviality of its image under $\\varrho_{G'}$ is equivalent to the validity of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F'$, as required to complete the proof of Theorem \\ref{MT result}.\n\n\n\n\n\nThe following algebraic result was used above and will also be useful in \\S\\ref{HHP}.\n\n\\begin{lemma}\\label{bley lemma}\nLet $A$ be a finite abelian group, $c$ a square-free positive integer and $(c_\\psi)_{\\psi\\in\\widehat{A}}$ a family of positive divisors of $c$. We fix an odd prime number $p$, we let $n_\\psi$ be equal to $p$ if $p$ divides $c$ but not $c_{\\psi}$, and we let $n_{\\psi}$ be equal to 1 otherwise.\n\nFix $i\\in\\{1,2\\}$ and assume that, for each positive divisor $d$ of $c$, there exists a subgroup $H_d$ which satisfies both of the following properties:\n\\begin{itemize}\\item[(i)] $d$ is divisible by an integer $c_{\\psi}$ if and only if $H_d\\subseteq\\ker(\\psi)$;\n\\item[(ii)] the order of $H_d$ divides $\\prod_{\\ell\\mid\\frac{c}{d}}(l+(-1)^i)$ (where $\\ell$ runs over primes dividing $\\frac{c}{d}$).\n\\end{itemize}\n\nThen the element $$\\sum_{\\psi \\in \\widehat{A}}((c_{\\psi}\/c) n_{\\psi})^i\\cdot e_{\\psi}$$ belongs to $\\ZZ_p[A]^\\times$\n\\end{lemma}\n\n\\begin{proof} Since $((c_{\\psi}\/c)n_{\\psi})^i$ is a $p$-adic unit for all $\\psi$ it suffices to prove that the inverse $x_i := \\sum_{\\psi \\in \\widehat{A}}(c_{\\psi}\/c)^{-i} n^{-i}_{\\psi}\\cdot e_{\\psi}$ of the given sum belongs to $\\ZZ_p[A]$.\n\nFor each divisor $\\delta$ of $c$ we set\n\\[ f_{p,i}(\\delta) := \\begin{cases} \\bigl(c\/\\delta\\bigr)^ip^{-i}, &\\text{ if $p$ divides $c\/\\delta$,}\\\\\n                          \\bigl(c\/\\delta\\bigr)^i, &\\text{ otherwise.}\\end{cases}\\]\n\nThen $x_i$ is equal to $\\sum_{\\psi \\in \\widehat{A}}f_{p,i}(c_{\\psi})\\cdot e_{\\psi}$ and so, when combined with assumption (i), the approach of Bley in \\cite[Prop. 3.1(a)]{Bley3} implies the required containment is true if for each positive divisor $d$ of $c$, the sum\n\\begin{equation}\\label{tricky sum} {\\sum}_{t \\mid \\frac{c}{d}} f_{p,i}\\left(dt\\right)\\mu(t)=\\sum_{\\{t:p\\nmid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^i\\mu(t)+\\sum_{\\{t:p\\mid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^ip^{-i}\\mu(t)\\end{equation}\nis divisible by $|H_d|$, where $\\mu(-)$ denotes the M\\\"obius function on square-free natural numbers.\n\n\nIf $d$ is such that $p$ does not divide $c\/d$, then the sum (\\ref{tricky sum}) is equal to\n\\[\\sum_{\\{t:p\\nmid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^i\\mu(t)=(\\frac{c}{d})^i\\sum_{t\\mid\\frac{c}{d}}t^{-i}\\mu(t)=(\\frac{c}{d})^i\\prod_{\\ell\\mid\\frac{c}{d}}(1-\\ell^{-i})=\\prod_{\\ell\\mid\\frac{c}{d}}(\\ell^{i}-1). \\]\n\n\nThe claimed result is thus true in this case since, using assumption (ii), $|H_d|$ divides (\\ref{tricky sum}) both if $i=1$ and also if $i=2$ because $\\ell+1$ divides $\\ell^2-1$ for each prime divisor $\\ell$ of $c\/d$.\n\nWe now assume that $p$ divides $c\/d$. In this case $p$ divides $c$ but not $d$ so the sum (\\ref{tricky sum}) is equal to\n\\begin{align*} &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(ds)\\mu(s)\n+ {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(dsp)\\mu(sp)\\\\\n= \\, &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(ds)\\mu(s)\n- {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(dsp)\\mu(s)\\\\\n=\\, &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} \\left(\\frac{c}{dsp}\\right)^i\\mu(s)\n- {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} \\left(\\frac{c}{dsp}\\right)^i\\mu(s)\\\\\n= \\, &0.\n\\end{align*}\nand the claimed result is clear.\n\n\\end{proof}\n\n\\subsubsection{} In this section we prove Corollary \\ref{bleycor}\n\n\nIf $L(A_\\QQ,1)$ is non-zero, then Kolyvagin has proved that $\\sha(A_\\QQ)$ and $A(\\QQ)$ are both finite and one knows that the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A_\\QQ$ is valid for all but finitely many primes $p$.\n\n\nTo prove Corollary \\ref{bleycor} it is therefore enough to fix a natural number $n$, assume to be given an odd prime $p'$ and a tamely ramified real abelian extension $F'\/\\QQ$ that has discriminant $d_{F'}$ prime to $N$ and is such that $L(A_{F'},1)$ is non-zero and all of the stated conditions are satisfied after replacing $p, F$ and $n$ by $p', F'$ and $n-1$, and then deduce from this assumption the assertion of Corollary \\ref{bleycor} for the fixed $n$. Indeed, the validity of Corollary \\ref{bleycor} would then follow from the previous paragraph by induction since ${\\rm BSD}_p(A_{\\QQ\/\\QQ})$(iv) is equivalent to $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A_\\QQ$ (see Remark \\ref{consistency remark}(iii)).\n\nWe hence fix a natural number $n$ as well as $p'$ and $F'$ with the assumed given properties.\nThen the results of Kato \\cite{kato} imply that $\\sha(A_{F'})$ and $A(F')$ are both finite and so we may fix a prime $p$ that is large enough to ensure that all of the following conditions are satisfied: the $p$-torsion subgroups of $A(F')$ and ${\\rm Sel}(A_{F'})^\\vee$ are trivial, $p$ is prime to the degree of $F'\/\\QQ$, to $d_{F'}$, to $N$, to $c(\\varphi_A)$ and to the Tamagawa numbers of $A$ at any place of $F'$ that divides $N$ and the reduction of $A$ at any place $v$ of $F'$ that divides $d_{F'}$ has no element of order $p$.\n\nIn addition, if we set $m := |A(\\QQ)_{\\rm tor}|$ and $G':=G_{F'\/\\QQ}$, then $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ[1\/2m][G']$ and so the interpolation property (\\ref{chiMT}) combines with the non-vanishing of $L(A_{F'},1)$ to imply that $\\Theta^{\\rm MT}_{A,F'}$ generates over $\\ZZ[1\/2m][G']$ a finite index submodule of\n $\\ZZ[1\/2m][G']$.\n\nThis means that we can also assume the prime chosen above to be such that both $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ_p[G']^\\times$ and the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F'$ is valid.\n\n\nNext we note that if $A$ does not have complex multiplication, then a result of Serre \\cite{serre} implies that for all sufficiently large primes $q$ the group\n$G_{\\QQ(A[q])\/\\QQ}$ is isomorphic to ${\\rm GL}_2(\\mathbb{F}_q)$ and so has order $(q^2-1)(q^2-q)\/2$, whilst if $A$ has complex multiplication by an order in an imaginary quadratic field $K$, then the theory of complex multiplication implies that for all sufficiently large $q$ the degree of $\\QQ(A[q])\/\\QQ$ is either $(q-1)^2\/w_K$ (if $q$ splits in $K$) or $(q^2-1)\/w_K$ (if $q$ is inert in $K$), where $w_K$ is the number of roots of unity in $K$.\n\nIt follows that for all sufficiently large primes $q$ the degree of $\\QQ(A[q])\/\\QQ(\\mu_q)$ is either $(q^2-1)q\/2$, $(q-1)\/w_K$ or $(q+1)\/w_K$.\n In particular, for sufficiently large $q$ the degree of $F'(\\mu_{q^n})\/\\QQ(\\mu_q)$ is not divisible by the degree of $\\QQ(A[q])\/\\QQ(\\mu_q)$ and so $\\QQ(A[q])$ is not contained in $F'(\\mu_{q^n})$.\n\nIn particular, we may assume that the prime $p$ fixed earlier has this property and then, by the Tchebotarev density theorem, it follows that there is an infinite set $\\mathfrak{L}$ of primes that each split completely in $F'(\\mu_{p^n})$ but not in $\\QQ(A[p])$.\n\nThen for any such prime $\\ell\\in\\mathfrak{L}$ one has $\\ell \\equiv 1$ modulo $p^n$ and, by the argument used prove \\cite[Th. 1.3]{Bley3}, also $a_\\ell \\not\\equiv 2$ modulo $p$ so that the order of $A(\\kappa_v) = A(\\mathbb{F}_\\ell)$ is prime to $p$ for any $\\ell$-adic place $v$ of $F'$.\n\nFor any given ordered subset $\\{\\ell_i: i \\in [n]\\}$ of $\\mathfrak{L}$ of cardinality $n$ and for each index $i$ we write $F_i$ for the unique subextension of $\\QQ(\\ell_i)$ of degree $p^n$.\n\nWe claim that the compositum $F$ of $F'$ with each of the fields $F_i$ has all of the  properties stated in Corollary \\ref{bleycor} with respect to the fixed $n$ and $p$.\n\nIt is certainly clear that this field satisfies all of the claimed properties except possibly for the assertion that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid.\n\nIn addition, our choice of $p$ implies the hypotheses of Theorem \\ref{MT result} are satisfied by the data $A, F\/\\QQ$ and $p$.\n\nIn particular, since the degree of $F\/F'$ is a power of $p$, and our choice of $p$ implies the $p$-part of the Birch and Swinnerton-Dyer Conjecture is valid for $A$ over $F'$, the criterion of Theorem \\ref{MT result}(ii) reduces us to showing that $\\Theta^{\\rm MT}_{A,F}$ belongs to ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ and $L(A_{F},1)$ is non-zero.\n\nNow the isomorphism (\\ref{comp iso}) is valid in this case and combines with our choice of $p$ to imply that $\\sha(A_{F})[p^\\infty]$ is trivial and hence that ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ is equal to $\\ZZ_p[G]$ and so contains $\\Theta^{\\rm MT}_{A,F}$.\n\nIn addition, the interpolation formula (\\ref{chiMT}) implies that $L(A_{F},1)$ is non-zero if and only if $\\Theta^{\\rm MT}_{A,F}$ is an invertible element of $\\QQ[G]$.\n\nTo complete the proof of Corollary \\ref{bleycor} it is therefore enough to show that $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]^\\times$.\n\nBut, since $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]$ and the order of $G_{F\/F'}$ is a power of $p$, it belongs to $\\ZZ_p[G]^\\times$ if and only if its image under $\\pi_{F\/F'}$ belongs to $\\ZZ_p[G']^\\times$.\n\nIn addition, since each prime $\\ell_i$ splits completely in $F'$, the distribution relation (\\ref{dist rel 2}) implies that\n\\[ \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}) = \\Theta^{\\rm MT}_{A,F'}\\cdot \\prod_{i=1}^{i=n}-(\\ell_i- a_{\\ell_i} + 1).\\]\nIn particular, since our choice of $p$ guarantees that both $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ_p[G']^\\times$ and for each $i$ one has\n\\[ \\ell_i - a_{\\ell_i} + 1 \\equiv 2-a_{\\ell_i} \\not\\equiv 0 \\,\\, \\text{modulo} \\,\\, p,\\]\nit follows that $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]^\\times$, as required.\n\n\n\\begin{remark}\\label{mod bley rem}{\\em The results of Bley in \\cite[Cor. 1.4 and Rem. 4.1]{Bley3} show that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid for an infinite family of abelian extensions $F\/\\QQ$ of exponent $p$ and rely on the explicit computations of Fearnley, Kisilevsky and Kuwata in \\cite{fkk} concerning the twists of modular elements by Dirichlet characters of prime order. The above argument avoids these computations by directly using the theorem of Kato. In addition, whilst the argument above only constructs fields $F$ in which $p$ is unramified, it is possible to modify the approach to construct families of abelian fields $F$ in which $p$ can in principle be tamely ramified and ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if $L(A_F,1)$ does not vanish. }\\end{remark}\n\n\n\\section{Heegner points}\\label{HHP}\n\nIn this section we again fix an elliptic curve $A$ that is defined over $\\QQ$ and has conductor $N$ and also assume that the modular parametrisation $\\varphi_A$ that was fixed in \\S\\ref{mod sect} is of minimal degree.\n\nWe also fix a quadratic imaginary field $K$ in which all prime divisors of $N$ split and a square-free product $c$ of primes that are both inert in $K$ and coprime to $N$. We write $K_c$ for the ring class field of $K$ of conductor $c$: this is an abelian extension of $K$ that is ramified exactly at the primes dividing $c$; moreover $K_c$ contains the Hilbert class field $K_1$ of $K$ and class field theory gives a natural isomorphism\n\\begin{equation}\\label{explicit iso} G_{K_c\/K_1} \\cong (\\mathcal{O}_K\/c\\mathcal{O}_K)^\\times\/(\\ZZ\/c\\ZZ)^\\times. \\end{equation}\n(cf. \\cite[\\S3]{gross_koly}).\n\nWe set $G_c := G_{K_c\/K}$ and $h_c := |G_c|$.\n\n\\subsection{Zhang's Theorem and a conjecture of Bradshaw and Stein}\\label{thm gzz} The above hypothesis on $K$ implies that the ideal $N\\mathcal{O}_K$ factorises as $\\mathcal{N}\\overline{\\mathcal{N}}$ where the natural map $\\ZZ\/N\\ZZ\\to \\mathcal{O}_K\/\\mathcal{N}$ is bijective. We write $\\mathcal{O}_c$ for the order $\\ZZ + c\\mathcal{O}_K$ in $K$ and set ${\\mathcal{N}}_c := \\mathcal{N}\\cap \\mathcal{O}_c$.\n\nFixing an embedding of $K$ into $\\CC$ the pair $(\\CC\/\\mathcal{O}_c,{\\mathcal{N}}_c^{-1}\/\\mathcal{O}_c)$ defines a CM elliptic curve equipped with a cyclic subgroup of order $N$ and the isomorphism class of this pair defines a point $x_c$ on $X_0(N)(K_c)$.\n\nWe define the `higher Heegner point' $y_c$ to be the element $\\varphi_A(x_c)$ of $A(K_c)$.\n\n\\subsubsection{\nAssuming $L(A_K,z)$ vanishes to order one at $z=1$, Zhang's generalisation \\cite{zhang01, zhang} of the seminal results of Gross and Zagier in \\cite{GZ} implies that for every $\\psi$ in $\\widehat{G_c}$ the function $L(A_K,\\psi,z)$ vanishes to order one at $z=1$, that the element $e_\\psi (y_c)$ of $\\CC\\cdot A(K_c)$ is non-zero and that the complex number $\\epsilon_{A,c,\\psi}$ defined by the equality\n\\begin{equation}\\label{e-def} L'(A_{K},\\check{\\psi},1) = \\epsilon_{A,c,\\psi}\\cdot\n\\frac{\\Omega^+\\Omega^-C}{c_\\psi\\sqrt{|d_K|}}\\cdot h_c\\cdot\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c}\\end{equation}\nsatisfies both\n\\begin{equation}\\label{stark ec} \\epsilon_{A,c,\\psi} \\in \\QQ(\\psi)^\\times \\,\\,\\,\\,\\text{ and }\\,\\,\\,\\, (\\epsilon_{A,c,\\psi})^\\omega = \\epsilon_{A,c,\\psi^\\omega} \\,\\,\\,\\,\\text{ for all $\\omega\\in G_{\\QQ(\\psi)\/\\QQ}$.}\\end{equation}\nHere we have set\n\\[ C := 4 \\cdot c_\\infty \\cdot c(\\varphi_A)^{-2}\\cdot \\bigl\\vert\\mathcal{O}_{K}^\\times\\bigr\\vert^{-2},\\]\nand written $c_\\psi$ for the least divisor of $c$ such that $\\psi$ factors through the restriction map $G_c\\to G_{c_\\psi}$ and $\\langle -,-\\rangle_{K_c}$ for the $\\CC$-linear extension of the N\\'eron-Tate height pairing for $A$ relative to the field $K_c$.\n\n\nTo investigate the conjecture ${\\rm BSD}(A_{K_c\/K})$ it is convenient to consider a version of the equality (\\ref{e-def}) that is adapted to truncated Hasse-Weil-Artin $L$-series.\n\nTo do this we follow Darmon \\cite{darmon} in defining the `regularised higher Heegner points of $A$ at level $c$' by setting\n\n\\[ z_c := \\sum_{m \\mid c}\\mu(m)y_m, \\,\\,\\text{ and }\\,\\,\\,\\,z'_c := \\sum_{m \\mid c}y_m.\\]\nFor each $\\psi$ in $\\widehat{G_c}$ we then define a non-zero complex number $\\epsilon_{A,\\psi}$ by means of the equality\n\\begin{equation}\\label{u-def} L'_{c}(A_{K},\\check{\\psi},1) = \\epsilon_{A,\\psi}\\cdot\n\\frac{\\Omega^+\\Omega^-C}{c_\\psi\\sqrt{|d_K|}}\\cdot \\frac{h_c}{(c\/c_\\psi)^2}\\cdot\\langle e_{\\psi}(z_c),e_{\\check\\psi}(z'_c)\\rangle _{K_c}\n\\end{equation}\nwhere the $L$-series $L_{c}(A_{K},\\check{\\psi},z)$ is truncated at all places of $K$ corresponding to rational prime divisors of $c$.\n\nThe next result describes the precise connection between the algebraic numbers $\\epsilon_{A,c,\\psi}$ and $\\epsilon_{A,\\psi}$ that are defined via the equalities (\\ref{e-def}) and (\\ref{u-def}). This result shows, in particular, that $\\epsilon_{A,\\psi}$ depends only on the character $\\psi$.\n\nIn the sequel, for each prime divisor $\\ell$ of $c$ we write $\\kappa_{(\\ell)}$ for the residue field of the unique prime of $K$ above $\\ell$.\n\n\n\\begin{lemma}\\label{independence} Fix $\\psi$ in $\\widehat{G_c}$. Set $c':= c_\\psi$ and write $\\phi$ for $\\psi$ regarded as an element of $\\widehat{G_{c'}}$. Set $a_{c,c'} := \\prod_{\\ell}a_\\ell$, where $\\ell$ runs over all prime divisors of $c\/c'$ and the terms $a_\\ell$ are as defined at the beginning of \\S\\ref{mod sect}.\n\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] $h_c\\cdot\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c} = a_{c,c'}^2\\cdot h_{c'}\\cdot\\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}$.\n\\item[(ii)] $\\epsilon_{A,c,\\psi} = a_{c,c'}^{-2}\\cdot\\epsilon_{A,c',\\phi}.$\n\\item[(iii)] $\\epsilon_{A,\\psi} = \\epsilon_{A,c',\\phi}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} We can clearly assume that $c \\not= c'$ and hence that $c > 1$.\n\nIn this case, if we set ${\\rm T}_{c,c'} := \\sum_{g \\in G_{K_c\/K_{c'}}}g$, then the norm-compatibility of Heegner points implies that\n\\begin{equation}\\label{nc heegner} {\\rm T}_{c,c'}(y_c) = a_{c,c'}\\cdot y_{c'}.\\end{equation}\n\nThis implies $e_{\\psi}(y_c) = (h_{c'}\/h_{c})a_{c,c'}\\cdot e_\\phi(y_{c'})$ and $e_{\\check\\psi}(y_c) = (h_{c'}\/h_{c})a_{c,c'}\\cdot e_{\\check\\phi}(y_{c'})$\nand hence\n\\begin{align*} h_c\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c} =\\, &h_c(h_{c'}\/h_{c})^2(a_{c,c'})^2\\cdot  \\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_c}\\\\\n=\\, &h_c(h_{c'}\/h_{c})(a_{c,c'})^2\\cdot  \\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}\\\\\n=\\, &(a_{c,c'})^2\\cdot h_{c'}\\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}.\\end{align*}\nwhere the second equality follows from the general result of \\cite[Chap. VIII, Lem. 5.10]{silverman}. This proves claim (i).\n\nClaim (ii) follows directly from claim (i) and the fact that the terms $L'(A_{K},\\check{\\psi},1)$ and $(\\Omega^+\\Omega^-C)\/(c_\\psi\\sqrt{|d_K|})$ that occur in (\\ref{e-def}) do not change if one replaces $\\psi$ by $\\phi$.\n\nTo prove claim (iii) we set $\\epsilon_{A,\\phi}:= \\epsilon_{A,\\psi}$. Then, since both $e_\\phi(z_{c'}) = e_\\phi(y_{c'})$ and $e_{\\check\\phi}(z'_{c'}) = e_{\\check\\phi}(y_{c'})$, an explicit comparison of the equalities (\\ref{e-def}) and (\\ref{u-def}) shows that it suffices to show (\\ref{u-def}) remains valid if one replaces $c$ by $c'$ and $\\psi$ by $\\phi$.\n\nWe now set $d:=c\/c'$. By a routine computation using (\\ref{nc heegner}) one then finds that\n\\begin{equation}\\label{trace heegner} {\\rm T}_{c,c'}(z_c) = \\prod_{\\ell \\mid d} (\\ell + 1 - a_\\ell)\\cdot z_{c'} \\,\\,\\,\\text{ and }\\,\\,\\,\n{\\rm T}_{c,c'}(z'_c) = \\prod_{\\ell \\mid d} (\\ell + 1 + a_\\ell)\\cdot z'_{c'}.\\end{equation}\n\nIn addition, for each prime divisor $\\ell$ of $c$ one has\n\\begin{equation}\\label{trace heegner2} |A(\\kappa_{(\\ell)})| = (\\ell + 1 - a_\\ell)(\\ell + 1 + a_\\ell)\\end{equation}\nand so\n\n\\begin{align*} \\frac{h_c}{c^2}\\langle e_{\\psi}(z_c),e_{\\check\\psi}(z'_c)\\rangle _{K_c} &=  \\frac{h_c}{c^2}(\\prod_{\\ell \\mid d}(\\ell + 1 - a_\\ell)(\\ell + 1 + a_\\ell))\\langle e_{\\phi}(z_{c'}),e_{\\check\\phi}(z'_{c'})\\rangle _{K_c}\\\\\n&= \\bigl(\\prod_{\\ell \\mid d}\\frac{|A(\\kappa_{(\\ell)})|}{\\ell^2}\\bigr)\\frac{h_{c'}}{(c')^2}\\langle e_{\\phi}(z_{c'}),e_{\\check\\phi}(z'_{c'})\\rangle _{K_{c'}},\\end{align*}\nwhere the second equality uses \\cite[Chap. VIII, Lem. 5.10]{silverman} and the fact $c = c'\\cdot\\prod_{\\ell\\mid d}\\ell$.\n\n\nThe right hand side of (\\ref{u-def}) therefore changes by a factor of $(\\prod_{\\ell \\mid d}|A(\\kappa_{(\\ell)})|\/\\ell^{2})^{-1}$ if one replaces $c$ by $c'$ and $\\psi$ by $\\phi$.\n\nTo show that the left hand side of (\\ref{u-def}) changes by the same factor we note that\n\\begin{align*} L'_{c'}(A_{K},\\check{\\phi},1)L'_{c}(A_{K},\\check{\\psi},1)^{-1} =\\, &L'_{c'}(A_{K},\\check{\\psi},1)L'_{c}(A_{K},\\check{\\psi},1)^{-1}\\\\ =\\, &\\prod_{\\ell\\mid d}P_\\ell(A_{K},\\check{\\psi},1)^{-1}\\end{align*}\nwhere $P_\\ell(A_{K},\\check{\\psi},t)$ denotes the Euler factor at (the unique prime of $K$ above) $\\ell$ of the $\\psi$-twist of $A$.\n\nNow $K_c$ is a dihedral extension of $\\QQ$ and so any prime $\\ell$ that is inert in $K$ must split completely in the maximal subextension of $K_c$ in which it is unramified. In particular, for each prime divisor $\\ell$ of $d$ this implies that $P_\\ell(A_{K},\\check{\\psi},t)$ coincides with the Euler factor $P_\\ell(A_{K},t)$ at $\\ell$ of $A_{K}$ and hence that\n\\[ P_\\ell(A_{K},\\check{\\psi},1) = P_\\ell(A_{K},1) = \\frac{|A(\\kappa_{(\\ell)})|}{{\\rm N}_{K\/\\QQ}(\\ell)} = \\frac{|A(\\kappa_{(\\ell)})|}{\\ell^2},\\]\nas required.\n\\end{proof}\n\n\\subsubsection{}\\label{bs conj section\n\nIf $c=1$, then for each character $\\psi$ in $\\widehat{G_c}$ one has $c_\\psi = 1$ and the results of Gross and Zagier in \\cite[see, in particular, \\S I, (6.5) and the discussion on p. 310]{GZ} imply directly that $\\epsilon_{A,c,\\psi}=1$.\n\nIn addition, for $c > 1$ the work of Zhang in \\cite{zhang01, zhang} implies for each $\\psi$ in $\\widehat{G_c}$ a formula for the algebraic number $\\epsilon_{A,c,\\psi}$.\n\nHowever, as observed by Bradshaw and Stein in \\cite[\\S2]{BS}, this formula is difficult to make explicit and is discussed in the literature in several mutually inconsistent ways.\n\nIn particular, it is explained in loc. cit. that the earlier articles of Hayashi \\cite{hayashi} and Jetchev, Lauter and Stein \\cite{JLS} together contain three distinct formulas for the elements $\\epsilon_{A,c,\\psi}$ that are mutually inconsistent and all apparently incorrect.\n\nIn an attempt to clarify this issue, in \\cite[Conj. 6]{BS} Bradshaw and Stein conjecture that for every non-trivial character $\\psi$ in $\\widehat{G_c^+}$ one should have\n\\begin{equation}\\label{bs conj} \\epsilon_{A,c,\\psi} = 1,\\end{equation}\nand Zhang has asserted that the validity of this conjecture can indeed be deduced from his results in \\cite{zhang} (see, in particular, \\cite[Rem. 7]{BS}).\n\nHowever, if $c > 1$, then Lemma \\ref{independence}(ii) implies $\\epsilon_{A,c,\\psi}$ is not always equal to $\\epsilon_{A,c_\\psi,\\psi}$ and hence that the conjectural equalities (\\ref{bs conj}) are in general mutually compatible only if one restricts to characters $\\psi$ with $c_\\psi = c$.\n\nFor further comments in this regard see Remark \\ref{bs conj rem} below.\n\n\\subsection{Heegner points and refined BSD} In this section we interpret the complex numbers $\\epsilon_{A,\\psi}$ defined above in terms of our refined Birch and Swinnerton-Dyer Conjecture.\n\n\n\nWe define an element of $\\CC[G_c]$ by setting\n\\[ \\epsilon_{A,c} := \\sum_{\\psi\\in \\widehat{G_c}}\\epsilon_{A,\\psi}\\cdot e_\\psi.\\]\nLemma \\ref{independence}(iii) combines with the properties (\\ref{stark ec}) to imply $\\epsilon_{A,c}$ belongs to $\\QQ[G_c]$.\n\nWe also define an element of $\\QQ[G]^\\times$ by setting\n\\[ u_{K,c}:= (-1)^{n(c)}\\sum_{\\psi\\in \\widehat{G_c}}(-1)^{n(c_\\psi)}\\cdot e_\\psi,\\]\nwhere $n(d)$ denotes the number of rational prime divisors of a natural number $d$.\n\n\\begin{theorem}\\label{h-rbsd} Let $F$ be an abelian extension of $K$ of conductor $c$ and set $G := G_{F\/K}$. Fix an odd prime $p$ and assume that all of the following conditions are satisfied:\n\\begin{itemize}\n\\item[$\\bullet$] the data $A$, $F\/K$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S\\ref{tmc}.\n\\item[$\\bullet$] $A(F)$ has no point of order $p$.\n\\item[$\\bullet$] The trace to $K$ of $y_1$ is non-zero.\n\\item[$\\bullet$] $p$ is unramified in $K$.\n\\end{itemize}\nSet $z_{F} := {\\rm Tr}_{K_c\/F}(z_c)$ and $z'_{F} := {\\rm Tr}_{K_c\/F}(z'_c)$. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)]\nIf ${\\rm BSD}_p(A_{F\/K})$(iv) is valid then every element of\n\\[ {\\rm Fit}^0_{\\ZZ_p[G]}\\left( \\bigl(A(F)_p\/\\langle z_F\\rangle\\bigr)^\\vee\\right)\\cdot  {\\rm Fit}^0_{\\ZZ_p[G]}\\left(\\bigl( A(F)_p\/\\langle z'_F\\rangle\\bigr)^\\vee\\right)\\cdot C\\cdot u_{K,c}\\cdot\\epsilon_{A,c} \\]\nbelongs to ${\\rm Fit}^1_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ and annihilates $\\sha(A_{F})[p^\\infty]$.\n\n\\item[(ii)] Assume that $F\/K$ is of $p$-power degree and that $p$ does not divide the trace to $K$ of $y_1$. Then ${\\rm BSD}_p(A_{F\/K})$(iv) is valid if and only if one has $${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty]) = \\ZZ_p[G]\\cdot C\\cdot u_{K,c}\\cdot \\epsilon_{A,c}.$$\n\\end{itemize}\n\\end{theorem}\n\n\n\\begin{proof} Since the extension $F\/K$ is tamely ramified we shall derive claim (i) as a consequence of the observation in Remark \\ref{new add2}.\n\n\n\nWe first note that the assumed non-vanishing of the trace to $K$ of $y_1=z_1$ combines with the trace compatibilities in (\\ref{trace heegner}) to imply that the elements $e_\\psi(z_c)$ and $e_\\psi(z'_c)$ are non-zero for all $\\psi$ in $\\widehat{G}$.\n\nTaken in conjunction with (\\ref{u-def}), this fact implies directly that each function $L_{S_{\\rm r}}(A,\\check{\\psi},z)$ vanishes to order one at $z=1$, where as in Remark \\ref{new add2} we have set $S_{\\rm r}=S_k^F\\cap S_k^F$. It also combines with the main result of Bertolini and Darmon in \\cite{BD} to imply that the $\\ZZ_p[G]$-modules generated by $z_c$ and $z_c'$ each have finite index in $A(K_c)$. This implies, in particular, that the idempotent $e_{(1)}$ is equal to $1$.\n\n\nSince every prime divisor of $c$ is inert in $K$ and then splits completely in the maximal subextension of $K_c$ in which it is unramified, the conductor of each character $\\psi$ in $\\widehat{G_c}$ is a divisor $c_\\psi$ of $c$ and the unramified characteristic $u_\\psi$ defined in \\S\\ref{mod GGS section} is equal to $(-1)^{n(c) + n(c_\\psi)}$.\n\nBy using \\cite[(21)]{bmw} one can then compute that for every $\\psi$ in $\\widehat{G_c}$ one has\n\\begin{align}\\label{gauss sums_eq}\n\\tau^\\ast \\bigl(\\QQ,\\psi\\bigr)\\cdot w_\\psi^{-1}=\\, &u_\\psi\\cdot \\tau \\bigl(\\QQ,\\psi\\bigr)\\cdot w_\\psi^{-1}\n\\\\ =\\, &u_\\psi\\sqrt{\\vert d_{K}\\vert} \\sqrt{{\\rm N}c_\\psi}\\notag\\\\\n =\\, &(-1)^{n(c) + n(c_\\psi)}c_\\psi\\sqrt{\\vert d_{K}\\vert}.\\notag\\end{align}\n\nIn addition, for each $\\psi$ in $\\widehat{G}$ one has\n\\[ \\Omega_A^\\psi= \\Omega^+\\Omega^-\\]\nand\n\\begin{align*} e_\\psi\\cdot h_{F\/K}(z_F,z_F') =\\, &\\sum_{g \\in G}\\langle g(z_F),z_F'\\rangle_{F}\\cdot \\psi(g)^{-1}e_\\psi\\\\\n=\\, & |G|\\langle e_\\psi(z_F),z_F'\\rangle_{F}\\cdot e_\\psi\\\\\n=\\, & |G|\\langle e_\\psi(z_F),e_{\\check\\psi}(z_F')\\rangle_{F} \\cdot e_\\psi\\\\\n=\\, & |G|(|G|\/h_c)\\langle e_\\psi(z_F),e_{\\check\\psi}(z_F')\\rangle_{K_c} \\cdot e_\\psi\\\\\n=\\, & h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c} \\cdot e_\\psi,\\end{align*}\nwhere in the last equality $\\psi$ and $\\check\\psi$ are regarded as characters of $G_c$.\n\nSetting $S_{p,{\\rm r}}=S_{\\rm r}\\cap S_k^p$ one may also explicitly compute, for $\\psi\\in\\widehat{G}$, that $m_\\psi:=\\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v,\\psi}$ is equal to $p^2$ if $p$ divides $c$ but not $c_\\psi$ and is equal to $1$ otherwise. We use this explicit description to extend the definition of $m_\\psi$ to all characters $\\psi\\in\\widehat{G_c}$.\n\nThese facts combine with (\\ref{u-def}) to imply that for any $\\psi\\in\\widehat{G}$ one has\n\n\\begin{align}\\label{explicit lt}&\\left(\\frac{L^{(1)}_{S_{\\rm r}}(A_{F\/K},1)\\cdot\\tau^*(F\/K)\\cdot\\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v}(F\/k)}{\\Omega_A^{F\/K}\\cdot w_{F\/k}\\cdot h_{F\/K}(z_F,z_F')}\\right)_\\psi\\\\= \\, &\\frac{L'_{c}(A_K,\\check{\\psi},1)\\cdot\\tau^*(\\QQ,\\psi)\\cdot m_\\psi}{\\Omega_A^\\psi\\cdot w_\\psi\\cdot h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c}}\\notag\\\\\n = \\, & \\frac{L'_{c}(A_{K},\\check{\\psi},1)(-1)^{n(c)+ n(c_\\psi)}c_\\psi\\sqrt{|d_K|}}{\\Omega^+\\Omega^-\\cdot h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c}} \\cdot m_\\psi\\notag\\\\\n = \\, & (-1)^{n(c)+ n(c_\\psi)}\\epsilon_{A,\\psi}\\cdot C \\cdot (c_\\psi\/c)^2m_\\psi \\notag\\\\% \\cdot h_c \\langle e_{\\chi}(z_c),e_{\\check\\chi}(z'_c)\\rangle_{K_c}\\notag\\\\\n = \\, & (-1)^{n(c)}(-1)^{n(c_\\psi)}\\epsilon_{A,\\psi}\\cdot C \\cdot (c_\\psi\/c)^2 m_\\psi. \\nota\n  \\end{align}\n\nTo deduce claim (i) from Remark \\ref{new add2} it is thus sufficient to show that the sum\n\\begin{equation}\\label{unit sum} \\sum_{\\psi\\in\\widehat{G_c}}(c_\\psi\/c)^2m_\\psi\\cdot e_\\psi\\end{equation}\n\nbelongs to $\\ZZ_p[G_c]^\\times$. This fact follows from the result of Lemma \\ref{bley lemma} with $A=G_c$ and $i=2$ (so that $n_\\psi^i=m_\\psi$) and, for each positive divisor $d$ of $c$, with the subgroup $H_d$ of $G_c$ specified to be $G_{K_c\/K_d}$. Indeed, this choice of subgroups satisfies the assumption (ii) of Lemma \\ref{bley lemma} because (\\ref{explicit iso}) implies that $|G_{K_c\/K_d}|$ is equal to $\\prod_{\\ell\\mid (c\/d)}(\\ell+1)$.\n\n\nTo prove claim (ii) it suffices to show that, under the given hypotheses, the equality in Theorem \\ref{bk explicit} is valid if and only if one has ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty]) = \\ZZ_p[G]\\cdot C\\cdot u_{K,c}\\cdot\\epsilon_{A,c}.$\n\nNow, in Theorem \\ref{bk explicit}, the set $S_{p,{\\rm w}}$ is empty since $F$ is a tamely ramified extension of $k =K$ and the set $S_{p,{\\rm u}}^\\ast$ is empty since we are assuming that $p$ is unramified in $K$.\n\nWe next note that $z_1 = z'_1 = y_1$ and hence that (\\ref{trace heegner}) implies\n\\[ {\\rm Tr}_{F\/K}(z_F) = {\\rm Tr}_{K_c\/K}(z_c) = \\mu_c\\cdot{\\rm Tr}_{K_1\/K}(z_1) = \\mu_c\\cdot {\\rm Tr}_{K_1\/K}(y_1)\\]\nwith $\\mu_c = \\prod_{\\ell \\mid c} (\\ell + 1 - a_\\ell)$ and, similarly, that\n\\[ {\\rm Tr}_{F\/K}(z'_F) = \\mu_c'\\cdot {\\rm Tr}_{K_1\/K}(y_1)\\]\nwith $\\mu'_c = \\prod_{\\ell \\mid c} (\\ell + 1 +a_\\ell)$.\n\nIn particular, since (\\ref{trace heegner2}) implies that $\\mu_c\\cdot \\mu'_c = \\prod_{\\ell\\mid c}|A(\\kappa_{(\\ell)})|$, our assumption that the hypotheses (H$_3$) and (H$_4$) hold for $F\/K$ means that $\\mu_c\\cdot \\mu'_c$ is not divisible by $p$. This fact in turn combines with our assumption that ${\\rm Tr}_{K_1\/K}(y_1)$ is not divisible by $p$ in $A(K)$ to imply that neither ${\\rm Tr}_{F\/K}(z_F)$ nor ${\\rm Tr}_{F\/K}(z'_F)$ is divisible by $p$ in $A(K)_p$.\n\n Since our hypotheses imply that $A(K)_p = A(F)_p^{G}$ is a free $\\ZZ_p$-module of rank one, it follows that ${\\rm Tr}_{F\/K}(z_F)$ and ${\\rm Tr}_{F\/K}(z'_F)$ are both $\\ZZ_p$-generators of $A(F)_p^G$, and hence, by Nakayama's Lemma, that $A(F)_p$ is itself a free rank one $\\ZZ_p[G]$-module that is generated by both $z_F$ and $z'_F$.\n\nTaken in conjunction with the explicit descriptions of cohomology given in (\\ref{bksc cohom}) (that are valid under the present hypotheses), these facts imply that the Euler characteristic that occurs in Theorem \\ref{bk explicit} can be computed as\n\\[ \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h_{A,F}) = {\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_F)[p^\\infty])\\cdot h_{F\/k}(z_F,z_F'),\\]\nwhere we have identified $K_0(\\ZZ_p[G],\\CC_p[G])$ with the multiplicative group of invertible $\\ZZ_p[G]$-lattices in $\\CC_p[G]$ (as in Remark \\ref{comparingdets}).\n\nWhen combined with the equality (\\ref{explicit lt}) these facts imply that the product $$\\mathcal{L}^*_{A,F\/K}\\cdot h_{F\/k}(z_F,z_F')^{-1},$$ where $\\mathcal{L}^*_{A,F\/K}$ is the leading term element defined in (\\ref{bkcharelement}), is equal to the projection to $\\ZZ_p[G]$ of $C\\cdot u_{K,c}\\cdot\\epsilon_{A,c}$ multiplied by the sum (\\ref{unit sum}).\n\nClaim (ii) is therefore a consequence of Theorem \\ref{bk explicit} and the fact that, as already observed above, the sum (\\ref{unit sum}) belongs to $\\ZZ_p[G_c]^\\times$.\n\\end{proof}\n\n\n\n                         \n\n\n\n\n\n\n\n\\begin{remark}\\label{bs conj rem}{\\em If $p$ is prime to all factors in $C$, then the hypotheses of Theorem \\ref{h-rbsd}(ii) combine with an argument of Kolyvagin to imply $\\sha(A\/K)[p^\\infty]$ vanishes (cf. \\cite[Prop. 2.1]{gross_koly}). This fact combines with the projectivity of $A(F)^\\ast$ to imply $\\sha(A\/F)[p^\\infty]$ vanishes and hence, via Theorem \\ref{h-rbsd}(ii), that ${\\rm BSD}_p(A_{F\/K})$(iv) is valid if and only if the product $u_{K,c}\\cdot\\epsilon_{A,c}$ projects to a unit of $\\ZZ_p[G]$. In the case that $F\/K$ is unramified, this observation was used by Wuthrich and the present authors to prove the main result of \\cite{bmw}. In the general case, it is consistent with an affirmative answer to the question of whether for every $\\psi$ in $\\widehat{G}$ one should always have\n\\[ \\epsilon_{A,\\psi} = (-1)^{n(c_\\psi)}?\\]\nWe observe that such an equality would, if valid, constitute a functorially well-behaved, and consistent, version of the conjecture of Bradshaw and Stein that was discussed in \\S\\ref{bs conj section}.}\\end{remark}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\textit{Chandra} X-ray observatory has found strong X-ray emission from large scale jets of \nmany radio loud quasars. Some of them, most notorious, PKS 0637-752 \n\\citep{schwartz2000,chartes2000} and 3C 273 \\citep{sambruna2001,marshall2001}, \nare hard to explain \nby synchrotron radiation from a high energy extension of radio-optical \nemitting electrons, \nsince the observed X-ray flux is far above the extension from radio-optical \nspectra and since the X-ray spectrum is harder than the optical one.\nThe most conventional synchrotron self-Compton (SSC) model requires \na very small magnetic field strength,\nwhich means a large departure from equi-partition between \nmagnetic field and relativistic electrons with an enormous jet power.\nThe latter has been regarded unlikely. \nIt is then considered that inverse Compton (IC) scattering of cosmic microwave \nbackground (CMB) photons may explain the X-ray emission provided that the jet is \nrelativistic and makes a small angle with the line of \nsight \\citep{tavecchio2000,cgc2001}.\n\\deleted{In this model, however, the intrinsic jet length becomes as large as Mpcs, \nrarely seen in the parent population of radio galaxies. }\nBoth SSC and IC\/CMB models predict that high energy extension of the \nX-ray spectrum reaches GeV energy range which can be detected with \\textit{Fermi} LAT \n\\citep{georganopoulos2006}. \nRecently,  \\textit{Fermi} LAT observations have been reported to put upper limits of \nthe $\\gamma$-ray flux for the jets of 3C 273 \\citep{mg2014}\nand PKS 0637-752 \\citep{meyer2015}. \nThey are an order of magnitude lower than the SSC and IC\/CMB predictions  \nso that these models are incompatible with observations. \n\nIn this situation, several alternative models should be considered. \nWithin the leptonic scenario, a separate population of electrons from \nthose emitting radio-optical photons, extending up to around 100 TeV is considered. \nThose electrons emit X-ray photons by synchrotron radiation. \nThe existence of such an electron \npopulation is rather ad-hoc although not impossible. \nIts origin is usually considered a separate particle acceleration process \nfrom that for the radio-optical emitting population.\nHowever, it is hard to imagine such an efficient acceleration mechanism, \nsince the high energy end of radio-optical \nemitting electrons is determined by the balance of acceleration and radiative cooling. \nActually, for PKS 0637-752, radio, optical, and X-ray emissions are spatially coincident and \nconcentrated in a few bright knots, the size of which is an order of kpc, \nand the broadband spectra imply a break around $10^{12}$-$10^{13}$ Hz which \nis naturally due to radiative cooling across the source.\n\nAlternatively, relativistic hadrons may be responsible for the X-ray emission.\n\\cite{bg2016} and earlier \\cite{aharonian2002} considered proton synchrotron emission \nassuming magnetic field stronger than 10 mG and a large energy density of \naccelerated protons up to more than 10 PeV. \nAlthough they argued that the proton and Poynting powers are modest, it is due to \ntheir adopted time scale. They assumed that the confinement time is as long as \n$10^7$-$10^8$ years,\nwhich is a few orders of magnitude longer than the light \ncrossing time of $10^3$-$10^4$ years. \nSince the jet does not involve a large inertia nor a large confining pressure, \nwe regard that the relevant time scale should be similar to the latter. \nThen, the required proton and Poynting powers become \nan order of $L_p \\approx 10^{50}$ erg s$^{-1}$ \nand $L_\\mathrm{Poy} \\approx 10^{49}$ erg s$^{-1}$, respectively.\nAt the same time, the energy density of relativistic electrons becomes\nabout six orders of magnitude smaller than that of the Poynting power. \n\nRelativistic protons also contribute to X-ray and $\\gamma$-ray emission \nthrough the production of high energy electrons and positrons by\nBethe-Heitler and photo-pion processes. \nThe latter processes were considered before the \\textit{Chandra} era by \\cite{mkb91}\nfor a possible mechanism of the X-ray emission from hot spots of radio galaxies.\n\\cite{aharonian2002} also examined these processes for the knot A1 of 3C 273 and \nconcluded that they are too inefficient.\nBut, the efficiency of these processes depends on the size of the knot as well as \nthe infrared photon spectrum.\n\\added{In fact, the energy density of mm and sub-mm radiation \nin \\cite{aharonian2002} is smaller than our model presented below.}\nThe proton models have been discussed for more compact emission regions of blazars and \nBethe-Heitler process can contribute to the same order, depending on the \nsoft photon spectral shape \\citep{pm2015}.\nThus, in this paper, we consider both Bethe-Heitler and photo-pion processes  \nand examine if these processes can explain the X-ray emission from PKS 0762-752.\n\nIn Section \\ref{sec:estimate} we make a rough estimate of physical quantities \nof the X-ray emitting large scale jet of PKS 0637-752, whose redshift is 0.651. \nIn Section \\ref{sec:model} we formulate the problem, \nand in Section \\ref{sec:results} numerical results are presented. \nIn Section \\ref{sec:conclusion} we draw conclusions.\n  \n\n\\section{Rough Estimate}  \\label{sec:estimate}\n\nWe first make a rough estimate of the physical quantities in order to \ncapture the essence of the problem. For simplicity, \nwe assume a single  uniform sphere of radius $R=R_\\mathrm{kpc}$ kpc for the emission region\nand ignore effects of relativistic beaming and redshift for the time being. \nAlthough the emission region is divided into a few knots in reality, \nwe here treat a combined emission region.\n\n\nObserved spectra at radio through optical frequencies suggest that \nthe synchrotron emission from primary electrons has a peak at infrared band \naround $10^{12}$ Hz with a luminosity $L_\\mathrm{syn}$ about \n$3\\times 10^{44}$ erg s$^{-1}$. \nThus,  energy density of synchrotron photons $u_\\mathrm{syn}$ is about \n\\begin{equation}\n  u_\\mathrm{syn}=\\frac{3L_\\mathrm{syn}}{4\\pi R^2c}\n  \\approx 3 \\times 10^{-10}R_\\mathrm{kpc}^{-2} \\,\\,  \\text{erg} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nThe number density of photons at radio frequencies may be approximated by \n\\begin{equation}\n  \\nu n_\\nu\\approx 10^6 \\left(\\frac{\\nu}{10^{10} \\, \\text{Hz}}\\right)^{-0.75}\n  R_\\mathrm{kpc}^{-2} \\,\\, \\text{cm}^{-3} \n\\end{equation}\nand that at optical frequencies \n\\begin{equation}\n  \\nu n_\\nu\\approx 10^{2}\\left(\\frac{\\nu}{10^{14} \\, \\text{Hz}}\\right)^{-1.25}\n  R_\\mathrm{kpc}^{-2} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nFor the magnetic field strength of $B=B_\\mathrm{mG}$ mG, \nthe energy density of magnetic field is \n\\begin{equation}\n  u_\\mathrm{mag}=4 \\times 10^{-8} B_\\mathrm{mG}^2 \\,\\, \\text{erg} \\,\\, \\text{cm}^{-3} ,\n\\end{equation}\nand the Poynting power is estimated as  \n\\begin{equation}\n  L_\\mathrm{Poy}=\\pi R^2 u_\\mathrm{mag}c \n  \\approx 4\\times 10^{46}B_\\mathrm{mG}^{2}R_\\mathrm{kpc}^{2} \n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1} .\n\\end{equation}\n\nThe Lorentz factor of electrons ranges from \n\\replaced{$\\gamma_{e , \\mathrm{min}}\\approx 3 \\times 10^3B_\\mathrm{mG}^{-0.5}$}\n{$\\gamma_{e , \\mathrm{min}}\\approx 2 \\times 10^3B_\\mathrm{mG}^{-0.5}\n (\\nu_\\mathrm{min}\/10^{10} \\, \\mathrm{Hz})^{0.5}$}\nto $\\gamma_{e, \\mathrm{max}}\\approx 10^6B_\\mathrm{mG}^{-0.5}$\nwith a broken power law spectrum. \nThe power law index of electrons is tentatively taken as 2.5 at low energies \nand 3.5 at high energies in accordance with the above photon spectra. \n\\deleted{The energy density of electrons is governed by the low energy end}\n\\added{The peak luminosity of $3 \\times 10^{44}$ erg s$^{-1}$  at $10^{12}$ Hz\nis emitted by primary electrons with \n$\\gamma_\\mathrm{br} \\sim 2 \\times 10^{4} B_\\mathrm{mG}^{-0.5}$ below which \nthe number spectrum is given by $n_e(\\gamma_e) = K_e \\gamma_e^{-2.5}$.\nOn the other hand, $L_\\mathrm{syn}$ is given by \n$\\sim (4 \\pi R^3\/3) c \\sigma_\\mathrm{T} u_\\mathrm{mag} \\gamma_\\mathrm{br}^2 \nn_e(\\gamma_\\mathrm{br}) \\gamma_\\mathrm{br}$,\nwhere $\\sigma_\\mathrm{T}$ is the Thomson cross section.\nFrom these relations we write $K_e$ in terms of $\\gamma_\\mathrm{br}$, $R$, and $B$.\nThe electron energy density is now given by\n$u_e \\sim 2 m_e c^2  K_e \\gamma_\\mathrm{min}^{-0.5}$ \n}\nand estimated as  \n\\begin{equation}\n  u_e \\approx  8  \\times 10^{-10} B_\\mathrm{mG}^{-1.5} R_\\mathrm{kpc}^{-3}\n  \\left(\\frac{\\nu_\\mathrm{min}}{10^{10} \\mathrm{Hz} }\\right )^{-0.25} \n  \\,\\, \\text{erg} \\,\\, \\text{cm}^{-3} .  \n  \\label{eq:ue}\n\\end{equation}\nIt may seem that \\replaced{inverse Compton}{SSC} luminosity can be large,\nif the magnetic field strength is smaller than 0.1 mG for a typical source size of 1 kpc. \nHowever, in this case, most of the inverse Compton \nemission is produced in the  MeV-GeV range. \nTo reproduce the observed X-ray flux, magnetic field needs to be as small as 0.01 mG.\nIn this case, the kinetic power of electrons given by\n\\begin{equation}\n  L_e= \\frac{4\\pi R^3u_e}{3}\\frac{c}{3R}\n  \\approx  3 \\times  10^{44} B_\\mathrm{mG}^{-1.5} R_\\mathrm{kpc}^{-1}\n  \\left(\\frac{\\nu_\\mathrm{min}}{10^{10} \\text{Hz} }\\right)^{-0.25}\n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1} \n\\end{equation}\nwould become very large, \n\\added{i.e., $L_e \\sim 3 \\times 10^{47}$ erg s$^{-1}$ for $B = 0.01$ mG\nand $\\nu_\\mathrm{min} = 10$ GHz.}\nHere, we take the escape time of electrons as $3R\/c$. \nHistorically, for this reason, the beamed IC\/CMB model was proposed, \nbut as noted in Section \\ref{sec:intro}, this model is now regarded unlikely. \nThe minimum power for explaining the radio-optical flux \nis realized at $B_\\mathrm{mG}\\approx 0.2$ for $R_\\mathrm{kpc}=1$\nwith $L_\\mathrm{Poy} \\approx L_e\\approx 2 \\times 10^{45} \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\n\n\n\n\\added{Theoretically,} \nthe break energy of the electron energy distribution, \\added{$\\gamma_b$,} \nis determined by the \nbalance between synchrotron cooling and escape; if we equate the cooling time \nwith escape time, we obtain\n\\begin{equation}\n  \\gamma_b \\approx 3 \\times 10^3B_\\mathrm{mG}^{-2}R_\\mathrm{kpc}^{-1} .\n\\end{equation}\nIf the break corresponds to the break of radio-optical spectrum at $10^{12}$ Hz, \nwe obtain the field strength of around 0.2 mG for $R=1 \\, \\text{kpc}$.\nThese considerations lead to $B_\\mathrm{mG}\\approx 0.1-0.3$ as an appropriate choice.\n \nThe maximum possible Lorentz factor of electrons is estimated by equating the \ncooling time with the  \ngyrotime and given by \n\\begin{equation}\n  \\gamma_{e, \\mathrm{lim}} \\approx 10^9 B_\\mathrm{mG}^{-1\/2}  .\n\\end{equation}\nThus, in principle it is possible to obtain $\\gamma_e$ \nas large as $10^8$ with which electrons emit synchrotron X-rays. \nHowever, since the observed X-ray spectrum is much flatter than the optical spectrum, \nsuch a high energy population should be separate from radio-optical emitting one and\nthe acceleration mechanism should be very efficient and distinct. \nAlternatively, such electrons may be supplied from photo-hadronic processes.\nIt should be noted that AGN jets are composed of protons and electrons\/positrons and \nthe inertia is likely to be dominated by protons \\added{\\citep{uchiyama2005}}, \nwhile the existence of electron-positron pairs is also suggested\nby various analysis of observations. \nProton acceleration \\added{may} also naturally \\deleted{takes} \\added{take} place \nand in principle the maximum energy of \nprotons can be as large as  $10^{20}$ eV for 1 mG field and 1 kpc size. \n\nTwo photo-hadronic processes can provide secondary high energy electrons\/positrons, i.e., \nphoto-pion production process and Bethe-Heitler process. \nThe former is through strong interaction with the cross section of about \n$3\\times 10^{-28} \\,\\, \\mathrm{cm}^2$ \nand the threshold energy of\n\\begin{equation}\n  \\gamma_{p, \\mathrm{th}} \\approx m_\\pi c^2 \\epsilon_\\mathrm{soft}^{-1}\n  = 3\\times 10^{12} \\left(\\frac{\\nu_\\mathrm{soft}}{10^{10} \\mathrm{Hz}}\\right)^{-1},\n\\end{equation}\nwhere $\\epsilon_\\mathrm{soft}=h \\nu_\\mathrm{soft}$ is the energy of a target photon. \nThus, for $\\gamma_p=10^{10}$, the energy of target photons should be larger than \n$3\\times 10^{12}$ Hz\nwith the number density about  $10^5 \\,\\, \\text{cm}^{-3}$ for $R_\\mathrm{kpc}=1$. \nThus, a proton interaction probability is around 0.1 for rectilinear propagation. \nCharged pions decay to produce electrons and positrons, while neutral pions decay into \ntwo $\\gamma$-rays, which interact with soft photons to produce electron-positron pairs. \n\\added{About 5 \\% of the inelastic energy goes into electron\/positrons.\nThis is because pion mass is about 15 \\% of proton mass, so that about 15 \\% of\nproton energy goes to pions near the threshold.\nThis energy is further distributed to 4 leptons almost equally,\n\\cite[e.g.,][]{ka2008,dm2009}.}\n\\deleted{Considering that about 5 \\% of the inelastic energy goes into electron\/positrons,  }\n\\added{Considering this,}\nwe estimate 0.5 \\% of the proton power can be used to produce electrons \nand positrons with the Lorentz factor of around $2.5 \\times 10^{11}$, \nwhich subsequently radiate synchrotron radiation below the TeV energy \nregion for $B=1 \\,\\text{mG}$. \nWhile photons with energy higher than TeV is optically thick to \nphoton-photon pair production, most synchrotron photons are emitted below TeV and \npair cascade process does not much develop. \nSince these electrons\/positrons rapidly cool to make \nthe energy distribution of a power law with an index of $-2$, \nthe resultant photon energy spectrum is a power law with an index of $-0.5$\nand the X-ray  luminosity is four orders of magnitudes smaller than the TeV luminosity. \nThus, if we would explain the X-ray observations with this mechanism, \nthe predicted GeV luminosity becomes around \n$3 \\times 10^{47} \\, \\text{erg} \\,\\, \\text{s}^{-1}$, \nwhich exceeds the \\textit{Fermi} LAT upper limit. \nRequired proton power is uncomfortably large \namounting to $3 \\times 10^{51} \\, \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\nHigher energy protons exacerbate the problem and lower energy protons \nhave a negligibly small interaction probability. \n\n\nBethe-Heitler process has a larger cross section of \n$3\\times 10^{-27} \\, \\text{cm}^2$ and the lower threshold energy of \n\\begin{equation}\n  \\gamma_{p, \\mathrm{th}}= m_e c^2\\epsilon_\\mathrm{soft}^{-1}\n  = 3\\times 10^{10} \\left(\\frac{\\nu_\\mathrm{soft}}{10^{10} \\, \\mathrm{Hz}}\\right)^{-1} ,\n\\end{equation}\nbut with a lower efficiency of energy transfer to electrons and positrons.\nFor $\\gamma_p =10^{10}$, target photon energy is larger than $3\\times 10^{10}$ Hz \nwith the number density $5\\times 10^5 \\,\\, \\text{cm}^{-3}$ for $R_\\mathrm{kpc}=1$, \nand on average a proton produces 5 electron-positron pairs before it escapes from the region.\nTaking the efficiency of 0.001, 0.5\\% of the proton power can be used to \npair production, roughly the same order as the photo-pion production. \nIn this case, however, the injection Lorentz factor of electrons and positrons is around \n\\deleted{$10^{10}$} \\added{$\\gamma_e \\sim \\gamma_p = 10^{10}$} \nand the synchrotron frequency is peaked at 100MeV for $B=1$ mG. \nSince the resultant synchrotron spectrum becomes a power law with an index of $-0.5$,\nthe X-ray luminosity  is about two orders of magnitudes lower than the 100 MeV \nluminosity. \nThe required proton power is around $10^{49} \\,\\, \\text{erg} \\, \\, \\text{s}^{-1}$, \nwhich is very large but not inconceivable considering the Poynting power is \nan order of $4 \\times 10^{46} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$. \nWhen the magnetic field strength is 0.1 mG, \nthe required proton power is $3\\times 10^{48} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$,\n\\added{because the beak frequency of synchrotron radiation by secondary leptons\ndecreases as $B$ decreases.}\nThe power of primary electrons becomes \n $10^{46} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$ while the Poynting power \nis $4 \\times 10^{44} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$. \nThus we regard about $B = 0.1 \\,\\text{mG}$ is a best guess. \n\\added{The proton power $\\sim 10^{49}$ erg s$^{-1}$ estimated above\n  is $\\sim 120 L_\\mathrm{Edd}$,\n  where $L_\\mathrm{Edd}$ is the Eddington luminosity with the black hole mass\n  $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\sun$ \\citep{ljg2006}.\n  It is to be noted that $M_\\mathrm{BH} = 7.8 \\times 10^9 M_\\sun$ has been \n  reported by \\cite{gcj2001}.  For this value of $M_\\mathrm{BH}$, \n  the proton power is $\\sim 10 L_\\mathrm{Edd}$.}\n\n\nSomewhat lower energy protons also contribute to Bethe-Heitler process; although the \ninteraction probability becomes lower, the injection Lorentz factor also becomes smaller, \nwhich makes the synchrotron peak lower.\nFor example, for $\\gamma_p=10^9$, the target photon energy is above $3\\times 10^{11}$ Hz\nwith the number density of $10^5 \\,\\, \\text{cm}^{-3}$, \nand the interaction probability of a proton becomes 1. \nThus 0.1\\% of the proton power is available. \nThe injection Lorentz factor of electrons\/positrons is also $10^9$, which emit \nsynchrotron radiation at 1 MeV for $B=1 \\, \\text{mG}$ or 0.1 MeV for $B=0.1 \\, \\text{mG}$.\nSo the proton power of $3 \\times 10^{48} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$ \ngives rise to the observed X-ray luminosity. \n\nIt is to be noted that for a larger $R_\\mathrm{kpc}$ the interaction probability \nof protons becomes small and required proton power accordingly increases; \na larger $R_\\mathrm{kpc}$ is unfavorable for the photo-hadronic model. \nFor these parameters, proton synchrotron luminosity is a few orders of magnitude \nsmaller than the synchrotron luminosity of secondary pairs.  \nIts peak is at $10^{19}$ Hz for $B=0.1\\, \\text{mG}$.\n\nSince the resultant emission spectra depend on details of the secondary \nelectron\/positron spectra and resultant radiative cooling, \nwe numerically investigate such details in the next section. \nBased on the present investigation, in this paper, \nwe concentrate on Bethe-Heitler process.\n\n\n\\section{The Model} \\label{sec:model}\n\nWe adopt a single zone model in which the size of the emission region of $R$\naround 1 kpc, magnetic field of $B$ around 0.1 mG, and the proton spectrum is \n\\begin{equation}\n  n_p \\added{(\\gamma_p)} = K_p \\gamma^{-p} ,\n  \\label{eq:proton-spec}\n\\end{equation}\nfor $\\gamma_{p, \\mathrm{min}}\\le \\gamma_p \\le \\gamma_{p, \\mathrm{max}}$. \nCanonically we take $p=2$, $\\gamma_{p, \\mathrm{min}}=1$,\nand $\\gamma_{p, \\mathrm{max}}=10^{10}$. \nSince photo-hadronic processes work only for large values of $\\gamma_p$, \nthe value of $\\gamma_{p, \\mathrm{min}}$ does not affect the resultant \nspectrum but only affects the energy density of protons \\added{logarithmically}.\nIf  proton energy distribution is concentrated in the range near $\\gamma_{p, \\mathrm{max}}$,\nthe power requirement below  will be relieved by an order of magnitude.  \nNote that the estimates of the proton power and proton energy density given in the \nprevious section are those for such a high energy population \nand an order of magnitude lower than those for $\\gamma_{p, \\mathrm{min}}=1$.\nIn contrast $\\gamma_{p, \\mathrm{max}}$ critically affects the results.\nThe energy density of protons is \n\\begin{equation}\n  u_p =10^{-3} K_p \\ln(\\gamma_{p, \\mathrm{max}}\/\\gamma_{p, \\mathrm{min}}) \\,\\,\n  \\text{erg} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nThe proton power is \n\\begin{equation}\n  L_p = \\frac{4\\pi R^3 u_p}{3}\\frac{c}{3R}\n  \\approx 4 \\times 10^{50} K_p\n  \\ln(\\gamma_{p, \\mathrm{max}}\/\\gamma_{p, \\mathrm{min}}) R_\\mathrm{kpc}^2 \n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1},\n\\end{equation}\nwhere we take the escape time of $3R\/c$.\n\n\nThe target photon spectra of photo-pion and Bethe-Heitler processes \nare based on the observed radio to optical photons,\ntaking into account of the cosmological redshift of 0.651.\nPrimary electrons responsible for the  radio-optical emission are \ndetermined so as to reproduce the radio-optical spectra. \nWe also calculate the inverse Compton scattering of primary electrons \noff radio-optical photons, \nwhose flux is generally orders of magnitude short of X-ray observations.\nIn numerical calculations we solve the kinetic equation of primary electrons\nwith the injection spectrum given by \n$q_\\mathrm{inj}(\\gamma_e) = K_e \\gamma_e^{-\\alpha_e} \\exp(-\\gamma_e\/\\gamma_{e, 0})$,\nwhere $\\gamma_e$ is the Lorentz factor of electrons and $K_e$, $\\alpha_e$, \nand $\\gamma_{e, 0}$ are parameters to fit the observed radio-optical spectrum.\nSince we consider only mildly relativistic beaming if any, we ignore IC\/CMB.  \nAlthough the number density of CMB photons is larger than  \nthat of radio-optical synchrotron photons when $R_\\mathrm{kpc}$ is larger than 10, \nthe efficiency of those processes becomes small for large $R$, \nso that our treatment is justified.\n  \n\nThe emission by high energy leptons produced by the hadronic processes is \ncalculated for  the lepton spectrum\nobtained by solving the kinetic equation given by\n\\begin{equation}\n  \\frac{d n_e (\\gamma_e)}{d t}=q_\\mathrm{BH}(\\gamma_e) + q_\\mathrm{pp}(\\gamma_e)\n  -\\frac{c n_e (\\gamma_e)}{3R}\n  -\\frac{d~}{d\\gamma_e} [\\dot{\\gamma}_e n_e (\\gamma_e)] ,\n  \\label{eq:e-kinetic-hadronic}\n\\end{equation}\nwhere $n_e(\\gamma_e)$ is the lepton density per unit interval of $\\gamma_e$.\nThe lepton injection rate is denoted by $q_\\mathrm{BH}(\\gamma_e)$ for Bethe-Heitler process, \nwhich is calculated according to the formulation  given by \\cite{ka2008},\n\\added{using a given proton spectrum with equation (\\ref{eq:proton-spec}).}\nLepton production via photo-pion processes is denoted by $q_\\mathrm{pp}(\\gamma_e)$.\nHowever, we do not include this term in numerical calculations \nbecause the leptons produced by photo-pion processes do not contribute much \nto the X-ray emission.\nHere $\\dot{\\gamma}_e$ denotes radiative cooling through synchrotron radiation \nand inverse Compton scattering.\nWe also set the lepton escape time to be $3 R\/c$.\nUsing the steady solution of equation (\\ref{eq:e-kinetic-hadronic}), \nwe calculate the emission spectra through synchrotron radiation \nand inverse Compton scattering. \n\n\n\n\\section{Results}  \\label{sec:results}\n\n\\begin{figure}[ht!]  \n  \\centering\\includegraphics[scale=0.5,clip]{figure1.eps}\n  \\caption{The production spectrum of electrons and positrons \n    for \\added{$p = 2$ and} $K_p = 1$ \\added{cm$^{-3}$.}\n    The synchrotron radiation spectrum by primary electrons \n    for $R = 1$ kpc and $B = 0.1$ mG (Fig. \\ref{fig:1kpc-no-beaming-exp})\n    is used as a target photon spectrum in lepton production.\n    The black line shows the pair production rate by Bethe-Heitler process,\n    the red and blue lines show the electron and positron production rates, \n    respectively, by photo-pion process.\n    \\label{fig:pair-production-spec}}\n\\end{figure}\n\n\\begin{figure}[ht!]  \n  \\centering\\includegraphics[scale=0.5,clip]{figure2.eps}\n  \\caption{The lepton energy spectrum for $R = 1$ kpc, $B = 0.1$ mG,\n    \\added{and $p=2$}.\n    The spectrum is calculated by equation (\\ref{eq:e-kinetic-hadronic})\n    for a steady state.\n    The injection of leptons is by Bethe-Heitler process.\n    \\label{fig:lepton-energy-spec}}\n\\end{figure}\n\n\n\n\nIn Figure \\ref{fig:pair-production-spec} we show the production spectrum of electrons \nand positrons through Bethe-Heitler and photo-pion processes\nfor a photon field relevant to PKS 0637-752 jet for the fiducial case $R_\\mathrm{kpc} = 1$\nand $B_\\mathrm{mG}  = 0.1$ with \\added{$p =2$} and $K_p=1$ \\added{cm$^{-3}$}.\n\\added{The pair production rate is calculated based on \\cite{ka2008}.\nThe target photons are synchrotron radiation by primary electrons,\nthe spectra of which are shown in Figure \\ref{fig:1kpc-no-beaming-exp} \nfor various parameters.}\n\\added{Figure  \\ref{fig:pair-production-spec} and Table \\ref{table:prod-rate-2-18} below\ndo not include pair production via the decay of neutral pions.\nThe gamma-rays produced by the decay of neutral pions have energy comparable to\nthe energy of electrons\/positrons produced by charged pions.\nIn collisions with soft photons the gamma-rays produce electron-positron pairs, \nand these pairs contribute mainly to TeV emission.}\nAs we described in Section \\ref{sec:estimate}, the production spectrum\nby Bethe-Heitler process has a rather broad number spectrum \ncentered on $10^6<\\gamma_e<10^{10}$,\nwhile those through photo-pion production have a \\deleted{peaked spectrum}\n\\added{peak} at $\\gamma_e \\approx 10^{11}$-$10^{12}$.\nThe energy injection rate for both processes concentrates on the high energy ends,\nwith the Bethe-Heitler process being an order of magnitude larger than that for the\nphoto-pion production.\n\\added{For $\\gamma_e \\gtrsim 10^{11.5}$, \nphoto-pion processes dominate the Bethe-Heitler process.\nLeptons with $\\gamma_e \\gtrsim 10^{11.5}$ emit synchrotron radiation at \n$\\nu \\gtrsim 10^{25}$ Hz and do not contribute to X-ray emission that is our\ninterest in this paper.}\n\n\nThe resultant steady state electron\/positron energy spectrum is shown \nin Figure  \\ref{fig:lepton-energy-spec}\nfor $R_\\mathrm{kpc}=1$, $B_\\mathrm{mG}=0.1$, \\added{and $p=2$}.\nThe value of $K_p$ is taken as $8.8 \\times 10^{-3}$ \\added{cm$^{-3}$} to reproduce \nthe observed X-ray flux.\nThe proton power amounts to $5 \\times 10^{49} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$.\nAlthough this seems to be too large, it can be reduced by an order of magnitude \nif $\\gamma_{p, \\mathrm{min}}$ is large enough or if the spectral index of protons \nis less than 2, i.e., when the energy of relativistic protons is \nconcentrated in the high energy end.  \nWe tabulate in Table \\ref{table:prod-rate-2-18}\nthe pair production rates for $p = 2$ and 1.8.\n\n\nThe resultant photon spectra are shown in Figure  \\ref{fig:1kpc-no-beaming-exp}.\nAs is seen, while an overall spectral shape is well reproduced, \noptical flux at $5 \\times 10^{14}$ Hz tends to be overproduced.\nAt this frequency primary electrons and secondary pairs equally contribute and \nthe resultant combined flux is a factor of two higher than the observed one.\nThis is due to radiative cooling of secondary pairs and rather a general feature.\nFor a range of the magnetic field strength this feature persists in the present model.\nPossible way out from this problem is discussed later.\nWhen the magnetic field becomes smaller, inverse Compton scattering \nof radio-optical synchrotron photons by primary electrons (SSC) becomes larger.\nIf the magnetic field is as small as 20$\\mu$G, SSC can work as an X-ray emission \nmechanism. \nIn this case, however, the electron power is as large \nas $3 \\times 10^{47} \\, \\text{erg} \\,\\, \\text{s}^{-1}$ with a large deviation \nfrom equi-partition.\nFurthermore, excessive production of optical photons by IC is inevitable.\nThus, SSC model does not work.\n\nThe secondary pairs scatter the synchrotron photons emitted by the primary electrons.  \nThis IC component does not affect the emission below\n$\\sim 10^{24} \\, \\text{Hz}$ but the peak in $\\nu F_\\nu$ appears in the TeV band.\nThe IC component is not much affected by the magnetic field strength\nbecause the spectra of the  (radio-optical) soft photons and the secondary pairs are fixed.\nIt is \\deleted{, however,} to be noted that \nour model does not include the absorption of $\\gamma$-rays\nby extragalactic background light.\n\\added{The TeV bump will be hard to be observed by CTA,\nbecause the expected flux is lower than the lower limit of CTA.}\n\n\n\\floattable\n\\begin{deluxetable}{c|rrrrrr}\n  \\tablecaption{Pair Production Rate for $R = 1$ kpc and  $B = 0.1$ mG\n  \\label{table:prod-rate-2-18}}\n  \\tablewidth{0pt}\n  \\tablehead{\n    \\colhead{} & \\multicolumn{2}{c}{$p=2$} \n    & \\colhead{} &  \\multicolumn{3}{c}{$p=1.8$}\n    \\\\\n    \\cline{2-3}\n    \\cline{5-7}\n    \\colhead{} &  \\colhead{$\\dot{n}_\\pm$} &  \\colhead{$q_\\pm$}\n    & \n    \\colhead{}  &   \\colhead{} &  \\colhead{$\\dot{n}_\\pm$} &  \\colhead{$q_\\pm$}\n    \\\\\n    \\colhead{} & \\colhead{(cm$^{-3}$ s$^{-1}$)} & \\colhead{(erg cm$^{-3}$ s$^{-1}$)} \n    &\n    \\colhead{}  &\\colhead{}   & \\colhead{(cm$^{-3}$ s$^{-1}$)} & \\colhead{(erg cm$^{-3}$ s$^{-1}$)} \n  }\n  \\startdata\n  B-H &  $9.4 \\times 10^{-21}$ \\phn & $4.0 \\times 10^{-18}$  \\phn \n & &  \n  &  $4.8 \\times 10^{-19}$ \\phn & $3.2 \\times 10^{-16}$  \\phn \n  \\\\ \n  $e^-$ &  $2.6 \\times 10^{-25}$ \\phn & $5.4 \\times 10^{-20}$ \\phn \n  & & \n  &  $1.9 \\times 10^{-23}$ \\phn & $3.9 \\times 10^{-18}$ \\phn \n  \\\\ \n  $e^+$ & $2.4 \\times 10^{-24}$ \\phn & $5.5 \\times 10^{-19}$ \\phn \n  & &\n  &   $1.5 \\times 10^{-22}$ \\phn & $4.0 \\times 10^{-17}$ \\phn \n  \\\\ \n  \\enddata\n  \\tablecomments{\n    $\\dot{n}_\\pm$: the electron\/positron production rate per unit volume,\n    $q_\\pm$: the energy production rate per unit volume in electron\/positron production.\n    B-H: Bethe-Heitler pair production, $e^-$ and $e^+$: photo-pion processes.\n    $K_p = 1$ \\added{cm$^{-3}$}, $\\gamma_{p, \\mathrm{min}}=1$,\n    and $\\gamma_{p, \\mathrm{max}} = 10^{10}$ are assumed.\n    All the values in the table are proportional to $K_p$.\n    }\n\\end{deluxetable}\n\n\n\\begin{figure}[t!]\n  \\centering\\includegraphics[scale=0.5,clip]{figure3.eps}\n  \\caption{The emission spectrum of PKS 0637-0752.\n    The black filled circles and crosses are data taken from \\cite{meyer2015} \n    and the crosses show the \\textit{Fermi} upper limit.\n    Models are calculated for $R =1$ kpc.\n    \\added{Solid lines are synchrotron radiation and SSC by primary electrons.\n      The bumps above $\\sim 10^{15}$ - $10^{16}$ Hz of the solid lines are SSC components.}\n    The blue solid line is for $B=0.05$ mG, the black solid line is \n    for $B=0.1$ mG, and the magenta solid line for $B=0.2$ mG.\n    \\added{Dashed and dot-dashed lines are emission by secondary pairs.}\n    The \\deleted{black} dashed line is for $p=2$ and $K_p = 8.8\\times 10^{-3}$ \n    \\added{cm$^{-3}$}\n    and the \\deleted{black} dot-dashed line is for $p=1.8$ and $K_p = 1.3 \\times 10^{-4}$\n    \\added{cm$^{-3}$}.\n    These spectra \\added{(dashed and dot-dashed)} are calculated with $B=0.1 \\,\\text{mG}$.\n    The bumps above $\\sim 10^{26} \\, \\text{Hz}$ are produced by inverse Compton scattering \n    of radio-optical photons off the secondary pairs produced by the Bethe-Heitler process.\n    \\label{fig:1kpc-no-beaming-exp}}\n\\end{figure}\n\n\nWe also examined a smaller size of $R_\\mathrm{kpc}=0.1$ and applied to \nthe knot WK8.9 as shown in Figure \\ref{fig:01kpc-spec}.\nWhen the magnetic field is $B_\\mathrm{mG}=0.3$, $K_p=0.18$ \\added{cm$^{-3}$} can \nreproduce the radio-optical and X-ray spectra, although \noverproduction of optical photons  still persists. \nThe proton power is around $10^{49} \\, \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\n\nFor a larger value of $R$, the required proton power increases; for example \nfor $R_\\mathrm{kpc} = 5$, $K_p =10^{-3}$ \\added{cm$^{-3}$} is needed amounting to \n$L_p =10^{50} \\,\\, \\text{erg}\\,\\, \\text{s}^{-1}$.\nThus, as for the emission region size, a small radius is favored in the energetics of protons. \nThese numerical results are consistent with a rather simple and optimistic estimate made \nin the previous section.\nThe predicted photon spectra show a roll-over at 10 MeV-GeV range \nand are compatible with the reported \\textit{Fermi} upper limits.\nSince the spectra are rather flat, the real problem is in the low energy \nend of the synchrotron emission. \nWe mostly skipped the contribution from electrons\/positrons of photo-pion origin.\nThey contribute mostly in the TeV range\nwith roughly similar luminosity to X-rays so that they do not affect the X-ray spectrum \nand the \\textit{Fermi} upper limit.\nWhen the maximum energy of protons is not so large, this component \ncan be totally ignored. \n\n\nSince the overproduction of optical flux is rather general, \nwe consider the reduction of the emission from the primary electrons.\nIn the above models we assumed the injection spectrum with the exponential cutoff.\nWhen we assume the power-law  injection spectrum without the exponential cutoff,\nthe optical emission is mainly from the secondary pairs.\nOur numerical result is shown in Figure \\ref{fig:1kpc-no-exp-cutoff}.\nSuch an abrupt super-exponential cutoff of primary electron \nenergy distribution may not be unlikely,\nwhen the acceleration is limited by cooling \\citep{krm98}.\n\nAn alternative idea to reduce the overproduction of the optical flux is \ntaking into account a mild relativistic beaming.\nIn this case, the photo-hadronic rates become less frequent\nby a factor of $\\delta^{3.75}$ due to the beaming effects for the same value \nof $\\gamma_p$,\n\\added{where $\\delta$ is the beaming factor.}\n\\added{(The scaling of $\\delta^{3.75}$ is obtained as follows.\nThe observed frequency $\\nu_\\mathrm{obs}$ and the frequency in the source \n$\\nu_s$ are related by $\\nu_\\mathrm{obs} = \\delta \\nu_s$. \nThe photon density at $\\nu_\\mathrm{obs}$ is given by\n$\\nu_\\mathrm{obs} n_{\\nu_\\mathrm{obs}} = \\delta^3 \\nu_s n_{\\nu_s}\n= A \\nu_\\mathrm{obs}^{-0.75} = A \\delta^{-0.75} \\nu_s^{-0.75}$, where $A$ is a constant.\nThen $\\nu_s n_{\\nu_s} = A \\delta^{-3.75} \\nu_s^{-0.75}$ follows.)\n}\nHowever, the source frame X-ray luminosity also decreases by a factor of $\\delta^4$, \nso that  the required proton power in the source frame does not much change,\n\\added{i.e., proportional to $\\delta^{0.25}$.}\nThe required kinetic power of protons increases by a factor of $\\delta^2$,\nif we set the bulk Lorentz factor $\\Gamma$ of the knot equal to $\\delta$.\nThe results for  $R_\\mathrm{kpc}=1$ and $B_\\mathrm{mG}=0.1$ are shown \nin Figure \\ref{fig:beaming-spec} for $\\delta = 3$.\nAs is seen, the overproduction of the optical flux can be \navoided in this case as well.\nThe appropriate value of $K_p=10^{-2}$ \\added{cm$^{-3}$} is similar to the non-beamed case,\nso that the required proton power becomes as high as \n$10^{51} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$, which seems to be unlikely.\nHowever, we note that milder value of $\\delta$ may reproduce the observed spectra , \nif the uncertainty of the ultraviolet flux exists by a factor of 1.5 or so.\n\n\n\n\\begin{figure}[ht!]     \n  \\centering\\includegraphics[scale=0.5,clip]{figure4.eps}\n  \\caption{Emission spectrum of the knot WK8.9 (the data are shown by filled green\n    circles taken from \\cite{meyer2015} ).\n    The size of the emission region is assumed to be $0.1$ kpc.\n    The red lines \\added{(solid and dashed)} are for $B = 0.3$ mG, \n    \\added{and} the black lines \\added{(solid and dashed)} are for $B=0.4$ mG.\n    \\deleted{and the green dot-dashed line is for $B=1$ mG.}\n    \\added{For $B = 1$ mG, only synchrotron radiation and SSC by primary electrons is shown\n      by the green dot-dashed line.}\n    To calculate the emission from pairs produced by Bethe-Heitler process, \n    $K_p = 0.18$ and 0.2 \\added{cm$^{-3}$} are assumed for $B=0.3$ and 0.4 mG, respectively.\n    \\label{fig:01kpc-spec}}\n\\end{figure}\n\n\n\\begin{figure}[ht!]     \n  \\centering\\includegraphics[scale=0.5,clip]{figure5.eps}\n  \\caption{\n    The injection of primary electrons without exponential cutoff\n    is assumed for $R = 1$ kpc.\n    The black line is for $B=0.1$ mG, the blue line is for $B=0.2$ mG, and the magenta line \n    is for  $B = 0.3$ mG.\n    The red line is the emission from pairs produced by Bethe-Heitler pair production for\n    $B= 0.1$ mG and $K_p=8 \\times 10^{-3}$ \\added{cm$^{-3}$}.\n    \\label{fig:1kpc-no-exp-cutoff}}\n\\end{figure}\n\n\n\\begin{figure}  [ht!]         \n  \\centering\\includegraphics[scale=0.5,clip]{figure6.eps}\n  \\caption{Beaming factor $\\delta = 3$ is assumed for $R =1$ kpc and $B = 0.1$ mG.\n    To calculate X-ray emission, $K_p = 1.1 \\times 10^{-2}$ \\added{cm$^{-3}$} is assumed.\n    \\label{fig:beaming-spec}}\n\\end{figure}\n\n\n \n\\section{Conclusion}  \\label{sec:conclusion}\n\nWe examined the  photo-hadronic  model of the X-ray emission from large scale jets \nof radio loud quasars specifically PKS 0637-752 and have found that Bethe-Heitler process is \neffective for the high energy electron\/positron injection. \nElectrons and positrons from photo-pion production  mainly radiate\nmulti-TeV photons by synchrotron radiation and do not much contribute to X-ray emission.\nFor an appropriate choice of parameters such as $R=1$ kpc and $B=0.1$ mG, \nthe required proton power is an order of $10^{49} \\, \\text{erg} \\,\\,  \\text{s}^{-1}$,\n\\added{which is $\\sim 120 L_\\mathrm{Edd}$ for $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\sun$},\nwhen the energy density of protons is concentrated in the region \nof $\\gamma_p =10^{9}$-$10^{10}$. \nCooling tail of these electrons and positrons radiate optical synchrotron emission,\nseparate from the primary electrons. \nTo avoid the overproduction of optical-ultraviolet flux,\neither the energy distribution of the primary electrons has a super-exponential  cutoff or \na mild degree of the relativistic beaming effect ($\\delta \\sim 3$) appears.\nFor the latter case, the required proton power tends to be large.\nThe Poynting and primary electron powers remain moderate. \nBecause the value of the beaming factor is not strongly constrained\n\\citep[e.g.,][]{meyer2015}, further work is needed to determine\nwhich mechanism is applicable to reduce the optical synchrotron emission.\n\nProton synchrotron radiation is a few orders of magnitude smaller than \nthe photo-hadronic model prediction.\nThus, photo-proton model is an alternative option to \nexplain the strong X-ray emission from large scale jets. \n\n\n\\added{The black hole mass of PKS0627-752 given by \\cite{ljg2006} is \n$6.5 \\times 10^8 M_\\odot$.  \nOn the other hand,  \\cite{gcj2001} gives $7.8 \\times 10^9 M_\\odot$.\nThe Eddington luminosity is \n$\\sim 8.2 \\times 10^{46}$ erg s$^{-1}$ and $\\sim 9.8 \\times 10^{47}$ erg s$^{-1}$ \nfor $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\odot$ and $7.8 \\times 10^9 M_\\odot$,\nrespectively.\nThus $10^{49}$ erg s$^{-1}$ is $\\sim 10$ - $100$ times larger than the Eddington \nluminosity.\nAs for other AGNs, very luminous AGNs have been observed.\nFor example, \\cite{gfv2009} showed that S5 0014+813 has\n$\\nu L_\\nu \\sim 10^{48}$ erg s$^{-1}$ in the optical and this corresponds to \n$0.17 L_\\mathrm{Edd}$ for $M_\\mathrm{BH} = 4 \\times 10^{10} M_\\sun$.\nSome authors, e.g., \\cite{sn2016}, on the other hand,  \nperformed radiation magnetohydrodynamical simulation of super-Eddington mass accretion.\nIn view of uncertainties of mass estimation and theoretical possibility \nof super-Eddington jet power, we believe our model is still viable, \nalthough the required proton power is very large.\n}\n\nFinally, our model predicts TeV emission by inverse Compton scattering of radio-optical\nphotons off pairs produced by the Bethe-Heitler process.\nEmission by electrons\/positrons produced by photo-pion processes will also \ncontribute to TeV emission.\n\n\n\n\\acknowledgments\n\\added{We are grateful to the referee for useful comments that improved the manuscript \nconsiderably. }\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nEntanglement shared among multiple parties is acknowledged as one of the fundamental resources driving the second quantum revolution~\\cite{DowlingMilburn2003}, for instance, as a basis of quantum network proposals~\\cite{EppingKampermannMacchiavelloBruss2017, PivoluskaHuberMalik2018, RibeiroMurtaWehner2018, BaeumlAzuma2017}, as a key resource for improved quantum sensing~\\cite{Toth2012} and quantum error correction~\\cite{Scott2004} or as generic ingredient in quantum algorithms~\\cite{BrussMacchiavello2011} and measurement-based quantum computation~\\cite{RaussendorfBriegel2001, BriegelRaussendorf2001}. Yet, its detection and characterisation are complicated by several factors: among them, the computational hardness of deciding whether any given system even exhibits any entanglement at all~\\cite{Gurvits2004} as well as the fact that the usual paradigm of local operations and classical communication (LOCC) lead to infinitely many types of entanglement~\\cite{VerstraeteDehaeneDeMoorVerschelde2002, OsterlohSiewert2005, DeVicenteSpeeKraus2013, SchwaigerSauerweinCuquetDeVicenteKraus2015, DeVicenteSpeeSauerweinKraus2017, SpeeDeVicenteSauerweinKraus2017, SauerweinWallachGourKraus2018} already for single copies of multipartite states. Significant effort has thus been devoted to devising practical means of entanglement certification from limited experimental data~\\cite{TothGuehne2005b, FriisVitaglianoMalikHuber2019}.\n\n\nOne of the principal challenges for the characterisation of multipartite entanglement lies in distinguishing between \\emph{partial separability} and its counterpart, \\emph{genuine multipartite entanglement} (GME)\\footnote{Note that the term was also coined for multipartite pure states with exclusively non-vanishing $n$-tangle in Ref.~\\cite{OsterlohSiewert2005}.}.\nHere, a multipartite state is called \\emph{partially separable} if it can be decomposed as a mixture of \\emph{partition-separable} states, i.e., of states separable with respect to some (potentially different) partitions of the parties into two or more groups, whereas any state that cannot be decomposed in this way has GME (see Fig.~\\ref{fig:GME structure} and Table~\\ref{tab:term}). One may further classify partially separable states as $k$-separable states according to the maximal number $k$ of tensor factors that all terms in the partially separable decomposition can be factorised into. If a state admits a decomposition where each term is composed of at least two tensor factors ($k=2$), the state is called \\emph{biseparable}. Thus, every partially separable state is $k$-separable for some $k\\geq2$, and hence (at least) biseparable.\nThis distinction arises naturally when considering the resources required to create a specific state:\nany biseparable state can be produced via LOCC in setups where all parties share classical randomness and subsets of parties share entangled states.\nOne of the counter-intuitive features of partially separable states is the possibility for bipartite entanglement across every possible bipartition\\footnote{An explicit example of a $k\/2$-separable (and thus biseparable) $k$-qubit state (for even $k$) with the bipartite entanglement between all neighbouring qubits in a linear arrangement can be found in~\\cite[footnote 30]{FriisMartyEtal2018}.}.\nConsequently, the notion of bipartite entanglement across partitions is insufficient to capture the notion of partial separability, and conventional methods, such as positive maps~\\cite{HorodeckiMPR1996, Peres1996}, cannot be straightforwardly applied to reveal GME (with new concepts for positive maps derived for that purpose in~\\cite{HuberSengupta2014, ClivazHuberLamiMurta2017}), which results in additional challenges compared to the \\textemdash~relatively \\textemdash~simpler scenario of detecting bipartite or partition entanglement (e.g., as in~\\cite{RodriguezBlancoBermudezMuellerShahandeh2021}).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{flower2.png}\n\\vspace*{-8mm}\n\\caption{\\textbf{GME and (partial) separability for three qubits}. All three-qubit states separable with respect to~one of the three bipartitions, $\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}$ (yellow), $\\mathcal{A}_{2}|\\mathcal{A}_{1}\\mathcal{A}_{3}$ (darker green), and $\\mathcal{A}_{3}|\\mathcal{A}_{1}\\mathcal{A}_{2}$ (background), form convex sets, whose intersection (turquoise) contains (but is not limited to) all fully separable states $\\mathcal{A}_{1}|\\mathcal{A}_{2}|\\mathcal{A}_{3}$ (dark blue). The convex hull of these partition-separable states contains all partially separable (the same as biseparable for tripartite systems) states. All states that are not biseparable are GME\\@. States with $k$-copy activatable GME are contained in the set of biseparable but not partition-separable states and are conjectured to form the lighter green areas, with those states for which GME is activatable for higher values of $k$ farther away from the border between GME and biseparability.\nThe horizontal line represents the family of isotropic GHZ states $\\rho(p)$, containing the maximally mixed state ($p=0$) and the GHZ state ($p=1$). The values $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ indicate $k$-copy GME activation thresholds, which we discuss in the following.}\n\\label{fig:GME structure}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{GME_activation_3b.pdf}\n\\vspace*{-4mm}\n\\caption{\\textbf{Activation of GME from biseparable states}. (a) Separable bipartite states remain separable, no matter how many copies are shared, e.g., if $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1}\\mathcal{A}_{2}}$ and $\\rho\\subtiny{0}{0}{\\mathcal{B}_{1}\\mathcal{B}_{2}}$ are separable with respect to~the bipartitions $\\mathcal{A}_{1}|\\mathcal{A}_{2}$ and $\\mathcal{B}_{1}|\\mathcal{B}_{2}$, then so is  $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1}\\mathcal{A}_{2}}\\otimes\\rho\\subtiny{0}{0}{\\mathcal{B}_{1}\\mathcal{B}_{2}}$. (b) In contrast, the joint state of multiple copies of biseparable states, e.g., $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}$, $\\rho\\subtiny{0}{0}{\\mathcal{B}_{1},\\mathcal{B}_{2},\\ldots,\\mathcal{B}_{N}}$, and $\\rho\\subtiny{0}{0}{\\mathcal{C}_{1},\\mathcal{C}_{2},\\ldots,\\mathcal{C}_{N}}$, can be GME with respect to~the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}\\mathcal{C}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}\\mathcal{C}_{2}|\\ldots|\\mathcal{A}_{\\!N}\\mathcal{B}_{\\!N}\\mathcal{C}_{\\!N}$.\n\\label{fig:GME activation}\n}\n\\end{figure}\n\n\\begin{table*}\n\\centering\n\\caption{\\label{tab:term}Summary of terminology on GME in this paper.}\n\\begin{tabular}{@{}ll@{}}\n\\toprule\nTerm & Meaning\\\\\n\\midrule\n$k$-separable &\\parbox{12cm}{convex combination of pure states, each of which is a product of at least $k$ projectors}\\\\\nbiseparable & synonymous with $2$-separable\\\\\npartially separable& $k$-separable for some $k>1$\\\\\npartition-separable & \\parbox{12cm}{separable for a specific partition of the multipartite Hilbert space, i.e., a convex combination of projectors, each of which is a product with respect to the same partition into subsystems}\\\\\nmultipartite entangled & entangled across all bipartitions\\\\\ngenuine multipartite entangled& non-biseparable\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table*}\n\nAn assumption inherent in the definitions above is that all parties locally act only on a single copy of the distributed state. \nHowever, in many experiments where quantum states are distributed among (potentially distant) parties, multiple independent but identically prepared copies of states are (or at least, can be) shared. For instance, exceptionally high visibilities of photonic states can only be achieved if each detection event stems from almost identical quantum states~\\cite{JoshieEtAl2020,WengerowskyJoshiSteinlechnerHuebelUrsin2018}. Adding noise to the channel then produces the situation we focus on in this article: multiple copies of noisy quantum states produced in a laboratory~\\cite{Ecker-Huber2019,HuEtAl2020}.\nEven limited access to quantum memories or signal delays then allows one to act on multiple copies of the distributed states, which is a recurring theme also in research on quantum networks~\\cite{YamasakiPirkerMuraoDuerKraus2018,NavascuesWolfeRossetPozasKerstjens2020, KraftDesignolleRitzBrunnerGuehneHuber2021}.\nCharacterising properties of GME in multi-copy scenarios is thus not only of fundamental theoretical interest but also crucial for practical applications that require GME to be distributed, such as conference key agreement~\\cite{MurtaGrasselliKampermannBruss2020}.\n\nHowever, we demonstrate here that, unlike the distinction between separable and entangled states, the distinction between biseparability and GME is not maintained in the transition from one to many copies; i.e., partial separability is not a tensor-stable concept.\nAs we show, for $N$ parties $1,\\ldots,N$, there exist multipartite quantum states $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}$ that are biseparable, but which can be \\emph{activated} in the sense that sharing two copies results in a GME state, i.e., such that the joint state $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}\\otimes \\rho\\subtiny{0}{0}{\\mathcal{B}_{1},\\mathcal{B}_{2},\\ldots,\\mathcal{B}_{N}}$ of two identical copies (labelled $\\mathcal{A}$ and $\\mathcal{B}$, respectively) is not biseparable with respect to the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}|\\ldots|\\mathcal{A}_{N}\\mathcal{B}_{N}$. (See Fig.~\\ref{fig:GME activation}.)\nThat such activation of GME is in principle possible had previously only been noted in~\\cite{HuberPlesch2011}, where it was observed that two copies of a particular four-qubit state that is itself almost fully separable can become GME\\@. \nHere, we systematically investigate this phenomenon of \\emph{multi-copy GME activation}. As the first main result, we show that the property of biseparability is not tensor stable in general by identifying a family of $N$-qubit isotropic Greenberger-Horne-Zeilinger (GHZ) states with two-copy activatable GME for all $N$.\nWe further demonstrate the existence of biseparable states within this family for which two copies are not enough to activate GME, but three copies are.\nMoreover, we show that the bound for partition-separability coincides with the asymptotic (in terms of the number of copies) GME-activation bound for isotropic GHZ states.\n\nMulti-copy GME activation is particularly remarkable \\textemdash~and may appear surprising at first \\textemdash~because it is in stark contrast to bipartite entanglement:\nTwo copies of states separable with respect to a fixed partition always remain partition-separable and can never become GME\\@.\nHowever, from the perspective of entanglement distillation \\textemdash~the concentration of entanglement from many weakly entangled (copies of) states to few strongly entangled ones \\textemdash~such an activation seems more natural.\nAfter all, if one party shares bipartite maximally entangled states with each other party, these could be used to establish any GME state among all $N$ parties via standard teleportation, thus distributing GME by sharing only two-party entangled states.\nNevertheless, such a procedure would require at least $N-1$ copies of these bipartite entangled states (in addition to a local copy of the GME state to be distributed), and already the example from~\\cite{HuberPlesch2011} suggests that one does not have to go through first distilling bipartite entangled pairs, followed by teleportation, but two copies can naturally feature GME already.\nWhile we have seen that the phenomenon of GME activation is more than just distillation, one may still be tempted to think that distillable entanglement is required for GME activation.\nIt is known that there exist bound entangled states \\textemdash~entangled states that do not admit distillation of entanglement no matter how many copies are provided.\nIn particular, all entangled states with positive partial transpose (PPT) across a given cut are undistillable since any number of copies is also PPT\\@.\nOne might thus suspect that GME activation should not be possible for biseparable states that are PPT across every cut and hence have no distillable entanglement (even if multiple parties are allowed to collaborate).\nAs another main result, we show that this is not the case by constructing a biseparable state that is PPT with respect to~every cut, yet two copies of the state are indeed GME\\@. \nTogether, our results thus support the following conjectures:\n\\begin{enumerate}[(i)]\n\\item{\\label{conjecture i}\nThere exists a hierarchy of states with $k$-copy activatable GME, i.e., for all $k\\geq2$ there exists a biseparable but not partition-separable state $\\rho$ such that $\\rho^{\\otimes k-1}$ is biseparable, but $\\rho^{\\otimes k}$ is GME\\@.\n}\n\\item{\\label{conjecture ii}\nGME may be activated for any biseparable but not partition-separable state (light green areas in Fig.~\\ref{fig:GME structure}) of any number of parties as $k\\rightarrow\\infty$.}\n\\end{enumerate}\n\nIn the following, we first provide the formal definitions for biseparability and GME in Sec.~\\ref{sec:sep and gme} before turning to the family of $N$-qubit isotropic GHZ states in Sec.~\\ref{sec:GME of isotropic GHZ states}. For all biseparable states in this family, we provide upper bounds on the minimal number of copies required to activate GME in Sec.~\\ref{sec:Multi-copy GME criterion}. In Sec.~\\ref{sec:Hierarchy of k-copy activatable states}, we then consider the case of three qubits ($N=3$), for which we can show that the bound on three-copy GME activation is tight in the sense that we identify all states in the family for which one requires at least three copies to activate GME, while two copies remain biseparable, and can also show that GME can indeed be activated for any biseparable but not partition-separable state in this family. Moreover, in Sec.~\\ref{sec:GME activation of PPT entangled states}, we construct an explicit example for two-copy GME activation from biseparable states with no distillable bipartite entanglement. Finally, we discuss the implications of our results and open questions in Sec.~\\ref{sec:Conclusion and Outlook}.\n\n\n\\section{Definitions of biseparability \\& GME}\\label{sec:sep and gme}\nWe summarise the formal definitions of biseparability and GME in this paper.\n(See also Table~\\ref{tab:term} for the summary of the definitions here.)\nFormally, a pure quantum state of an $N$-partite system with Hilbert space $\\mathcal{H}\\suptiny{0}{0}{(N)}=\\bigotimes_{i=1}^{N}\\mathcal{H}_{i}$ is separable with respect to~a $k$-partition $\\{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{k}\\}$, with $\\mathcal{A}_{i}\\subset\n\\{1,2,3,\\ldots,N\\}$ and $\\bigcup_{i=1}^{k} \\mathcal{A}_{i}= \\{1,2,3,\\ldots,N\\}$ such that $\\bigotimes_{i=1}^{k}\\mathcal{H}_{\\mathcal{A}_{i}}=\\mathcal{H}\\suptiny{0}{0}{(N)}$, if it can be written as\n\\begin{align}\n    \\ket{\\Phi\\suptiny{0}{0}{(k)}}  &=\\,\\bigotimes\\limits_{i=1}^{k}\\,\\ket{\\phi_{\\mathcal{A}_{i}}},\\quad\\ket{\\phi_{\\mathcal{A}_{i}}}\\in\\mathcal{H}_{\\mathcal{A}_{i}}\n    \\,.\\label{pure}\n\\end{align}\nWhen generalising to density matrices, it is common not to specify all possible partitions, but to use the notion of \\emph{$k$-separability} instead: \nA density operator is called \\emph{$k$-separable} if it can be decomposed as a convex sum of pure states that are all separable with respect to~\\emph{some} $k$-partition, i.e., if it is of the form (see, e.g., the review~\\cite{FriisVitaglianoMalikHuber2019})\n\\begin{align}\n    \\rho\\suptiny{0}{0}{(k)} &=\\,\n    \\sum\\limits_{i} p_{i} \n    \\ket{\\Phi_i\\suptiny{0}{0}{(k)}}\\!\\!\\bra{\\Phi_i\\suptiny{0}{0}{(k)}}\n    \\,.\\label{ksep}\n\\end{align}\nNote that the lack of tensor stability of partial separability shown in the following also implies that the related concept of $k$-producibility~\\cite{GuehneToth2009,Szalay2019} is not tensor stable. Crucially, each $\\ket{\\Phi_i\\suptiny{0}{0}{(k)}}$ may be $k$-separable with respect to~a different $k$-partition. Consequently, $k$-separability does not imply separability of $\\rho\\suptiny{0}{0}{(k)}$ with respect to~a specific partition, except when $\\rho\\suptiny{0}{0}{(k)}$ is a pure state or when $k=N$. In the latter case the state is called \\emph{fully separable}.\nTo make this distinction more explicit, we refer to all (at least) biseparable states that are actually separable with respect to~some bipartition as \\emph{partition-separable}.\nAt the other end of this separability spectrum one encounters \\emph{biseparable states} ($k=2$), while all states that are not at least biseparable (formally, $k=1$) are called \\emph{genuinely $N$-partite entangled}. We will here use the term GME for the case $k=1$.\nThe operational reason for this definition of GME is easily explained: any biseparable state of the form of Eq.~(\\ref{ksep}) can be created by $N$ parties purely by sharing partition-separable states of the form of Eq.~(\\ref{pure}) and some classical randomness. \nIn addition, this conveniently results in a convex notion of biseparability (as illustrated for the example in Fig.~\\ref{fig:GME structure}) amenable to entanglement witness techniques, which inherently rely on convexity.\n\n\n\\section{GME of isotropic GHZ states}\\label{sec:GME of isotropic GHZ states}\nTo overcome the difficulty in analysing GME, the crucial technique here is to use states in $X$-form, i.e., those with nonzero entries of density operators only on the main diagonal and main anti-diagonal with respect to~the computational basis.\nLet us now consider a family of mixed $N$-qubit states, \\emph{isotropic GHZ states}, given by\n\\begin{align}\n    \\rho(p) &=\\,p\\,\\ket{\\mathrm{GHZ}_{N}\\!}\\!\\!\\bra{\\mathrm{GHZ}_{N}\\!}\\,+\\,(1-p)\\,\\tfrac{1}{2^{N}}\\mathds{1}_{2^{N}}\\,,\n    \\label{eq:GHZ with white noise}\n\\end{align}\nobtained as convex combination of the $N$-qubit maximally mixed state $\\tfrac{1}{2^{N}}\\mathds{1}_{2^{N}}$ and a pure \n$N$-qubit GHZ state\n\\begin{align}\n    \\ket{\\mathrm{GHZ}_{N}\\!}  &=\\,\\tfrac{1}{\\sqrt{2}}\\bigl(\\ket{0}^{\\otimes N}+\\ket{1}^{\\otimes N}\\bigr).\n\\end{align}\nwith real mixing parameter $p\\in[-1\/(2^{N}-1),1]$.\nSince states in this family are in $X$-form with respect to~the $N$-qubit computational basis, we can straightforwardly calculate the \\emph{genuine multipartite} (GM) \\emph{concurrence}, an entanglement measure for a multipartite state defined in terms of a polynomial of elements of its density matrix~\\cite{HashemiRafsanjaniHuberBroadbentEberly2012,MaChenChenSpenglerGabrielHuber2011}. For any $N$-qubit density operator $\\rho_{X}$ in $X$-form, i.e., \n\\begin{align}\n    \\rho_{X}=\\begin{pmatrix} \\tilde{a} & \\tilde{z}\\,\\tilde{d} \\\\ \\tilde{d}\\,\\tilde{z}^{\\dagger} & \\tilde{d}\\,\\tilde{b}\\,\\tilde{d} \\end{pmatrix},\n\\end{align}\nwhere $\\tilde{a}=\\diag\\{a_{1},\\ldots,a_{n}\\}$,  $\\tilde{b}=\\diag\\{b_{1},\\ldots,b_{n}\\}$, and $\\tilde{z}=\\diag\\{z_{1},\\ldots,z_{n}\\}$ are diagonal $n\\times n$ matrices with $n=2^{N-1}$, $a_{i},b_{i}\\in\\mathbb{R}$ and $z_{i}\\in\\mathbb{C}$ for all $i=1,2,\\ldots,n$, and $\\tilde{d}=\\operatorname{antidiag}\\{1,1,\\ldots,1\\}$ is antidiagonal,\nthe GM concurrence is given by\n\\begin{align}\n    C_{\\mathrm{GM}}(\\rho_{X}) &=\\,2\\max\\bigl\\{0,\\max_{i}\\{|z_{i}|-\\sum\\limits_{j\\neq i}^{n}\\sqrt{a_{j}b_{j}}\\}\\bigr\\},\n    \\label{eq:GM concurrence}\n\\end{align}\nand provides a necessary and sufficient condition for GME whenever $C_{\\mathrm{GM}}>0$. \nIn the case of the state $\\rho(p)$ from Eq.~(\\ref{eq:GHZ with white noise}), we have $a_{i}=b_{i}=\\tfrac{1-p}{2^{N}}+\\delta_{i1}\\tfrac{p}{2}$ and $z_{i}=\\delta_{i1}\\tfrac{p}{2}$, such that\n\\begin{align}\n    C_{\\mathrm{GM}}\\bigl[\\rho(p)\\bigr] &=\\,\\max\\{0,|p|-(1-p)(1-2^{1-N})\\}.\n\\end{align}\nThus, $\\rho(p)$ is GME if and only if\n\\begin{align}\n    p   &>\\,p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{2^{N-1}-1}{2^{N}-1}\\,,\n\\end{align}\ni.e., if and only if $p$ surpasses the single-copy threshold $p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}$.\nConversely, we can be certain that $\\rho(p)$ is not GME for $p\\leq (2^{N-1}-1)\/(2^{N}-1)$, and hence at least biseparable.\n\n\n\\section{Multi-copy GME criterion}\\label{sec:Multi-copy GME criterion}\nOur first goal is then to check if two copies of $\\rho(p)$ are GME\\@.\nSince the GM concurrence is an entanglement monotone, $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes k}\\bigr]$ is monotonically non-decreasing as $k$ increases~\\cite{MaChenChenSpenglerGabrielHuber2011};\nthat is, if we have $C_\\mathrm{GM}\\bigl[\\rho(p)\\bigr]=0$ for $\\rho(p)$ in $X$-form, it holds that $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]\\geq 0$ in general.\nHowever, \nusing $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]> 0$\nas a necessary and sufficient criterion for GME \nis not an option in this case, \nsince ${\\rho(p)}^{\\otimes 2}$ may not be of $X$-form even if a single copy is, and we therefore generally cannot directly calculate $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]$. \nThe crucial idea here is to make use of the fact that stochastic LOCC (SLOCC) can never create GME from a biseparable state.\n\nTo construct a sufficient GME criterion,\nwe therefore use a map $\\mathcal{E}_{\\circ}$ implementable via SLOCC~\\cite{LamiHuber2016}, which, for any two density operators $\\rho$ and $\\sigma$ acting on $\\mathcal{H}$,\nmaps the state $\\rho\\otimes\\sigma$  acting on $\\mathcal{H}^{\\otimes 2}$ to\n\\begin{align}\n    \\mathcal{E}_{\\circ}[\\rho\\otimes\\sigma] &=\\,\\frac{\\rho\\circ\\sigma}{\\textnormal{Tr}(\\rho\\circ\\sigma)}\\quad\\text{on }\\mathcal{H},\n\\end{align}\nwhere the right-hand side is a density operator acting on $\\mathcal{H}$, and ``$\\circ$'' denotes the Hadamard product (or Schur product), i.e., the component-wise multiplication of the two matrices.\nWhat is useful for us here is that the Hadamard product of two $X$-form matrices results in an $X$-form matrix. Consequently, we can directly calculate the GM concurrence for the state resulting from applying the `\\emph{Hadamard-product map}' $\\mathcal{E}_{\\circ}$ to two copies of an originally biseparable state.\nIf the GM concurrence of $\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]$ is nonzero, we can conclude that two copies of $\\rho(p)$ are GME, even if a single copy is not.\nTo decide whether $\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]$ is GME or not, i.e., whether the GM concurrence is nonzero or not, we can ignore the normalization and just consider $\\rho(p)\\circ\\rho(p)=\\rho(p)^{\\circ 2}$. Moreover, in the maximization over the index $i$ in Eq.~(\\ref{eq:GM concurrence}), the maximum is obtained for $i=1$. We can thus conclude that $\\rho(p)^{\\otimes 2}$ is GME if\n\\begin{align}\n    |z_{1}^{2}|-\\sum\\limits_{j\\neq 1}^{n}\\sqrt{a_{j}^{2}b_{j}^{2}}\\,=\\,\\tfrac{p^{2}}{4}-(2^{N-1}-1)\\bigl(\\tfrac{1-p}{2^{N}}\\bigr)^{2}\\,>\\,0,\n    \\label{eq:nonzero GM concurrence 2 copies}\n\\end{align}\nwhich translates to the condition $p\/(1-p)>\\sqrt{2^{N-1}-1}\/2^{N-1}$, and in turn can be reformulated to the condition\n\\vspace*{-2mm}\n\\begin{align}\n    p   &>\\,p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{\\sqrt{2^{N-1}-1}}{2^{N-1}+\\sqrt{2^{N-1}-1}}.\n    \\label{eq:GME treshhold 2 copies}\n\\end{align}\nAs we see, we have $p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}>p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}$ for all $N\\geq3$, confirming that \\emph{there exist biseparable states} with values $p<p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}$ \\emph{for which two copies are GME}, i.e., such that $p>p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}$. \n\nMoreover, we can now concatenate multiple uses of the SLOCC map $\\mathcal{E}_{\\circ}$. For instance, we can identify the threshold value $p\\suptiny{0}{0}{(3)}_{\\mathrm{GME}}$ of $p$ at which the state $\\mathcal{E}_{\\circ}\\bigl[\\rho(p)\\otimes\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]\\bigr]$ resulting from $2$ applications of $\\mathcal{E}_{\\circ}$ to a total of $3$ copies of $\\rho(p)$ is GME, or, more generally, the corresponding threshold value $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ for which $k$ copies result in a GME state after applying the map $\\mathcal{E}_{\\circ}$ a total of $k-1$ times. From Eq.~(\\ref{eq:nonzero GM concurrence 2 copies}) it is easy to see that these threshold values are obtained as\n\\begin{align}\n    p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{\\sqrt[k]{2^{N-1}-1}}{2^{N-1}+\\sqrt[k]{2^{N-1}-1}}.\n\\end{align}\n\n\n\\section{Hierarchy of $k$-copy activatable states}\\label{sec:Hierarchy of k-copy activatable states}\nThe threshold values $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ provide upper bounds on the minimal number of copies required to activate GME\\@: a value $p$ satisfying $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}<p<p\\suptiny{0}{0}{(k-1)}_{\\mathrm{GME}}$ for $k\\geq2$ implies that $k$ copies are enough to activate GME\\@.\nBut since the map $\\mathcal{E}_{\\circ}$ (does not create and) may reduce GME, it does not imply that $k$ copies are actually needed; up to this point, there is a possibility that two copies are all it takes.\n\nHowever, at least for the case of three qubits ($N=3$) and three copies ($k=3$), we find that this is not the case. That is, for all isotropic three-qubit GHZ states with $p\\le p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}(N=3)=\\sqrt{3}\/(4+\\sqrt{3})$, we find that two copies are still biseparable, and thus at least three copies are required to activate GME\\@. The explicit biseparable decomposition of two copies of the states in this range is presented in Appendix~\\ref{sec:appendix two copy bisep}. Although it does not constitute conclusive proof, this result nevertheless supports our first conjecture, repeated here for convenience:\n\n\\noindent\n\\emph{Conjecture~(\\ref{conjecture i}):\\ There exists a hierarchy of states with $k$-copy activatable GME, i.e., for all $k\\geq2$ there exists a biseparable but not partition-separable state $\\rho$ such that $\\rho^{\\otimes k-1}$ is biseparable, but $\\rho^{\\otimes k}$ is GME\\@.}\n\nThe conjectured existence of a hierarchy of biseparable states with $k$-copy activatable GME means that states become less and less `valuable' as the number of copies $k$ required to obtain GME increases. At the same time, it is also clear that all partition-separable states cannot be used to activate GME because separability with respect to~any fixed partition is tensor stable. But it is not clear where exactly the boundary between activatable and non-activatable states really lies (see Fig.~\\ref{fig:GME structure}).\n\nTo shed light on this question, let us again examine the isotropic GHZ states from Eq.~(\\ref{eq:GHZ with white noise}) with regards to partition-separability with respect to~the bipartition separating the first qubit $\\mathcal{A}_{1}$ from the remaining $N-1$ qubits (collected in $\\tilde{\\mathcal{A}}_{2}$), i.e., $\\mathcal{A}_{1}|\\tilde{\\mathcal{A}}_{2}$. Using this partition, we can write\n\\begin{align}\n    \\rho(p) &=p\\ket{\\Phi^{+}}\\!\\!\\bra{\\Phi^{+}}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}\n    \\!+\\!\\tfrac{1-p}{2^{N}}\\,\\mathds{1}_{\\!\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\n    \\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}\n    \\!+\\!\\tfrac{1-p}{2^{N}}\\,\\mathds{1}_{\\!\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}\\nonumber\\\\\n    &=\\,\n    \\tfrac{1+p}{2}\\,\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}\\,\n    +\\tfrac{1-p}{2}\\,\\tfrac{1}{2^{N-1}}\\,\n    \\mathds{1}_{\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\n    \\mathds{1}_{\\tilde{\\mathcal{A}}_{2}^{\\perp}},\n    \\label{eq:GHZ with white noise rewritten}\n\\end{align}\nwhere $\\ket{\\Phi^{+}}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}=\\tfrac{1}{\\sqrt{2}}\\bigl(\\ket{0}_{\\mathcal{A}_{1}}\\ket{\\tilde{0}}_{\\tilde{\\mathcal{A}}_{2}}+\\ket{1}_{\\mathcal{A}_{1}}\\ket{\\tilde{1}}_{\\tilde{\\mathcal{A}}_{2}}\\bigr)$ with $\\ket{\\tilde{i}}_{\\tilde{\\mathcal{A}}_{2}}=\\bigotimes_{j=2}^{N}\\ket{i}_{\\mathcal{A}_{j}}$ for $i=0,1$, $\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}=\\sum_{i=0,1}\\ket{\\tilde{i}}\\!\\!\\bra{\\tilde{i}}$ and $\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}=\\mathds{1}_{2^{N-1}}-\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}$. From this decomposition, it becomes clear that the state can be written as a convex sum of a two-qubit state $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$ (where the second qubit lives on the two-dimensional subspace of $\\tilde{\\mathcal{A}}_{2}$, spanned by the states $\\ket{\\tilde{i}}$ for $i=0,1$) and diagonal terms proportional to $\\mathds{1}_{\\mathcal{A}_{1}}\\otimes\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}$ with support in a subspace $\\tilde{\\mathcal{A}}_{2}^{\\perp}$ orthogonal to $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$. The latter diagonal terms trivially have a separable decomposition with respect to~the bipartition $\\mathcal{A}_{1}|\\tilde{\\mathcal{A}}_{2}$. For the two-qubit state $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$, the PPT criterion offers a necessary and sufficient separability criterion, and one easily finds that the partial transpose of $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$ is non-negative if $p\\leq p_{\\mathrm{crit}}\\coloneqq 1\/(1+2^{N-1})$ (see Appendix~\\ref{appendix:PPT criterion for isotropic GHZ states}). Further taking into account its qubit exchange symmetry, we thus find that $\\rho(p)$ is partition-separable with respect to~any bipartition for $p\\leq p_{\\mathrm{crit}}$. At the same time, we find that $\\lim_{k\\rightarrow\\infty}p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}(N)=p_{\\mathrm{crit}}$, which implies that any isotropic GHZ state with $p>p_{\\mathrm{crit}}$ features $k$-copy activatable GME, at least asymptotically as $k\\rightarrow\\infty$, and is thus also not partition-separable. This leads us to our second conjecture, also repeated here for convenience:\n\n\\noindent\n\\emph{Conjecture~(\\ref{conjecture ii}):\\ \nGME may be activated for any biseparable but not partition-separable state of any number of parties as $k\\rightarrow\\infty$.}\n\nConjecture~(\\ref{conjecture ii}) holds for isotropic GHZ states. But does it hold in general?\n\n\n\\section{GME activation from PPT entangled states}\\label{sec:GME activation of PPT entangled states}\nA situation where one might imagine Conjecture~(\\ref{conjecture ii}) to fail is the situation of biseparable (but not partition-separable) states with PPT entanglement across every bipartition, as discussed in Sec.~\\ref{sec:introduction}.\nFor isotropic GHZ states, however, the PPT criterion across every cut coincides exactly with the threshold $p_{\\mathrm{crit}}$ for biseparability (and GME activation), as one can confirm by calculating the eigenvalues of the partial transpose of $\\rho(p)$ (see Appendix~\\ref{appendix:PPT criterion for isotropic GHZ states}).\nWe thus turn to a different family of states, for which this is not the case.\n\nSpecifically, as we show in detail in Appendix~\\ref{appendix:PPT entangled GME activation}, we construct a family of biseparable three-party states \n\\begin{align}\n    \\rho_{\\mathcal{A}_{1}\\mathcal{A}_{2}\\mathcal{A}_{3}\n    } &=\\,\n    \\sum\\limits_{\\substack{i,j,k=1\\\\ i\\neq j\\neq k\\neq i}}^{3}p_{i}\\ \\rho_{\\mathcal{A}_{i}}\\otimes\\rho_{\\mathcal{A}_{j}\\mathcal{A}_{k}}\\suptiny{0}{0}{\\mathrm{PPT}}\n\\end{align}\nwhere the $\\rho_{\\mathcal{A}_{j}\\mathcal{A}_{k}}\\suptiny{0}{0}{\\mathrm{PPT}}$ are (different) two-qutrit states with PPT entanglement across the respective cuts $\\mathcal{A}_{j}|\\mathcal{A}_{k}$ for $j\\neq k\\in\\{1,2,3\\}$ and $\\sum_{i}p_{i}=1$. Via LOCC, three copies (labelled $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$, respectively) of this state $\\rho_{\\mathcal{A}_{1}\\mathcal{A}_{2}\\mathcal{A}_{3}}$ can be converted to what we call \\emph{PPT-triangle states} of the form\n\\begin{equation}\n    \\rho_{\\mathcal{A}_{2}\\mathcal{A}_{3}}\\suptiny{0}{0}{\\mathrm{PPT}}\\otimes \\rho_{\\mathcal{B}_{1}\\mathcal{B}_{3}}\\suptiny{0}{0}{\\mathrm{PPT}}\\otimes \\rho_{\\mathcal{C}_{1}\\mathcal{C}_{2}}\\suptiny{0}{0}{\\mathrm{PPT}}.\n\\end{equation}\nUsing a GME witness based on the lifted Choi map (cf.~\\cite{HuberSengupta2014, ClivazHuberLamiMurta2017}), we show that there exists a parameter range where these PPT-triangle states are GME.\nTherefore, it is proved that GME activation is possible even from biseparable states only with PPT entanglement across every bipartition.\n\n\n\\section{GME activation and shared randomness}\n\nProvided that our conjectures are true, incoherent mixing (access to shared randomness) can lead to situations where the number of copies needed for GME activation is reduced. In the extreme case, and this is true even based only on the results already proven here (and thus independently of whether or not the conjectures turn out to be true or not), the probabilistic combination of partition-separable states (without activatable GME) can results in a state \\textemdash\\ a biseparable isotropic GHZ state \\textemdash\\ which has activatable GME. Although this may at first glance appear to be at odds with the usual understanding of bipartite entanglement, which cannot arise from forming convex combinations of separable states, we believe this can be understood rather intuitively if we view incoherent mixing as a special case of a more general scenario in which one may have any amount of information on the states that are shared between different observers. As an example, consider the following situation:\n\nThree parties, labelled, $1$, $2$ and $3$, share two identical (as in, the system and its subsystems have the same Hilbert space dimensions and are represented by the same physical degrees of freedom) tripartite quantum systems, labelled $\\mathcal{A}$ and $\\mathcal{B}$, in the states $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}$ and $\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$, respectively, where we assume that $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}$ is separable with respect to the bipartition $\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}$ and $\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$ is separable with respect to the bipartition $\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}$. Clearly, both of these systems and states individually are biseparable, but if the parties have full information about which system is which, e.g., the first system is $A$ and the second system is $B$, then the joint state $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}\\otimes\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$ can be GME with respect to the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}|\\mathcal{A}_{3}\\mathcal{B}_{3}$. In this sense, two biseparable systems can yield one GME system. Now, let us suppose that the parties do not have full information which system is in which state. For simplicity, let us assume that either system may be in either state with the same probability $\\tfrac{1}{2}$. Then the state of either of the systems is described by the convex mixture $\\rho_{\\mathrm{mix}}=\\tfrac{1}{2}\\rho_{A_{1}|A_{2}A_{3}}+\\tfrac{1}{2}\\rho_{B_{1}B_{2}|B_{3}}$, where we have kept the labels $A$ and $B$, but they now refer to the same subsystems, i.e., $A_{i}=B_{i}$ for all~$i$. The state $\\rho_{\\mathrm{mix}}$ may not be partition separable anymore, but is certainly still biseparable. In particular, it may have activatable GME, even though neither $\\rho_{A_{1}|A_{2}A_{3}}$ nor $\\rho_{B_{1}B_{2}|B_{3}}$ do. For the sake of the argument let us assume that the latter is indeed the case and that GME is activated for $2$ copies in this case, such that $\\rho_{\\mathrm{mix}}^{\\otimes 2}$ is GME. That means, if one has access to both systems, $A$ and $B$, even without knowing which system is in which state, one would end up with GME. However, the additional randomness with respect to the case where one knows exactly which state which system is in results in an increased entropy of $\\rho_{\\mathrm{mix}}^{\\otimes 2}$ with respect to $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}\\otimes\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$, and thus represents a disadvantage with respect to the latter case.\n\nIn general, it is therefore not problematic that the conjectures, if true, would imply that incoherent mixtures of $k$-activatable states may result in $k'$-activatable states with $k'<k$. Instead, this can be considered as a sign that scenarios with multiple copies of multipartite quantum states give rise to features that are not captured by convex structures on the level of the single-copy state space.\n\n\\section{Conclusion and outlook}\\label{sec:Conclusion and Outlook}\nOur results show that a modern theory of entanglement in multipartite systems, which includes the potential to locally process multiple copies of distributed quantum states, exhibits a rich structure that goes beyond the convex structure of partially separable states on single copies. While we conjecture that asymptotically, an even simpler description might be possible, i.e., separability in multipartite systems collapses to a simple bipartite concept of separability, we show that two copies are certainly not sufficient for reaching this simple limit, thus leaving the practical certification with finite copies a problem to be studied.\n\nIndeed, our results show that GME is a resource with a complex relationship to bipartite entanglement in the context of local operations and shared randomness (cf.~\\cite{SchmidRossetBuscemi2020}). An array of important open questions arises from our results, which can thus be considered to establish an entirely new direction of research: first and foremost, this includes the quest for conclusive evidence for or against our conjectures. Besides determining whether these conjectures are ultimately correct or not, it will be of high interest to determine which properties (of the biseparable decompositions) of given states permit or prevent GME activation with a certain number of copies. Another open question is the minimal local dimension necessary for GME activation from biseparable states with PPT entanglement across every cut. Furthermore, from a practical point of view, it will be desirable to develop a theory of $k$-copy multipartite entanglement witnesses that are non-linear expressions of density matrices and allow for a more fine-grained characterisation of multipartite entanglement in networks with local memories.\nFinally, although separable states and shared classical randomness are free under LOCC, i.e., under a conventional choice of free operations in the resource theory of bipartite entanglement, our results suggest that convex combinations of different partition-separable states with shared classical randomness can be used as a resource for GME activation in multi-copy scenarios; that is, it may not be straightforward to study GME activation within the usual resource-theoretical framework under LOCC\\@. \nIn view of this situation, it would be interesting for future research to establish a new framework for understanding such a complicated aspect of multipartite entanglement as GME activation by, e.g., considering non-convex quantum resource theories where classical randomness can be used as a resource~\\cite{Kuroiwa2020generalquantum,PhysRevA.104.L020401}.\n\n\\begin{acknowledgments}\nWe acknowledge support from the Austrian Science Fund (FWF) through the START project Y879-N27, the project P 31339-N27, and the Zukunftskolleg ZK03. H.Y.\\ was supported by JSPS Overseas Research Fellowships and JST, PRESTO Grant Number JPMJPR201A, Japan.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1fixed_with_article_titles_full_names_new}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}