{"text":"\\section{Introduction}\n\nFix a continuous, absolutely irreducible, 2-dimensional, odd, mod $p$ representation\n$ {\\bar {\\rho} } :\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q}) \\rightarrow \\mathrm{GL}_2({\\mathbb F})$ with ${\\mathbb F}$ a finite field of\ncharacteristic $p$.\nWe denote by $N( {\\bar {\\rho} } )$ the (prime to $p$) Artin conductor of $ {\\bar {\\rho} } $, and \n$k( {\\bar {\\rho} } )$ the weight of $ {\\bar {\\rho} } $ as defined in \\cite{Serre3}. Serre has conjectured in \\cite{Serre3} that such a $ {\\bar {\\rho} } $ {\\it arises} (with respect to some fixed embedding $\\iota:\\overline{\\mathbb{Q}} \\hookrightarrow \\overline{\\mathbb{Q}_{p}}$) from a newform of weight $k( {\\bar {\\rho} } )$ and level $N( {\\bar {\\rho} } )$. \n\nIf $ {\\bar {\\rho} } $ is unramified outside $p$, we say that it is of level 1.\nThis corresponds to the case \nwhen $N( {\\bar {\\rho} } )$ is $1$. \n\n\n\\subsection{The main result}\n\n\\begin{theorem}\\label{main}\n  A $ {\\bar {\\rho} } $ of level one arises from $S_{k( {\\bar {\\rho} } )}(SL_2(\\mathbb{Z}))$ with respect\n  to an embedding $\\iota:\\overline{\\mathbb{Q}} \\hookrightarrow \\overline{\\mathbb{Q}_{p}}$.\n\\end{theorem}\n\nThe theorem settles the conjecture stated \nin article 104 of \\cite{Serre2} which is in the case of level 1.\nWe summarise the history of the conjecture. The level 1  \nconjecture was first \nmade by Serre in September 1972 (just after the Antwerp meeting)  when he\nwrote to Swinnerton-Dyer about it, and asked him to mention this\nconjecture (or problem) in his Antwerp text (see \\cite{SWD}, p.9). This is\nprobably the first appearance (1973) of this conjecture in print.\nSerre wrote about these conjectures to Tate on May 1st, 1973.\nTate replied to Serre first on June 11, and then on July 2, 1973: in the second letter, he \nproved the conjecture for $p = 2$.\n\n\nThe proof of Theorem \\ref{main} builds on the ideas of an earlier work of\nWintenberger and the author, see \\cite{KW}. There the above\ntheorem was proved for primes $p=5,7$ (it being known \nearlier conditionally under GRH for $p=5$ by \\cite{Brue}), it being already known\nfor $p=2,3$ because of a method of Tate, see \\cite{Tate} for $p=2$, which was \nlater applied to the case of $p=3$ by Serre, see page 710 of\n\\cite{Serre2}. In \\cite{KW} it was shown how\nmodularity lifting theorems would yield Serre's conjecture when proved in\nsufficient generality, and the conjecture was proven in level 1 for\nweights $2,4,6,8,12,14$.\nThe main contribution of this paper is a method to \nprove the level 1 case of the\nconjecture using only known modularity lifting theorems, thus completing the proof of the level 1 case of Serre's conjecture. We need lifting theorems when either the $p$-adic lift is crystalline at $p$ of weight $k$ (i.e., Hodge-Tate weights $(k-1,0)$) $\\leq p+1$ (and when the weight is $p+1$ the lift is ordinary at $p$), or at $p$ the lift is of Hodge-Tate weights $(1,0)$ and \nBarsotti-Tate over $\\mathbb{Q}_p(\\mu_p)$. \n\n\nHenceforth $p$ will be an odd prime.\nWe use the inductive method proposed in Theorem 5.1 of \\cite{KW}\nto prove the level 1 case of the conjecture. \nThe main new idea of this paper is a ``weight reduction'' \ntechnique. This  allows us to carry out \nthe inductive step in a manner that is different from the one contemplated in \n{\\it loc.\\ cit.\\ }\n\nThe cases of the conjecture for small weights in level 1 proved in \\cite{KW}\nwere dealt with using the\nresults of Fontaine, Brumer and Kramer, and Schoof (\\cite{Fontaine}, \\cite{BK} and \\cite{Schoof1}), together with modularity\nlifting results of the type pioneered by Wiles.  \n\nIn this paper we use the results of \\cite{KW} for weights $2,4,6$, and after that prove the level\none case without making any further use of results classifying abelian varities\nover $\\mathbb{Q}$ with certain good reduction properties. Thus in the end we see that the\nonly such results we use are those showing that there is no semistable\nabelian variety\nover $\\mathbb{Q}$ with good reduction outside 5 (see \\cite{Fontaine}, \\cite{Schoof1} and \\cite{BK}). \nWe do make use of course of modularity lifting results. These are \ndue to Wiles, Taylor, Breuil, Conrad, Diamond, Fujiwara, Kisin, Savitt, Skinner et al\n(see \\cite{Wiles}, \\cite{TW},  \n\\cite{Fujiwara}, \\cite{CDT},\\cite{BCDT}, \\cite{SW2}, \\cite{SW3}, \\cite{Savitt}, \\cite{Kisin}). In particular, besides the basic method \nof Taylor and Wiles in \\cite{Wiles} and \\cite{TW}, we need crucially the results of Skinner and Wiles in \\cite{SW2},\n\\cite{SW3}, and the result of Kisin in \\cite{Kisin}.  Although Kisin proves a very general modularity lifting theorem for potentially Barsotti-Tate lifts (at $p$) when the residual representation is non-degenerate, i.e.,  \nirreducible on restriction to $G_{\\mathbb{Q}(\\sqrt{{ (-1)}^{p-1 \\over 2}p})}$, in this paper the main theorem of \\cite{Kisin} is used only in the \ncase when the $p$-adic lift being considered is (locally at $p$) Barsotti-Tate over $\\mathbb{Q}_p(\\mu_p)$. The residually degenerate cases are handled by quoting the results of Skinner and Wiles in \\cite{SW2}, \\cite{SW3} which may be applied as in these cases the lifts that need to be proved modular are ordinary up to a twist. The ordinarity is  a consequence of a  result\nof Breuil and M\\'ezard (Proposition 6.1.1 of \\cite{BM}), and Savitt (see \\cite{Savitt1}, Theorems 6.11 and 6.12), which is vital for us. \n\nAs in \\cite{KW}, we use crucially the\npotential version of Serre's conjecture proved by Taylor in \\cite{Tay1} and \\cite{Tay2}, and a deformation theoretic result of B\\\"ockle in the appendix to \\cite{[K03]}.\n\nTheorem \\ref{main} yields the following corollaries (see Section \\ref{cors}),\nthe first needing also the method of ``killing ramification'' of Section 5.2 of \\cite{KW}:\n\n\\begin{cor}\\label{cond}\n  If $ {\\bar {\\rho} } $ is  an irreducible, odd, 2-dimensional, mod $p$ representation of $G_\\mathbb{Q}$ with $k( {\\bar {\\rho} } )=2$, \n  $N( {\\bar {\\rho} } )=q$, with $q$ prime, and $p>2$, then\n  it arises from $S_2(\\Gamma_1(q))$.\n\\end{cor}\n\n\\begin{cor}\\label{finite}\n There are only finitely many isomorphism classes of continuous semisimple\n odd representations $ {\\bar {\\rho} } :G_\\mathbb{Q} \\rightarrow GL_2(\\overline{\\mathbb{F}_{p}})$\n that are unramified outside $p$.\n\\end{cor}\n\n\n\nOur theorem, when combined with modularity lifting theorems also implies that if $\\rho:G_\\mathbb{Q} \\rightarrow GL_2({O})$ is an irreducible $p$-adic representation unramified outside $p$ and at $p$ crystalline of Hodge Tate weights $(k-1,0)$ with $k$ even and \neither $2 \\leq k \\leq p-1$ (even the weight $p+1$ case can be deduced, after the work in \\cite{BLZ}: \nsee Lemma \\ref{rubbish} below) or $\\rho$ is ordinary at $p$, then $\\rho$ arises from $S_k(SL_2(\\mathbb{Z}))$. It is quite likely that the restriction on weights in the non-ordinary cases\ncan be eased to allowing weights up to $2p$ provided that residually the representation is non-degenerate. This will probably follow\nfrom ongoing work of Berger on Breuil's conjecture in \\cite{Breuill}, and the modifications of the Taylor-Wiles system carried out in \\cite{Kisin}.\nTheorem \\ref{main} also implies that\na $\\mathrm{GL}_2$-type semistable abelian variety over $\\mathbb{Q}$ with good reduction outside a prime $p$ is a factor of $J_0(p)$. Such a result was earlier used in \\cite{KW} in the case when $p=5,7,11,13$ being a special case of the results proven in \\cite{Schoof1}, \\cite{BK}: now we can recover these results in the case of $\\mathrm{GL}_2$-type abelian varieties when $p>5$. \n\n\\subsection{Sketch of proof}\n\nWe give a \nrough sketch of the proof of Theorem \\ref{main}, starting with some general comment about the method used. As in Theorem 5.1 of \\cite{KW}, the method is inductive with respect to the prime which is the residue characteristic, but as said earlier the inductive step is carried out differently. The method uses in an essential way the method of \n``congruences between Galois representations'' which was introduced in Section 4 of \\cite{KW}\nto prove the cases of the conjecture in level 1 and weights $6,8,12,14$, and is refined here. In Section 4 of {\\em loc.\\ cit.\\ }   congruences were produced\nbetween $p$-adic representations of $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$ \nthat were crystalline of weight $p+1$ at $p$ and semistable of weight 2 at $p$\n(the analog for modular forms being a result of Serre: see Th\\'eor\\`eme 11 of \narticle 97 of \\cite{Serre2}).\nThe  method in Section 2 of \\cite{KW} can be used to prove more results about such congruences which parallel \nresults that are well-known for congruences between modular forms. For instance\nwe can now in principle prove analogs for Galois representations of \nthe ``type changing'' arguments of Carayol in \\cite{car} for modular forms (some instances are carried out in \nSection 3 of the paper, and used in the proof of Theorem \\ref{main} to ``change types'' at a prime different from the residue characteristic), or the level raising results for\nmodular forms of Ribet (see  \\cite{Ribet1}). We do not use the Galois-theoretic\nanalog of the latter in this paper, but this and other such ``level raising'' results for Galois representations will be crucial in future work. \n\nWe fix an embedding $\\iota_p:\\overline{\\mathbb{Q}} \\hookrightarrow \\overline{\\mathbb{Q}_{p}}$ for each prime $p$. \nWe say a residual mod $p$ representation $ {\\bar {\\rho} } $ or a $p$-adic \nrepresentation $\\rho$, is {\\it modular} if it is either reducible (in the $p$-adic case we assume irreducibility) or it arises from a newform with respect to the embedding $\\iota_p$.\nWe say that a compatible system $(\\rho_{\\lambda})$ is modular if it arises from a newform.\n\nAssuming we have proved Serre's conjecture for level 1 modulo the $n$th prime $p_n$, we prove it mod $P_{n+1}$ where $P_{n+1}$ is the least non-Fermat prime $>p_n$. This is (more or less!) the inductive method \nto attack the level 1 case of Serre's conjecture \nproposed in Theorem 5.1 of \\cite{KW}.\nVia the methods of \\cite{KW}, see also Lemma \\ref{rubbish} and Corollary \\ref{trivial} below, this also means that one knows (the level 1 case of) Serre's conjectures for $ {\\bar {\\rho} } $ of any residue characterictic $p$ bigger than $p_n$ for all weights up to $P_{n+1}+1$. Then we repeat the process starting with $P_{n+1}$ instead of $p_n$.\n\n\nIn \\cite{KW} to prove weights $6,8,12,14$ the minimal lifting result in Section 2 of \\cite{KW} was used twice to get 2 different compatible systems whose interplay (via the residual representations at a certain place of the 2 compatible systems being isomorphic up to semisimplification: we say that 2 such compatible systems are {\\em linked}) proved the modularity\nof $ {\\bar {\\rho} } $. Here 3 compatible systems are considered instead which arise from $ {\\bar {\\rho} } $ directly or indirectly.\nThe lifts constructed are not always minimal. The inductive step is different\nfrom the one proposed in \\cite{KW} \nin that the residual modularity is essentially used only for representations in  smaller weights of the same residual characteristic which has already been \ninductively established earlier. Another prime $\\ell$ is used as a foil in an auxiliary fashion to achieve this reduction of weight.\n\nMore precisely, starting with a $ {\\bar {\\rho} } $ mod $p:=P_{n+1}$ such that\n$p_{n}+1 < k( {\\bar {\\rho} } )\\leq p+1$ of level 1, we first reduce to considering \n$ {\\bar {\\rho} } $ which are ordinary at $p$ (by Lemma \\ref{weights} below), and then construct \na minimal lifting (see Proposition \\ref{p} below) of $ {\\bar {\\rho} } $ to a $p$-adic representation that is unramified outside $p$ and of weight 2 at $p$ (it is \nBarsotti-Tate \nover $\\mathbb{Q}_p(\\mu_p)$ if $k( {\\bar {\\rho} } ) \\neq p+1$, and otherwise semistable of weight 2). We then get a compatible system $(\\rho_{\\lambda})$ (see Proposition \\ref{c}) such that $\\rho$ is\npart of this system for a $\\lambda$ above $p$. \nWe choose an odd prime $\\ell$ such that $\\ell^r||p-1$ (hence the assumption that $p$ is not a Fermat prime), \nand consider the residual representation $\\overline \\rho_{\\lambda}$ for $\\lambda$ now a prime above $\\ell$. \n\nNow in the cases when the residual modularity is not known for this mod $\\ell$ representation \n(the residual modularity will be known if the image is solvable, or the representation is unramified at $p$) we \nconstruct another lifting (see Proposition \\ref{q} which plays in some sense the analog of a lemma of Carayol \\cite{car}, or \nde Shalit's lemma as in \\cite{Wiles} and \\cite{TW}, for Galois representations) \n$\\rho'$ of this mod $\\ell$ representation that is Barsotti-Tate at $\\ell$, and at $p$ is non-minimal with \n``nebentype'' that is well-chosen, and is unramified outside $\\ell,p$. \n(If the mod $\\ell$ modularity were known, we conclude the modularity of $(\\rho_\\lambda)$ \nby a modularity lifting theorem applied at $\\ell$: the modularity lifting result in \nthis case would be the one contained in \\cite{Wiles}, \\cite{TW}, \\cite{SW2} and \\cite{SW3}.)\nWe construct another compatible \nsystem $(\\rho'_{\\lambda})$ of which $\\rho'$ is a member, and such that at \na prime above $p$, because of the well-chosenness of the nebentype at $p$ in the $\\ell$-adic lift, the residual representation \nis already known to be modular by the inductive hypothesis (which includes the case when the representation \nis reducible as these by convention are also called modular). This control of the weight of the mod $p$ \nresidual representation arising from $(\\rho'_{\\lambda})$ is due \nto a result of Breuil and M\\'ezard in \\cite{BM}, and Savitt \\cite{Savitt1}. Then the fact that we are \nin a position to apply known modularity lifting results (see \\cite{SW2}, \\cite{SW3}, \\cite{Kisin}) \nis again because of \\cite{BM} which says that all lifts of reducible 2-dimensional mod $p$ representation of $G_{\\mathbb{Q}_p}$\nthat become Barsotti-Tate over $\\mathbb{Q}_p(\\mu_p)$ are up to twist ordinary. (This is one important reason why the \nstrategy here does not need modularity lifting results beyond the known range.)\n\nThese known modularity lifting theorems then prove modularity of \n$(\\rho'_{\\lambda})$, and then another use of modularity lifting theorems (of \\cite{Wiles}, \\cite{TW}, \\cite{SW2}, \\cite{SW3}) \nprove the modularity of $(\\rho_{\\lambda})$, as the 2 compatible systems are linked mod the prime above $\\ell$ fixed by $\\iota_\\ell$, \nand thus $ {\\bar {\\rho} } $ is modular. We need a third kind of compatible system $(\\rho''_{\\lambda})$\nconstructed in \\cite{KW} to conclude now that all level 1 representations modulo a prime $\\geq k( {\\bar {\\rho} } )-1$ \nand of the weight of $ {\\bar {\\rho} } $ are modular (see Lemma \\ref{rubbish} and \nCorollary \\ref{trivial} below). \n \n\nThe entire strategy of the paper roughly uses that $p_n$ \nis at least two-thirds as large as $p_{n+1}$ (using classical Chebyshev estimates: \nsee Section \\ref{cheb}). \n\nThus schematically the argument may be summarised as follows:\n\\begin{itemize}\n\n\\item Start with a mod $p$ representation\n$ {\\bar {\\rho} } $,\n\n\\item lift it to $\\rho$ (using Proposition \\ref{p}), \n\n\\item construct a compatible system\n$(\\rho_{\\lambda})$ (using Proposition \\ref{c}) such that for the prime above $p$ fixed by $\\iota_p$ the corresponding representation\nis $\\rho$, \n\n\\item consider the residual representation at a prime $\\lambda$ above \na ``suitable'' prime $\\ell$ ($\\lambda$ fixed by $\\iota_\\ell$), \n\n\n\\item lift this to a  ``good'' lift  $\\rho'$  (using Proposition \\ref{q}), \n\n\\item make it part of a compatible system \n$(\\rho'_{\\lambda})$ (using Proposition \\ref{p}),  \n\n\\item consider the residual representation at the above $p$ fixed by $\\iota_p$\narising from the system $(\\rho'_{\\lambda})$, \n\n\\item inductively this is known to be modular as $\\rho'$ is a ``good'' lift, \n\n\\item modularity lifting theorems imply $(\\rho'_{\\lambda})$ is modular, \n\n\\item another application of lifting theorems gives  $(\\rho_{\\lambda})$ is modular, and hence $ {\\bar {\\rho} } $ is modular. \n\\end{itemize}\n\n\\noindent (This is in the ``generic'' case, as sometimes the procedure yields success earlier.) After this we use a third kind of compatible system \n$(\\rho''_{\\lambda})$ constructed in Sections 2 and 3 of \\cite{KW}, to deduce that once Serre's conjecture is known in weight $k$ for a prime $p$ then it is known in weight $k$ for all primes $\\geq k-1$ (see Corollary \\ref{trivial} below).\n\n\nJust as in \\cite{KW}, besides modularity lifting results, the potential version of Serre's conjectures \nproven by Taylor in \\cite{Tay1} and \\cite{Tay2}, and a result of B\\\"ockle\nin the appendix to \\cite{[K03]}, is crucial in constructing (minimal and non-minimal) liftings \n$\\rho$ of various kinds (see Sections 2 and 3), and then making them part of a compatible system \n$(\\rho_\\lambda)$ (see Section 4)  whose refined properties are then obtained by arguments \nof Dieulefait and Wintenberger,\n\\cite{D} and \\cite{Wint}.\n\n\n\\section{Liftings}\n\nFor $F$ a field, $\\mathbb{Q} \\subset F \\subset \\overline{\\mathbb{Q}}$,\nwe write $G_{F}$ for the Galois group of $\\overline{\\mathbb{Q}} \/ F$.\nFor $\\lambda$ a prime\/place of $F$,\nwe mean by $D_{\\lambda}$ (resp., $I_{\\lambda}$) a decomposition\n(resp., inertia) subgroup of $G_F$ at  $\\lambda$.\nWe have fixed embeddings $\\iota_p, \\iota_\\infty$ of $\\overline{\\mathbb{Q}}$ in its completions $\\overline{\\mathbb{Q}_{p}}$ and $\\mathbb{C}$ in the introduction.\nDenote by $\\chi_p$ the $p$-adic\ncyclotomic character, and $\\omega_p$ the Teichm\\\"uller lift of the mod $p$ cyclotomic character $\\overline \\chi_p$ \n(the later being the reduction mod $p$ of $\\chi_p$). By abuse of notation we also denote by $\\omega_p$ the $\\ell$-adic character $\\iota_\\ell\\iota_p^{-1}(\\omega_p)$ for any prime $\\ell$: this should not cause confusion as from the context it will be clear where the character is valued. For a number field $F$ we denote the restriction of a character of $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$  to $G_F$ by the same symbol.\n\nLet $p$ an odd prime.\nFix  $ {\\bar {\\rho} } : G_{\\mathbb{Q}}\\rightarrow \\mathrm{GL}_2 (\\overline{\\mathbb{F}_{p}} )$\nto be an odd irreducible representation.\nWe assume that the Serre\nweight $k( {\\bar {\\rho} } )$ is such that $2 \\leq k( {\\bar {\\rho} } ) \\leq p+1$. (Note that\nthere\nis always\na twist of $ {\\bar {\\rho} } $ by some power of the mod $p$ cyclotomic character\n$\\overline{\\chi _p}$\nthat has weights in this range.) We denote by ${\\rm Ad}^0( {\\bar {\\rho} } )$ the $G_{\\mathbb{Q}}$-module arising from the adjoint action on the trace 0 matrices of $M_2({\\mathbb F})$.\n\nLet ${\\mathbb F} \\subset \\overline{\\mathbb{F}_{p}}$ be a finite field\nsuch that the image of $\\overline{\\rho}$ is contained in\n$\\mathrm{GL}_2 ({\\mathbb F} )$, and let $W$ be the Witt vectors $W({\\mathbb F} )$.\nBy a {\\it lift} of $\\overline{\\rho}$, we mean a continuous representation\n$\\rho : G_{\\mathbb{Q}}\\rightarrow\n\\mathrm{GL}_2 ({V})$, where $ V$ is the ring of integers of a\nfinite\nextension of the field of fractions of  $W$, such that\nthe reduction of $\\rho$ modulo the maximal ideal of ${V}$ is\nisomorphic to $\\overline{\\rho}$. Liftings when considered up to equivalence, i.e., up to conjugation by matrices that are residually the identity, are referred to as deformations.\nWe say that $\\rho$ is minimal at a prime $\\ell \\neq p$ if it  is minimally\nramified at  $\\ell$ in the terminology of \\cite{[D97]}.\n\n\nLet ${\\varepsilon}$ be the Teichm\\\"uller lift of the  character \n${\\rm det}( {\\bar {\\rho} } )\\overline \\chi_p^{1-k( {\\bar {\\rho} } )}$ whose restriction to any open subgroup of $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$ we denote by the same symbol. \n\n\n\\subsection{The method of producing liftings of \\cite{KW}}\\label{liftings}\n\nIn this section we will produce liftings with certain prescribed local properties\nof $ {\\bar {\\rho} } $ (one of the properties being unramified almost everywhere) using the methods of Section 2 of \\cite{KW}. We assume \nthat $ {\\bar {\\rho} } $ has non-solvable image and $k( {\\bar {\\rho} } ) \\neq p$.\nBy Lemma 2.6 of \\cite{KW}, this also means that for any totally real field $F$,\n $ {\\bar {\\rho} } |_{G_F}$ has non-solvable image, and as $2 \\leq k( {\\bar {\\rho} } ) \\leq p+1$ and $\\neq p$,  if $F$ is unramified at $p$, \n$\\wp$ a place of $F$ above $p$, $ {\\bar {\\rho} } |_{I_{\\wp}}$ is non-scalar.\n\n\nTo orient the reader we say a few words about \nthe way the method of {\\it loc.\\ cit.\\ } gets used here. We wish to point out that the method there is flexible enough to produce liftings with the desired local properties, provided the local calculations work out.\n\nWe \nproduce liftings\nwith  prescribed properties as in Propositions \\ref{p} and \\ref{q} below, \nby proving, as in \\cite{KW},\nthat a deformation ring $R$ (over a suitable ring of integers $O$ of a finite extension of $\\mathbb{Q}_p$, with uniformiser $\\pi$)  which parametrises (equivalence classes of) liftings of $ {\\bar {\\rho} } $\nwith these prescribed properties is flat \nover $O$. (The prescribed properties will be deformation conditions in the sense of \\cite{Mazur} and thus the problem will be representable globally by a \nuniversal ring $R$.) To do this one shows first that $R\/(\\pi)$ is finite, which\nby Lemma 2.4 of {\\it loc.\\ cit.\\ }, as explained below,\nis equivalent to showing that $R_F\/(\\pi)$ is finite, where $R_F$ is\na certain deformation ring for $ {\\bar {\\rho} } |_{G_F}$ and $F$ is some totally real\nfield. \n\nThe finiteness of $R_F\/(\\pi)$ is established by identifying $R_F$ to\na Hecke algebra $\\mathbb{T}_F$ which we know is finite over $\\mathbb{Z}_p$. \nA suitable $F$ with this property is produced by using Taylor's results\nin \\cite{Tay1}, \\cite{Tay2}: for instance $F$ is unramified at $p$, and in the cases below $R_F$ can be taken to be a minimal deformation ring (although $R$ need not be a minimal deformation ring, see Proposition \\ref{q} for instance).\nHere the minimality of $R_F$ at the local defining conditions at \nplaces $\\lambda$ not above $p$ does not need futher comment: for $\\lambda$ above $p$ the minimality condition is as explained below the same as the minimal condition at $p$ when defining $R$, and is different in the case\nof Propositions \\ref{p} and \\ref{q}.\n\nTo see that $R_F$ can be taken to be the the minimal deformation ring in the \nproofs of Propositions \\ref{p} and \\ref{q} below we indicate the argument. Denote the universal representation $\\rho_R$ and $\\rho_{R_F}$ corresponding\nto the deformation problem that $R$ and $R_F$ represent (with $R_F$ the minimal deformation ring), and $\\overline \\rho_R$ and $\\overline \\rho_{R_F}$ it's reduction mod $\\pi$. Then using the properties of the lifts prescribed in \nPropositions \\ref{p} and \\ref{q} below, for $\\ell \\neq p$, the order of ${\\overline \\rho_R} (I_\\ell)$ is the same as the order of $ {\\bar {\\rho} } (I_\\ell)$. This is seen by using the proof of the claim after Lemma 2.4 of \\cite{KW}. \n\nThis gives that\n${\\overline \\rho_R}|_{G_F}$ is a specialisation of $\\overline \\rho_{R_F}$. \nThe finiteness of $R_F$ as a $O$-module yields that the latter has finite image and hence so does the former, which by Lemma 2.4 of \\cite{KW} gives that\n$R\/(\\pi)$ is of finite cardinality. \n\nWe give some more details about the choice of $F$ and the identification of $R_F$ to\na Hecke algebra $\\mathbb{T}_F$ (see also proof of Theorem 2.2 of \\cite{KW}).\nWe choose the totally real field $F$, Galois over $\\mathbb{Q}$ and of even degree, so that when $ {\\bar {\\rho} } $ is ordinary at $p$ (resp., supersingular by which we mean locally irreducible at $p$), $F$ is unramified at $p$ \n(resp., split at $p$),\n$ {\\bar {\\rho} } |_{G_F}$ is unramified outside places above $p$, {\\it and} such that there is a cuspidal automorphic representation $\\pi$ for $GL_2(\\mathbb{A}_F)$\nthat is discrete series of parallel weight $(2,\\cdots,2)$ at infinity (resp.,\nof parallel weight $(k( {\\bar {\\rho} } ),\\cdots,k( {\\bar {\\rho} } ))$ at infinity) is unramified at all places not above $p$, and is ordinary at places $\\wp$ above $p$ \nof conductor dividing $\\wp$ (resp., unramified at places $\\wp$ above $p$), and unramified at $p$ when $k( {\\bar {\\rho} } )=2$,\nthat gives rise to $ {\\bar {\\rho} } |_{G_F}$ with respect to (w.r.t.) the \nembedding $\\iota_p$. In the ordinary case (in Proposition \\ref{q})\nwe also use the existence of $F$ unramified at $p$, \nGalois over $\\mathbb{Q}$ and of even degree, such that $ {\\bar {\\rho} } |_{G_F}$ is unramified outside places above $p$, {\\it and} such that there is a cuspidal automorphic representation $\\pi$ for $GL_2(\\mathbb{A}_F)$\nthat is discrete series of parallel weight $(k( {\\bar {\\rho} } ),\\cdots,k( {\\bar {\\rho} } ))$ at infinity and is unramified at all finite places (see \\cite{Tay1} and Proposition 2.5 of \\cite{KW}), that gives rise to $ {\\bar {\\rho} } |_{G_F}$ with respect to (w.r.t.) the \nembedding $\\iota_p$.\n\nThe existence of a $F$ with all these properties \nfollows from Lemma 1.5 and Corollary 1.7 of \\cite{Tay1}, and Proposition 2.5 of \\cite{KW}, in the ordinary case, and Theorem 5.7 of \\cite{Tay2} in the supersingular case, and uses as an ingredient the level-lowering up to base change method of \\cite{SW1}. (The evenness of $[F:\\mathbb{Q}]$ is not mentioned in \\cite{Tay1} in the ordinary case,\nbut certainly may be ensured by a further quadratic base change.) \n\nIn the case of Proposition \\ref{p}, in which case $F$ \nis unramified at $p$, the minimal deformation ring $R_F$ \nparametrises lifts $\\rho$ of $ {\\bar {\\rho} } |_{G_F}$ that are unramified away from $p$, and\nat places $\\wp$ above $p$ are such that  $\\rho|_{I_{\\wp}}$ is of the form \n$$\\left( \\begin{array}{cc}\n    \\omega_p^{k-2}\\chi_p  & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ if $ {\\bar {\\rho} } |_{I_{\\wp}}$\nis of the form $$\\left( \\begin{array}{cc}\n    \\overline \\chi_p ^{k-1} & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ with $2 \\leq k \\leq p-1$, with the further condition that if $k( {\\bar {\\rho} } )=2$,  $\\rho|_{I_{\\wp}}$ is Barsotti-Tate, and of determinant (the restiction of) ${\\varepsilon} \\omega_p^{k-2}\\chi_p$. In the case of Proposition \\ref{q}, in which case $F$ may be taken to be split at $p$ when $ {\\bar {\\rho} } |_{D_p}$ is supersingular, the minimal deformation ring $R_F$ \nparametrises lifts $\\rho$ of $ {\\bar {\\rho} } |_{G_F}$ that are unramified away from $p$, and at places $\\wp$ above $p$ the representation is crystalline of weight $k( {\\bar {\\rho} } )$, and the lifts have determinant ${\\varepsilon} \\eta_q^i \\chi_p^{k( {\\bar {\\rho} } )-1}$\nusing the notation of Proposition \\ref{q}. (We can also ensure that ${\\varepsilon}|_{G_F} $ or ${\\varepsilon} \\eta_q^i|_{G_F}$ is trivial if we want when the Serre weight is even:\nthe Serre weight will always be even in all applications below.)\n\nIt remains to recall the identification of $R_F$ to suitable Hecke algebra $\\mathbb{T}_F$.\nIn the  case of Proposition \\ref{p} below, the Hecke algebra $\\mathbb{T}_F$ is a ($\\mathbb{Z}_p$-)algebra cut out by the \nHecke action on cusp forms for $GL_2(\\mathbb{A}_F)$ that are of weight $(2,\\cdots,2)$, \nunramified outside $p$, and at places $\\wp$ above $p$ of conductor dividing $\\wp$ (and unramified if $k( {\\bar {\\rho} } )=2$), and\nof central character corresponding to (restriction of) ${\\varepsilon} \\omega_p^{k-2}$ by class field theory, with respect to the embedding $\\iota_p$.\nIn the case of Proposition \\ref{q} below, the Hecke algebra $\\mathbb{T}_F$ is a ($\\mathbb{Z}_p$-)algebra \ncut out by the \nHecke action on cusp forms for $GL_2(\\mathbb{A}_F)$ that are of weight $(k( {\\bar {\\rho} } ),\n\\cdots,k( {\\bar {\\rho} } ))$, unramified at all finite places, and of central character \ncorresponding to, using it's notation, $\\eta_q^i {\\varepsilon} \\chi_p^{k( {\\bar {\\rho} } )-2}$ by class field theory. \n(The fact that the $\\mathbb{T}_F$ is non-zero is a consequence of the results of \nTaylor in \\cite{Tay1} and \\cite{Tay2} that we have recalled.) \n\nThe identification $R_F\\simeq \\mathbb{T}_F$\nis proved using \\cite{Fujiwara} in the ordinary case, and  \nSection 3 of \\cite{Tay2} in the supersingular case. ({\\small \nAs \\cite{Fujiwara} may not be widely available, note that in the ordinary case,\nwe are in a situation where we do have a minimal modular lift of $ {\\bar {\\rho} } |_{G_F}$ that is ordinary at places above $p$, and such that $ {\\bar {\\rho} } |_{G_F}$ has nonsolvable image, and $F$ is unramified at $p$. Thus the deduction of the isomorphism \n$R_F \\simeq \\mathbb{T}_F$ is by now standard.})\n\nNote that we are allowing $p=3$, which is a case excluded in\nsome sections of \\cite{Tay2}: thus we say a few words to justify why we still have the results of \\cite{Tay2} available. We exploit the fact that we know \nthat by our assumption \n$ {\\bar {\\rho} } |_{G_F}$ is not solvable for any totally real field $F$ (Lemma 2.6 of \\cite{KW}), and as\n$\\overline \\chi_3$ restricted to $F$ has order 2, \n we thus have an auxiliary prime $r$, as guaranteed\nby Lemma 3  of \\cite{DT} or Lemma 4.11 of \\cite{[DDT]}, which handles non-neatness problems. (For instance, this allows us to pass from the results in Section 4 of \\cite{Tay2}\nto those of Section 5 requiring only that $p>2$, and also to have available the results of Sections 2 and 3 of \\cite{Tay2} requiring only that $p>2$.)\n\n\nThe finiteness of $R\/(\\pi)$ leads to $R$ being  flat (finite, complete intersection) over $ O$ if we know that $R \\simeq {O}[[X_1,\\cdots,X_r]]\/(f_1,\\cdots,f_s)$ with $s \\leq r$.  By the crucial Proposition 1 of B\\\"ockle's appendix to \\cite{[K03]}, this will follow from some purely local information about the kind of lifts $R$ parametrises. (The local conditions will always be deformation conditions in the sense of \\cite{Mazur}.) If $R$ parametrises (equivalence classes of) lifts unramified outside a fixed set of primes, of a certain fixed determinant, only the following local properties at each prime $\\ell$ need be checked:  the corresponding local deformation ring $R_{\\ell}$ should be a flat, complete intersection  over $O$, of relative dimension \n${\\rm dim}{_{\\mathbb F}} H^0(D_{\\ell},{\\rm Ad}^0( {\\bar {\\rho} } ))$ \nwhen $\\ell \\neq  p$, and of relative dimension \n${\\rm dim}{_{\\mathbb F}} H^0(D_{p},{\\rm Ad}^0( {\\bar {\\rho} } ))+1$ \nwhen $\\ell =  p$.  We check these local conditions in Propositions \\ref{p} and \\ref{q} showing that  they follow from results of B\\\"ockle, Ramakrishna and Taylor, see \\cite{Boe}, \\cite{Boe1}, \\cite{Tay3}, \\cite{[R02]} (in all the cases below the local deformation ring turns out to be smooth). At primes $\\ell \\neq p$ \nwhere no ramification is allowed the deformation ring is directly checked to be smooth of relative dimension ${\\rm dim}{_{\\mathbb F}} H^0(D_{\\ell},{\\rm Ad}^0( {\\bar {\\rho} } ))$. At primes $\\ell \\neq p$ at which the residual representation is ramified and \nwhere the corresponding local deformations that are \nallowed are minimal the ring is smooth of the required dimension as checked in Section 3 of \\cite{Boe}, \\cite{[R02]} (see the \n`` local at $\\ell \\neq p$'' section) and  \\cite{Tay3} (see E1 to E3). \nThus below we only check the local condition at the residual characteristic \n(where the results are again found in \\cite{Boe}, \\cite{[R02]} \nand \\cite{Tay3}), \nand at a prime $\\ell$ where the deformations allowed are not minimal.\n(Note that in \\cite{Tay3} for the local versal \ndeformation rings below, \nat $p$, or when $\\ell \\neq p$ and the deformations allowed are minimal, \nthe unobstructedness of the ring\nis checked, and it is shown that the tangent space is of dimension ${\\rm dim}{_{\\mathbb F}} H^0(D_{\\ell},{\\rm Ad}^0( {\\bar {\\rho} } ))+\\delta_{\\ell p}$, which implies that the versal ring is smooth of relative dimension ${\\rm dim}{_{\\mathbb F}} H^0(D_{\\ell},{\\rm Ad}^0( {\\bar {\\rho} } ))+\\delta_{\\ell p}$ over $W$.)\n\n\\subsection{Minimal $p$-adic weight 2 \nlifts of $ {\\bar {\\rho} } $}\\label{weight2}\n\nWe consider (just for this subsection) only\n$ {\\bar {\\rho} } $ such that\n$ {\\bar {\\rho} } $ is ordinary at $p$, i.e., $ {\\bar {\\rho} } |_{I_p}$ has non-trivial\ncovariants, and we also assume as before $k( {\\bar {\\rho} } ) \\neq p$.\nIn this subsection we consider lifts \n$\\rho$ of $ {\\bar {\\rho} } $ whose determinant is ${\\varepsilon}\\omega_p^{k( {\\bar {\\rho} } )-2}\\chi_p$.\n\n\nWe say that $\\rho$ is minimal of weight 2 at $p$ (as in E3 and E4 of\n\\cite{Tay3})  if \nits determinant is ${\\varepsilon}\\omega_p^{k( {\\bar {\\rho} } )-2}\\chi_p|_{D_p}$, and assuming $ {\\bar {\\rho} } |_{I_p}$\nis of the form\n$$\\left( \\begin{array}{cc}\n    \\overline \\chi_p ^{k-1} & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ with $2 \\leq k \\leq p-1$, then $\\rho|_{I_p}$ is of\nthe form\n$$\\left( \\begin{array}{cc}\n    \\omega_p^{k-2}\\chi_p  & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ with the further condition that when $k( {\\bar {\\rho} } )=2$, $\\rho|_{I_p}$ is Barsotti-Tate. \n\nIf a lift $\\rho$ satisfies this condition at $p$ and is minimal at all prime $\\neq p$ (and thus necessarily\nhas determinant fixed as above), then we say that it is minimal\nof weight 2.\n\n\\begin{prop}\\label{p}\nLet $p$ be a prime $> 3$.\nLet $\\overline{\\rho} : G_{\\mathbb{Q}}\\rightarrow\n\\mathrm{GL}_2 ({\\mathbb F})$ be an odd absolutely irreducible representation.\nWe suppose that $2 \\leq k( {\\bar {\\rho} } ) \\leq p+1$ and\n $k( {\\bar {\\rho} } )\\not= p$ and with $ {\\bar {\\rho} } $ ordinary at $p$. Then\n$ {\\bar {\\rho} } $ has  a lift $\\rho$ that is minimal of weight 2. (Its determinant\nis necessarily ${\\varepsilon}\\omega_p^{k( {\\bar {\\rho} } )-2}\\chi_p$.)\n\\end{prop}\n\n\\begin{proof} If the image of $ {\\bar {\\rho} } $ is solvable we are done using \nthat Serre's conjecture is known in this case even in its refined form (see \\cite{Ribet}) and the fact that if $ {\\bar {\\rho} } $ arises from a mod $p$ ordinary \neigenform in $S_{k( {\\bar {\\rho} } )}(\\Gamma_1(N),\\overline{\\mathbb{F}_{p}})$  ($(N,p)=1$), then it also arises from an ordinary eigenform in\n$S_2(\\Gamma_1(N) \\cap \\Gamma_0(p),\\overline{\\mathbb{F}_{p}}({\\overline \\chi_p}^{k( {\\bar {\\rho} } )-2}))$ (see Proposition 8.13 of \\cite{Gross} or\nSection 6 of \\cite{Edix}). So we\nnow assume that the image of $ {\\bar {\\rho} } $ is not solvable.\n  \nThe proof follows immediately from the method of proof of Theorem 2.1 of\n\\cite{KW}, as noted in Section \\ref{liftings}, on noting the\nfollowing local fact: consider the versal deformation ring $R_p$\n  that parametrises deformations of $ {\\bar {\\rho} } |_{D_p}$\n(to $W$-algebras that are complete Noetherian local (CNL)\nrings with residue field ${\\mathbb F}$ as usual)\nthat on inertia $I_p$ have the form $$\\left( \\begin{array}{cc}\n    \\omega_p^{k-2}\\chi_p  & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ have fixed determinant the image of the character \n${\\varepsilon}\\omega_p^{k-2}\\chi_p|_{D_p}$, and in the case of weight $k( {\\bar {\\rho} } )=2$ the deformation is Barsotti-Tate. Then $R_p$ is a complete intersection, flat over $W({\\mathbb F})$\nof relative dimension $1+{\\rm dim}_{{\\mathbb F}}(H^0(D_p,{\\rm Ad}^0( {\\bar {\\rho} } ))$: \nin fact it is even smooth \nof relative dimension $1+{\\rm dim}_{{\\mathbb F}}(H^0(D_p,{\\rm Ad}^0( {\\bar {\\rho} } ))$ as checked\n  in E3 and  E4 of \\cite{Tay3} (see also \\cite{[R02]}).  This, by Section \\ref{liftings}, gives the flatness of the ring $R$ over $W$ that parametrises (equivalence classes of) \nlifts of $ {\\bar {\\rho} } |_{D_p}$ that are minimal of weight 2 and hence we are done.\n\\end{proof}\n\n\\noindent{\\bf Remark:} In this paper we use this theorem only for representations unramified outside $p$. We also do not need to use the proposition when the image is solvable. The condition of ordinarity may be removed using\nresults in \\cite{BM}, \\cite{Savitt1} (Proposition 6.1.2(iii) of the former, \nTheorem 6.22 of latter), together with the modification of Taylor-Wiles systems\nin \\cite{Kisin}.\n\n\\subsection{A Galois theoretic analog of Carayol's lemma}\\label{type}\n\nIn this subsection we prove a Galois theoretic analog of Lemme 1 of \\cite{car}.\nConsider $ {\\bar {\\rho} } : \\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q}) \\rightarrow GL_2({\\mathbb F})$ that is continuous, odd, irreducible,  $2 \\leq k( {\\bar {\\rho} } ) \\leq p+1$ and $k( {\\bar {\\rho} } ) \\neq p$, \nand consider an odd prime $q$ which we assume is ramified in\n$ {\\bar {\\rho} } $.\nFurther assume that $ {\\bar {\\rho} } |_{I_q}$ is of the form\n$$\\left( \\begin{array}{cc}\n    \\overline \\chi  & *  \\\\\n    0 & 1\n\\end{array} \\right),$$ where $\\overline \\chi$ arises from a mod $p$ character of ${\\rm Gal}(\\mathbb{Q}_q(\\mu_q)\/\\mathbb{Q}_q)$.\nLet $\\chi$ be its Teichm\\\"uller lift. This will be a power of the character $\\iota_p\\iota_q^{-1}(\\omega_q)$\nwhich we recall that by our conventions is again denoted by $\\omega_q$.\n\n\nAssume that $p^r||q-1$ ($r>0$) and consider $\\eta_q=\\omega_q^{ {q-1} \\over p^r }$: this is (for this subsection)\n a character with values in $\\overline{\\mathbb{Q}_{p}}^*$.\nWe denote the corresponding global characters which factor through\n${\\rm Gal}(\\mathbb{Q}(\\zeta_q)\/\\mathbb{Q})$ by the same symbol. We enlarge ${\\mathbb F}$ so that it contains all the ${q-1} \\over p^r$th roots of 1.\nBelow, we denote by $O$ the ring of integers of $W({\\mathbb F})(\\mu_{q-1})$ and by $\\pi$ a uniformiser of $O$.\n\nIn this section the minimality condition at $p$ we will consider\nis of being crystalline of weight $k( {\\bar {\\rho} } )$.\n\n\\begin{prop}\\label{q}\n  Let $p$ be an odd prime, fix a $ {\\bar {\\rho} } $ as above (in particular $k( {\\bar {\\rho} } )\\neq     p$ and $ {\\bar {\\rho} } |_{I_q}$ has the form above), and assume that $ {\\bar {\\rho} } $ does not have solvable\nimage. Fix an integer $i$. For some $V$ that is the ring of \nintegers of a finite extension of $\\mathbb{Q}_p$, there is a $V$-valued lift $\\rho$ of $ {\\bar {\\rho} } $ of determinant\n${\\varepsilon}\\chi_p^{k( {\\bar {\\rho} } )-1}\\eta_q^i$, that is minimal at primes outside $p,q$, is minimal\nat $p$ (crystalline of weight $k( {\\bar {\\rho} } )$), and at $q$, $\\rho|_{I_q}$ is of the from\n$$\\left( \\begin{array}{cc}\n     \\chi \\eta_q^i  & *  \\\\\n    0 & 1\n\\end{array} \\right).$$ We say that such a lifting has nebentype $\\chi \\eta_q^i$ at $q$.\n\\end{prop}\n\n\\begin{proof}\n This follows by the arguments in Section 2 of \\cite{KW}, as noted in \nSection \\ref{liftings}, \nfrom the following 2 local\nfacts: \n\n- The local deformation ring $R_p$ which parametrises (equivalence classes of)\nlifts of $ {\\bar {\\rho} } |_{G_p}$\nto $O$-algebras that are crystalline of weight $k( {\\bar {\\rho} } )$, and of fixed determinant, is smooth over $ O$ of dimension ${\\rm dim}{_{\\mathbb F}} H^0(D_p,{\\rm Ad}^0( {\\bar {\\rho} } ))+1$.\nThis is proved in \\cite{[R00]}, \\cite{Tay3} (see also discussion in Section 2\nof \\cite{KW} and Proposition 2.3 of \n\\cite{KW} for the case of $k( {\\bar {\\rho} } )=p+1$).\n\n- Consider the versal ring $R_q$ that parametrises (equivalence classes of) lifts of\n$ {\\bar {\\rho} } |_{D_q}$ to CNL $O$-algebras with residue field ${\\mathbb F}$ \nof determinant (the restriction to $D_q$ of) ${\\varepsilon}\\chi_p^{k( {\\bar {\\rho} } )-1}\\eta_q^i|_{D_q}$, and such\nthat\n$\\rho|_{I_q}$ is of the from\n$$\\left( \\begin{array}{cc}\n     \\chi\\eta_q^i  & *  \\\\\n    0 & 1\n\\end{array} \\right).$$  That such a ring exists (i.e., the conditions we are defining, which can be interpreted as a condition of ordinarity as we are fixing determinants, are deformation conditions) follows easily from our assumption that $q$ is ramified in $ {\\bar {\\rho} } $ (see for instance Section 6.2 of \\cite{dsl}). The key fact we need is that $R_q$ is a complete intersection, which is flat over $ O$, and of relative\ndimension ${\\rm dim}_{{\\mathbb F}}H^0(D_q,{\\rm Ad}^0( {\\bar {\\rho} } ))=1$. (In fact, it is even smooth.) The asserted dimension of the cohomology group is easily checked using\nthat $ {\\bar {\\rho} } $ is ramified at $q$.\n\nWe now prove that $R_q$ is smooth over $O$ of relative dimension 1.\nThis essentially follows from Section 2 of \\cite{Boe} \n(see Section 2, and in particular Theorem 3.8 and Lemma 3.10: the deformation problem we are describing here \nis one obtained by specializing the $T$ in \nTheorem 3.10 (iii) of {\\it loc.\\ cit.\\ } to a specific value and twisting).\nWe sketch an argument to be more self-contained. \n\nThe dimension over ${\\mathbb F}$  of the mod $\\pi$ Zariski tangent space of $R_q$ is 1. This follows from the calculations in Section 1 of \\cite{Wiles}, \nsee also Section 4.3 of \\cite{deshalit} for an exposition in semistable cases, as the dual number lifts that arise from $R_q$ are the same as the minimal lifts in\n\\cite{Wiles}. Thus to show that $R_q$ is in fact smooth of relative \ndimension 1 over O it will be enough to show that there are infinitely many \nnon-equivalent $O$-valued lifts $\\rho_q$ of $ {\\bar {\\rho} } |_{D_q}$ \nof the required kind which we now proceed to show. \nAny lift will be tamely ramified and thus will be specified by lifting the image of $ {\\bar {\\rho} } (\\sigma_q)$ \nand $ {\\bar {\\rho} } (\\tau_q)$, where $\\sigma_q,\\tau_q$\nare generators of Galois group of the maximal tamely ramified extension of $\\mathbb{Q}_q$, and the \nonly relation these satisfy  is $\\sigma_q\\tau_q\\sigma_q^{-1}=\\tau_q^q$. When $\\overline \\chi$ is non-trivial, \nand thus $ {\\bar {\\rho} } |_{I_q}$ may be assumed split and \n$ {\\bar {\\rho} } |_{D_q}$ is diagonal, \nby inspection we get infinitely many lifts to diagonal matrices. \n\nIn the case when $\\overline{\\chi}$ and $\\chi$ are  trivial, and hence $ {\\bar {\\rho} } |_{I_q}$ is unipotent, and not-trivial by \nassumption, again a simple calculation yields infinitely many $O$-valued lifts.\nThere will be 2 cases corresponding to $\\chi':=\\eta_q^i$ being trivial or non-trivial.\nWhen it is trivial this is covered by E3 of \\cite{Tay3} (this is the only case when the lifts considered locally at $q$ do not have abelian image). Otherwise we choose a $\\sigma=\\sigma_q$ and a $\\tau=\\tau_q$ such that\n$ {\\bar {\\rho} } (\\sigma)$ is $$\\left( \\begin{array}{cc}\n     r  & b  \\\\\n    0 & r\n\\end{array} \\right)$$ (note that by the relation $\\sigma\\tau\\sigma^{-1}=\\tau^q$, the characteristic polynomial \nof $ {\\bar {\\rho} } (\\sigma)$ is forced to have double roots as $q$ is 1 mod $p$ and \n$ {\\bar {\\rho} } $ is tamely ramified at $q$), and $ {\\bar {\\rho} } (\\tau)$ is the matrix $$\\left( \\begin{array}{cc}\n     1  & a  \\\\\n    0 & 1\n\\end{array} \\right).$$  We want to construct infinitely many $O$-valued lifts $\\rho_q$ of determinant \n${\\varepsilon}\\chi_p^{k( {\\bar {\\rho} } )-1}\\eta_q^i|_{D_q}$, and such that\n$\\rho_q|_{I_q}$ is of the from\n$$\\left( \\begin{array}{cc}\n     \\chi'  & *  \\\\\n    0 & 1\n\\end{array} \\right).$$ We seek $$\\rho_q(\\sigma)=\\left( \\begin{array}{cc}\n     \\alpha  & \\gamma  \\\\\n    0 & \\beta\n\\end{array} \\right),$$ say $A$, and $$\\rho_q(\\tau)=\\left( \\begin{array}{cc}\n     \\chi'(\\tau)  & a'  \\\\\n    0 & 1\n\\end{array} \\right),$$ say $B$. \nHere  $\\alpha\\beta={\\varepsilon}(\\sigma)\\chi_p^{k( {\\bar {\\rho} } )-1}(\\sigma)\\eta_q^i(\\sigma)$, $\\alpha,\\beta$ reduce to $r$, $a'$ reduces to $a$ (and hence is a unit), $\\gamma$ \nto $b$. \n\nWe explicitly produce these lifts by the following calculation. (The version \nof the calculation we present here is suggested by B\\\"ockle.)\nWe can assume by conjugation that $a$ and $a'$ are equal to one. \nAs the order of $B$ divide $q-1$ (as $\\chi'$ is not trivial of order dividing $q-1$), the relation \n$\\rho_q(\\sigma)\\rho_q(\\tau)\\rho_q(\\sigma)^{-1}=\\rho_q(\\tau)^q$\nis therefore equivalent to $AB=BA$, which yields\n     $\\alpha-\\beta=\\gamma(\\chi'(\\tau)-1)$\n  as the only relation. Combining this with the equation \n$\\alpha\\beta=\\psi:=\\epsilon(\\sigma)\n\\chi_p^{k( {\\bar {\\rho} } )-1}(\\sigma)\\eta_q^i(\\sigma)$\n  gives the quadratic equation\n     $\\beta^2-\\beta\\gamma(\\chi'(\\tau)-1)-\\psi=0$ for $\\beta$.\n  Since $\\chi'(\\tau)-1$ lies in the maximal ideal, it follows easily that \n  for each $\\gamma$ (reducing to $b$)\n  there is a unique solution $\\beta$ that is congruent to $r$ \n  modulo the maximal ideal of $ O$, and hence there is a unique $\\alpha$ depending on $\\gamma$.\n  Thus one has a 1-parameter family (in $\\gamma$) of lifts \n$\\rho_q$ of the required type, thus proving that $R_q$ is smooth over $O$ of relative dimension 1.\n\n\\vspace{3mm}\n\nAfter these 2 facts, from Section \\ref{liftings}, we deduce that\nthe global deformation ring $R$ which parametrises (equivalence classes of) lifts of $ {\\bar {\\rho} } $\n\n- of the given determinant ${\\varepsilon}\\chi_p^{k( {\\bar {\\rho} } )-1}\\eta_q^i$, \n\n- that at $q$ are of the given form,\n\n- are minimal at primes $\\neq q,p$ in the sense of the section above, \n\n- at $p$ the lift is crystalline of weight $k( {\\bar {\\rho} } )$ \n(and thus finite flat when the residual representation has weight 2),\n\n\\noindent is a finite flat complete intersection (ffci) over $O$, and hence we get a lift of the desired kind. \n\n\n\\end{proof}\n\n\n\\noindent{\\bf Remark:} We will apply this proposition only when $ {\\bar {\\rho} } $ has weight 2, and \nis unramified outside $p,q$, and the lifts that need to be constructed\nhave non-trivial nebentype at $q$. In the proposition, \nwhen $ {\\bar {\\rho} } $ is ordinary we could also have allowed the \ndeformations at $p$ to be minimal of weight 2, and hence \nBarsotti-Tate over $\\mathbb{Q}_p(\\mu_p)$ when the weight is not $p+1$, and semistable of weight 2 otherwise.\n\nAs we have seen, the proofs of Proposition \\ref{p} and \\ref{q} follow easily from the method of proof of Theorem 2.2 of \\cite{KW} after \nsome computations of local deformation rings. \nThe point is that because of the method of Section 2 of \\cite{KW}, we can prove results about congruences of Galois representations to parallel many of the results known for congruences between modular forms.  The method should allow one to prove in many more cases the analog of the results in \\cite{DT} and\n\\cite{inv}, these are level raising results for modular forms, for Galois representations: this is reduced \nto some local computation. The local computations in the ``$(p,p)$ case'' (needed for the analog of \\cite{inv}) are likely to be involved, while those in the $\\ell \\neq p$ case may be easier in many cases, and could be deduced for instance from \\cite{Boe} when locally the residual representation at the place $\\ell$ is not scalar. \n\n\\section{Compatible systems}\\label{comp}\n\nWe explain how to make the lifts $\\rho$ of Proposition \\ref{p} and \\ref{q}\npart of a compatible system. As in Section 3 of\n\\cite{KW}, the proof uses the method of \\cite{Tay2} (see proof of Theorem 6.6 of \\cite{Tay2}), and the refinements in \\cite{D} and \\cite{Wint}.\n\n\\begin{prop}\\label{c}\n \n(i)  Assume $ {\\bar {\\rho} } $ is as in Proposition \\ref{p}: so it is ordinary of weight $2 \\leq k( {\\bar {\\rho} } ) \\leq p+1$ and $k( {\\bar {\\rho} } ) \\neq p$, and $p>3$.\nGiven a minimal lift $\\rho$  of $ {\\bar {\\rho} } $ as in Proposition \\ref{p}, there is a\n(weakly) compatible system $(\\rho_{\\lambda})$ where $\\lambda$ runs through all\nplaces of a number field $E$, and $\\rho$ is a member of the\ncompatible system above the prime of $p$ fixed by $\\iota_p$. Further \n$\\rho_{\\lambda}$ for $\\lambda$ above the prime $\\ell$ ($>2$) fixed by $\\iota_\\ell$, that is not ramified in $ {\\bar {\\rho} } $ (and hence $\\neq p$), \nthe representation is Barsotti-Tate at $\\ell$, unramified outside the primes ramified in $ {\\bar {\\rho} } $, and \nthe inertial Weil-Deligne (WD) parameter at $p$ of $\\rho_{\\lambda}$ \n(i.e., if $(\\tau,N)$ is the WD parameter, with $\\tau$ a $F$-semisimple representation of the Weil group and $N$ \na nilpotent matrix, we consider only $(\\tau|_{I_q},N)$) is the same as that of\n$\\rho$ at $p$.\n \n(ii) Now we assume $ {\\bar {\\rho} } $ is as in Proposition \\ref{q}: thus \n$ {\\bar {\\rho} } $ does not have solvable image (but $p=3$ is allowed), it has the behaviour at a prime $q$ as in  Proposition \\ref{q}, but we make the additional assumption that \n$k( {\\bar {\\rho} } )=2$. \nConsider a minimal lift $\\rho$ as in Proposition \\ref{q}, thus $\\rho$\nis Barsotti-Tate at $p$, and we assume that the nebentype at $q$, \n$\\chi\\eta_q^i$, is non-trivial. The inertial parameter at $q$ of $\\rho$ is $(\\omega_q^j \\oplus 1,0)$ ($ 1 \\leq j \\leq q-2$) where \n$\\omega_q^j:=\\chi\\eta_q^i$. There is a (weakly) compatible system $(\\rho_{\\lambda})$ where $\\lambda$ runs\nthrough all places of a number field $E$, and $\\rho$ is a member of\nthe compatible system at the place above  $p$ fixed by $\\iota_p$. Further $\\rho_{\\lambda}$ for\n$\\lambda$ a prime above $q$ that is determined by $\\iota_q$, is unramified at all primes $\\neq\np,q$ outside which $ {\\bar {\\rho} } $ is unramified, is unramified at $p$, and at $q$\nis Barsotti-Tate over $\\mathbb{Q}_q(\\mu_q)$.  Assume that $\\overline{\\rho_\\lambda}$\nhas non-solvable image. Then $\\overline{\\rho_\\lambda}$ has weight $j+2$, \nor its twist by ${\\overline \\chi_q}^{-j}$\nhas weight $q+1-j$. \n\\end{prop}\n\n\\begin{proof}\nThis follows from the results in \\cite{Tay1} and \\cite{Tay2}, using the arguments in Section 3 of\n\\cite{KW} (see Theorem 3.1 of \\cite{KW}).  \n\nThe existence of a weakly compatible system $(\\rho_{\\lambda})$, of which $\\rho$ is a member, follows easily from the proof of Theorem 6.6 of \\cite{Tay2} (which uses Brauer's theorem on writing representations of finite groups as a virtual sum of representations induced from characters of solvable subgroups, and base change results of Arthur and Clozel in \\cite{AC}). ({\\small We recall the construction of Taylor in \\cite{Tay1} and \\cite{Tay2}. In both (i) and (ii) we may assume that the image of $ {\\bar {\\rho} } $ is non-solvable. Then the lift $\\rho$ constructed is such that there is a totally real field $F$, Galois over $\\mathbb{Q}$, such that $\\rho|_{G_F}$ arises from a holomorphic, cuspidal automorphic representation $\\pi$ of $\\mathrm{GL}_2(\\mathbb{A}_F)$ with respect to the embedding $\\iota_p$. Using Brauer's theorem\nwe get subextensions $F_i$ of $F$ \nsuch that $G_i=\\mathrm{Gal} (F\/F_i)$ is solvable, characters $\\chi_i$ of $G_i$ with values in $\\overline \\mathbb{Q}$ (that we embed in $\\overline \\mathbb{Q}_p$ using $\\iota_p$), such that $1_{G}=\\sum_{G_i}n_i{\\rm Ind}_{G_i}^{G} \\chi_i$. Using \\cite{AC} we also\nget holomorphic cuspidal automorphic representations $\\pi_i$ of $\\mathrm{GL}_2(\\mathbb{A}_{F_i})$ such that\nif $\\rho_{\\pi_i,\\iota_p}$ is the representation of $G_{F_i}$ corresponding to $\\pi_i$ w.r.t. $\\iota_p$, then $\\rho_{\\pi_i,\\iota_p}=\\rho|_{G_{F_i}}$. Thus $\\rho=\\sum_{G_i}n_i{\\rm Ind}_{G_{F_i}}^{G_\\mathbb{Q}} \\chi_i\\otimes \\rho_{\\pi_i,\\iota_p}$.\nNow for any prime $\\ell$ and any embedding \n$\\iota:\\overline \\mathbb{Q} \\rightarrow \\overline \\mathbb{Q}_\\ell$, we define  the virtual representation\n$\\rho_{\\iota}=\\sum_{G_i}n_i{\\rm Ind}_{G_{F_i}}^{G_\\mathbb{Q}} \\chi_i\\otimes \\rho_{\\pi_i,\\iota}$ of $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$\nwith the $\\chi_i$'s now regarded as $\\ell$-adic characters via the embedding  $\\iota$. We check that $\\rho_{\\iota}$ is a true representation by computing its inner product. The representations $\\rho_\\iota$ together constitute the weakly compatible system we seek.})\n\nThus we concentrate below on proving some of the finer ramification properties claimed for $\\rho_{\\lambda}$\nat the prime of the same residue characteristic as $\\lambda$.\n\nFor instance the property at $p$ asserted in (i) follows directly from\nTaylor's results in \\cite{Tay1} (see Lemma 1.5 and Corollary 1.7 of it) which show that $\\rho|_{G_F}$ is modular for some $F$ that is unramified at $p$, i.e., it arises (w.r.t.\nthe embedding $\\iota_p$) from a  Hilbert modular form $f$ for $F$.\nIn fact \\cite{Tay1} also shows that $f$ is ordinary at all primes above $p$ (w.r.t. $\\iota_p$), and at such primes \nis either principal series of conductor dividing $p$ with ``nebentype'' $\\omega_p^{k-2}$ (i.e.,\nthe automorphic representation corresponding to $f$ is such that at primes $\\wp$ above $p$ the corresponding\nlocal component is the principal series $\\pi(\\psi_1,\\psi_2)$, with $\\psi_1$ restricted to the units given by the character  corresponding to  $\\omega_p^{k-2}$ by local class field theory and $\\psi_2$ unramified), or Steinberg at $p$ (the latter only in the case $k( {\\bar {\\rho} } )=p+1$). The reader may also consult proof of Proposition 2.5 of \\cite{KW} for more details, especially in the ordinary case.\n\nWe turn to proving (ii).\nWe know by \\cite{Tay1} and \\cite{Tay2} that there is a totally real field $F$\n(which we may assume to be Galois over $\\mathbb{Q}$) \nover which $\\rho|_{G_F}$ is modular. \nIf $F'\\subset F$ is the fixed field of a  decomposition group above $q$, then there is a prime $Q$ above $q$ in $F'$ which is a split prime. We deduce  using \\cite{AC}, \\cite{Carayol} and \\cite{Tay}, \nthat $\\rho|_{G_F'}$, and hence \n$\\rho_\\lambda|_{G_{F'}}$,  arises from a Hilbert modular form $f$ which locally at \n$Q$ is a ramified principal series of conductor $Q$, whose nebentypus at $Q$ is $\\omega_q^j:=\\chi\\eta_q^i$. This also proves the assertion about the inertial parameter of $\\rho$ at $q$.\n\nWe now focus on the properties of $\\rho_{\\lambda}$ asserted at $q$\n(as outside $q$ the proof is the same as in Section 3 of \\cite{KW}).(We may assume that $F$ is of even degree over $\\mathbb{Q}$ as we are done otherwise by \\cite{Saito}.) We first prove that $\\rho_\\lambda$ when restricted to\n$\\mathbb{Q}_q(\\mu_q)$ is Barsotti-Tate (which uses the assumption\nthat $\\chi\\eta_q^i$ is non-trivial). To see this, we work over a totally real field \n$F''$ that is a solvable extension of $\\mathbb{Q}$ which when completed at all \nplaces above $q$ is $\\mathbb{Q}_q(\\mu_q)$ (see Lemma 2.2 of \\cite{Tay3}), \nand apply to $(\\rho_\\lambda|_{G_{F''}})$ the same arguments as those in proof of Theorem 3.1 of \\cite{KW} for $(\\rho_\\lambda)$, but instead of using \nTh\\'eor\\`eme 1 (ii) of  \\cite{Breuil-cong} we use \na result of \\cite{Raynaud} (see Proposition 2.3.1), which we may \nas we are in the weight 2 case. \n({\\small Recall that $\\rho|_{G_{F'}}$ arises from a weight 2 Hilbert modular form $f$ for $GL_2(\\mathbb{A}_{F'})$ that when base changed to the composite $F'F''$ of $F'$ and $F''$, which is a solvable extension of $F'$, becomes\nunramified at places above $q$. This uses results of \\cite{Tay1}, \\cite{Tay2}, \n\\cite{Carayol}, \\cite{Tay}, \\cite{AC} as in Section 3 of \\cite{KW}.\nThis then gives the required statement by deducing that \n$\\rho_{\\lambda}|_{G_{F''F'}}$ mod $\\lambda^n$ is finite flat at primes above $q$ by Th\\'eor\\`eme 1 (i) of \\cite{Breuil-cong},\nand then using  Proposition 2.3.1 of \\cite{Raynaud}, instead of using Th\\'eor\\`eme 1 (ii) of \\cite{Breuil-cong} which got used in proof of Theorem 3.1 of \\cite{KW}. Note that the completion at a prime above $Q$ of $F'F''$ is $\\mathbb{Q}_q(\\mu_q)$.}) \n\nNow assume that the image of $\\overline \\rho_\\lambda$ is non-solvable.\nIt is not hard to see that \n${\\overline \\rho_\\lambda}|_{G_{F'}}$, which we know is modular, also arises from  a Hilbert modular \nform $f'$, congruent to $f$ mod place fixed by $\\iota_q$, that is square integrable at a finite place $\\alpha$ and at $Q$ of conductor $Q$ and of nebentypus $\\chi\\eta_q^i$ at $Q$. \n({\\small To get such a $f'$ congruent to $f$ is standard: \nWe use the level raising techniques of \\cite{Tay}, using the proof of Theorem 2 of {\\it loc.\\ cit.\\ } to find a \n$\\alpha$, prime to $q$,  such that using notation of Theorem 1 of {\\it loc.\\ cit.\\ } the valuation under $\\iota_p$ of $\\Theta_f(T_{\\alpha}^2-S_{\\alpha}(\\mathbb{N} \\alpha +1)^2)$\nis bigger than that of $E_f((\\mathbb{N} \\alpha +1))$ (for instance $\\mathbb{N}$ stands for the norm from $F'$ to $\\mathbb{Q}$). Then we prove the existence of the desired $f'$ which is square integrable at a finite place $\\alpha$ and at $Q$ is fixed by $U_1(Q)$ which gives rise to ${\\overline \\rho_\\lambda}|_{G_{F'}}$, using  the proof of Theorem 1 of {\\it loc.\\ cit.\\ } with $\\lambda=\\alpha$ in the notation there.\nThat $f'$ has nebentype $\\omega_q^j=\\chi\\eta_q^i$ at $Q$ follows by considering determinants and central characters.})\n\nThen we use the results of \\cite{Saito} to conclude that at $Q$ the inertial WD parameter of $\\rho_\\lambda|_{G_{F'}}$ \nis the same as that of $\\rho|_{G_{F'}}$ which we know to be $(\\omega_q^j \\oplus 1, 0)$\n(note that as $Q$ is a split prime of \n$F'$, local information at $Q$ of $\\rho_\\lambda|_{G_{F'}}$ gives the local information of $\\rho_\\lambda$\nat $q$).  After this, to get the information about \nweights, we use Proposition 6.1.1 of \\cite{BM} and Theorem 6.11 of \\cite{Savitt1},\nas we have that $\\rho_\\lambda|_{G_{F'}}$\nis irreducible by Lemma 2.6 of \\cite{KW}.\n\\end{proof}\n\n\n\n\n\\section{Chebyshev's estimates on primes}\\label{cheb}\n\nIn the proof of Theorem \\ref{main} below, \nwe will need some estimates on prime numbers proven by Chebyshev (we learnt about \nthe following precise form of Chebyshev's estimate from a message of J-P.~Serre, and from R.~Ramakrishna). \n\nIf $\\pi(x)$ is the prime counting function, then if $x >30$\n$$A({ x \\over {\\rm log}(x)}) \\leq \\pi(x) \\leq B({ x \\over {\\rm log}(x)})$$ where $A=0.921...$ and \n$B \\over A$ is ${6 \\over 5}=1.2$ (see \\cite{Chebyshev1} and \\cite{Chebyshev2}, and also page 21 of \\cite{Ellison}). From this we easily deduce that\nif we fix a real number $a>1.2$, and denote by $p_n$ the $n$th prime which we\nassume $> {\\rm max}(30,a^{{6} \\over {5a-6}})$, then $p_{n+1} \\leq ap_n$. \n\nIn the arguments below this estimate will be relevant for particular values of $a$. Given $p_n>2$ we consider an odd (prime power) \ndivisor $\\ell^r=2m+1$ of \n$P_{n+1}-1$, where $P_{n+1}$ is either $p_{n+1}$, or $p_{n+2}$ if $p_{n+1}$ is a Fermat prime,  \nand divide the integers in the interval $[0,P_{n+1}-1]$ into blocks of size ${{P_{n+1}-1} \\over {\\ell^r}}$. We need to  ensure that\n$p_n+1 \\geq {\\rm max}({{m+1} \\over {2m+1}}({P_{n+1}-1}  )+2,\nP_{n+1}-({{m} \\over {2m+1}}({P_{n+1}-1}  )))$. A computation shows that\nthis is ensured by requiring that\n$${{P_{n+1}}\\over p_n} \\leq {2m+1 \\over {m+1}} - ({m \\over {m+1}})({1 \\over {p_n}}).$$\nWe consider $p_n \\geq 31$.\n\nAn inspection shows that there is always a $P_{n+1}$ as required up to $p_n=1000$: use $a$ in the Chebyshev estimate to be ${44 \\over 30}={3 \\over 2} -{1 \\over 30}$ and rule out Fermat primes causing problems in that range (the only ones are $5,17,257$ and for $p_n=251$, the prime preceding $257$, we can use $P_{n+1}=263$).\nAfter that the Chebyshev estimate used with $a=\\sqrt{1.499}$ (note $1.499={3 \\over 2} - {1 \\over 1000}$, and that after $3,5$ no two successive primes can both be Fermat primes) \ngives the existence of $P_{n+1}$ of the required kind if $p_n$ is at least $21591$. For primes $1000<p_n \\leq 21591$ we can again get the desired prime $P_{n+1}$ using $a={44 \\over 30}$ as there are no Fermat primes in this range.\nThus we conclude that we always have a $P_{n+1}$ as desired \nonce $p_n \\geq  31$.\n\n(We thank Bo-Hae Im and Faheem Mitha for help with the calculations of this section.)\n\n\\section{Four lemmas}\n\nWe have the following lemma which we owe, \ntogether with its proof which we reproduce {\\it verbatim}, to Wintenberger\n(it was in an early version of \\cite{KW}). Below by a mod $p$ representation $ {\\bar {\\rho} } $ being semistable\nwe mean that for all the primes ramified in $ {\\bar {\\rho} } $ that are $\\neq p$, the \nramification is unipotent, and by $ {\\bar {\\rho} } $ being dihedral we mean its projective image is a dihedral group.\n\n\\begin{lemma}\\label{dihedral}(Wintenberger)\n (i) A mod $p$ dihedral, semistable representation $ {\\bar {\\rho} } $\nwhich is odd and irreducible, with $p$ odd, exists if  and only if $p$ is 3 mod $4$, and\nthe class number of the imaginary quadratic field $\\mathbb{Q}(\\sqrt{ -p })$\nis non-trivial (i.e., $p$ is 3 mod 4 and $p \\neq 3, 7, 11, 19,\n43, 67,163$). Such a $ {\\bar {\\rho} } $ is split at $p$ and has a twist with Serre weight\n$k={ {p+1} \\over 2}$.\n (ii) If $ {\\bar {\\rho} } $ is a dihedral representation  induced from the quadratic\nsubfield of $\\mathbb{Q}(\\mu_p)$,\nand $ {\\bar {\\rho} } $ is locally irreducible at $p$ of weight at most $p+1$, then its\nweight is $p+3 \\over 2$.\n \n\\end{lemma}\n\n\\begin{proof} (i) First note that such a dihedral $ {\\bar {\\rho} } $\nis unramified outside $p$. Let the projective\nimage $H$ of $ {\\bar {\\rho} } $ be a dihedral group $D_{2t}$ of order $2t$, $t>1$.\nAs $H$ has order prime to $p$, the image of inertia $I_p$ in $H$ is cyclic\n: denote by $\\tau _p$ a generator of the image of $I_p$ in $H$, and by $i_p$ the order of $\\tau_p$.\nNote that $t$ is odd. For if $t$ were even, $D_{2t}$ would have\na quotient which is $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/2\\mathbb{Z}$ which would give\na character of order $2$ unramified everywhere.\nAs $t$ is odd, $D_{2t}$ has only one\ncharacter of order $2$, say $\\eta$. It corresponds to a quadratic\nfield $K$ which is ramified at $p$. As $\\eta (\\tau _p )=-1$,\nand $t$ is odd, $\\tau _p$ is of order $2$.\nLet $L$ be the field cut up by the representation\nof $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$ in $H$. Then $L\/K$ is unramified everywhere.\n\nLet $c\\in \\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$ be a complex conjugation. As $ {\\bar {\\rho} } $ is odd,\n$c$ has non trivial image $\\gamma$  in $H$. As $t$ is odd,\n$\\eta (\\gamma)\\neq 1$ and\nthe quadratic field  $K$ is imaginary. This implies\n$p \\equiv 3$ modulo $4$, and that $t$ is a divisor of $h_K$,\nthe class number of $K$ (which is known to be odd). Let $\\Delta _p$ be the image of the decomposition group\nat $p$ in $H$. As  $\\Delta _p$ is a quotient of the Galois\ngroup of $\\mathbb{Q}_{p,unr}(\\sqrt{p} )\/\\mathbb{Q}_p$ (wher $\\mathbb{Q}_{p,nr}$ is the maximal unramified extension of $\\mathbb{Q}_p$), $\\Delta _p$ is\nabelian. As the subgroup $\\Gamma_p$ of $H$ of order $2$\ngenerated by  $\\tau _p$ is its own centralizer ($t$ is odd),\none sees that $\\Delta _p=\\Gamma_p$. This implies that $ {\\bar {\\rho} } (D_p)$ is abelian.Thus $I_p$ acts under\n$ {\\bar {\\rho} } $ via characters of level $1$.\nLet us assume that we have twisted $ {\\bar {\\rho} } $ by the suitable\npower of the cyclotomic character so that\nthe Serre weight $k$ of $\\rho$  satisfies $2\\leq k\\leq p+1$.\nThen $I_p$ acts with characters $1$ and $\\chi_p ^{k-1}$.\nAs the image of $I_p$ in $H$ is of order $2$, one has $k-1= {p-1 \\over 2}$\nand $k={p+1 \\over 2}$.\n\n(ii) This follows because the image of inertia $I_p$ in the projective \nimage of $ {\\bar {\\rho} } $, which we know to be cyclic, is forced to be of order 2.\n\\end{proof}\n\n\\noindent{\\bf Remark:} We use part (i) of Lemma \\ref{dihedral} only to\ndeduce that a dihedral $ {\\bar {\\rho} } $ which is unramified outside $p$ is up to twist\nordinary at $p$. Note that once we know that in the proof $t$ is odd, we can also see this by deducing that locally at $p$\nthe fixed field of the kernel of $ {\\bar {\\rho} } $ cannot have an unramified quadratic subfield.\n\\vspace{3mm}\n\nWe also have a useful lemma which follows immediately from Serre's definition of weights:\n\n\\begin{lemma}\\label{weights}\n  Assume $\\tau:G_{\\mathbb{Q}_p} \\rightarrow GL_2({\\mathbb F})$ is such that its weight $k \n\\neq 2$ is $<p$. Then if $\\tau$ is irreducible,\n  and $k-1=p-k'$ with $k'$ non-negative, $\\tau \\otimes \\overline{\\chi_p}^{k'}$ has weight $k'+2=p+3-k$. If $\\tau|_{I_p}$ is\n  split, $\\tau \\otimes \\overline{\\chi_p}^{1-k}$ has weight $p+1-k$.\n\\end{lemma}\n\nThe following lemma is crucial for us and is due to Breuil and M\\'ezard, \\cite{BM}, Proposition 6.1.1, and Savitt, \\cite{Savitt1}, Theorem 6.11:\n\n\\begin{lemma}\\label{breuil}\nIf a representation $\\rho: G_{\\mathbb{Q}_p} \\rightarrow GL_2({ O})$ \nbecomes Barsotti-Tate over $\\mathbb{Q}_p(\\mu_p)$, then if residually the representation is reducible, then $\\rho$ itself is reducible. More precisely, a twist of\n$\\rho$ by some power of $\\omega_p$ (the Teichmuller lift of $\\overline \\chi_p$) is ordinary.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from Proposition 6.1.1 of \\cite{BM}, and Theorem 6.11 of \\cite{Savitt1} as then we know that some twist of $\\rho$ by a power of $\\omega_p$ will be ordinary (i.e., in case (i) or (iii) of {\\it loc.\\ cit.\\ } which note are related to each other by twists, and are Cartier dual).\n\\end{proof}\n\n\\vspace{3mm}\n\n\\noindent{\\bf Remark:} The particular case of the lemma above for representations arising from modular forms is in Proposition 8.13 of \\cite{Gross} or Section 6 of \\cite{Edix}.\n{\\small A non-ordinary newform in $S_2(\\Gamma_1(Np))$\nwhich gives rise to a  $ {\\bar {\\rho} } $ that is reducible at $p$ is such that its $p$th Hecke eigenvalue $a_p$ has valuation 1 and its complex conjugate form is ordinary.} Further, if a mod $p$ newform in $S_2(\\Gamma_1(Np))$ ($(N,p)=1$) has nebentype at $p$ given by \n$\\overline{\\chi_p}^j$ ($0 \\leq j \\leq p-2$), then the associated mod $p$ representation either has Serre weight $j+2$ or its twist by $\\overline{\\chi_p}^{-j}$ has weight $p+1-j$. (We are grateful to Bas Edixhoven for helpful correspondence about this.)\n\n\\vspace{3mm}\n\nThe following lemma is a simple consequence of modularity lifting theorems \nof \\cite{Wiles}, \\cite{TW}, \\cite{SW2}, \\cite{SW3}, \\cite{DFG}, \\cite{Tay1},\\cite{Tay2} and the work in \n\\cite{BLZ}.\n\n\\begin{lemma}\\label{rubbish}\n  Let $p$ be an odd prime. If $\\rho:G_\\mathbb{Q} \\rightarrow GL_2({O})$ is an \n  irreducible $p$-adic representation unramified outside $p$ and at $p$ \n  crystalline of Hodge Tate weights $(k-1,0)$ with $k$ even and $2 \\leq k \\leq p+1$,\n  and if the residual representation $ {\\bar {\\rho} } $ is modular,  \n  then $\\rho$ arises from  $S_k(SL_2(\\mathbb{Z}))$.\n\\end{lemma}\n\n\\begin{proof}\n  Using Lemma \\ref{dihedral} we see that if $ {\\bar {\\rho} } $ is irreducible, and restricted to $G_{ \\mathbb{Q}(\\sqrt{{(-1)}^{p-1 \\over 2}}p)  }$\n  is reducible, then $ {\\bar {\\rho} } |_{I_p}$ is ordinary and distinguished.\n  Thus the only case of the lemma not covered in the literature (see \\cite{Wiles}, \\cite{TW}, \\cite{SW2}, \\cite{SW3},\\cite{DFG})\n  is when $\\rho$ is possibly of weight $p+1$ and is not ordinary. In fact this case does not occur, as using Corollary\n  4.1.3 of \\cite{BLZ}, we see that in this case the residual representation is \nsuch that $ {\\bar {\\rho} } |_{D_p}$ is irreducible and of Serre weight 2. But this contradicts the fact that\n  there is no Serre-type $ {\\bar {\\rho} } $ of level 1 and weight 2 proved in Theorem 4.1 of \\cite{KW}.\n\\end{proof}\n\n\\noindent{\\bf Remark:} Using \\cite{BLZ} it must be easy to prove a lifting theorem when at $p$ the lifts are crystalline of weight $p+1$ even without assuming that the lift is unramified outside $p$. As we do not need it in this paper, we do not do this. Note that oddness of the residual representation forces $k$ to be even (see Proposition 2 and Corollary 2 of \\cite{Wint}).\n\n\n\\begin{cor}\\label{trivial}\n  (i) Given an odd  prime $p$, if all 2-dimensional, \n  mod $p$, odd, irreducible representations $ {\\bar {\\rho} } $\n  of a given weight $k( {\\bar {\\rho} } )=k \\leq p+1$, and unramified\n  outside $p$, are known to be modular,\n  then for any prime $q \\geq k-1$, all 2-dimensional, mod $q$, odd, irreducible representations $ {\\bar {\\rho} } '$\n  of weight $k( {\\bar {\\rho} } ')=k$, and unramified outside $q$, are modular.\n  \n(ii) If the level 1 case of Serre's conjecture is known for a prime $p>2$,\n  then for any prime $q$ the level 1 case of Serre's conjectures is known\n  for all 2-dimensional, mod $q$, odd, irreducible representations $ {\\bar {\\rho} } $\n  of weight $k( {\\bar {\\rho} } ) \\leq p+1$.\n\\end{cor}\n\n\\begin{proof} This is proved implicitly in \\cite{KW}.\nIt is enough to prove the first statement. We may assume $q>3$.\nBy Theorem 2.1 of \\cite{KW} lift $ {\\bar {\\rho} } '$ to a representation $\\rho'$ which is unramified outside $q$ and crystalline at $q$ of weight $k$. By Theorem 3.1\nof {\\em loc.\\ cit.\\ }, $\\rho'$ is part of a compatible system $(\\rho'_\\lambda)$\nsuch that at a place $\\lambda$ above $p$ the representation is unramified outside $p$ and crystalline of weight $k$ at $p$. The hypothesis of the corollary and Lemma \\ref{rubbish}, then gives the modularity of $(\\rho'_\\lambda)$ and hence that of $ {\\bar {\\rho} } '$.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{main}}\n\n\\subsection{Small weights}\n\nWe now prove our main theorem up to weights 32. (We do not really have to do so many weights before giving the general argument, \nbut this seems good preparation for that and also verification of the general strategy in concrete cases.) When we choose a place $\\lambda$ above a prime $\\ell$ this is always chosen to be with respect to the\nembedding $\\iota_\\ell: \\overline{\\mathbb{Q}} \\hookrightarrow \\overline{\\mathbb{Q}_{\\ell}}$  fixed once and for all a while ago.\n\nFor weights 2,4,6 this has already been proved in \\cite{KW}.  Although the cases of weights 8,12,14 are also done in \\cite{KW}\nwe redo them to illustrate that after weight 6 our inductive method takes over.\nOur arguments below prove that the $ {\\bar {\\rho} } $ being considered arises from some level and weight, and then that it arises from weight $k( {\\bar {\\rho} } )$ and level $N( {\\bar {\\rho} } )=1$ follows from the results in \\cite{Ribet}, \\cite{Edix}.\n(Also note that the Serre weight in all cases considered is even and so \nthe residual mod $p$ representations when restricted to $I_p$ are never \nscalar. We often use the fact implicitly below that if $ {\\bar {\\rho} } $ is modular so is any twist of it by an abelian character, and the same fact for twists of $p$-adic representations $\\rho$ by finite order characters.)\n\n-- Consider the\nweight 8 case. It's enough to prove using Corollary \\ref{trivial}\nthat an irreducible $2$-dimensional, mod $7$ representation $ {\\bar {\\rho} } $ of level $1$ and\nweight $8$ is modular. (If the image is solvable we are done.) Such a representation is ordinary at $7$. By Proposition \n\\ref{p}, lift $ {\\bar {\\rho} } $ to a weight 2,\nsemistable at $7$, $7$-adic representation $\\rho$ that is unramified outside $7$. Using part (i) of \nProposition \\ref{c} get a compatible system $(\\rho_\\lambda)$ and reduce it mod a prime above 3 determined by $\\iota_3$ to get $ {\\bar {\\rho} } '$. Note that \nby Proposition \\ref{c} (i), $k( {\\bar {\\rho} } ')=2$, and $ {\\bar {\\rho} } '$ is\nunramified outside 3 and 7.  If $ {\\bar {\\rho} } '$ has solvable image we are done, as in that case we have a representation to which we can apply known \nmodularity lifting results to conclude that the compatible system\n$\\rho_{\\lambda}$ is modular (\\cite{Wiles}, \\cite{TW}, \\cite{SW2}, \\cite{SW3}) and hence so is $ {\\bar {\\rho} } $:\nthese modularity lifting results may be applied\nas by part (ii)   of Lemma \\ref{dihedral}, $ {\\bar {\\rho} } '$ cannot be both reducible when restricted\nto $G_{\\mathbb{Q}(\\sqrt{-3})}$ and irreducible locally at $3$. Similarly \nif $ {\\bar {\\rho} } '$ is unramified at $7$ we are again done\nas we know by page 710 of \\cite{Serre2} that the residual mod $3$ representation is then reducible.\nOtherwise use Proposition \n\\ref{q}, to\nget a 3-adic lift $\\rho'$ of $ {\\bar {\\rho} } '$ with nebentype $\\omega_7^2$ at 7 ($\\omega_7^4$ would also work). Then use Proposition\n\\ref{c} to get a compatible system $(\\rho'_{\\lambda})$ with $\\rho'$ the member of this compatible system at the place corresponding to $\\iota_3$, \nand consider a residual representation $ {\\bar {\\rho} } '_7$ arising from this system at a place \n$\\lambda$ above 7 fixed by $\\iota_7$. We know\nby Lemma \\ref{breuil} (Proposition 6.1.1 of \\cite{BM})\nthat if the residual mod $7$ representation $ {\\bar {\\rho} } '_7$ locally at $7$ is reducible then the $7$-adic\nrepresentation is also locally reducible. Thus in this case up to twisting \nby a power of $\\omega_7$ we may\nassume the $7$-adic representation is ordinary at $7$ by Proposition 6.1.1 of \\cite{BM}: the representation will also be $I_7$-distinguished as the residual weights are even.\n\nIf $ {\\bar {\\rho} } '_7$ has solvable image, and hence known to be modular, we are done by applying results of \\cite{SW2}, \\cite{SW3} and \\cite{Kisin}, which can be applied as\nin the ordinary cases the representation will be $I_7$-distinguished\n(and that in the dihedral case the representation is ordinary at $7$ by part (i) of Lemma \\ref{dihedral}), and we conclude that \n$(\\rho'_{\\lambda})$ is modular.\n\nNow assume that $ {\\bar {\\rho} } '_7$ has non-solvable image. We get a\nresidual representation whose Serre weight (up to twisting) is either 4\nor $7+3-4=6$  by Proposition 6.1.1 of \\cite{BM}, Theorems 6.11 and 6.12 of \n \\cite{Savitt1} (as explained in Proposition \\ref{c} (ii)), \nand we know the residual modularity for such weights (in fact we also know that in these cases again that $ {\\bar {\\rho} } '_7$ is reducible,\nalthough for uniformity of treatment we do not use this).  \nNow we again use modularity lifting results in \\cite{SW2}, \\cite{SW3}, \\cite{Kisin} as we just used and conclude that \n$(\\rho'_{\\lambda})$ is modular.\n\nHence so is the residual representation \n$ {\\bar {\\rho} } '$, and hence by another application of modularity lifting theorem \nwe conclude that the first compatible system $(\\rho_{\\lambda})$ is modular (as the compatible systems $(\\rho_{\\lambda})$ and $(\\rho'_{\\lambda})$ are linked at the place above 3 fixed by $\\iota_3$), and hence so is $ {\\bar {\\rho} } $ (which in this case means that it does not exist!).\n\n(Now we will be more succinct, and skimp some of the details before we get to the general arguments of the next section, as it's much of the same thing!)\n\n-- Weights 10 and 12: Consider a representation $ {\\bar {\\rho} } $ mod 11 that is irreducible and of\nweight 10 or 12. Its enough by Corollary \\ref{trivial} to prove this to be modular to conclude that for any prime at least 11, any $ {\\bar {\\rho} } $ of level 1 and \nweight 10 or 12 is modular. (If the image is solvable we are done.) By Lemma \\ref{weights} locally at 11 we may assume that the representation is (non semisimple and)\nordinary, and hence we get a weight 2 minimal lifting $\\rho$ unramified outside 11. \nThen by Propositions \\ref{p} and  \\ref{c} make $\\rho$ part of a \ncompatible system $(\\rho_{\\lambda})$, \nsuch that the representation in the system corresponding to $\\iota_{11}$ is $\\rho$,\nand then consider the residual representation $\\overline{\\rho_{5}}$ \nat a prime $\\lambda$ above 5 fixed by $\\iota_5$.  We are done if the image\nof $\\overline{\\rho_{5}}$ is solvable (or unramified at 11 and hence solvable by the weight 2, level 1 case proved in \\cite{KW}) arguing just as in the previous case. \nOtherwise using Proposition \\ref{q} construct lift of $\\overline{\\rho_{5}}$ with nebentype $\\omega_{11}^4$ ($\\omega_{11}^6$ would also work) at 11, and we can do this for either of the 2 weights 10 or 12 being considered.\nArgue as before and get a compatible system $(\\rho'_{\\lambda})$ which residually \nat the  prime above 11 fixed by $\\iota_{11}$ \nwill have weight (up to twisting by some power of $\\overline \\chi_{11}$)\neither 6 or 8 (the same weights if we had chosen lifting with nebentype $\\omega_{11}^6$  when applying Proposition \\ref{q}), and then we are done as before.\n\n\n--Weights $14,16,18,20$: It will be enough to show that a mod $19$ representation $ {\\bar {\\rho} } $ \n(irreducible, odd, $2$-dimensional of level one as always) of any of these weights is modular. (If the image is solvable we are done.) As we have \ndealt with weights up to $12$, we may assume by Lemma \\ref{weights} that $ {\\bar {\\rho} } $ is ordinary at $19$.\nWe again construct a weight 2 minimal lift $\\rho$ of $ {\\bar {\\rho} } $ and get a compatible system $(\\rho_{\\lambda})$ (by Propositions \\ref{p} and \\ref{c}(i)) and consider a residual representation at the place above $3$\ndetermined by $\\iota_3$ (again we are done if the residual mod 3 representation has either solvable image or is unramified at 19), and apply Proposition \\ref{q}\nto get a lifting $\\rho'$ of the mod 3 representation  that is unramified outside $3,19$, Barsotti-Tate at 3, and has nebentype\n$\\omega_{19}^8$ at 19 ($\\omega_{19}^{10}$ would also work). Then make it part of a compatible system $(\\rho'_{\\lambda})$ (using Proposition \\ref{c} (ii)) \nwhich residually at a prime above 19 will \nhave weight (up to twisting by some power of $\\overline \\chi_{19}$) either 10 or 12 (the same if the nebentype $\\omega_{19}^{10}$ had been chosen). Again we can argue as \nbefore, knowing Serre's conjecture in level 1 for weights 10 and 12,  and we are done.\n\n\n-- Weights $22,24,26,28,30$: It will be enough to show that a mod 29 representation $ {\\bar {\\rho} } $ (irreducible, odd, 2-dimensional of level one as always) of any of these weights is modular. (If the image is solvable we are done.) As we have dealt with weights up to 20, \nwe may assume by Lemma \\ref{weights} that $ {\\bar {\\rho} } $ is ordinary at 29.\nThis time we lift $ {\\bar {\\rho} } $ to a minimal lift $\\rho$ of weight 2\n(by Proposition \\ref{p}), make $\\rho$ part of \na  compatible system $(\\rho_{\\lambda})$ \n(by Proposition \\ref{c}(i)) and \nconsider a residual representation at a prime above 7 determined by $\\iota_7$\n(that we may assume is ramified at $29$ and has non-solvable image). Apply Proposition \\ref{q}\nto get a lifting $\\rho'$ of the mod $7$ representation  that is unramified outside $3,19$, Barsotti-Tate at $7$, and has nebentype\n$\\omega_{29}^{16}$ at 29 when the weight is one of 22,26,30, or nebentype\n$\\omega_{29}^{14}$ at 29 when the weight is 24 or 28. Using Proposition \\ref{c} (ii)\nmake it part of a compatible system $(\\rho'_{\\lambda})$ which residually at a prime above 29 will have weight (up to twisting by some power of $\\overline \\chi_{29}$) 18 or 14 (if the \nweight of $ {\\bar {\\rho} } $ is one of  22,26,30), or weight 16 (if the weight of $ {\\bar {\\rho} } $ was either 24 or 28). Again we can argue as \nbefore, knowing Serre's conjecture in level 1 for weights 14,16,18,  and we are done.\n\n-- Weight 32: It will be enough to show that a mod 31 representation $ {\\bar {\\rho} } $ (irreducible, odd, 2-dimensional of level one as always) of weight 32 is modular. (If the image is solvable we are done.) It's the same argument as before. We use as a foil the prime 5, and in the end construct a weight 2 compatible system $(\\rho'_{\\lambda})$, whose modularity yields that of $ {\\bar {\\rho} } $, \nof nebentype $\\omega_{31}^{16}$ at 31\nsuch that at a prime above 31 the residual representation has weight (up to twisting by some power of $\\overline \\chi_{31}$)\neither 18 or 16. As we know Serre's conjecture in level 1 for these weights we can conclude.\n\n\\subsection{The general argument}\n\nNow we give the general argument (which the reader \nmust surely have guessed the gist of). This, together with the previous section, will prove Theorem \\ref{main}.\n\nAssume we have proven the level 1 case of Serre's \nconjecture mod $p_n$ where $p_n$ is a prime $\\geq 31$. This implies by Corollary \\ref{trivial} (ii)\nthat for any prime $q$ we know Serre's conjecture for any level 1 \nmod $q$  representation $ {\\bar {\\rho} } $ that is odd, irreducible, 2-dimensional\nof weight $ k( {\\bar {\\rho} } ) \\leq p_n+1$.\n \nBy the arguments in Section \\ref{cheb}, we can find a prime $P_{n+1}>p_n$, \nwhich is not a Fermat prime, and an odd prime power $\\ell^r=2m+1$ that divides $P_{n+1}-1$ exactly such that\n\\begin{equation}\\label{*}\n{{P_{n+1}} \\over {p_n}} \\leq \n{2m+1 \\over m+1} - ({m \\over m+1})({1 \\over p_n}).\n\\end{equation} (This inequality holds for any integer $m \\geq 1$.)\nConsider any weight $k$ such that $p_n+2 \\leq k \\leq P_{n+1}+1$: \ngiven such a $k$ \nwe would like\nto prove that any 2-dimensional irreducible, odd, mod $P_{n+1}$ representation $ {\\bar {\\rho} } $ of $\\Galois( \\bar{ {\\mathbb Q}}\/{\\mathbb Q})$ of level 1 and weight $k( {\\bar {\\rho} } )=k$ is modular. \nBy Corollary \\ref{trivial}, this will prove Serre's level 1 conjecture\nfor any 2-dimensional, odd irreducible, mod $q$ representation \nof Serre weight $k$, where $q  \\geq k-1$, and also the level 1 conjecture mod all primes $\\leq P_{n+1}$ \nonce we have done this for all weights $p_n+2 \\leq k \\leq P_{n+1}+1$. Then we continue with $P_{n+1}$, treating it like $p_n$ etc.\nWe denote $P_{n+1}$ by $p$ for notational \nsimplicity. (This is the inductive method \nto attack the level 1 case of Serre's conjecture \nproposed in Theorem 5.1 of \\cite{KW}.)\n\nBy  Lemma \\ref{weights}, and as we know the conjecture for weights $\\leq p_n+1$, and using (\\ref{*}),\nwe can assume that $ {\\bar {\\rho} } $ is ordinary at $p$. \nConstruct a minimal weight 2 \nlift $\\rho$ of $ {\\bar {\\rho} } $ using Proposition \\ref{p}. (If $ {\\bar {\\rho} } $ is solvable we already know its modularity, so we may assume it is not, although we need not.) \nThis lift is unramified outside $p$ and minimal of weight 2 at $p$.\nUsing part (i) of Proposition \\ref{c} make it part of a compatible system $(\\rho_{\\lambda})$, so that for a place above $p$ fixed by the embedding $\\iota_p$ the corresponding representation is $\\rho$. \nNow reduce the system modulo a prime $\\lambda$ above \n$\\ell$ with $\\lambda$ determined by the embedding $\\iota_{\\ell}$. \nDenote the residual  mod $\\ell$ representaion by $ {\\bar {\\rho} } _\\ell$, and note that by part (i) of Proposition \\ref{c}, $k( {\\bar {\\rho} } _\\ell)=2$.\nIf $ {\\bar {\\rho} } _\\ell$ had solvable image we would be done. This is because by Lemma \\ref{dihedral} (i), \nin the case when the representation $ {\\bar {\\rho} } _\\ell$ restricted to $G_{\\mathbb{Q}(\\sqrt{{(-1)}^{\\ell-1 \\over 2}\\ell})}$ is reducible, \n$ {\\bar {\\rho} } _\\ell|_{D_\\ell}$ is ordinary (as $\\ell>2$ and hence $\\ell+3 \\over 2$$>2$), \nand in the cases when either this happens or $ {\\bar {\\rho} } _\\ell$ is reducible, and thus ordinary at $\\ell$, \n$ {\\bar {\\rho} } _\\ell$ restricted to $I_{\\ell}$ is distinguished.  As $\\ell$ is an odd prime, \nthis allows us to apply the modularity lifting theorems \nproved in \\cite{Wiles}, \\cite{TW}, \\cite{SW2} and \\cite{SW3}, to prove the modularity of \n$\\rho_\\lambda$ and hence of $ {\\bar {\\rho} } $.\nIf the mod $\\ell$ representation were unramified at \n$p$, using the result in \\cite{KW} for weight 2, we \nwould see that the representation is reducible and \nagain we would be done.\n\nOtherwise apply Proposition \\ref{q}\n(with $p$ of that proposition, the present $\\ell$, and the $q$ there the present $p$!) \nto choose a lifting $\\rho'$ of this residual mod $\\ell$ representation $ {\\bar {\\rho} } _\\ell$, unramified outside $\\ell,p$, \nBarsotti-Tate at $\\ell$,   and such  \nthat the nebentype at $p$ is $\\omega_p^j$  \nwith $j$ in the interval $({m \\over 2m+1} ({p-1}),{m+1 \\over 2m+1} ({p-1})]$\n(recall that we have set $\\ell^r=2m+1$).\nUsing part (ii) of Proposition \\ref{c}, make $\\rho'$ part of a compatible system $(\\rho'_{\\lambda})$ and consider a place above $p$\ndetermined by the embedding $\\iota_p$. We get a $p$-adic representation $\\rho''$ at that place \nthat is unramified outside $p$, and such that\nat $p$ the representation is Barsotti-Tate over (the degree $p-1 \\over 2$ subfield of) $\\mathbb{Q}_p(\\mu_p)$ by part (ii) of Proposition \\ref{c}. \n\nNow note that if the representation residually, say $ {\\bar {\\rho} } ''$, were reducible locally at $p$, then $\\rho''$ at $p$ itself would be ordinary up to twisting by \na power of $\\omega_p$ by \nLemma \\ref{breuil} above (which is a consequence of \nProposition 6.1.1 of \\cite{BM}, and Theorems 6.11  of \n \\cite{Savitt1}), and would also\nhave distinct characters on the diagonals when restricted to $I_p$ (as the weight is even), i.e., is $I_p$-distinguished. \n\nFurther if $ {\\bar {\\rho} } ''$ had solvable image, then  as by \n Lemma \\ref{dihedral} above we know that if $ {\\bar {\\rho} } ''$ is reducible when restricted to $G_{\\mathbb{Q}(\\sqrt{{(-1)}^{p-1 \\over 2}p})}$\nthen it's ordinary at $p$, and also $I_p$-distinguished, we can apply the modularity lifting theorems of \n\\cite{SW2},\\cite{SW3} and \\cite{Kisin} to conclude the modularity of $\\rho''$.\nThe main theorem of \\cite{Kisin}  \nis used in the non-ordinary case,\nwhen we invoke a modularity lifting result when the lift is potentially Barsotti-Tate at $p$ over a tamely ramified extension, in fact over $\\mathbb{Q}_p(\\mu_p)$, and the residual representation is irreducible when restricted to\n$G_{\\mathbb{Q}(\\sqrt{{(-1)}^{p-1 \\over 2}}p)}$.\n\n\nSo we can now assume that $ {\\bar {\\rho} } ''$ has non-solvable image. \nWe see by the last line of part (ii)  of Proposition \\ref{c},\nthat $k( {\\bar {\\rho} } '')$  is either $j+2$ or $k( {\\bar {\\rho} } '' \\otimes \\overline \\chi_p^{-j})=p+1-j$. Note that by our choice of $j$, $j+2$ is contained \nin in the interval $({ m \\over 2m+1} ({p-1})+2,({ m+1 \\over 2m+1}) ({p-1})+2]$ and $p+1-j$\nis in the interval\n$[p+1-(({ m+1 \\over 2m+1}) ({p-1})), p+1-(({ m \\over 2m+1}) ({p-1})))$.\n But (\\ref{*}) gives that both these intervals are contained in the interval $[2,p_n+1]$  as noted in Section \\ref{cheb}.\nHence we know the modularity of $ {\\bar {\\rho} } ''$ as we know by  hypothesis the modularity of irreducible, mod $p$, odd, 2-dimensional, unramified outside $p$ \nrepresentations of weights $\\leq p_n+1$.\nWe can now apply the modularity lifting theorems of \n\\cite{Kisin} to conclude that $\\rho''$ is modular.\n\nThus the compatible system $(\\rho_\\lambda')$ is modular. Note that $(\\rho_\\lambda)$ \nand $(\\rho_\\lambda')$ are linked at the prime above $\\ell$ determined by $\\iota_\\ell$, i.e., \nat the prime \nabove $\\ell$ fixed by $\\iota_\\ell$, the residual representations \narising from the 2 systems are isomorphic (and have non-solvable image). Thus applying the modularity lifting theorems which were the first ones to be proven, i.e., in \\cite{Wiles} and \\cite{TW}, \nwe conclude that $(\\rho_{\\lambda})$ is modular. Hence $ {\\bar {\\rho} } $ is modular of some weight $k$ and level, and then of weight $k( {\\bar {\\rho} } )$ and level 1 \nby \\cite{Ribet}, \\cite{Edix}.\n\n\\vspace{3mm}\n\n\\noindent{\\bf Remark:} It will be of interest to see, both in \\cite{KW} and the present paper, if the residually solvable cases can also be handled internally by the method of the papers themselves, rather than using the fact that Serre's conjecture is known for them.\n\nWe also remark that the proof does use an auxiliary  prime $\\ell \\neq p$, but no ramification is introduced at this prime, unlike in the \nmore traditional use of auxiliary primes. \n\n \n\\section{Corollaries}\\label{cors}\n\n\\subsection{Proof of corollaries}\n\nThe case of $q=2$ of Corollary \\ref{cond} is dealt with in Theorem 4.1(ii) \nof \\cite{KW}. (The case $p=3$ is excluded there, but is dealt with\nby the remarks about $p=3$ in Section \\ref{liftings}.) Note that in the case $q=2$ we are using results of the type proven in \\cite{Fontaine}, \\cite{BK}, \\cite{Schoof1}.\n\nCorollary \\ref{cond} when $q \\neq 2$ follows from Theorem \\ref{main} on using the method of ``killing ramification'' in Section 5.2 of \\cite{KW}. We give very briefly the proof.\nFirst apply the minimal lifting result of \\cite{KW}, Theorem 2.1 of Section 2,\nto construct first a minimal $p$-adic lift $\\rho$ of $ {\\bar {\\rho} } $ and then apply Theorem 3.1 of \n{\\it loc.\\ cit.\\ } to get a compatible system $(\\rho_\\lambda)$\nof which $\\rho$ is a part.\nConsider the prime above $q$ determined by $\\iota_q$ and reduce the representation which is a member of $(\\rho_\\lambda)$ \nat this prime (this is the method of ``killing ramification'' of Section 5.2 of \n\\cite{KW}). Residually we get a mod $q$ representation that is unramified outside $q$, for which we know Serre's conjecture by the main theorem of this paper. Then the lifting theorems recalled above, which may be applied because of  Lemma \\ref{dihedral} and Lemma \\ref{breuil} (the latter due to Breuil, M\\'ezard and Savitt), imply that the compatible system $(\\rho_\\lambda)$ is modular and hence so is $ {\\bar {\\rho} } $.\n\nCorollary \\ref{finite} follows as we know that a mod $p$ irreducible 2-dimensional representation\nof $G_\\mathbb{Q}$ that arises from $S_k(SL_2(\\mathbb{Z}))$ (for any integer $k\\geq     2$) also arises from $S_2(\\Gamma_1(p^2))$. \n\n\n\n\\subsection{Quantitative refinements?} \n\nCorollary \\ref{finite} is a folk-lore consequence of Serre's conjecture and \nis there implicitly at the end of Tate's  article \\cite{Tate}.\nIt will be of interest to get quantitative refinements of Corollary \\ref{finite}. \n\n After the corollary it is easy to see that for a fixed prime $p$ the number\n$N(2,p)$ of\nisomorphism classes of continuous semisimple\n odd representations $ {\\bar {\\rho} } :G_\\mathbb{Q} \\rightarrow GL_2(\\overline{\\mathbb{F}_{p}})$\n that are unramified outside $p$ is bounded by $Cp^3$ for a constant $C$ independent of $p$. This is seen as by the main theorem of the paper we are counting the number of distinct level 1 mod $p$ Hecke eigensystems, and then using formulas for the dimension of $S_k(SL_2(\\mathbb{Z}))$, and the fact that, up to twist by powers of $\\overline \\chi_p$, all mod $p$ forms have weight $\\leq p+1$, we are done. \n\nWe would guess that $N(2,p)$ is asymptotic to ${1 \\over {48}} p^3$ with $p$.\n\nSerre has pointed out that another easy corollary of Theorem \\ref{main} is that any  $ {\\bar {\\rho} } $  in Corollary \\ref{finite} can be\nwritten over a finite field  ${\\mathbb F}_{p^e}$  with  $e \\leq {\\rm sup}(1,{p+1 \\over 12})$.\n\n\\section{Some remarks}\n\nThe arguments in this paper were arrived at when thinking of how to extend the results in \\cite{KW} \nto prove Serre's conjecture in the level 1 case for \n(finitely many) more weights, upon Schoof telling us that he could prove that all semistable abelian varieties over $\\mathbb{Q}$ with good reduction outside 17 (resp., 19) are isogenous to powers of $J_0(17)$ (resp., $J_0(19)$). Using the method of \\cite{KW} this immediately proved modularity of $ {\\bar {\\rho} } $ of level 1 and weights 18 and 20. With further thought, we could do weights 10 and 16. The trick we came up with to do weight 16\nled to the method presented in this paper. \n\n\nThe trick to do weight 10, which we have not used in the\npresent paper, was as follows: it's enough to prove that a mod $11$ representation of weight 10 is modular, and enough to assume that it's ordinary at 11. Now we can get a minimal lifting of $ {\\bar {\\rho} } $ that is crystalline, and ordinary, of weight 20 at $11$ by using the arguments of proof of Theorem 2.1 of \\cite{KW}, and then putting it in a compatible system and \nreducing it mod $19$ we deduce modularity by the weights $2$ and $20$ results.\nThe trick for weight 16 is subsumed, and superseded (as we at first still made\nuse of Schoof's result for the prime 19), in the \nmethod of the paper. \n\nThe weaning away (after weight 6) \nfrom results classifying semistable abelian varieties \n$A$ with good reduction outside a prime $p$ was necessary, as such results can only be proven for small $p$. For large $p$ such $A$ may not be isogenous to products of $\\mathrm{GL}_2$-type abelian varieties, and even those of the latter type cannot be proved to be factors of $J_0(p)$  by arguments that use discriminant bounds. For a related reason the method of Tate cannot be used\nto prove Theorem \\ref{main} for large primes $p$ (perhaps $p=5$ is the limit\nof this method, even assuming the GRH: see \\cite{Tate} and the discussion in \\cite{Brue}).\n\nIn \\cite{KW} a possible, and even plausible, path to the general case of Serre's conjecture was mapped out\nin its last section. The method was to use induction on primes in 2 different ways. For the level 1 case (see Theorem 5.1 there), the induction was on the prime which was the residue characteristic, the starting point being the cases of the conjecture proved for $p=2,3$ by Tate and Serre. \nWe have used this method in the present paper.\nFor the reduction to the level 1 case (see Theorem 5.2 there; this is the method of ``killing ramification''),\nthe induction was on the number of primes ramified in the residual representation, the starting point being the level 1 case.  \n\nBut the path seemed to be blocked by \nformidable obstacles. This paper reaches \nthe level 1 case by sidestepping (and not overcoming any of) these technical obstacles. The method here, combined \nwith the proposed method of reduction of the general case \nto the level 1 case in the last section of \\cite{KW}, will probably be useful for general\nlevel $N$, and ease some of the technical difficulties. \n\n\n\n\n\n\n\\vspace{5mm}\n\n\\noindent{\\bf Acknowledgements:} We are grateful to Ren\\'e Schoof for \ntelling us about his unpublished results \nfor semistable abelian varieties over $\\mathbb{Q}$ with good reduction outside 17 and 19: although in this paper we have \nnot directly made use of these, our attempt to apply them to Serre's conjecture led to the method of the paper.\nWe would like to thank Gebhard B\\\"ockle for very helpful correspondence about Section 2 of the paper. \nWe are grateful to Gebhard B\\\"ockle, Kevin Buzzard, Jean-Pierre Serre and Jean-Pierre Wintenberger for very helpful comments on the mansucript. We thank Christophe Breuil for pointing out the reference \\cite{Savitt1}. We thank Jean-Pierre Serre for telling us about the history of his conjecture.  The author was partially supported\nby NSF Grant DMS - 0355528.\n\n\\nocite{*}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:introduction}\nGeneral matrix multiplication(GEMM), as one of the most important numerical algorithm in dense linear algebra, has been exhaustively studied over the years\\cite{goto2008anatomy, wang2013augem, van2015blis}. Many famous BLAS(Basic Linear Algebra Subprograms) libraries, like Intel MKL\\cite{intel}, OpenBLAS\\cite{wang2013augem}, BLIS\\cite{van2015blis}, and ARMPL\\cite{armpl}, already implemented high-performance GEMM. GEMM is used to compute $C=\\alpha A\\times B + \\beta C$. Here $C$, $A$, $B$ are $M\\times N$, $M\\times K$, and $K \\times N$ matrices, respectively.\n\nIn recent years, small GEMM becomes more and more important in many fields, such as machine learning\\cite{hinton2018matrix}, sparse matrix\\cite{borvstnik2014sparse}, and fluid dynamics\\cite{wozniak2016gimmik}. Many CNNs algorithms use the small matrix on their fully connected layers\\cite{CNNinFClayer, wang2021high}. \nCaffe \\cite{caffe} is a famous deep learning framework. \nComparing to BLIS that has not optimized small GEMM, the performance of Caffe utilized optimized BLIS can obtain a performance improvement of 17\\%\\cite{2017AcceleratingMLbyBLIS}.\nBy using the optimized implementation of small GEMM, Geoffrey Hinton reduced the number of parameters by a factor of 15 to 310K compared with their baseline CNN with 4.2M parameters \\cite{hinton2018matrix}.\nIn this paper, we define small GEMM as, $\\sqrt[3]{MNK}\\le 80$, when transposition of input matrices is not TN(TN will be explained in TABLE \\ref{tab:ALL GENERATED KERNELS}), or $\\sqrt[3]{MNK}\\le 32$ when transposition of input matrices is TN. This definition will be explained in Section \\ref{sec:performance}.\n\nTraditional implementation and optimization methods of GEMM mainly have three steps: block step, pack step and compute step. Block step tiles matrices into a series of small blocks based on features of the hardware, e.g., TLB, size of the L2 cache. Pack step packs these small blocks based on kernel size to ensure continuity of memory access during kernel calculation. Compute step uses one high-performance kernel with boundary processing to compute matrix multiplication. Because input matrices are relatively large, pack step can massively reduce cache miss and TLB miss, and costs of boundary processing can be neglected.\n\nHowever, traditional implementation and optimization methods of GEMM, as described above, cannot achieve optimal performance for small GEMM. Here are two reasons for this. First, the overhead of pack step in small GEMM is too high, as shown in Section \\ref{sec:performance}. The advantages of pack step are no longer significant, but it results in high extra memory access overhead. Second, the costs of boundary processing are not neglected for small GEMM. Therefore, designing and implementing a method without pack steps and boundary processing is very necessary for achieving high performance of small GEMM.\n\nThis paper proposes an input-aware adaptive tuning framework(IAAT) for small GEMM to achieve near-optimal performance. IAAT has two stages, the install-time stage and the run-time stage. The install-time stage is responsible for auto-generating high-performance assembly kernels of different sizes. This stage automatically tunes kernels based on features of hardware to achieve optimal performance. The run-time stage's core is the input-aware adaptive tile algorithm, which tiles input matrices into some small blocks. This stage plays the role of runtime tuning by tiling matrix during program execution. Our performance evaluation shows that IAAT can achieve near-optimal performance when the size of input matrices is small as shown in Section \\ref{sec:performance}.\n\nOur contributions are summarized as follows:\n\\begin{itemize}\n    \\item We propose a template-based high-performance code auto-generation method to generate high-performance kernels for GEMM of different sizes in assembly language.\n    \\item We design an input-aware adaptive algorithm to divide input matrices into blocks in runtime to obtain a near-optimal solution. \n    \\item We implement a high-performance input-aware adaptive tuning framework(IAAT) for small GEMM based on ARMv8.\n\\end{itemize}\n\nThe remainder of this paper is organized as follows. Section \\ref{sec:relatedwork} presents related works. Section \\ref{sec:framework} provides an overview of the framework. Section \\ref{sec:installtime} and Section \\ref{sec:runtime} introduces the details of two stages of IATT. Section \\ref{sec:performance} presents the experimental results. Finally, Section \\ref{sec:conclusions} concludes the paper.\n\n\\section{Related Works}\n\\label{sec:relatedwork}\nMatrix multiplication has been optimized over years. Researchers utilize different methods and technologies to improve various matrix multiplication, such as tall and skinny matrix multiplication\\cite{chen2019tsm2, rivera2021tsm2x, li2021autotsmm}, batches of matrix multiplication\\cite{abdelfattah2020matrix,abdelfattah2019fast}, parallel matrix multiplication\\cite{kang2020hpmax} and so on. \nFor example,  tall and skinny matrix multiplication kernels are optimized by a flexible, configurable mapping scheme and outperform on an NVIDIA Volta GPGPU\\cite{ernst2021performance}.\nLionel Eyraud-Dubois\\cite{eyraud2018using} uses more general allocations to perform matrix multiplication on a heterogeneous node based on task-based runtime systems. \n\nSmall GEMM are becoming more and more important in recent years. The optimization of small GEMM is introduced by many libraries, like LIBXSMM\\cite{libxsmm}, BLIS\\cite{blislib, blis2019}. LIBXSMM uses a code generator that has a built-in architectural model to auto-generate code. And the code runs well without requiring an auto-tuning phase by utilizing just-in-time compilation. BLIS uses the method of optimizing skinny matrix to optimize the small matrix and works well\\cite{blis2019}. \n\nHowever, current methods and implementations of small GEMM cannot achieve near-optimal performance on ARMv8 platform. LIBXSMM and BLIS only focused on x86 CPU. Besides, BLIS tile algorithm cannot improve the performance of small GEMM to optimal performance of small GEMM\\cite{blis2019}. And BLIS only implemented the small GEMM for single-precision and double-precision but not single-precision complex and double-precision complex. Distinguish from LIBXSMM and BLIS, we optimize all types of small GEMM for ARMv8 platform.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig\/codeframework.pdf}\n\\caption{The overall IAAT}\n\\label{fig:code framework}\n\\end{figure}\n\n\\section{Framework}\n\\label{sec:framework}\nThis section introduces the input-aware adaptive tuning framework(IAAT), as shown in Fig.\\ref{fig:code framework}, with two stages, the install-time stage and the run-time stage to achieve near-optimal performance for small GEMM.\n\n\\subsection{The Install-Time Stage}\n\\label{subsec:framework_installtimestage}\nThe install-time stage auto-generates hundreds of kernels of different sizes. \nPack step of the traditional method of GEMM makes data access continuous. So the traditional method of GEMM only needs one kernel to accomplish computation for different transpositions. After removing pack step, we have to use hundreds of kernels for different matrix sizes, types and transpositions. These kernels need lots of work to write by hand. Therefore, IAAT uses auto-generation to generate high-performance kernels in the install-time stage. This stage automatically tunes kernels based on features of the hardware to achieve optimal performance. The install-time stage utilizes four components to generate kernels:\n\n\\begin{itemize}\n\\item \\textbf{Computational Template Designer} abstracts typical computing patterns of matrix multiplication as templates.\n\\item \\textbf{Kernel Generator} designs a kernel generation algorithm, which utilizes templates from compute template designer to generate basic kernels of different sizes. \n\\item \\textbf{Register Allocator} allocates SIMD registers for kernels based on the size of kernel and SIMD register features.\n\\item \\textbf{Kernel Optimizer} optimizes kernels from kernel generator to approach full potential power of the hardware.\n\\end{itemize}\n\n\\begin{table}[htbp]\n\\caption{ALL GENERATED KERNELS}\n\\footnotesize\n\\begin{center}\n\\setlength\\tabcolsep{4pt}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n & \\textbf{NN} & \\textbf{NT} & \\textbf{TN} & \\textbf{TT} \\\\ \\hline\n\\textbf{\\rotatebox[origin=c]{270}{SGEMM}} & \\begin{tabular}[c]{@{}l@{}}16$\\times$\\{1,2,3,4\\}\\\\ 12$\\times$\\{1,2,...,6\\}\\\\ 8$\\times$\\{1,2,...,8\\}\\\\ 4$\\times$\\{1,2,...,13\\}\\\\ 3$\\times$\\{1,2,...,13\\}\\\\ 2$\\times$\\{1,2,...,13\\}\\\\ 1$\\times$\\{1,2,...,13\\}\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}16$\\times$\\{1,2,3,4\\}\\\\ 12$\\times$\\{1,2,...,8\\}\\\\ 8$\\times$\\{1,2,...,8\\}\\\\ 4$\\times$\\{1,2,...,20\\}\\\\ 3$\\times$\\{1,2,...,24\\}\\\\ 2$\\times$\\{1,2,...,28\\}\\\\ 1$\\times$\\{1,2,...,32\\}\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,3,4\\}\\\\ 3$\\times$\\{1,2,3,4,5\\}\\\\ 2$\\times$\\{1,2,...,7\\}\\\\ 1$\\times$\\{1,2,...,10\\}\\end{tabular}                    & \\begin{tabular}[c]{@{}l@{}}\\{1,2,3,4\\}$\\times$16\\\\ \\{1,2,...,6\\}$\\times$12\\\\ \\{1,2,...,8\\}$\\times$8\\\\ \\{1,2,...,13\\}$\\times$4\\\\ \\{1,2,...,13\\}$\\times$3\\\\ \\{1,2,...,13\\}$\\times$2\\\\ \\{1,2,...,13\\}$\\times$1\\end{tabular} \\\\ \\hline\n\\textbf{\\rotatebox[origin=c]{270}{DGEMM}} & \\begin{tabular}[c]{@{}l@{}}8$\\times$\\{1,2,3,4\\}\\\\ 4$\\times$\\{1,2,...,8\\}\\\\ 3$\\times$\\{1,2,...,8\\}\\\\ 2$\\times$\\{1,2,...,15\\}\\\\ 1$\\times$\\{1,2,...,15\\}\\end{tabular}                                         & \\begin{tabular}[c]{@{}l@{}}8$\\times$\\{1,2,3,4\\}\\\\ 4$\\times$\\{1,2,...,8\\}\\\\ 3$\\times$\\{1,2,...,8\\}\\\\ 2$\\times$\\{1,2,...,20\\}\\\\ 1$\\times$\\{1,2,...,20\\}\\end{tabular}                                         & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,3,4\\}\\\\ 3$\\times$\\{1,2,3,4,5\\}\\\\ 2$\\times$\\{1,2,...,7\\}\\\\ 1$\\times$\\{1,2,...,10\\}\\end{tabular}                    & \\begin{tabular}[c]{@{}l@{}}\\{1,2,3,4\\}$\\times$8\\\\ \\{1,2,...,8\\}$\\times$4\\\\ \\{1,2,...,8\\}$\\times$3\\\\ \\{1,2,...,15\\}$\\times$2\\\\ \\{1,2,...,15\\}$\\times$1\\end{tabular}                     \\\\ \\hline\n\\textbf{\\rotatebox[origin=c]{270}{CGEMM}} & \\begin{tabular}[c]{@{}l@{}}8$\\times$\\{1,2,3,4\\}\\\\ 4$\\times$\\{1,2,...,9\\}\\\\ 3$\\times$\\{1,2,...,9\\}\\\\ 2$\\times$\\{1,2,...,12\\}\\\\ 1$\\times$\\{1,2,...,20\\}\\end{tabular}                                         & \\begin{tabular}[c]{@{}l@{}}8$\\times$\\{1,2,3,4\\}\\\\ 4$\\times$\\{1,2,...,8\\}\\\\ 3$\\times$\\{1,2,...,8\\}\\\\ 2$\\times$\\{1,2,...,12\\}\\\\ 1$\\times$\\{1,2,...,20\\}\\end{tabular}                                         & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,...,9\\}\\\\ 3$\\times$\\{1,2,...,9\\}\\\\ 2$\\times$\\{1,2,...,12\\}\\\\ 1$\\times$\\{1,2,...,20\\}\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}\\{1,2,3,4\\}$\\times$8\\\\ \\{1,2,...,9\\}$\\times$4\\\\ \\{1,2,...,9\\}$\\times$3\\\\ \\{1,2,...,12\\}$\\times$2\\\\ \\{1,2,...,20\\}$\\times$1\\end{tabular}                      \\\\ \\hline\n\\textbf{\\rotatebox[origin=c]{270}{ZGEMM}} & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,3,4\\}\\\\ 3$\\times$\\{1,2,3,4\\}\\\\ 2$\\times$\\{1,2,...,7\\}\\\\ 1$\\times$\\{1,2,...,10\\}\\end{tabular}                                                              & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,3,4\\}\\\\ 3$\\times$\\{1,2,3,4\\}\\\\ 2$\\times$\\{1,2,...,7\\}\\\\ 1$\\times$\\{1,2,...,10\\}\\end{tabular}                                                              & \\begin{tabular}[c]{@{}l@{}}4$\\times$\\{1,2,3,4\\}\\\\ 3$\\times$\\{1,2,3,4\\}\\\\ 2$\\times$\\{1,2,...,7\\}\\\\ 1$\\times$\\{1,2,...,10\\}\\end{tabular}                      & \\begin{tabular}[c]{@{}l@{}}\\{1,2,3,4\\}$\\times$4\\\\ \\{1,2,3,4\\}$\\times$3\\\\ \\{1,2,...,7\\}$\\times$2\\\\ \\{1,2,...,10\\}$\\times$1\\end{tabular}                                           \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\footnotesize\nWe define the kernel for different matrix types and different transpositions. The SGEMM\/DGEMM\/CGEMM\/ZGEMM represent single-precision matrix multiplication, double-precision matrix multiplication, single-precision complex matrix multiplication, double-precision complex matrix multiplication. Each type has four transpositions, NN, NT, TN, and TT. For example, NT means matrix A is not transposed and matrix B is transposed. We will also use abbreviations like SGEMM\\_TN, which means input matrix type is single and matrix A is transposed and matrix B isn't transposed. We acquiescence the matrix in column-major order.\n\\label{tab:ALL GENERATED KERNELS}\n\\end{table}\n\nTABLE \\ref{tab:ALL GENERATED KERNELS} shows all kernels we defined in this paper, which are completely auto-generated. All these kernels construct basic computation of small GEMM and form a kernel array, which is directly invoked by the run-time stage.\n\n\\subsection{The Run-Time Stage}\nThe run-time stage tiles input matrices A, B, and C into blocks and generates a near-optimal small GEMM kernel executing plan.\nThe costs of boundary processing can be neglected for GEMM. However, for small GEMM, the costs of boundary processing are high and cannot be neglected. To reduce or eliminate boundary processing, an algorithm is required, which can tile input matrices into optimal blocks with less boundary processing. The core of the run-time stage is the input-aware adaptive tile algorithm. This algorithm tiles input matrices into optimal blocks according to the size of kernels from the install-time stage. These blocks are tuned according to input matrix sizes, types and transpositions. Therefore, the run-time stage plays the role of runtime tuning. Then this stage connects the kernel to form a sequence of kernels, which is called the kernel executing plan. Finally, IAAT computes the small GEMM based on this kernel executing plan.\n \n\\section{The Install-Time Stage}\n\\label{sec:installtime}\nThis section focuses on the install-time stage, which auto-generates hundreds of kernels of different sizes. Below we introduce four components of the install-time stage as shown in Fig.\\ref{fig:code framework}.\n\n\\subsection{Computational Template Designer}\nTo construct main calculation of GEMM kernel, we introduce computational template designer. The computational template designer extracts typical computing patterns of matrix multiplication as templates, which are shown in TABLE \\ref{tab:The Kernel Computational Templates}.\n\\begin{itemize}\n\\item \\textbf{sfmlas} and \\textbf{dfmlas} represent a vector-scalar multiply-add operation.\n\\item \\textbf{sfmlav} and \\textbf{dfmlav} represent a vector-vector multiply-add operation.\n\\item \\textbf{sfmlss} and \\textbf{dfmlss} represent a vector-scalar multiplication and subtraction.\n\\item \\textbf{sfnegv} and \\textbf{dfnegv} are used to invert values in register.\n\\item \\textbf{sfcmlas} and \\textbf{dfcmlas} represent a vector-scalar complex multiply-add operation.\n\\item \\textbf{sfcmlav} and \\textbf{dfcmlav} represent a vector-vector complex multiply-add operation.\n\\end{itemize}\n\n\\begin{table}[htbp]\n\\caption{Kernel Computational Templates}\n\\scriptsize\n\\begin{center}\n\\begin{tabular}{|l|l|}\n\\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfmlas}(out, in1, in2, index):\\\\\\  \nfmla out.4s, in1.4s, in2.s{[}index{]} \n\\end{tabular}                        & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfmlas}(out, in1, in2, index):\\\\\\  \nfmla out.2d, in1.2d,  in2.d{[}index{]}\n\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfmlav}(out, in1, in2):\\\\\\  \nfmla out.4s, in1.4s, in2.4s\n\\end{tabular}                       & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfmlav}(out, in1, in2):\\\\\\  \nfmls out.2d, in1.2d,  in2.2d\n\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfmlss}(out, in1, in2, index):\\\\\\  \nfmls out.4s, in1.4s, in2.s{[}index{]}\n\\end{tabular}                       & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfmlss}(out, in1, in2, index):\\\\\\  \nfmla out.2d, in1.2d, in2.d{[}index{]}\n\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfnegv}(out, in1):\\\\\\ \nfneg out.4s, in1.4s\n\\end{tabular}                       & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfnegv}(out, in1):\\\\\\  \nfneg out.2d, in1.2d\n\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfcmlas}(out, in1, in2, index, rot[2]):\\\\\\  \nfcmla out.4s, in1.4s,  in2.s{[}index{]}, rot[0] \\\\\\ \nfcmla out.4s, in1.4s,  in2.s{[}index{]}, rot[1]\n\\end{tabular}                       & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfcmlas}(out, in1, in2, rot[2]):\\\\\\  \nfcmla out.2d, in1.2d,  in2.2d, rot[0] \\\\\\ \nfcmla out.2d, in1.2d,  in2.2d, rot[1]\n\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}\\textbf{sfcmlav}(out, in1, in2, rot[2]):\\\\\\  \nfcmla out.4s, in1.4s,  in2.4s, rot[0] \\\\\\ \nfcmla out.4s, in1.4s,  in2.4s, rot[1]\n\\end{tabular}                       & \n\\begin{tabular}[c]{@{}l@{}}\\textbf{dfcmlav}(out, in1, in2, rot[2]):\\\\\\  \nfcmla out.2d, in1.2d,  in2.2d, rot[0] \\\\\\ \nfcmla out.2d, in1.2d,  in2.2d, rot[1]\n\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\label{tab:The Kernel Computational Templates}\n\\end{center}\n\\end{table}\n\n\\subsection{Kernel Generator}\n\\label{subsec:kernel generator}\nKernel generator is responsible for generating kernels. These kernels are used to compute $C_c=A_c\\times B_c+Cc$. Here $A_c$, $B_c$, and $C_c$ are blocks of input matrices A, B, and C. And they are $m_c \\times k_c$,  $k_c \\times n_c$, and $m_c \\times n_c$ matrices, respectively. The algorithm of kernel generator takes size of $C_c$ as input and outputs high-performance kernel in assembly language.\n\nKernel generator generates two kinds of subkernels for ping-pang operation. The ping-pang operation is an optimization method that split the multiplication into two stages, M1 and M2 stages. There are two types of ping-pang operations. In the first type, each stage of ping-pang operation multiplies a column of block $A_c$ and a row of block $B_c$ and loads the next column of block $A_c$ and next row of block $B_c$. In the second type, each stage multiplies a column of block $A_c$ and a row of block $B_c$, M1 stage loads the next column of block $A_c$ and two rows of block $B_c$, and M2 stage loads the next column of block $A_c$. And the performance difference between these two types is not too much.\n\nThe kernel generator algorithms for various input matrix types and transpositions are similar. We only discuss SGEMM\\_NN kernel generator shown in Algorithm \\ref{alg:kernel generation of SGEMMNN}.\n\nSGEMM\\_NN kernel generator generates two subkernels in lines 6-12 and 14-19. The first subkernel loads a column of $A_c$ and two rows of $B_c$ in lines 6-7 and the second subkernel loads a column of $A_c$ in line 14. Each subkernel multiplies a column of $A_c$ and a row of $B_c$ by utilizing \\textbf{sfmlas} in lines 8-12 and 15-19.\n\nAfter two kinds of subkernels of SGEMM\\_NN are generated, the kernel generator invokes these two subkernels in a loop on the $k_c$ dimension and completes the generation of SGEMM\\_NN kernel.\n\n\\begin{algorithm}[h]\n\\caption{kernel generator of SGEMM\\_NN}\n\\label{alg:kernel generation of SGEMMNN}\n\\begin{algorithmic}[1]\n\t    \\REQUIRE $m_c$, $n_c$: the size of the input kernel\n\t    \\ENSURE $kernel$\n\t    \\STATE $Cregs \\gets \\{C_1, C_2,..., C_{m_{\\left \\lceil m_c\/4 \\right \\rceil}n_c}\\}$\n\t    \\STATE $A1regs \\gets \\{A_1, A_2,..., A_{\\left \\lceil m_c\/4 \\right \\rceil}\\}$\n\t    \\STATE $A2regs \\gets \\{A_{\\left \\lceil m_c\/4 \\right \\rceil+1},$ $A_{\\left \\lceil m_c\/4 \\right \\rceil+2},$ $...,$ $A_{2\\left \\lceil m_c\/4 \\right \\rceil}\\}$\n\t    \\STATE $Bregs \\gets \\{B_1, B_2,..., B_{n_c}\\}$\n\t    \\STATE \/\/first subkernel\n\t    \\STATE load next column of block $A_c$ to $A2regs$\n\t    \\STATE load two rows of block $B_c$ to $Bregs$\n\t    \\FOR{$i$ $\\gets$ $0$ $to$ $n_c$}\n\t        \\FOR{$j$ $\\gets$ $0$ $to$ $\\left \\lceil m_c\/4 \\right \\rceil$}\n\t            \\STATE \\textbf{sfmlas}($Cregs{[}i\\left \\lceil m_c\/4 \\right \\rceil+j{]}$, $A1regs{[}j{]}$, $Bregs{[}i{]}$, 0)\n\t        \\ENDFOR\n\t    \\ENDFOR\n\t    \\STATE \/\/second subkernel\n\t    \\STATE load next column of block $A_c$ to $A1regs$\n\t    \\FOR{$i$ $\\gets$ $0$ $to$ $n_c$}\n\t        \\FOR{$j$ $\\gets$ $0$ $to$ $\\left \\lceil m_c\/4 \\right \\rceil$}\n\t            \\STATE \\textbf{sfmlas}($Cregs{[}i\\left \\lceil m_c\/4 \\right \\rceil+j{]}$, $A2regs{[}j{]}$, $Bregs{[}i{]}$, 1)\n\t        \\ENDFOR\n\t    \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Register Allocator}\n\\label{subsec:register allocator}\nThe allocation of registers is very important for the performance of small GEMM. Hence, we need to define the strategies of register allocation for different kernels. The work of our paper is mainly carried out on ARMv8 platform, which contains 32 128-bit SIMD registers.\n\nThe basic idea behind the register allocator is to divide all registers into three groups. $A_c$ register group contains two columns of $A_c$; $B_c$ register group contains two rows of $B_c$ for ping-pang operation; $C_c$ register group holds the whole block $C_c$.\n\nAllocation of the $A_c$ register group has four main strategies, ANTwoCC, ATEachCTwo, ATEachCOne, and ATTwoRR. \n\\begin{itemize}\n    \\item \\textbf{ANTwoCC} is for loading two columns of $A_c$ to registers. It allocates $2\\left \\lceil m_c\/elenum \\right \\rceil$ registers, the $elenum$ means the number of elements that a register can store.\n    \\item \\textbf{ATEachCTwo} is for loading first two data of each column of transposed $A_c$ to two registers. It allocates a total of $2m_c$ registers.\n    \\item \\textbf{ATEachCOne} is for loading first two data of each column of transposed $A_c$ to one register. It requires a total of $m_c$ registers for single-precision, double-precision, and single-precision complex. As for double-precision complex, it requires a total of $2m_c$ registers. \n    \\item \\textbf{ATTwoRR} is for loading two rows of transposed $A_c$ to registers. It allocates $2\\left \\lceil m_c\/elenum \\right \\rceil$ registers.\n\\end{itemize}\n\nThe strategies of allocating $B_c$ register group are BTTwoCC, BNEachCTwo, BNEachCOne, and BNTwoRR corresponding to ANTwoCC, ATEachCTwo, ATEachCOne, and ATTwoRR. This is because load methods of $A_c$ are the same as load methods of $B_c$.\n\nThe strategy of allocating the C register group is allocating $\\left \\lceil m_c\\times n_c\/elenum \\right \\rceil$ registers.\n\nThe register allocator has one special strategy for TN transposition that allocates $2m_c$ registers for $A_c$ and $2n_c$ registers for $B_c$. This transposition makes memory access to $A_c$ and $B_c$ discontinuous. So we cannot vectorize small GEMM for this transposition. Therefore, the methods of loading data are load data from each column of $A_c$ by columns and load data from each column of $B_c$ by columns.\n\n\\subsection{Kernel Optimizer}\nAfter kernels are generated, kernels will be optimized as follows.\n\n\\paragraph{Instruction Choice} Computational template designer utilizes the FMA instruction instead of $mul$ or $add$ because there usually are fused multiply-add(FMA) units in hardware. Besides, we prioritize the $ldp$ and $ldr$ instructions because these two instructions are relatively high-performance.\n\n\\paragraph{Instruction Order} The loading instructions are interspersed among the computing instructions. It makes better use of the instruction pipeline to avoid pipeline stalling.\n\n\\paragraph{Ping-Pang Operation} As described in Subsection \\ref{subsec:kernel generator}, this optimization utilizes computing instruction to hide the delay of loading instructions.\n\n\\section{The Run-Time Stage}\n\\label{sec:runtime}\nThis section introduces the run-time stage. This stage first tile input matrices and then construct a kernel executing plan to compute small GEMM. The input-aware adaptive tile algorithm is the core of this stage.\n\n\\subsection{Input-Aware Adaptive Tile Algorithm}\nThe input-aware adaptive tile algorithm first tiles input matrix C into some small blocks. Each block has the same size as one of the generated kernels. Then, this algorithm tiles matrices A and B based on tiled blocks of C. This algorithm is based on the three principles listed below.\n\n\\begin{algorithm}[h]\n\t\\caption{SGEMM\\_NN Tile Algorithm}\n\t\\label{alg:SGEMMNNtileAlg}\n\t\\footnotesize\n\t\\begin{algorithmic}[1]\n\t    \\REQUIRE $M,N,K$: the sizes of input matrices, $kernels$: array of all sorted SGEMM\\_NN kernels from TABLE \\ref{tab:ALL GENERATED KERNELS}\n\t    \\ENSURE $blocksC$[], $blocksA$[], $blocksB$[]\n\t    \\IF {$N \\le 13$}\n\t        \\STATE $m[] \\gets (m_1, I)$, $m_1$ is the largest $m_c$ of kernel that's $n_c$ is equal $N$ and $I$ is an integer and make sure the $m_1I \\le M$\n\t        \\STATE $n[] \\gets [(N, 1)]$\n            \\IF{$m_1I < M$}\n                \\STATE $m$.append($(M-m_1I, 1)$)\n                \\STATE $n$.append($[(N, 1)]$)\n            \\ENDIF\n        \\ELSE\n            \\IF{$M < 8$}\n                \\STATE $m[] \\gets TileSingleDim(M, [1,2,3,4])$\n                \\STATE $n[] \\gets [TileSingleDim(N, [1,2,...,13])]$\n                \\IF{size of $m$ == 2}\n                    \\STATE $n.append([TileSingleDim(N, [1,2,...,13])])$\n                \\ENDIF\n            \\ELSIF{$M == 9$}\n                \\STATE $m[] \\gets (4, 1), (3, 1), (2, 1)$\n                \\STATE $n[] \\gets [TileSingleDim(N, [1,2,...,13])],$ $[TileSingleDim$ $(N,$ $[1,2,...,13])],$ $[TileSingleDim(N,$ $ [1,2,...,13])]$\n            \\ELSIF{$M < 12$}\n                \\STATE $m[] \\gets (8, 1), (M - 8, 1)$\n                \\STATE $n[] \\gets [TileSingleDim(N,[1,2,...,8])],$ $[TileSingleDim$ $(N,$ $[1,2,...,13])]$\n            \\ELSIF{$M == 12$}\n                \\STATE $m[] \\gets (12, 1)$\n                \\STATE $n[] \\gets [TileSingleDim(N, [1,2,3,4,5,6])]$\n            \\ELSE\n                \\STATE $m_1[] \\gets (4, \\lfloor M\/4 \\rfloor)$\n                \\STATE $m_2[] \\gets (M-4\\lfloor M\/4 \\rfloor, 1)$\n                \\STATE $n_2[] \\gets [TileSingleDim(N, [1,2,...,13])]$\n                \\IF{$M-4\\lfloor M\/4 \\rfloor == 1$}\n                    \\STATE $m_1[] \\gets (4, \\lfloor M\/4 \\rfloor-1)$\n                    \\STATE $m_2[] \\gets (3, 1), (2, 1)$\n                    \\STATE $n_2[] \\gets$ $[TileSingleDim(N,$ $[1,2,...,8])],$ $[TileSingleDim$ $(N,$ $[1,2,...,13])]$\n                \\ENDIF\n                \\STATE $m8\\gets ExtendTo8(m_1)$\n                \\STATE $m16\\gets ExtendTo16(m_1)$\n                \\STATE $n8[]$ and $n16[]$ $\\gets$ tile $N$ by $m8[]$ and $m16[]$\n                \\STATE $blocksC1\\gets Combine(m8, n8)$\n                \\STATE $blocksC2\\gets Combine(m16, n16)$\n                \\STATE $blocksC\\gets CompareLessMemops(blocksC1, blocksC2)$\n                \\STATE $blocksC$.append(combine($m_2$, $n_2$))\n                \\STATE return\n            \\ENDIF\n\t    \\ENDIF\n\t    \\STATE $blocksC \\gets Combine(m, n)$\n\t    \\STATE $blocksA[] \\gets$ tile matrix A according to $blocksC[]$\n\t    \\STATE $blocksB[] \\gets$ tile matrix B according to $blocksC[]$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\paragraph{Bigger Block Size} Smaller blocks cause matrices A and B to be repeatedly loaded more times. The larger the block size, the lower the number of repetitions.\n\n\\paragraph{Minimal Memops} Different tiling methods have the same amount of computing instructions but different numbers of loading instructions. Therefore, the optimal tiling method is tiling matrices into blocks with the fewest loading instructions. The tiled blocks for matrix C are supposed to be $m_0 \\times n_0$, $m_1 \\times n_1$, ..., $m_i \\times n_i$. The $m_i \\times n_i$ is size of tiled block. This tiling method have a total of $(m_0+n_0+m_1+n_1...+m_a+n_a)K+2mn$ data to access from L2 cache to register. So the value of $(m_0+n_0+m_1+n_1...+m_a+n_a)$ should be preserved to a bare minimum.\n\n\\paragraph{SIMD Friendly} The dimension of block, that data is continuous, can be divisible by the length of SIMD register.\n\nThe pseudo-code of SGEMM\\_NN tile algorithm are shown in Algorithm \\ref{alg:SGEMMNNtileAlg}. The outline of this algorithm is below:\n\nWhen $N \\le 13$, we let $n_c=N$ and make $m_c$ the maximum value that $m_c$ can be taken in lines 1-7. When $N > 13$, we first tile $M$ into multiples of 4 and use 1, 2, 3 to supplement the deficiency, and\nthen tile $N$ into maximum value that $n_c$ can be taken according to the result of $M$'s tile in lines 9-42. Besides, when $M > 12$, $M$ can be tiled by 8 or 16 and we compare which one is better by counting the number of loading instructions and choose that. Then, we combine the two tiled dimensions $m[]$ and $n[]$ into $blocksC$. Finally, we tile matrices A and B into $blocksA$ and $blocksB$ according to blocks of matrix C.\n\n$TileSingleDim$ algorithm, as shown in line 10, is for tiling a single dimension. It takes two input parameters: the length that you want to tile, and the array of lengths that you used to tile. This algorithm outputs array $(dim, nums)$ means $dim$ is repeated $nums$ times. We tile $dim$ into $nums_1I+nums_2...+nums_i$ and the bigger $nums_1$, the better. And if $nums_i$ is too small, this algorithm will average $nums_{i-1}$ and $nums_i$.\n\nFor various types and transpositions, the specific tile algorithm is changed slightly. But the basic ideas are consistent as shown above.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[traditional tiling method]{\n\\includegraphics{fig\/traditionalkernelallocationsamplegraph.pdf}\n\\label{fig:fig1}\n}\n\\quad\n\\subfigure[new tiling method]{\n\\includegraphics{fig\/newkernelallocationsamplegraph.pdf}\n\\label{fig:fig2}\n}\n\\caption{Schematic sketch of tiling method for a $15 \\times 15$ SGEMMNN matrix}\n\\end{figure}\n\nFor SGEMM\\_NN $15\\times 15\\times K$ matrix, the traditional tiling method is showed in Figure \\ref{fig:fig1}. This method needs to load $105k+450$ data from L2 cache to register. And, our method tile SGEMM\\_NN is showed in Figure \\ref{fig:fig2}. This tiling method needs to load $72K+450$ data. The amount of data loaded by the traditional method is 45\\% more than that of our method.\n\n\\subsection{Kernel Executing Plan}\nAfter input matrices are tiled, IAAT constructs a kernel executing plan by connecting kernels, which correspond to the sizes of tiled blocks. Finally, IAAT executes this plan to compute small GEMM.\n\n\\section{Performance Evaluation}\n\\label{sec:performance}\nIn this section, we analyze small GEMM's performance on ARMv8 platform as listed in TABLE \\ref{tab:Experimental Environment Of ARMv8 platform}. We compared IAAT with currently state-of-the-art BLAS libraries: OpenBLAS, ARMPL, and BLIS. GEMM in these libraries is well optimized. Our work supports four data types: single-precision, double-precision, single-precision complex, and double-precision complex. Each data type supports four transpositions: NN, NT, TN, TT. Thus, we compared 16 kinds of small GEMMs. We use Equation \\ref{equ:sd} to evaluate performance of SGEMM and DGEMM and Equation \\ref{equ:cz} to evaluate performance of CGEMM and ZGEMM.\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Experimental Environment Of ARMv8 platform}\n\\renewcommand{\\arraystretch}{1.2}\n\\setlength{\\tabcolsep}{10pt}\n\\begin{tabular}{lcc}\n\\hline\n\\multicolumn{1}{l}{Hardware} & CPU      & Kunpeng920 \\\\\n\\hline\n&Arch.    & ARMv8.2    \\\\\n&Freq.    & 2.6GHz     \\\\\n&SIMD     & 128bits    \\\\\n&L1 cache & 4MiB       \\\\\n&L2 cache & 32MiB      \\\\\n\\hline\n\\multicolumn{1}{l}{Software} &Compiler & GCC7.5     \\\\\n&OpenBLAS & 0.3.13    \\\\\n&ARMPL    & 21.0       \\\\\n&BLIS     & 0.81       \\\\\n\\hline\n\\end{tabular}\n\\label{tab:Experimental Environment Of ARMv8 platform}\n\\end{table}\n\n\\begin{eqnarray}\nGFLOPS&=&\\frac{2\\times M \\times N\\times K}{t} \\label{equ:sd}\\\\\nGFLOPS&=&\\frac{2\\times 4 \\times M \\times N\\times K}{t} \\label{equ:cz}\n\\end{eqnarray}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=3.4in]{perf\/pack.pdf}\n\\caption{Pack step cost proportion}\n\\label{fig:pack}\n\\end{figure}\n\nFig.\\ref{fig:pack} shows proportion of pack step cost in traditional implementation of GEMM. It shows that the proportion of pack step cost can reach 67\\% when input matrices are very small. As the size of input matrices increases, the proportion decreases exponentially. When input matrices are large enough, the proportion is near 3\\%.\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[NN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmsnn.pdf}\n\\end{minipage}\n\\label{fig:snn}\n}\n\\hspace{-0.5cm}\n\\subfigure[NT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmsnt.pdf}\n\\end{minipage}\n\\label{fig:snt}\n}\n\\hspace{-0.5cm}\n\\subfigure[TN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmstn.pdf}\n\\end{minipage}\n\\label{fig:stn}\n}\n\\hspace{-0.5cm}\n\\subfigure[TT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmstt.pdf}\n\\end{minipage}\n\\label{fig:stt}\n}\n\\centering\n\\caption{Performance evaluation of IAAT vs. OpenBLAS, BLIS, ARMPL for SGEMM}\n\\label{fig:sgemm}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[NN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmdnn.pdf}\n\\end{minipage}\n\\label{fig:dnn}\n}\n\\hspace{-0.5cm}\n\\subfigure[NT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmdnt.pdf}\n\\end{minipage}\n\\label{fig:dnt}\n}\n\\hspace{-0.5cm}\n\\subfigure[TN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmdtn.pdf}\n\\end{minipage}\n\\label{fig:dtn}\n}\n\\hspace{-0.5cm}\n\\subfigure[TT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmdtt.pdf}\n\\end{minipage}\n\\label{fig:dtt}\n}\n\\centering\n\\caption{Performance evaluation of IAAT vs. OpenBLAS, BLIS, ARMPL for DGEMM}\n\\label{fig:dgemm}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[NN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmcnn.pdf}\n\\end{minipage}\n\\label{fig:cnn}\n}\n\\hspace{-0.5cm}\n\\subfigure[NT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmcnt.pdf}\n\\end{minipage}\n\\label{fig:cnt}\n}\n\\hspace{-0.5cm}\n\\subfigure[TN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmctn.pdf}\n\\end{minipage}\n\\label{fig:ctn}\n}\n\\hspace{-0.5cm}\n\\subfigure[TT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmctt.pdf}\n\\end{minipage}\n\\label{fig:ctt}\n}\n\\centering\n\\caption{Performance evaluation of IAAT vs. OpenBLAS, BLIS, ARMPL for CGEMM}\n\\label{fig:cgemm}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[NN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmznn.pdf}\n\\end{minipage}\n\\label{fig:znn}\n}\n\\hspace{-0.5cm}\n\\subfigure[NT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmznt.pdf}\n\\end{minipage}\n\\label{fig:znt}\n}\n\\hspace{-0.5cm}\n\\subfigure[TN]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmztn.pdf}\n\\end{minipage}\n\\label{fig:ztn}\n}\n\\hspace{-0.5cm}\n\\subfigure[TT]{\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=4.1cm]{perf\/gemmztt.pdf}\n\\end{minipage}\n\\label{fig:ztt}\n}\n\\centering\n\\caption{Performance evaluation of IAAT vs. OpenBLAS, BLIS, ARMPL for ZGEMM}\n\\label{fig:zgemm}\n\\end{figure*}\n\nFig.\\ref{fig:sgemm} shows performances of NN, NT, TN, TT of SGEMM of IAAT, OpenBLAS, ARMPL, and BLIS. When input matrices are small, IAAT is faster than OpenBLAS, ARMPL, and BLIS for all transpositions. When $M=N=K\\le 80$ and transposition is NN, as shown in Fig.\\ref{fig:snn}, IAAT is on average 1.81, 2.3, and 20.17 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is NT, as shown in Fig.\\ref{fig:snt}, IAAT is on average 1.81, 2.29, and 20.19 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 32$ and transposition is TN, as shown in Fig.\\ref{fig:stn}, IAAT is on average 1.65 times faster than OpenBLAS. When $M=N=K>32$ and transposition is TN, as shown in Fig.\\ref{fig:stn}, IAAT is only faster than OpenBLAS when sizes of input matrices are multiples of 4. However, when $M=N=K\\le 100$ and transposition is TN, IAAT is faster than ARMPL and BLIS and is on average 2.15 and 11.57 times, respectively. \nWhen $M=N=K\\le 80$ and transposition is TT, as shown in Fig.\\ref{fig:stt}, IAAT is on average 1.73, 2.55, and 18.76 times faster than OpenBLAS, ARMPL, and BLIS, respectively.\n\nFig.\\ref{fig:dgemm} shows performances of NN, NT, TN, TT of DGEMM of IAAT, OpenBLAS, ARMPL, and BLIS. When input matrices are small, IAAT is faster than OpenBLAS, ARMPL, and BLIS for all transpositions. When $M=N=K\\le 80$ and transposition is NN, as shown in Fig.\\ref{fig:dnn}, IAAT is on average 1.48, 1.66, and 15.0 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is NT, as shown in Fig.\\ref{fig:dnt}, IAAT is on average 1.43, 1.66, and 14.56 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is TN, as shown in Fig.\\ref{fig:dtn}, IAAT is on average 1.32, 1.47, and 12.78 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is TT, as shown in Fig.\\ref{fig:dtt}, IAAT is on average 1.43, 1.64, and 14.54 times faster than OpenBLAS, ARMPL, and BLIS, respectively.\n\nFig.\\ref{fig:cgemm} shows performances of NN, NT, TN, TT of CGEMM of IAAT, OpenBLAS, ARMPL, and BLIS. When input matrices are small, IAAT is faster than OpenBLAS, ARMPL, and BLIS for all transpositions. When $M=N=K\\le 80$ and transposition is NN, as shown in Fig.\\ref{fig:cnn}, IAAT is on average 1.31, 1.30, and 13.24 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is NT, as shown in Fig.\\ref{fig:cnt}, IAAT is on average 1.37, 1.44, and 13.55 times faster than OpenBLAS, ARMPL, and BLIS, respectively. \nWhen $M=N=K\\le 64$ and transposition is TN, as shown in Fig.\\ref{fig:ctn}, IAAT is on average 1.16, 1.33, and 13.68 times faster than OpenBLAS, ARMPL, and BLIS, respectively. \nWhen $M=N=K\\le 80$ and transposition is TT, as shown in Fig.\\ref{fig:ctt}, IAAT is on average 1.27, 1.46, and 12.94 times faster than OpenBLAS, ARMPL, and BLIS, respectively.\n\nFig.\\ref{fig:zgemm} shows performances of NN, NT, TN, TT of ZGEMM of IAAT, OpenBLAS, ARMPL, and BLIS. When input matrices are small, IAAT is faster than OpenBLAS, ARMPL, and BLIS for all transpositions. When $M=N=K\\le 80$ and transposition is NN, as shown in Fig.\\ref{fig:znn}, IAAT is on average 1.09, 1.3, and 9.62 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is NT, as shown in Fig.\\ref{fig:znt}, IAAT is on average 1.09, 1.32, and 9.6 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is TN, as shown in Fig.\\ref{fig:ztn}, IAAT is on average 1.11, 1.32, and 9.69 times faster than OpenBLAS, ARMPL, and BLIS, respectively. When $M=N=K\\le 80$ and transposition is TT, as shown in Fig.\\ref{fig:ztt}, IAAT is on average 1.1, 1.34, and 9.6 times faster than OpenBLAS, ARMPL, and BLIS, respectively.\nei\nIn addition to the above performance description, we still observe the following three phenomena.\n\nFirstly, as shown in all Fig. \\ref{fig:sgemm}, \\ref{fig:dgemm}, \\ref{fig:cgemm}, and \\ref{fig:zgemm}, all performance curves of IAAT are very steep When input matrices are small and tend to be smooth along with increase of size. All performance curves of IAAT relate to the proportion of pack step, as shown in Fig.\\ref{fig:pack}. As mentioned in Section \\ref{sec:introduction}, performance improvement of small GEMM comes from removing pack steps. The greater proportion of pack step cost, the higher performance improvement. For example, when $M=N=K\\le 64$, the proportion of pack step of SGEMM\\_NN drops, when $M=N=K>64$, the curve of proportion is smooth. The corresponding performance curve of SGEMM\\_NN, as shown in \\ref{fig:snn}, rises when $M=N=K\\le 64$ and stops rising when $M=N=K>64$. The others curve are the same as the curve of SGEMM\\_NN.\n\nSecondly, performance of TN transposition is not as good as other transpositions as shown in Fig.\\ref{fig:sgemm}, \\ref{fig:dgemm}, \\ref{fig:cgemm} and \\ref{fig:zgemm}. Because data is not continuous and vectorized computation is not feasible in TN transposition, register allocator has to allocate individual registers for each element of block $C_c$. Therefore, blocks of matrix C occupy too many registers, which causes kernel sizes of SGEMM\\_TN smaller than other transpositions. We have to tile input matrices into smaller blocks than other transpositions. The number of loading instructions increases significantly. As the size of input matrices increases, the advantage of small GEMM for TN transposition will vanish.\n\nThirdly, the curves of performance of IAAT, as shown in Fig.\\ref{fig:sgemm}, is wavy. The performance of four transpositions reaches wave crests when the size of input matrices is multiples of 4 and it falls into wave troughs when sizes of input matrices are not multiples of 4. Here are two reasons for this phenomenon. First, Kunpeng920 platform has two fused multiply-add(FMA) units for single but Kunpeng920 cannot fully utilize units. Kunpeng920 can issue two FMA instructions or issue one FMA instruction and one loading instruction at the same time. \nTherefore, for a kernel of any size, the lower the proportion of loading instructions, the closer the performance is to the peak performance.\nWhen the size of the kernel is multiples of 4, the kernel can achieve better performance. When the sizes of input matrices are not multiples of 4, the tile algorithm tiles matrix into blocks, which corresponds to the low performance of kernel. Second, it is because only when the size of input matrices is multiples of 4, small GEMM can make full use of the registers. As one register can store four floats, other sizes of input matrices lead to insufficient register utilization. Therefore, the implementation of small GEMM for input matrices whose size is multiples of 4 has better performance. Similar waves occur in CGEMM as shown in Fig.\\ref{fig:cgemm}, which has the same reasons as SGEMM. Besides, compared with SGEMM and CGEMM, curves of DGEMM and ZGEMM, as shown in Fig.\\ref{fig:dgemm} and Fig.\\ref{fig:zgemm}, are more smooth and the performance curve of DGEMM and ZGEMM reaches wave crest when the size of input matrices is multiples of 2. Because Kunpeng920 platform has one fused multiply-add(FMA) unit for double, which can be fully utilized. Besides, it is also because the size of data type of DGEMM and ZGEMM is bigger than that of SGEMM and CGEMM. DGEMM and ZGEMM can make better use of registers than SGEMM and CGEMM.\n\nConsider the above performance results, we conclude that our implementation is faster than the others library when the size of input matrices is small enough. As shown by above performance analysis, we define small GEMM, as $\\sqrt[3]{MNK}\\le 80$, when transposition of input matrices is not TN, or $\\sqrt[3]{MNK}\\le 32$ when transposition of input matrices is TN as mentioned in Section \\ref{sec:introduction}.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper, we propose the input-aware adaptive tuning framework(IAAT) for small GEMM with two stages: the install-time stage and the run-time stage. The install-time stage auto-generates assembly kernels for ARMv8 platform and the run-time stage tiles input matrices into blocks. Finally, IAAT constructs a kernel executing plan by connecting kernels, which corresponds to the sizes of tiled blocks. As shown in the experiment, IAAT utilizes code generation and adaptive tuning to achieves near-optimal performance for small GEMM. IAAT fits the situation where computes matrix multiplication with the same size repeatedly. Our future work will focus on extending IAAT to other platforms.\n\n\\section*{Acknowledgements}\nWe would like to express our gratitude to all reviewer's constructive comments for helping us polish this article. This work is supported by the National Key Research and Development Program of China under Grant Nos.2017YFB0202105, the National Natural Science Foundation of China under Grant No.61972376 and the Natural Science Foundation of Beijing under Grant No.L182053.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \\label{sec-intro}\n\nBulk metallic glasses are strongly out-of-equilibrium industrially relevant materials with exceptional physical and mechanical properties~\\cite{Hufnagel2016}. Produced by a rapid quench from the under-cooled liquid, the amorphous state emerges during the glass transition temperature ($T_{\\mathrm{g}}$) regime with an extensive configurational entropy. However, this entropy is considerably lower than that of the liquid indicating strong constraints on the available atomic configurations~\\cite{Debenedetti2001}. Understanding these constraints and how they are overcome is central to a proper control of both structure and mechanical properties --- a parameter space that is explored via structural relaxation and rejuvenation of the glass. Increasingly, such structural tuning is assessed via the excess enthalpy storage of the glass, which is amongst the highest of any metallic material~\\cite{Sun2016,Kuechemann2017,Kuechemann2018,Ding2019}.\n\nTwo structural asymptotes have remained omnipresent in both the experimental and theoretical efforts to understand glass tunability. The first is crystallization due to annealing protocols close to $T_{\\mathrm{g}}$, and the second is relaxation towards the ideal glass state, characterized by a sub-extensive configurational entropy~\\cite{Kauzmann1948}. Generally these are considered to be quite different and unrelated structural states. Due to undesirable mechanical properties and experimental difficulties, as-produced glasses tend to be away from the latter extreme, whereas metallic-glass matrix composites (multi-phase structures containing crystalline and possibly amorphous regions) embrace the former. On the other hand, approaching the ideal glass limit, the low temperature limit of the undercooled liquid, poses interesting theoretical questions as first pointed out by Kauzmann~\\cite{Kauzmann1948} and also the alluring possibility of a glass structure which is in a true meta-equilibrium state.\n\nOnce a glassy structure is obtained after quenching a melt, there will be a natural tendency for the amorphous solid to relax. The resulting structural evolution is seen in small changes in the pair distribution function (PDF) at radial distances beyond the nearest neighbour peak(s) that characterize the short-range order (SRO). The more distant order has been referred to as medium-range order (MRO). A common framework to understand the developing MRO is with respect to the packing of SRO icosahedral content~\\cite{Sheng2006}. An atom is said to be icosahedrally coordinated if its nearest neighbour spatial environment can be characterized by a Wigner-Seitz or Voronoi generated twelve faced polyhedron where each face consists of three edges --- the icosahedron. This originates from the more fundamental unit structure of the tetrahedron~\\cite{Chaudhari1978}. When isolated, this atomic arrangement can usually minimize the energy of all six bonds. However it is not possible to pack such tetrahedra in a space-filling way whilst maintaining this minimum ground-state energy and therefore bonds become frustrated, both in terms of energy and volume. Optimal space-filling packing of icosahedra at the length-scale of the MRO (and beyond) is a difficult problem involving defected icosahedra and more general Voronoi polytopes~\\cite{Frank1952,Nelson1983a,Nelson1983b}, underlying the very close connection between local structure, bond energy and atomic volume~\\cite{Ding2017}.\n\nThe relationship between dynamical heterogeneities and structure in the under-cooled liquid-metal regime gives additional insight into the nature of this structural relaxation. In particular the connection to amorphous disorder, which is normally considered to have little in common with those local structures ultimately giving rise to crystalline order~\\cite{Charbonneau2016}. This aspect may be articulated as the competition of local structures characterized by five- (icosahedral) and six-fold bond symmetries (close-packed)~\\cite{Taffs2016}. Common equilibrium crystal structures are close-packed and therefore dominated by six-fold bond-order. However there do exist a class of crystal structures for which the five-fold bond symmetry dominates. These are the so-called topologically close packed phases discovered by Frank and Kasper~\\cite{Frank1958}. For binary systems common realizations are the cubic (C15), the two hexagonal (C14 and C36) laves structures, and the A15 structure. Atomistic simulations of model binary glasses have revealed connected fragments of the Laves C15 structure for both Lennard-Jones (LJ) force models~\\cite{Pedersen2010} (using the Wahnstr{\\o}m parameterization~\\cite{Wahnstrom1991}), and material specific force models for the experimentally realizable CuZr binary alloy~\\cite{Zemp2014,Zemp2016,Ryltsev2016}. Experimental evidence for such local icosahedral structures mainly relies on a combination of x-ray diffraction and the reverse Monte-Carlo method. For the CuZr alloy, for a range of concentration, this method suggests extended structures of connected icosahedral units~\\cite{Wang2008} possibly inter-dispersed amongst liquid-like structures~\\cite{Li2009}. For the case of the Cu$_{50}$Zr$_{45}$Al$_{5}$ ternary alloy, reverse Monte Carlo simulations in combination with fluctuation electron microscopy suggested the existence of a population of both icosahedral and crystal-like (six-fold bonded) atomic structures~\\cite{Hwang2012}. The LJ work of Pedersen {\\em et al}~\\cite{Pedersen2010} found that such Laves fragments took the form of two dimensional ring-structures referred to as Frank-Kasper backbone polyhedrons. Earlier work also showed that at sufficiently high temperatures, within the undercooled liquid regime, the Wahstr{\\o}m LJ potential can also crystallize into the C15 Laves phase~\\cite{Pedersen2006}.\n\nPast MD works probing the nano-second timescale~\\cite{Ding2014,Derlet2017,Derlet2017a,Derlet2018}, have shown that below $T_{\\mathrm{g}}$, the cohesive energy decreases with respect to increasing icosahedral content. Indeed, Ref.~\\cite{Derlet2018}, which considers the model Wahnstr{\\o}m LJ binary glass, has demonstrated that glass relaxation in terms of both energy and pressure\/volume originates directly from the conversion of non-icosahedral environments to icosahedral environments, via structural excitations characterized by thermally activated collective string-like atomic displacements~\\cite{Derlet2017,Derlet2017a}. Such a geometry of structural evolution has also been observed in the dynamical heterogeneities of the under-cooled liquid~\\cite{Donati1998,Schroeder2000,Gebremichael2004,Vogel2004,Kawasaki2013}.\n\nIn the present work we employ atomistic simulation to investigate the non-negligible structural relaxation in a model binary glass which occurs at time-scales up to approximately $80$ $\\mu$sec. We show that that the linear relation between icosahedral content and energy remains valid over this time-scale and therefore to glassy structures significantly more relaxed than previously considered. We find that this very general relaxation trajectory can lead to emergent structural length-scales reflecting both the nano-composite and ideal-glass structural asymptotes, suggesting that the ideal glass and crystallization might be intimately related if not associated with the same structural state. We argue that this perspective offers a new approach to structural tuning of nano composite (nc) amorphous-crystalline microstructures, which we here reveal for binary metallic glass formers. In sec.~\\ref{sec-simulation} we outline our simulation approach, in sec.~\\ref{sec-results} we detail the major simulation results, and in sec.~\\ref{sec-discussion} we discuss them within the framework of temperature-time-transition (TTT) diagrams that are now accessible within the time-frame of an atomistic simulation. Finally, we formulate a number of open questions resulting from this work and in sec.~\\ref{sec-conc} we summarize and conclude.\n\n\\section{Simulation Methodology} \\label{sec-simulation}\n\nAs already noted, the physics of structural frustration can be captured by simple pair potential force models such as the Lennard-Jones (LJ) potential~\\cite{Rodney2011}, whereas for metal-specific applications embedded atom~\\cite{Daw1984} or second-moment~\\cite{Finnis1985} model potentials may be used. This demonstrates the more involved force models are not essential to understanding the underlying structural glass physics~\\cite{Derlet2017a,Derlet2018} --- glassy physics is a universal phenomenon~\\cite{Debenedetti2001,Kauzmann1948,Goldstein1969} which can be equally well simulated using an appropriate LJ parameterization. Because of this, and due to their more-rapid numerical evaluation, we use a model binary LJ system parameterized by Wahnstr\\\"{o}m~\\cite{Wahnstrom1991} to investigate its microstructural relaxation behaviour at the timescale of many tens of $\\mu$sec. For two atoms of type $a$ and $b$, separated by a distance $r$, the interaction potential energy is given by,\n\\begin{equation}\nV_{ab}(r)=4\\varepsilon\\left(\\left(\\frac{\\sigma_{ab}}{r}\\right)^{12}-\\left(\\frac{\\sigma_{ab}}{r}\\right)^{6}\\right),\n\\end{equation}\nwhere $\\sigma_{22}=5\/6\\sigma_{11}$ and $\\sigma_{12}=\\sigma_{21}=11\/12\\sigma_{11}$. The atoms of type 1 may be considered as the larger atom type. The atomic masses of the two atom types are arbitrarily chosen such that $m_{1}\/m_{2}=2$. For a molecular dynamics iteration, a time step of $0.002778\\sigma_{11}\\sqrt{m_{1}\/\\varepsilon}$ is used. The distance unit is taken as $\\sigma=\\sigma_{11}$ and the energy unit as $\\varepsilon$. For this work, the potential is truncated to a distance 2.5$\\sigma$. Atoms of type 1 may be considered as the larger atom. All molecular dynamics (MD) simulations are done using the LAMMPS software~\\cite{LAMMPS}, and atomic scale analysis via the OVITO visualization software~\\cite{Stukowski2010}.\n\nWhen metallic units (representative of say, CuZr) are taken for $\\varepsilon$ and $\\sigma$, an MD time-step is of the order of a femto-second. Thus one billion MD iterations corresponds to approximately one micro-second. Throughout the remainder of this paper, simulation times will be measured with respect to one billion MD iterations, which in turn is approximated as one micro-second.\n\nA model binary glass system is created via an NVT temperature quench from a close-to-equilibrium liquid state~\\cite{Derlet2017,Derlet2017a}. Following either pressure or cohesive energy, a change in slope with respect to temperature indicates a glass transition is reached and the amorphous solid regime is entered. The temperature at which this occurs is referred to as the fictive glass transition temperature ($T_{\\mathrm{f}}$). For MD, $T_{\\mathrm{f}}$ is usually taken as a reference glass transition temperature, $T_{\\mathrm{g}}$. Experimentally, $T_{\\mathrm{f}}$ is seen as the glass transition temperature without any additional annealing protocol. Such a temperature protocol, sometimes with a pressure protocol, embodies the standard method to produce a glass using atomistic simulation~\\cite{Rodney2011}.  At a somewhat lower temperature than $T_{\\mathrm{f}}$, the temperature quench may be halted and long-time constant temperature simulations are performed, followed by a further quench to zero temperature. This annealing step is known to produce more relaxed samples~\\cite{Derlet2017a}. The present work considers only fixed volume simulations, the so-called NVT ensemble. Recent work has shown that when performed at appropriately rescaled temperatures, both fixed volume and fixed pressure (NPT ensemble) sample production protocols result in similar relaxation trajectories and thus similar final structural states with zero temperature volumes that differ by only $\\sim0.2\\%$. See Ref.~\\cite{Derlet2018} for more details.\n\nIn the first instance, we consider a glass sample consisting of 500 atoms, at a fixed volume per atom equal to 0.767$\\sigma^{3}$. This sample is initially produced using the linear-quench protocol just discussed. Such a small atom number is chosen to maximize computational throughput, but is large enough to accommodate the local structural excitations (LSEs) that mediate general structural relaxation~\\cite{Swayamjyoti2014,Derlet2017,Derlet2017a}. The precise details of the quench protocol are identical to Ref.~\\cite{Derlet2017a}, resulting in the present sample having the same value of $T_{\\mathrm{f}}$ as that of Ref.~\\cite{Derlet2017a}.\n\n\\section{Results} \\label{sec-results}\n\n\\subsection{Long time $T<T_{\\mathrm{f}}$ relaxation} \\label{ssec-lts}\n\n\\begin{figure*}\n\t\\subfloat[]{\\includegraphics[width=0.4\\linewidth]{fig1a.pdf}}\n\t\\subfloat[]{\\includegraphics[width=0.4\\linewidth]{fig1b.pdf}} \\\n\t\\subfloat[]{\\includegraphics[width=0.8\\linewidth,trim=1.5cm 1cm 1cm 1cm,clip]{fig1c.pdf}}\n\t\\caption{a) Energy per atom and icosahedral content evolution over a time period of 80 $\\mu$sec at an annealing temperature $0.95T_{\\mathrm{f}}$. b) Scatter plot of energy per atom versus icosahedral content. Also shown is data for a larger sample of 32000 atoms~\\cite{Derlet2017a}, a geometrically constructed nano-composite Laves phase with two characteristic grain sizes, and the results of a hybrid-MD\/MC simulation starting from a chemically disordered Laves crystal. c) Atomic displacement maps between configuration separated by 1 million MD iterations (0.001 $\\mu$sec) taken at 0.001, 0.01, 0.1, 1.0 and 10 $\\mu$sec along the relaxation shown in a). The displacement vectors are coloured according to their magnitude with red representing displacement magnitudes greater than 1.25$\\sigma$ and blue a magnitude of zero --- see linear color bar for intermediate magnitudes.}\n\t\\label{Fig1}\n\\end{figure*}\n\nAt a temperature of $0.95T_{\\mathrm{f}}$ the linear temperature protocol is halted and constant temperature NVT simulations are performed for 80 $\\mu$sec. Fig.~\\ref{Fig1}a plots the cohesive energy per atom with the thermal component removed (via a conjugate gradient relaxation), indicating an initially rapid energy drop within the first few micro-seconds followed by a slower decrease in energy involving significant fluctuations in energy. At longer timescales, relaxation appears arrested, although strong fluctuations persist. This figure also displays the corresponding icosahedral content showing an anti-correlation with the cohesive energy confirming its usefulness as a structural measure. This icosahedral content consists almost exclusively of the smaller atoms. It is emphasized that most published atomistic glass simulations employed samples relaxed at time-scales below a few hundred nano-seconds. The present work therefore gives insight into relaxation at time-scales three orders of magnitude larger than that considered in the past.\n\nFig.~\\ref{Fig1}b is a scatter plot of the cohesive energy per atom and icosahedral content, and displays an approximately linear relation. Data is also shown for a larger sample of 32000 atoms~\\cite{Derlet2017,Derlet2018}, demonstrating this linear relation is independent of the structural realization. The data-points of this larger sample are taken from relaxation occurring at a range of temperatures much lower than the $T_{\\mathrm{f}}$, showing that this line represents a general relaxation trajectory of an evolving amorphous solid for this LJ-system. The increased scatter of the 80 $\\mu$sec simulations originates from the smaller size of the present glass sample. \n\nFig.~\\ref{Fig1}c plots the atomic displacement vectors between two configurations separated by one million MD iterations (0.001 $\\mu$sec) at the times 0.001, 0.01, 0.1, 1.0 and 10.0 $\\mu$sec of the relaxation. These structures have all been relaxed using a conjugate gradient method to remove the effects of thermal motion, and are thus at a local energy minima. The figures show that in the early stages of the relaxation, significant atomic displacement occurs arising from multiple LSE activity within the 0.001 $\\mu$sec time interval, and that as the relaxation progresses, such LSE events become increasingly rare. \n\nFig.~\\ref{Fig2} plots the pair distribution function for the first configuration and the configuration at 35 $\\mu$sec in Fig.~\\ref{Fig1}a, revealing a weak peak-like structure beyond the nearest neighbour SRO-peak for the more relaxed structure. Such signatures of strong MRO are known to reflect icosahedral content~\\cite{Sadoc1973} and are often seen in experiment~\\cite{Dmowski2011,Mauro2014,Tong2015}, but rarely in simulation due to their sub-$\\mu$sec relaxation time scales. The sub-PDFs indicate how this MRO arises from the spatial arrangement of the small icosahedrally coordinated atoms, and the larger atoms clustering around these. This developing spatial order is best seen by attaching nearest neighbour bonds to the icoshadrally coordinated atoms and visualizing only these bonds via OVITO~\\cite{Stukowski2010}. The inset panel in Fig.~\\ref{Fig2} does this and shows a structure containing repeated ring-like mosaics of Frank-Kasper polyhedra joined together by tetrahedrally packed atoms. Such structures signify the defected C15 Laves back-bone-polyhedra\/crystals seen in the past works~\\cite{Pedersen2010,Zemp2014,Zemp2016,Ryltsev2016} discussed in Sec.~\\ref{sec-intro}. Beyond 35 $\\mu$secs, the pair distribution function does not change significantly.\n\n\\begin{figure}\n\t\\begin{center}\n\t\\includegraphics[width=0.55\\linewidth,trim=0cm 2cm 0cm 2cm]{fig2.pdf}\n\t\\end{center}\n\t\\caption{The pair distribution function of the initial and 35 $\\mu$sec structures associated with Fig.~\\ref{Fig1}a. Also shown are the contributions due to only icosahedrally coordinated atoms and their neighbouring smaller atoms, of the final structure. The inset visualizes the final structure by only showing the icosahedrally coordinated atoms and the bonds between them.}\n\t\\label{Fig2}\n\\end{figure}\n\nThus the linearity between energy and icosahedral content seen in past work clearly extends to larger values of icosahedral content and lower values of energy --- a structural regime that includes system spanning (for our small sample) icosahedral content in the form of Frank-Kasper C15 Laves crystal fragments.\n\n\\subsection{Possible structural asymptotes} \\label{ssec-ncl}\n\nThe results of sec.~\\ref{ssec-lts} motivates the question as to what type of micro-structure does the relaxation embodied in Figs.~\\ref{Fig1}a-b tend towards? It is again emphasized that the linear trajectory seen in Fig.~\\ref{Fig1}b is also seen at temperatures significantly lower than $T_{\\mathrm{f}}$~\\cite{Derlet2017a}. That is, the relaxation seen at the temperature $0.95T_{\\mathrm{f}}$ is the one and the same as that occurring at temperatures well below this glass transition temperature regime.\n\nTo help answer this question, we construct a chemically disordered Laves C15 phase in which some of the larger atoms have been replaced with smaller atoms to accommodate the 50:50 composition of our present glassy system. Within the restrictions of the C15 crystallography, we choose a system size comparable to the 500 atom sample of the previous section. This ensures that any developing microstructural length-scale cannot exceed that of the well relaxed 50:50 glassy sample. To investigate the dynamics at the same temperature and volume per atom, we use hybrid MD\/MC, where at every 100 MD steps we perform one MC particle exchange between atoms of different type. Doing so, eventually results in a segregation to two crystalline structures --- a Laves C15 structure with composition 67:33 and some chemical disorder, and an ordered close-packed structure consisting primarily of the larger atoms ---  see inset of fig.~\\ref{Fig3}a. The dynamics of this process are mapped onto our energy versus icosahedral plot as red points in Fig.~\\ref{Fig1}b. This energy and that in fig.~\\ref{Fig3}a have the thermal component removed. Two clusters of points are seen, both having an icosahedral content of the order of 50\\%. Such a high icosahedral percentage is expected since all small atoms are icosahedrally coordinated within the perfect Laves C15 structure. The higher energy cluster of points in fig.~\\ref{Fig3}a represent the initial chemically disordered C15 structure, whereas the lower energy cluster is indicative of the final structural state the hybrid simulation reaches. \n\n\\begin{figure}\n\t\\begin{center}\n\t\\subfloat[]{\\includegraphics[width=0.45\\linewidth,trim=3cm 9.5cm 3cm 10cm]{fig3a.pdf}}\n\t\\subfloat[]{\\includegraphics[width=0.45\\linewidth]{fig3b.pdf}}\n\t\\end{center}\n\t\\caption{a) Plot of energy per atom versus hybrid MD\/MC iteration number for an initially chemically disordered laves phase with a 50:50 composition of small:large atoms. The inset plots (left) the initial structure and (right) the final structure, the latter of which shows a close-packed structure consisting of only the larger atoms (high-lighted in yellow), and a less chemically disordered laves crystal with a 67:33 atom type composition. b) Geometrically constructed nano composite Laves structure with C15 crystallites embedded in a amorphous matrix existing predominantly of the larger atom. Only icosahedrally coordinated atoms are shown with a surface mesh enclosing regions containing such atoms.}\n\t\\label{Fig3}\n\\end{figure}\n\nThe remarkable result in Fig.~\\ref{Fig1}b is that the final cluster of data points aligns with the intersection of the relaxation trajectory of the glasses, suggesting glassy relaxation could tend towards a structure characterized by a two-phase crystalline composite consisting of Laves C15 crystals and a close-packed region of larger atoms --- with a micro-structural length-scale comparable to the current system sizes of about 500 atoms. \n\nInspired by experimentally observed two-phase nano-structures~\\cite{Calin2003,Pekarskaya2003,Wu2017}, a nano-structure is artificially constructed consisting of randomly orientated Laves C15 crystals embedded within an amorphous matrix of only the larger atoms. This is done by exploiting geometrical methods normally used for computer generated nanocrystalline metals~\\cite{DAC2008,Froseth2005}. This procedure uses the Voronoi tessellation method to construct a Laves C15 grain network, which initially results in planar interface regions of finite thickness into which an disordered arrangement of the larger atoms are inserted at a density compatible with the global volume per atom and global 50:50 composition. These samples are then relaxed using hybrid MD\/MC NVT to produce micro-structures that are stable at the $\\mu$sec-timescale. The energy and icosahedral values of the final nano-composite Laves (ncL) structure are shown in Fig.~\\ref{Fig1}b, for two average C15 crystallite sizes corresponding to about 400 ($r\\simeq4\\sigma)$ and 1200 ($r\\simeq6\\sigma$) atoms. It is seen that these artificially constructed candidate structures again line up with the relaxation trajectory of the corresponding glasses, for which the smaller grain sized sample has a higher energy than that of the well relaxed glass. The larger grain sized sample is however at a lower energy and higher icosahedral composition than the glass, suggesting that the associated grain size reflects a more realistic micro-structural length-scale towards which the glass relaxes. Fig.~\\ref{Fig3}b visualizes this smaller grain-size sample, where only the icosahedrally coordinated atoms are shown. To aid in visualization, a surface mesh bounds the volume of these atoms, encompassing the C15 grains and a neural-like network of icosahedral connectivity between the C15 grains, which has developed during the hybrid MD\/MC relaxation simulations.\n\nFurther simulations demonstrate that a simulation cell consisting only of the larger atoms at a volume per atom comparable to the average volume per atom of the amorphous interface region, rapidly crystallizes into a system-spanning closed packed structure. Thus the mono-atomic amorphous interface region of our nanophase material is stabilized by the surrounding Laves crystals, suggesting a strongly constrained glassy structure. Moreover, detailed inspection of the thermally-activated atomic displacements at $T_{\\mathrm{f}}$ shows string-like excitations concentrated mainly at the interface between the Laves crystal and the amorphous region, but not within the latter.\n\n\\subsection{$T>T_{\\mathrm{f}}$ relaxation and variation of chemical content} \\label{ssec-lts1}\n\nA similar annealing protocol for the 50:50 sample is now performed at a temperature equal to 1.2$T_{\\mathrm{f}}$ (here the temperature quench from the melt has been halted at 1.2$T_{\\mathrm{f}}$ instead of 0.95$T_{\\mathrm{f}}$). The energy per atom and icosahedral content per atom as a function of time is plotted in Fig.~\\ref{Fig4}a. At this higher temperature, initially there is little relaxation when compared to the first $\\mu$sec of the lower temperature constant temperature anneal of Fig.~\\ref{Fig1}a. Brief excursions to lower energy structures are however seen, and at approximately 1.2 $\\mu$sec a significant irreversible reduction in structural energy occurs. The anti-correlation with icosahedral content is again seen (Fig.~\\ref{Fig4}b) and the resulting relaxation trajectory is shown in Fig.~\\ref{Fig4}c. Inspection of the figure reveals the same linear correlation as seen in Fig.~\\ref{Fig1}b. Indeed, from a general structural perspective there is little difference between that of the final state arising from this higher temperature and that of the structure at a comparable energy per atom arising from the 80 $\\mu$sec anneal, where in both cases the icosahedrally coordinated atoms arise from fragments of the C15 Laves crystal structure.\n\n\\begin{figure*}\n\t\\begin{center}\n\t\\subfloat[]{\\includegraphics[width=0.45\\linewidth]{fig4a.pdf}}\n\t\\subfloat[]{\\includegraphics[width=0.45\\linewidth]{fig4b.pdf}} \\\n\t\\subfloat[]{\\includegraphics[width=0.45\\linewidth]{fig4c.pdf}}\n\\end{center}\n\t\\caption{a) Energy per atom and b) icosahedral content evolution over a time at an annealing temperature $1.2T_{\\mathrm{f}}$ for the small:large atom type compositions 33:67, 50:50 and 67:33. c) Corresponding scatter plots of energy per atom and percentage icosahedral content. Also shown are the isolated data points of the perfect laves structure and the nano composite laves structure of Fig.~\\ref{Fig3}b evaluated at the corresponding volumes.}\n\t\\label{Fig4}\n\\end{figure*}\n\nApplying the present results to a variation of chemistry, suggests that a 67:33 composition of small to large atoms would exhibit a relaxation trajectory which asymptotes to the perfect C15 Laves crystal or poly-crystal. For a direct validation of this we redo the $T>T_{\\mathrm{f}}$ isotherm NVT simulations now using a 67:33 composition, but at the fixed volume for which the average volume available to the small and large atoms remains the same as that of the 50:50 simulations~\\footnote{At the fixed volume of $0.76\\sigma^{3}$, the average volume per atom (derived from a Voronoi tessellation) of the different sized atoms is $\\Omega_{\\mathrm{small}}=0.579\\sigma^{3}$ and $\\Omega_{\\mathrm{large}}=0.955\\sigma^{3}$. The global volume for an $N=N_{\\mathrm{small}}+N_{\\mathrm{large}}$ atom system, is then taken as  $N_{\\mathrm{small}}\\Omega_{\\mathrm{small}}+N_{\\mathrm{large}}\\Omega_{\\mathrm{large}}$.}. Fig.~\\ref{Fig4} displays the energy and icosahedral content as a function of time, and at approximately 0.6 $\\mu$sec an abrupt change occurs demonstrating significant structural relaxation. This may also bee seen in the resulting relaxation trajectory shown in fig.~\\ref{Fig4}c, where also the energy and icosahedral data-point corresponding to the perfect Laves structure at the same volume per atom is shown. As with the 50:50 relaxation trajectory there is alignment between the glass and the suggested asymptotic structure, confirming our initial hypothesis of a fundamental connection between glass relaxation and the Laves crystal structure.\n\nTo further investigate the effect of stoichiometry, we also perform similar simulations for the 33:67 composition which, by our current reasoning, would result in small C15 crystallites surrounded by a dominate mono-atomic amorphous phase of the larger atom. This structural asymptote scenario is however unlikely since we have already demonstrated that the remaining mono-atomic region (without a high density of laves crystal surfaces) rapidly crystallizes to a close-packed structure. Inspection of the corresponding energy versus icosahedral trajectory in Fig.~\\ref{Fig4}c does indeed demonstrate an abrupt cessation of the trajectory in terms of the icosahedral structural measure. Inspection of the atomic scale processes underlying this behaviour reveals an initial segregation of the smaller atoms into clusters involving several atoms and the creation of icosahedral content. This is followed by a system-spanning ordering to a chemically disordered cubic structure dominant in FCC\/HCP content, and negligible icosahedral content (see Fig.~\\ref{Fig4}b).\n\nThe similar gradients seen in Fig.~\\ref{Fig4}c are a result of the appropriately scaled global volumes which ensure that in all three compositions the same average volume is available to the differently sized atoms. This reiterates the importance of local volume. When the 67:33 and 33:67 simulations are done at the fixed volume of the 50:50 simulations, similar trajectories are seen (with the 67:33 trajectory again asymptoting to the Laves crystal limit), but with gradients which now depend on composition.\n\n\\subsection{Global and local energy} \\label{ss-eos}\n\nTo gain a better understanding of the energetic origin of the observed behaviour, Fig.~\\ref{Fig5} displays the equation of state (EOS) curves for the Wahnstr{\\o}m LJ potential for a variety of relevant crystal phases. The black vertical line indicates the fixed volume at which the present simulations were performed, and it is seen that the Laves, the intermetallic bcc-b2 and the bcc-C11b phases have comparable energies at this chosen volume per atom. Note that there is also little difference between the EOS curves of the hcp and fcc phases of each atom type, which is to be expected for a short-range pair potential.\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.55\\linewidth]{fig5.pdf}\n\t\\end{center}\n\t\\caption{Plot of equation-of-state curves for a number of pure and binary alloy phases. Also included are scatter plots of local potential energy versus local volume for the 80 micro-second annealed glass and the nano composite Laves structure at a fixed volume of 0.767$\\sigma^{3}$ --- indicated by the black vertical line, the thicker red part of which indicates the energy per atom range of the glass. The two inverted black triangles indicate the volume and energy per atom of the equilibrium laves C15 phase at a similar fixed volume.}\n\t\\label{Fig5}\n\\end{figure*}\n\nThe Laves C15 EOS curve indicates that a significant reduction in energy could be achieved by forming this phase at a reduced volume, and that the remaining larger atoms of non-stoichiometric compositions (such as 50:50) could become hcp\/fcc by increasing their volume per atom. This possibility may be quantified by writing the energy of this dual phase as\n\\begin{equation}\nE_{\\mathrm{Laves-hcp}}=\\frac{3}{4}E_{\\mathrm{Laves}}\\left(V_{\\mathrm{Laves}}\\right)+\\frac{1}{4}E_{\\mathrm{hcp}}\\left(V_{\\mathrm{hcp}}\\right) \\label{Eqn1}\n\\end{equation}\nwhere the volume per atom of each phase must satisfy the global fixed volume per atom, V,\n\\begin{equation}\nV=\\frac{3}{4}V_{\\mathrm{Laves}}+\\frac{1}{4}V_{\\mathrm{hcp}}. \\label{Eqn2}\n\\end{equation}\nUsing the EOS curves of Fig.~\\ref{Fig5}, a minimum of Eqn.~\\ref{Eqn1} is found to be $-7.7\\varepsilon$ which occurs at $V_{\\mathrm{Laves}}=0.69\\sigma^{3}$ and $V_{\\mathrm{hcp}}=1.00\\sigma^{3}$. Such an energy does not include the cost of the interface energy and therefore represents the limit where the microstructural length-scales have diverged and the interface region becomes negligible compared to the volume occupied by the Laves and hcp\/fcc phases. Indeed, drawing a line between the energies of the two ncL micro-structures (which are distinguished by the two grain sizes) and extrapolating this to the regime of an icosahedral content of $50\\%$ yields an energy close too and slightly higher than the bulk value of $-7.7\\varepsilon)$. Together, these results give insight into the origin of the energetics of the dynamics seen in Fig.~\\ref{Fig1}b. Whilst certainly influenced by the bulk energies, the interface energy between the laves or laves-like regions and an approximately mono-atomic close-packed or amorphous structure plays a significant role in setting the overall micro-structural energy scale and therefore the emergent length scales of the well relaxed amorphous solid. \n\nFurther insight can be gained by inspecting the distribution of local energies and volumes in the considered systems. For the perfect Laves phase, the local energy and volume are indicated by the black inverted triangles in Fig.~\\ref{Fig5} with the low\/high volumes representing the small\/large atoms. Also included in Fig.~\\ref{Fig5} is a scatter plot of the local energies of the final glass structure of the 80 $\\mu$sec anneal, and of the artificially constructed ncL microstructure (grain size $6\\sigma$). Both structures show local energies that are centered close to the equilibrium local energies of the perfect Laves crystal --- this becomes more clear when a density plot with respect to local energy and volume is made (not shown). This is particularly the case for the smaller atoms which are predominantly in icosahedral environments. The remaining atoms not associated with the Laves structures tend to have higher energies and volumes, especially the larger atoms in the amorphous interface of the ncL microstructure. Thus the local energies and volumes of the amorphous structure, derived from a direct quench from the melt and a long-time anneal are similar to the artificially generated nano-phase Laves structure generated in Fig.~\\ref{Fig3}b.\n\n\\section{Discussion} \\label{sec-discussion}\n\nThe temperature-time-transformation (TTT) diagram provides a framework in which to quantify when either glass formation or crystallization occurs. Fig.~\\ref{Fig6} details a schematic of such a diagram with a normalized temperature regime on the vertical axis. The quantitative details depend strongly on the chemistry of the melt. Traditional quenching of a glass-forming metallic liquid requires cooling rates less than the critical value $(dT\/dt)_{\\mathrm{crit}}$, such that the temperature-time trajectory does not intersect the so-called nose of the crystalline region. Exceptional glass formers have low $(dT\/dt)_{\\mathrm{crit}}$ of the order of 1 K\/s, whereas mono-atomic metallic glasses may require quench rates up to 10$^{14}$ K\/s to bypass the crystallization nose~\\cite{Mao2014}.\n\nThe crystallization nose in Fig.~\\ref{Fig6} extends over a finite time interval, underlying a transition (as time evolves) from a dominant glassy structure to a dominant equilibrium polycrystalline structure. On the glassy side of the nose this encompasses the micro-structural regime of nanocrystalline composites ~\\cite{Calin2003,Pekarskaya2003,Wu2017} --- a microstructure characterized by nanocrystals of the equilibrium crystalline structure typically embedded in a glassy matrix. To follow a particular isotherm above the glass transition temperature regime, after a heating or quenching, is experimentally challenging due to the short accessible time-window prior to the crystallization nose, and requires exceptionally stable glassy alloys with a crystallization nose located at several tens of seconds~\\cite{Loffler2000}. Alternatively, new high-speed calorimetric instrumentation can now be used to access the shorter time-scales of a TTT diagram~\\cite{Schawe2019}. \n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.9\\linewidth,trim=0cm 4cm 2cm 4cm]{fig6.pdf}\n\t\\end{center}\n\t\\caption{Schematic of a temperature-time-transformation (TTT) diagram showing the glass and crystalline regions which can be accessed during a temperature quench protocol from the liquid state. The region indicated by red represents a possible intermediate regime in which out-of-equilibrium nano composite structures are encountered that are not consisting of the equilibrium polycrystalline phases of the inner nose region.}\n\t\\label{Fig6}\n\\end{figure*}\n\nThe consequences of following such isotherms are best illustrated for the two different regimes, $T<T_{\\mathrm{f}}$ and $T>T_{\\mathrm{f}}$. The first case considers the structural trajectory of a glassy solid, as done in the simulations of sec.~\\ref{ssec-lts}. The typical timescales associated with MD places the onset of the $\\mu$sec MD isotherm far to left of the TTT diagram. At the beginning of the isotherm, the glass is unrelaxed and the connectivity of SRO is low. With increasing time, the icosahedral content increases, leading to the development of MRO and internal length scales which increase as a function of annealing time. The second isotherm in Fig.~\\ref{Fig6} at $T>T_{\\mathrm{f}}$, is motivated by the idea that shorter time-scales are needed to approach the crystallization nose. The results of sec.~\\ref{ssec-lts1} indeed demonstrate this, where within the order of 1$\\mu$sec, significant relaxation associated with the creation of icosahedral SRO and MRO is seen, producing structures not so different from the 80 $\\mu$sec anneal of sec.~\\ref{ssec-lts} at $T<T_{\\mathrm{f}}$ via a similar relaxation trajectory (Figs.~\\ref{Fig1}b and \\ref{Fig4}c).\n\nThe existence of glassy microstructural length scales, typically referred to as spatial heterogeneities, has been demonstrated in numerous experiments at both the nanometer and micrometer scale~\\cite{Liu2018,Qiao2019,Ross2017,Liu2011}, but their structural origin remains unclear. Based on Figs.~\\ref{Fig1}, \\ref{Fig4}c and~\\ref{Fig6} it becomes apparent that numerous final glassy or nc materials can be realized by isothermal processing and varying initial composition. Spatial heterogeniety length-scales can be tuned along an isotherm, and are expected to maximize at a length-scale set by the forming nc-phase. Whilst only shown for binary model glass formers, it could be that recent reports of unusual nano-phase metallic glass matrix composites~\\cite{Rupert2015,Wu2017} are a result of the here traced isothermal evolution of an out-of-equilibrium two-phase structure, where no immediate crystalline phase is available to all constituent elements of the glass. Thus, the relaxation into a nc phase is expected to be less probable with increasing chemical complexity, since this increases the probability for the remaining amorphous matrix --- now at a different composition than the initially glass forming system --- to find crystallization pathways. The mechanisms discussed here should be distinguished from the micro-structural length-scales arising in eutectic-like compositions for which segregation effects can occur due to spinoidal decomposition in the under-cooled liquid state~\\cite{Zhao2014}.\n\nFor the geometrically constructed nc Laves micro-structure (Fig.~\\ref{Fig3}b), the surrounding grain structure appears to stabilize the amorphous matrix. In this context it is noteworthy that out-of-equilibrium processing of binary CuZr alloys via ball milling and subsequent annealing has produced nc structures~\\cite{Rupert2015} that may form due to the reasons discussed here. In this case, an isothermal annealing at 90$\\%$ of the melting temperature lead to the formation of a stable glassy interface. The origin of this experimental observation can now be framed within a chemical segregation effect that promoted the formation of inter-grain compositions not able to form a thermodynamically favored crystalline phase. In fact, the original work reporting nanometre-scale glassy boundaries in nanocrystalline CuZr argued that an enrichment of Zr-atoms in the boundary regions promoted its formation~\\cite{Rupert2015}. This is very much in agreement with the here observed amorphous matrix that forms via the complete depletion of the smaller atoms for the equiatomic composition.\n\nThe observation that the relaxation trajectory seen in Fig.~\\ref{Fig1}b occurs at temperatures well below $T_{\\mathrm{f}}$ (See refs.~\\cite{Derlet2017a,Derlet2018}) as well as temperatures above $T_{\\mathrm{f}}$ (fig.~\\ref{Fig4}), and that the trajectory is inclusive of geometrically constructed Laves nano-composite structures, suggests that in the low temperature amorphous solid regime, structural relaxation is also driven by the formation of an extended network of Laves back-bone structures which might eventually lead to Laves crystallite structures. This would lead us to a well-relaxed glassy state which is not fully amorphous and has structural length-scales well above that of the SRO and traditional MRO scale.\n\nThe appropriate historical context lies in the seminal work of Kauzmann~\\cite{Kauzmann1948} which was concerned with the low temperature limit of the under-cooled liquid and predates the potential energy landscape picture of Goldstein~\\cite{Goldstein1969}. As the temperature reduces the non-vitrified meta-stable liquid increasingly exhibits structural configurations that are neither representative of the equilibrium crystal nor of the equilibrium liquid --- the so-called ``ambiguous regions'' of phase space. A key concept was that these ambiguous regions entail dynamics in which the timescales associated with equilibrium crystal nucleation and atomic mobility converge. The finite temperature at which this occurs characterizes the Kauzmann temperature, $T_{k}$, at which the configurational entropy would approach that of the crystal and become sub-extensive. Three structural scenarios were envisaged: 1) the equilibrium crystal structure emerges thus entirely avoiding the first order nature of the equilibrium liquid-to-solid transition, 2) the emergence of a new state of ``high order'' and 3) an amorphous structure with a sub extensive configurational entropy. Scenario 1) was dismissed since experiment suggested that the free energy of the meta-stable under-cooled liquid does not converge to that of the relevant equilibrium crystal phase (or phases). Scenario, 2), was not considered viable since it was difficult to conceive of a ``plausible structure''. Kauzmann ultimately chose 3) entailing that a glass was indistinguishable from the liquid at $T_{k}$ --- the ideal glass.\n\nIf one assumes that structural relaxation below $T_{\\mathrm{f}}$ leads to Laves crystallites, the present work suggests two possible variants of scenario 2): i) the emerging Laves C15 phase is an equilibrium crystal phase of the empirical potential for the atom-type composition considered, or ii) it is a non-equilibrium crystal phase. \n\nIf it is ii), then the TTT diagram will need to distinguish between nc regimes consisting of equilibrium or out-of-equilibrium crystallite phases. Indeed, one could envisage that the equilibrium regime would dominate at higher temperatures, $T>T_{\\mathrm{f}}$ and towards the melting temperature regime, whereas the latter becomes relevant at lower temperatures, $T<T_{\\mathrm{f}}$ and close to the glass transition temperature regime. The scenario of the Laves phase being non-equilibrium is interesting since it suggests the ``high order'' structure of Kauzmann has little to do with the relevant equilibrium crystal phase(s), but rather is intimately connected to the reduction of frustration associated with glass relaxation as suggested by the works of Frank and Kasper~\\cite{Frank1952,Frank1958}, Chaudri and Turnbull~\\cite{Chaudhari1978}, and Nelson and co-workers~\\cite{Nelson1983a,Nelson1983b}. If, however, it is i) and the Laves C15 phase is an equilibrium crystal phase, then the structure of ``high order'' nano-composite partially involves an equilibrium crystal phase (mixed with a glassy interface region) and no such modification of the TTT diagram would be needed.\n\nWhilst the binary phase diagram of the LJ Wahstr{\\o}m potential is not known, the zero-temperature equation of state curves displayed in Fig.~\\ref{Fig5} and discussed in sec.~\\ref{ss-eos} indicate a purely energetic reason for the formation of the Laves crystal phase. Free-energy equation of state curves will ultimately be needed to answer whether or not this is the equilibrium phase, however it is noteworthy to point out that a similar energetic reason for the creation of a Laves phase is seen for the embedded atom method CuZr potential when producing analogous equation of state curves (not shown) ---  a material specific model system that also exhibits the growth of Laves fragments~\\cite{Zemp2014,Zemp2016,Ryltsev2016}. For these simulations, there exists an experimental counterpart for which the binary phase diagram is precisely known~\\cite{Okamot02009}. For compositions close to the inter-metallic 50:50 composition, the equilibrium phase has little to do with the Laves phase and much more to do with the B2 BCC intermetallic phase. On the other hand, for Eutectic-like compositions near 67:33, the Laves C15 structure can be an equilibrium phase. Thus whether or not the TTT diagram needs to be modified to include a nano-composite regime involving non-equilibrium structures will depend on the chemical content. Of course the used CuZr empirical potential might be unable to describe this regime of equilibrium behaviour correctly, but it is encouraging that there exists experimental evidence for the formation of Laves fragments in the CuZr binary glass system~\\cite{Wang2008,Li2009,Hwang2012}. \n\nAlternatively, one could envisage that structural relaxation below $T_{\\mathrm{f}}$ does not lead to Laves C15 crystallites within the experimental timescale, but rather to a disordered and interconnected collection of Frank-Kasper backbone structures with a strong signature of MRO --- the possible asymptote to the ideal glass. In other words kinetic arrest below $T_{\\mathrm{f}}$ limits the accessible regime of the relaxation trajectory of Fig.~\\ref{Fig1}b and \\ref{Fig4}c. In this scenario, the TTT diagram could again be modified to represent the emerging structural length scales associated with this asymptote, indeed the nano-composite microstructural picture might still have relevance, although now based on the details of an entirely non-crystalline structure. Indeed the possibility of such kinetic arrest provides a more precise formulation of questions concerned with the constraints inherent to a glass structure --- a theme discussed in Sec.~\\ref{sec-intro}. In particular, do such constraints (what ever they are) preclude glassy structural evolution that eventually ends with the formation of Laves crystallites? Using the language suggested by Nelson and co-workers, is the disorded disclination network of a well-relaxed glass able to transform to the ordered disclination network of a topological close-packed structure, such as the C15 phase? Might the amorphous solid be topologically protected from such evolution? \n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.9\\linewidth,trim=2cm 2cm 2cm 2cm]{fig7.pdf}\n\t\\end{center}\n\t\\caption{Visualization of nearest neighbour bond network for the well relaxed glass (top panels) and the well relaxed laves\/closepacked crystal structure of Fig.~\\ref{Fig3}a (bottom panels). Here each bond is coloured according to the local defect disclination structure, where purple\/red\/blue\/yellow\/green\/orange represent $n$-fold bonds with $n=3\/4\/5\/6\/7\/8$. For both structures, the right hand panels (b and d) omits the, disclination free, 5-fold symmetric bonds. The three layers in c) containing a low density of blue bonds, represents the close-packed structure of the larger atoms.}\n\t\\label{Fig7}\n\\end{figure*}\n\nFigs.~\\ref{Fig7}a plots the nearest neighbour bond network of the final configuration of the relaxation entailed in Fig.~\\ref{Fig1}a. Each bond is colored according to the number of common neighbours, $n$, of the two atoms that make up the bond, and may be referred to as either a $n$-fold symmetric bond --- corresponding to the number of distorted tetrahedrons packed around the bond. In the figure the blue bonds represent the five symmetric\/disclination free bonds, whereas the yellow bonds represent the six-fold\/disclinated bonds underlying the close-packed structure. Fig.~\\ref{Fig7}b omits the disclination free bonds and would in principle represent the disclination network of the glass structure --- this should be compared to figure 3 of Ref.~\\cite{Nelson1983b} which is a schematic of a glass containing significantly less disclination structures. The figures show a disordered network of blue and yellow bonds reflecting the glass's MRO and network of Frank-Kasper backbone polyhedra. Fig.~\\ref{Fig7}c-d similarly visualize the approximate Laves\/close packed structure displayed in Fig.~\\ref{Fig3}a. Inspection of these figures reveals an extended and ordered network of both five-fold symmetric and six-fold symmetric bonds within the Laves crystal regime. \n\nThe work of Nelson~\\cite{Nelson1983b,Mermin1979,NelsonBook} indicates that the algebra corresponding to the connectivity of the glassy disclination network is non-abelian, which entails strong constraints on the available disclination configurations to the extent that the entanglement seen in Figs.~\\ref{Fig7}a-b (say) might be topologically protected against structural evolution towards long-range order. This should be contrasted with the dislocation network of a crystal, whose defect algebra is abelian (additive) and thus far less constrained in terms of the available configurations structural evolution could lead to. This perspective is certainly an interesting line of further research.\n\n\\section{Summary and concluding remarks} \\label{sec-conc}\n\nMolecular dynamics have been used to investigate model binary glass relaxation at temperatures above and below the glass transition temperature regime. The approximately linear relation between energy per atom and the fraction of atoms in a local icosahedral environment is found to extend to higher icosahedral compositions and therefore to significantly more relaxed amorphous structures. We identify this relation as a relaxation trajectory that:\n\\begin{enumerate}\n\\item occurs during isothermal protocols both above and below the glass transition temperature regime.\n\\item reflects a structural evolution in which the medium range order increases through a growing population of inter-connected Frank-Kasper polyhedra fragments.\n\\item apart from an energy shift, remains the same for a range of chemical compositions when the atom types have similar accessible characteristic volumes.\n\\item encompasses geometrically constructed nano-composite microstructures consisting of Laves C15 crystallites of a particular size embedded in an amorphous matrix consisting primarily of the larger atoms, and a micro-structural energy and length scale that is determined also by the interface energy.\n\\end{enumerate}\nThe mono-atomic amorphous interface region of the nano-composite is found to be strongly constrained, and therefore stabilized, by the surrounding Laves C15 crystallites. \n\nTaken together, the above trajectory implies a structural relaxation path of a binary glass that naturally leads to emergent structural length-scales, or spatial heterogeneities which might involve the crystallization of the Laves phase, or more generally topological close-packed phases. Such crystal structures are dominated by five-fold symmetric bonds which are closely associated with the local minimization of structural frustration within the glass. Thus the evolution of such spatial heterogeneities may now be understood as a manifestation of medium-range-ordered evolution that might eventually lead to nanometer-sized crystalline structures. Alternatively, via fig.~\\ref{Fig1}b, the emerging spatial heterogeneities will strongly correlate with the amount of energy stored within the glassy structure. This suggests, a regime of so-far unidentified and unexploited nano-composite structures, hidden in the lower-temperature regime below the nose in a temperature-time-transformation diagram, which will be a sensitive function of binary (or ternary) composition, annealing protocol and the stability of any remaining glassy matrix phase.\n\n\\section{Acknowledgments}\n\nThe present work was supported by the Swiss National Science Foundation under Grant No. 200021-165527 \n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\n\n\n\\section{Introduction}\nSupervision in machine learning is traditionally provided in the form of ground-truth labels that are assigned to every instance of a training data set. From this true assignment, a model can learn to predict a label for any other instance it is later presented with. But what if true labels are missing? Research on machine learning has explored a multitude of settings \\cite{hernandez2016weak} with weaker forms of supervision, e.g. noisy labels \\cite{menon2018learning,natarajan2013learning,scott2013classification,ghosh2015making}, (partial) equivalence relations \\cite{wagstaff2001constrained,kumar2007classification}, zero-shot \\cite{Romera-Paredes\/Torr\/2015} or few-shot learning \\cite{wang2020generalizing}, multiple-instance learning \\cite{foulds2010review,dietterich1997solving}, and learning from label proportions \\cite{Stolpe\/Morik\/2011a}, just to name a few.\n\nModern gamma ray astronomy addresses its lack of labels differently: the field relies on sophisticated simulations as alternative ground-truth data providers \\cite{Bockermann\/etal\/2015a,anderhub2013design}. However, these simulations are computationally demanding; reducing the amount of simulated training data yields considerable savings of time and energy.\n\nIn this paper, we explore the potential of weak supervision for \\emph{gamma hadron classification} \\cite{bock2004methods,Buschjaeger\/etal\/2020a}, a central learning task in gamma ray astronomy. By learning directly from the unlabeled data that a telescope produces anyway, we solve this task without the need for costly simulations. Namely, we learn a strong classifier by adopting one of the features, namely the origin of gamma and hadron particles, as a noisy weak label.\n\nDiscussing our approach within the theoretical framework of noisy labels even reveals a more general contribution of our work: unlike existing approaches in noisy label learning \\cite{natarajan2013learning,scott2013classification,ghosh2015making}, we do not require any knowledge about the class-wise noise rates.\nIn fact, all we assume is that the noisy-positive class contains more truly-positive instances than the noisy-negative class. Based on this weak assumption, we propose an interpretable optimization criterion for classifier training in general.\n\nSec.~\\ref{sec:background} presents our contributions against the background of noisy label learning. Gamma ray astronomy is detailed in  Sec.~\\ref{sec:physics}. Sec.~\\ref{sec:algorithms} proposes two algorithms for noisy supervision: we either cluster all unlabeled instances and make a post-hoc assignment from clusters to classes or we split the feature space via decision tree induction. Sec.~\\ref{sec:experiments} shows that these weakly supervised algorithms compete with the strongly supervised state of the art. We connect our findings to other weak learning paradigms in Sec.~\\ref{sec:related} and conclude with Sec.~\\ref{sec:conclusion}.\n\n\n\n\n\\section{Background: Learning Under Class-Conditional Label Noise}\n\\label{sec:background}\nIn learning under random label noise, we only have access to training labels $\\widehat{y} \\in \\{+1, -1\\}$ that are noisy in the sense of being randomly flipped versions of the ``clean'' ground-truth labels $y \\in \\{+1, -1\\}$. In particular, the class-conditional random label noise model (CCN) \\cite{menon2015learning,natarajan2013learning,scott2013classification,blum1998combining} states that the labels are flipped according to probabilities $p_+$ and $p_-$ that depend on the true class $y$, but not on the features. Feasible learning under CCN is conventionally established by assuming that the noisy labels are correct on average, i.e. that\n\\begin{equation}\n  p_+ + p_- < 1,\n    \\quad\\text{where}\\quad\n    p_i = \\mathbb P(\\widehat{Y} = -i \\mid Y = i).\n    \\label{eq:ccn}\n\\end{equation}\n\n\n\nTheoretic studies \\cite{menon2015learning,natarajan2013learning,scott2013classification,blum1998combining} have shown that even an \\emph{optimal} classifier can be learned under CCN. Remarkably, the only difficulty lies in finding the decision threshold $\\theta \\in \\mathbb R$; an optimal scoring function $h : \\mathcal X \\rightarrow \\mathbb R$, which is to be thresholded at $\\theta$, can be learned directly from the noisy labels.\n\n\\begin{remark}\n  Assume that we knew the true probability density $\\mathbb P(X \\mid \\widehat{Y})$ of the features conditioned on the noisy labels. According to Scott et al. \\cite[see Proposition~1]{scott2013classification}, there is a one-to-one correspondence between noisy thresholds $\\theta\\in\\mathbb R$ and clean thresholds $\\lambda\\in\\mathbb R$, i.e. \\mbox{$\\forall\\,x\\in\\mathcal X$} \\,\\mbox{$\\forall\\,\\lambda\\geq 0$} \\,\\mbox{$\\exists\\;\\theta\\geq 0$} such~that\n\n  %\n  \\begin{displaymath}\n    \\frac{\\mathbb P(x \\mid Y=+1)}{\\mathbb P(x \\mid Y=-1)} > \\lambda\n      \\;\\Leftrightarrow\\;\n      \\frac{\\mathbb P(x \\mid \\widehat{Y}=+1)}{\\mathbb P(x \\mid \\widehat{Y}=-1)} > \\theta.\n  \\end{displaymath}\n\n  Due to this one-to-one correspondence, noisy and clean classifiers have same receiver operator characteristic (ROC) if they are based on the true class-conditional densities of the clean and noisy data. Depending on the data and the model, we might even be able to learn a good approximation of $\\mathbb P(X \\mid \\widehat{Y})$ solely from noisy data; all we need to do then is to find a decision threshold that is optimal with respect to the clean labels.\n\\end{remark}\n\nIn fact, the area under the ROC curve is immune to CCN \\cite{menon2015learning}, also when the classification model is not based on estimates of the densities $\\mathbb P(X \\mid \\widehat{Y})$.\n\n\n\n\\subsection{Finding the Optimal Decision Threshold under CCN}\nOther measures, such as accuracy, can be optimized in two steps: first, we fit a scoring function from the noisy labels and then set the decision threshold to a value that is optimal in terms of the clean labels \\cite{menon2015learning}. Analogously, we can employ a label-dependent weighting of the loss function already during training \\cite{natarajan2013learning,ghosh2015making}. However, finding an optimal threshold or optimal loss weights requires precise knowledge of the noise rates $p_+$ and $p_-$.\n\nIf $p_+$ and $p_-$ are unknown, we can resort to estimating them from the noisy data \\cite{menon2015learning,scott2013classification}. Without access to clean labels, this approach requires additional assumptions, such as the existence of clean labels at a point in feature space where the other clean class has zero probability \\cite{menon2015learning}. In the context of gamma ray astronomy, we conceive such assumptions as being overly restrictive. Under class imbalance, the estimation of noise rates becomes even more difficult \\cite{mithal2017rapt}.\n\nAlternatively, if we had access to a small amount of clean labels, we could tune the decision threshold directly on this clean set \\cite{blum1998combining}. However, cleanly labeled real data is not available to gamma ray astronomers.\n\n\n\n\\subsection{Our Contributions}\nFor gamma hadron classification, astronomers have proposed to determine the decision threshold $\\theta$ of a scoring function by maximizing a \\emph{criterion} that resembles a hypothesis test.\nIn the more general scope of CCN, we demonstrate several qualities of this strategy:\n\\begin{itemize}\n  \\item straightforward: choosing $\\theta$ by computing and maximizing the criterion is easy with standard optimization tools; noise rates do not need to be estimated.\n  \\item interpretable: the criterion resembles a hypothesis test which tells us whether learning is actually feasible with the given noisy labels.\n  \\item general: we assume $p_-$ to be known, but do not introduce assumptions otherwise needed \\cite{menon2015learning} to estimate an unknown $p_+$ from noisy data.\n  \\item versatile: the criterion is not limited to finding $\\theta$; we propose learning algorithms which simultaneously fit a scoring function and a threshold (i.e. a complete classifier) through maximizing the criterion.\n\\end{itemize}\n\nAdopting a term from astronomy, we call this criterion the \\emph{significance of detection}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Classification in Gamma Ray Astronomy}\n\\label{sec:physics}\nModern gamma-ray astronomy utilizes imaging air Cherenkov telescopes \\cite{actis2011design,tridon2010magic,anderhub2013design} to gather information about cosmic gamma ray emitters, e.g. supernova remnants and active galactic nuclei.\nThese telescopes record the interactions of cosmic gamma rays within Earth's atmosphere. The properties of the gamma rays, and therefore some important properties of their cosmic origin, can be reconstructed from these recordings.\n\nMachine learning plays a crucial role in the reconstruction of the telescope data \\cite{Bockermann\/etal\/2015a,Noethe\/2020a}, as it tackles the assignment of particle properties (labels) to telescope recordings (instances).\nOne fundamental prediction task is to separate the interesting gamma ray observations from \\emph{hadronic particle} observations \\cite{bock2004methods,Buschjaeger\/etal\/2020a}. This binary classification problem emerges from the fact that imaging air Cherenkov telescopes are not only triggered by gamma rays; hadronic particles exhibit a similar behavior in the atmosphere and are therefore recorded, too.\nConsequently, they must be filtered out of the data sample, to not aggravate the actual analysis of cosmic gamma ray sources\n\nThe state of the art in this \\emph{gamma hadron classification} task is to learn a classifier with strong supervision, i.e. with large amounts of simulated data including ground-truth labels. Here, the need for simulations arises from the fact that the real telescope recordings do not have ground-truth labels and manually labeling the recordings is not feasible. To ensure valid prediction models, astronomers are setting high standards for the simulation: it must resemble the entire detection process, from cascading particle interactions within the atmosphere up to artifacts from the telescope hardware and camera electronics. Consequently, running these simulations is time- and energy-consuming. Moreover, closing the gap between simulated and real telescope data requires to manually fine-tune countless parameters of the simulator, an undertaking that is prone to errors.\n\n\n\n\n\n\n\n\\subsection{Detection of Gamma Ray Emitters}\nOne purpose of gamma hadron classifiers is to detect the cosmic sources of gamma radiation: if the telescope keeps pointing at a sky coordinate where a gamma ray source is assumed, does it ``see'' this gamma ray source? Seeing sources in the data is not only aggravated by the presence of hadronic particles, but also by the presence of a cosmic gamma ray background. In fact, background gamma rays can originate from every position in the sky. To detect a gamma ray emitter therefore means to confirm that this emitter produces a rate of gamma rays that outstands the background rate of gamma radiation.\n\nAn elegant way of simultaneously observing a gamma ray source and its background is to operate the telescope in the \\emph{Wobble} mode \\cite{fomin1994new} from Fig.~\\ref{fig:wobble}.\nThis default mode of operation places the source on a ring around the camera center. The region around this position is called the \\emph{On} region; it measures the rate of gamma rays that originate in the assumed source.\nThe additional \\emph{Off} regions on the same ring simultaneously measure the background rate, provided that no other gamma ray source falls into any of them.\nThe FACT telescope \\cite{anderhub2013design}, for instance, places six regions on a $0.6^{\\circ}$ ring inside its $4.5^{\\circ}$ field of view.\nEvery 20 minutes, another region is used as the \\emph{On} region, which causes the telescope to ``wobble'' from region to region while observing a fixed position in the sky.\nA gamma ray source is successfully detected when the \\emph{On} region exhibits a rate of gamma rays that is considerably larger than the average rate exhibited by the \\emph{Off} regions.\n\n\n\\begin{figure}\n  \\centering\n  \\include{img\/wobble}\n  \\caption{%\n    Illustration of the \\emph{Wobble} mode for source detection.\n    This mode observes the assumed source position, i.e. the \\emph{On} region, simultaneously with multiple background regions. The FACT telescope \\cite{anderhub2013design} employs five background regions, but other segmentations of the camera plane are also feasible.\n  }\n  \\label{fig:wobble}\n\\end{figure}\n\n\n\n\n\\subsection{Significance of Detection}\nFormally, source detection is established as a hypothesis test. This test aims at rejecting the null hypothesis that the \\emph{On} region does \\emph{not} exceed the background rate. The statistical significance of such a rejection can then be interpreted as a measure of certainty: if this value is above some threshold, we can declare a successful detection. Moreover, values that exceed this threshold by large indicate that less data would have been needed for a detection; if the data set is fixed, the significance of detection therefore also measures the \\emph{detection efficiency} of the telescope and the data analysis.\n\nIn this paper, we employ the test statistic by Li and Ma \\cite{Li\/etal\/1983a}. Let $n^\\gamma_\\mathrm{on}$ and $n^\\gamma_\\mathrm{off}$ denote the number of gamma rays which originate in the \\emph{On} and \\emph{Off} regions, respectively. To compare the average rates in these regions, we also need a scaling factor $\\alpha = \\sfrac{A_\\mathrm{on}}{A_\\mathrm{off}}$, which is the ratio of the \\emph{On} area and all \\emph{Off} areas. For instance, the five equi-sized background regions of the FACT telescope, see Fig.~\\ref{fig:wobble}, induce a factor $\\alpha = \\sfrac{1}{5}$. The significance of detection is given by\n\\begin{align}\n  f(n^\\gamma_\\mathrm{on}, n^\\gamma_\\mathrm{off}) = \\bigg[&\n         2n^\\gamma_\\mathrm{on} \\cdot \\ln\\left( \\frac{1+\\alpha}{\\alpha} \\cdot \\frac{n^\\gamma_\\mathrm{on}}{n^\\gamma_\\mathrm{on}+n^\\gamma_\\mathrm{off}} \\right)\n         \\label{eq:lima} \\\\%\n    +\\,& 2n^\\gamma_\\mathrm{off} \\cdot \\ln\\left( (1  + \\alpha)          \\cdot \\frac {n^\\gamma_\\mathrm{off}}{n^\\gamma_\\mathrm{on}+n^\\gamma_\\mathrm{off}} \\right)\n  \\bigg]^{1\/2} \\nonumber\n\\end{align}\n\nThis statistic measures the significance of detection in the unit of ``sigmas'' ($\\sigma$). Values of five and more sigma, i.e. $f(n^\\gamma_\\mathrm{on}, n^\\gamma_\\mathrm{off}) > 5$, are usually considered successful detections in gamma ray astronomy.\n\nNote that hadronic instances are completely ignored in Eq.~\\ref{eq:lima}. This fact results in a trade-off between precision and recall of the gamma hadron classifier: if all instances were classified as hadrons, both $n^\\gamma_\\mathrm{on}$ and $n^\\gamma_\\mathrm{off}$ would be zero and so would be the statistic; if all instances were classified as gammas, it would be unlikely that the \\emph{On} region contains significantly more positive events than the \\emph{Off} regions, so that the resulting statistic would be low, too. In contrast, a perfect classifier would achieve high significance values, provided that the \\emph{On} region contains a gamma ray source indeed.\n\nThis line of thought has motivated astronomers to leverage Eq.~\\ref{eq:lima} in choosing the decision threshold of their gamma hadron classifiers. First, such a classifier is trained, usually with strong supervision, i.e. with a labeled training set that is provided by a costly simulation. Second, the decision threshold of the trained classifier $h : \\mathcal X \\rightarrow \\mathbb R$ is chosen such that the value of Eq.~\\ref{eq:lima} is maximized on real-world data, for which we know the regions of origin, but not the labels.\n\nFormally, let $X = (\\vec{x}_1, \\dots, \\vec{x}_n, \\; \\vec{x}_{n+1}, \\dots, \\vec{x}_{n+m})$ be the concatenation of $n$ unlabeled real-world instances from the \\emph{On} region and $m$ instances from the \\emph{Off} region. Choose the decision threshold $\\theta^\\ast$ such that\n\\begin{equation}\n  \\theta^\\ast = \\argmax_{\\theta \\in \\mathbb R} \\, f\\left(\n    \\sum_{i=1}^n \\mathds{1}_{h(\\vec{x}_i) > \\theta},\n    \\sum_{i=1}^m \\mathds{1}_{h(\\vec{x}_{n+i}) > \\theta}\n  \\right)\n  \\label{eq:threshold-tuning}\n\\end{equation}\n\n\n\\subsection{Noisy Labels from On and Off Information}\nWe establish noisy label learning for gamma hadron classification by identifying the \\emph{On} and \\emph{Off} regions as noisy labels for gamma rays and hadronic particles. This identification is justified if the \\emph{On} region pictures a gamma ray source: only then, we can expect a correspondence between clean labels $y$ and noisy labels $\\widehat{y}$. Namely, we expect that the \\emph{On} region (our noisy-positive class) contains more gamma rays (truly-positive instances) than the average \\emph{Off} region (our noisy-negative class). To ensure that this assumption holds, we train with observations of known gamma ray sources.\n\\begin{figure}\n  \\centering\n  \\include{img\/noise-model}\n  \\caption{%\n    The class-conditional random noise model $y \\mapsto \\widehat{y}$ in the context of weakly supervised gamma hadron classification. We employ the \\emph{On} and \\emph{Off} regions as noisy labels for gamma rays and hadronic particles with noise rates $p_+$ and $p_-$.\n  }\n  \\label{fig:noise-model}\n\\end{figure}\n\nFig.~\\ref{fig:noise-model} presents our noisy label interpretation of \\emph{On} and \\emph{Off} annotations with the resulting class-wise noise rates $p_+$ and $p_-$. In gamma ray astronomy, we are facing $p_- \\approx \\frac{A_\\mathrm{on}}{A_\\mathrm{on} + A_\\mathrm{off}}$ in particular because the hadronic particles can originate in any sky position. Assuming that the \\emph{On} region contains more gamma rays than the average \\emph{Off} region translates to $p_+ < \\frac{A_\\mathrm{off}}{A_\\mathrm{on} + A_\\mathrm{off}}$, which nicely fits the standard CCN assumption from Eq.~\\ref{eq:ccn}.\nWe employ our noisy labels in three distinct ways:\n\\begin{itemize}\n  \\item We follow existing work on CCN by learning a scoring function from the noisy labels, which is then thresholded via an estimation of the noise rates. In particular, we consider the thresholding techniques by Menon \\cite{menon2015learning} and Mithal \\cite{mithal2017rapt} in our experiments.\n  \\item We learn a scoring function from the noisy labels again, but set the threshold by maximizing the significance of detection. This approach is similar to the one from Eq.~\\ref{eq:threshold-tuning}, with the difference that $h$ is trained with noisy labels rather than ground-truth.\n  \\item We learn a full classification model directly from the significance of detection, without separating the scoring function from the threshold. We develop the algorithms for this approach in the next section.\n\\end{itemize}\n\n\n\n\n\\section{Weakly-Supervised Algorithms based~on~the~Significance of Detection}\n\\label{sec:algorithms}\nWe propose two weakly supervised machine learning algorithms for binary classification under class-conditional label noise. Both algorithms learn by maximizing the significance of detection directly, thus overcoming the separation of the scoring function and the decision threshold that is usual in noisy label learning. While our algorithms are inspired by gamma hadron separation, they apply to any learning task that is aggravated by class-conditional label noise.\n\nFor generality, we omit the notion of the areas $A_\\mathrm{on}$ and $A_\\mathrm{off}$, which have defined $p_-$ and the $\\alpha$ factor in the astronomy domain. In equivalence to assuming the values of such areas, we instead assume $p_-$ to be known. About $p_+$, contrastingly, we assume no knowledge. Re-arranging the area-based definitions of $\\alpha$ and $p_-$ yields the $\\alpha$ factor from Eq.~\\ref{eq:lima} as\n\\begin{equation}\n  \\alpha \\;=\\; \\frac{p_-}{1-p_-}.\n\\end{equation}\n\nRemarkably, the hypothesis test from Eq.~\\ref{eq:lima} is now able to detect violations of Eq.~\\ref{eq:ccn}, the standard CCN assumption. Namely, if a significance outcome is small, say below a threshold of $5\\sigma$ for example, we know that $p_+$ does not obey Eq.~\\ref{eq:ccn} with sufficient certainty. In this case, learning from the given noisy labels is not feasible. This matter of interpretation is in contrast to existing methods \\cite{menon2015learning,mithal2017rapt}, which take Eq.~\\ref{eq:ccn} for granted.\n\n\n\n\n\n\n\n\n\\subsection{K-Means Detection}\nOur first approach is centered around the k-means clustering algorithm, a standard unsupervised method in the toolbox of data science \\cite{Hastie\/etal\/2003a}. The underlying assumption, as for supervised classification, is that the feature distributions of the two classes differ. Since clustering algorithms group similar feature vectors in the same clusters, some of these clusters are likely to contain an excess of one class or the other.\n\nLet $X=\\bigcup_{c=1}^k X_c$ with $X_c \\cap X_{c'} = \\emptyset$ for all $c \\not = c'$ be the partitioning of all observations, as computed by $k$-means or any other cluster partitioning algorithm. Moreover, let $n_\\mathrm{on}^{(c)}$ and $n_\\mathrm{off}^{(c)}$ be the number of noisy-positive and noisy-negative instances in each cluster, respectively. We achieved satisfactory results with a standard $k$-means++ implementation from scikit-learn \\cite{Pedregosa\/etal\/2011}, which applies a greedy initialization strategy \\cite{Arthur\/Vassilvitskii\/2007a} and minimizes the squared Euclidean distance between the observations and the nearest of $k$ cluster centroids using Lloyd's algorithm. The only hyper-parameter of this approach is $k$, the number of clusters.\n\nFrom such a partitioning, we compute the optimal assignment of clusters to \\emph{clean} classes by maximizing the significance of detection from Eq.~\\ref{eq:lima}. Namely, we find the optimal assignment $a^\\ast \\in \\{0,1\\}^k$, such that\n\\begin{equation}\n  a^\\ast =\n\t\\argmax\\limits_{a \\in \\{0,1\\}^k} f\\left(\\sum\\limits_{c=1}^k a_c n_\\mathrm{on}^{(c)}, \\, \\sum\\limits_{c=1}^k a_c n_\\mathrm{off}^{(c)}\\right).\n  \\label{eq:assignment}\n\\end{equation}\n\nFor computational efficiency, we relax this problem to a continuous, convex parameter space $a\\in[0,1]^k$. This relaxation does not change the optimal solution $a^*$, but allows us to apply first-order constrained optimization methods to efficiently find the optimal assignment.\n\n\nAt inference time, we determine the class of an event $\\vec{x}$ by first assigning it to its nearest cluster centroid, thereby determining its cluster $c(\\vec{x})$. Then, we return the clean class $a^*_{c(\\vec{x})} \\in \\{0,1\\}$ that is associated with this cluster.\n\n\n\n\\subsection{Decision Tree Detection}\nOur second approach builds on decision tree induction. We modify this popular method for supervised classification such that it directly maximizes the significance of detection with noisy-labeled training data.\n\nLike classic decision trees, our algorithm works by recursively partitioning the training set in a greedy fashion. At each node, we split the data by comparing a single feature to a learned threshold. To find the best split, we test every possible value in the data and evaluate the significance of detection for both sides of the partition. When the value does not increase, as compared to not splitting, we return. Alternatively, we return when a user-defined maximum tree depth is reached.\n\nWe denote by $X[x_{\\cdot j} \\leq \\theta]$ all examples in the current node where the $j$-th feature is less or equal $\\theta$ and analogously define $X[x_{\\cdot j} > \\theta]$ the complement. For each candidate value of $j$ and $\\theta$, we evaluate Eq.~\\ref{eq:lima} to derive the optimal split as\n\\begin{align}\n\tj^*, \\, \\theta^* &= \\argmax_{j, \\theta} \\; \\max\\left\\{f_j^{\\leq\\theta}, \\; f_j^{\\vphantom\\leq>\\theta}\\right\\}, \\label{eq:split}\\\\\n\t\\nonumber\\text{where} \\enspace f_j^{\\leq\\theta} &= f\\left(n_\\mathrm{on}(X[x_{\\cdot j} \\leq \\theta]), \\, n_\\mathrm{off}(X[x_{\\cdot j} \\leq \\theta])\\right) \\\\\n\t\\nonumber\\text{and} \\enspace f_j^{>\\theta} &= f\\left(n_\\mathrm{on}(X[x_{\\cdot j} > \\theta]), \\, n_\\mathrm{off}(X[x_{\\cdot j} > \\theta])\\right).\n\\end{align}\n\nThe best split according to Eq.~\\ref{eq:split} can be evaluated efficiently by sorting the data according to each of the features. Considering only the maximum of $f_j^{\\leq\\theta}$ and $f_j^{\\vphantom\\leq>\\theta}$ amounts to the fact that a greedy maximization of Eq.~\\ref{eq:lima} does not require balanced splits: if the other side of a split results in a low sigma value, we can either discard this side without harming the overall significance of detection or we can split this side further in a later step.\n\nAs an alternative to Eq.~\\ref{eq:split}, we can conceive a slightly modified criterion that balances both sides of each split. It can be shown that the following alternative is equivalent to maximizing the traditional information-gain split criterion of decision trees with noisy labels:\n\\begin{equation}\n\tj^*, \\, \\theta^* = \\argmax_{j, \\theta} \\; \\left(f_j^{\\leq\\theta}\\right)^2 + \\left( f_j^{\\vphantom\\leq>\\theta}\\right)^2 \\label{eq:split2}\n\\end{equation}\n\n\n\n\n\n\\subsection{Ensemble Variants}\nBoth algorithms lend themselves for application in an ensemble. We choose a bagging approach \\cite{Breiman\/1996a} as with classical random forests \\cite{Breiman\/2001a}. Instead of a single model, we compute $T$ models and employ randomization techniques to obtain $T$ different decision functions.  We use the following sources of randomization:\n\\begin{itemize}\n  \\item In the K-Means ensemble, we base each of the $T$ clusterings on a random subset of the features as well as a boostrapped sample of the events, i.e. we sample $n$ events with replacement.\n  \\item In the Decision-Forest ensemble, we use a bootstrapped sample of the events and choose the optimal splits only on a random subset of features of size $\\lfloor\\sqrt d \\rfloor$ that we sample at each split.\n\\end{itemize}\n\nWe combine the $T$ models by returning the mean value of their classification outputs. \nTo tune the decision threshold of the ensemble methods, we optimize Eq.~\\ref{eq:threshold-tuning}.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn the following, we investigate the performance of our proposed methods in comparison with existing methods from CCN learning and in comparison with strongly supervised methods.\nThe implementation of our methods and experiments is published on GitHub\\footnote{See \\url{https:\/\/github.com\/tudo-ls8\/unsupervised_fact\/}}.\n\n\n\n\\subsection{Validation Protocol for the Telescope Data}\n\\label{sec:validation}\nDue to the lack of labeled real-world data in gamma ray astronomy, we can report the real-world performance of our models only in terms of their significance of detection, as according to Eq.~\\ref{eq:lima}, but not in terms of supervised measures such as accuracy or F1 score. This matter also has an implication on the way we cross-validate our results: we repeatedly train and predict on disjoint splits of the data, as usual. However, we store the predictions of each holdout set, rather than computing separate scores for the folds. This way, we obtain predictions for all instances in the data, but each prediction is based on a model that has not seen the instance during training. The reported values of the significance of detection are then obtained with the full data set.\n\nFor reliable results, the training and hold-out sets of the data should be as independent as possible. Since recording conditions, like the brightness of the night sky, change during the course of data taking, we group the data based on an attribute that corresponds to the time of the day. We found that this grouping yields the lowest values of the significance of detection, giving us the most cautious estimates of model performances.\n\n\n\\subsection{Detection of the Crab Nebula}\nWe use the open data sample of the FACT collaboration.\\footnote{\\url{https:\/\/factdata.app.tu-dortmund.de\/}}. This data contains five nights of wobble mode observations, totalling 757,993 instances, where the telescope is pointed at the Crab Nebula, a bright supernova remnant. Each instance in the dataset corresponds to a camera recording of an event. We use 22 handcrafted, high-level features extracted from the camera recordings\\cite{Bockermann\/etal\/2015a}. These features include statistics like the number of pixels that recorded photons and geometric traits \\cite{Hillas\/1985} that describe the shape of the event on the camera surface. We expect that the vast majority of events are induced by hadronic particles and that only around 0.1\\% to 1.0\\% of them are gamma ray events. The data contains 4,332 \\emph{On} events and 17,229 \\emph{Off} events.\n\nWe compare our methods to two different approaches for learning with class-conditional label noise, namely to the approaches by Menon et al \\cite{menon2015learning} and by Mithal et al \\cite{mithal2017rapt}, the latter of which is designed for the particular case of imbalanced class frequencies. Both methods start by learning any standard classifier from weak labels, which is decision trees and random forests in our experiments. In a second step, both methods estimate the class-wise noise rates from the noisy-labeled data and adapt their decision thresholds according to these estimates.\n\nWe further establish a comparison to the gamma hadron predictions from the default and fully supervised pipeline\\footnote{\\url{https:\/\/github.com\/fact-project\/open_crab_sample_analysis}}. These predictions are produced by a random forest classifier that is trained on 240,000 simulated and ground-truth labeled instances with balanced classes. \n\\begin{table}\n\\caption{Significance scores for the detection of the Crab Nebula (see Eq.~\\ref{eq:lima}, higher is better). Our algorithms are compared to existing CCN methods and to the strongly supervised state-of-the-art. This table presents only the best performances; additional results are given in the supplementary material.}\n\\label{table:insample_results}\n\\centering\n\\addtolength{\\tabcolsep}{-1pt}   \n\\begin{tabular}{@{}llcc@{}}\n\\toprule\n\\multicolumn{2}{l}{Model}         & Single Model & Ensemble $(T=100)$ \\\\ \\midrule\n\\multicolumn{2}{l}{\\textbf{k-Means} (Eq.~\\ref{eq:assignment})}   & &          \\\\\n         & k=48                   &\\textbf{19.5799 $\\pm$ 0.3708}&$23.1176 \\pm 0.1974$\\\\\n         & k=256                  &$17.7766 \\pm 0.5713$&\\textbf{24.3246 $\\pm$ 0.4966}\\\\\n\\multicolumn{2}{l}{\\textbf{LiMa Tree} (Eq.~\\ref{eq:split})} & & \\\\\n         & max\\_depth=4           &\\textbf{21.1666 $\\pm$ 0.0000}&$23.3015 \\pm 0.3321$\\\\\n         & max\\_depth=6           &$18.4227 \\pm 0.0000$&\\textbf{25.1085 $\\pm$ 0.2236}\\\\\n\\multicolumn{2}{l}{\\textbf{Noisy Tree} (Eq.~\\ref{eq:split2})} & & \\\\         \n         & max\\_depth=4           &\\textbf{24.2899 $\\pm$ 0.0000}&$25.3015 \\pm 0.3321$\\\\\n         & max\\_depth=8           &$21.0170 \\pm 0.0000$&\\textbf{26.3175 $\\pm$ 0.2479}\\\\\n\\multicolumn{2}{l}{\\textbf{Menon Tree} \\cite{menon2015learning}} & & \\\\         \n         & max\\_depth=4           &\\textbf{24.2899 $\\pm$ 0.0000}&$25.2321 \\pm 0.1812$\\\\\n         & max\\_depth=7           &$18.3020 \\pm 0.0000$&\\textbf{26.0165 $\\pm$ 0.2950}\\\\\n\\multicolumn{2}{l}{\\textbf{Mithal Tree} \\cite{mithal2017rapt}} & & \\\\         \n         & max\\_depth=4           &\\textbf{24.2899 $\\pm$ 0.0000}&$25.0858 \\pm 0.2089$\\\\\n         & max\\_depth=8           &$19.7144 \\pm 0.0000$&\\textbf{26.4465 $\\pm$ 0.1804}\\\\\n\\multicolumn{2}{l}{\\textbf{Supervised} (SOTA)} & & \\\\         \n         & DT \/ RF                &\\textbf{24.6506 $\\pm$ 0.0000}&\\textbf{26.2526$\\pm$ 0.2024}\\\\\n\\bottomrule\n\\end{tabular}\n\n\\end{table}\n\nWe test a number of different hyper-parameters per model and repeat each setup ten times to report average values and standard deviations, as displayed in Table~\\ref{table:insample_results}. We see that the noisy-label approach to gamma hadron classification with random forests performs as good as the traditional approach that is based on costly, simulated training data with full supervision. This remarkable result could shape the way in which future telescopes approach their analysis pipelines: it demonstrates that we can benefit from real recordings for training, not only as background events, as it has been done before \\cite{Acciari\/etal\/2020}, but also for learning about the gamma class. \nGamma hadron classification could become even more data driven than today and less sensitive to imperfections and biases that are exhibited by the simulations.\n\nOnce having decided to use noisy labels, the choice of the particular method for tuning the decision threshold is only a secondary concern, as all of the tested approaches yield high significance scores between 25.0 and 26.4 sigmas. The noisy random forest (Eq.~\\ref{eq:split2}), Menon forest \\cite{menon2015learning} and Mithal forest \\cite{mithal2017rapt} even outperform the supervised model on average, albeit by a slim margin.\n\nThe plausibility of these scores is verified by another experiment in which we violate the assumption that the \\emph{On} region contains more gamma events than any of the \\emph{Off} regions. Namely, we artificially remove the \\emph{On} region and declare one of the \\emph{Off} regions as being the new \\emph{On} region. All our methods now return approximately zero sigmas, which means that the violation of assumptions is successfully detected. The plausibility of the scores in Tab.~\\ref{table:insample_results} is further supported by the observation that our weakly supervised predictions achieve high correlations with the strongly supervised state-of-the-art and high classification performances on simulated, ground-truth test data. We further detail these plausibility checks in the appendices~\\hyperref[sec:apx:violated]{A} and~\\hyperref[sec:apx:agreement]{B}.\n\n\n\n\n\n\n\n\n\\subsection{Results in Other Application Domains}\nOur proposed methods are not limited to gamma ray astronomy. In fact, methods based on the significance of detection can be applied in any application domain where we expect substantially more true positives in the noisy-positive class than we expect in the noisy-negative class. We support this claim with experiments on standard imbalanced datasets where we artificially inject class-dependent noise into the labels. We focus on the setting $p_- = \\frac{1}{2}$ and consequently set $\\alpha=1$ in Eq.~\\ref{eq:lima}. Hence, the null-hypothesis that we intend to reject is that the noisy-positive class and the noisy-negative class contain the same number of true positives.\n\nIn this experiment, we evaluate random forests that are trained on noisy labels and use 10-fold stratified cross-validation to estimate the F1 score. We optimize the decision thresholds on the out-of-bag predictions, such that we do not need a separate calibration dataset. Like before, we compare our significance-based thresholds to the thresholds that have been proposed earlier \\cite{menon2015learning,mithal2017rapt}.\nWe try different levels of label noise, as presented in Tab.~\\ref{table:classification_results} and in appendix~\\ref{sec:apx:other}. We use all datasets available in the \\texttt{imblearn} library \\cite{lemaitre2017imbalanced} with at most 100 features and at least 200 samples for the minority class.\n\nWe see that our two methods, which are based on significance-based thresholds, outperform the existing methods on 8 out of 10 datasets. Our forest based on Eq.~\\ref{eq:split2} has the highest average F1 score. On the other datasets, our methods do not dramatically deviate from the winner, whereas both competitors exhibit some catastrophic failures. The critical difference diagram \\cite{demsar2006statistical} in Fig~\\ref{fig:ccd} confirms that the Noisy Forest significantly outperforms the existing two approaches.\n\n\\begin{table}\n\\caption{Averaged F1 scores (20 trials, higher is better) on imbalanced data sets with noise levels $p_+=0.1$ and $p_-=0.5$.}\n\\label{table:classification_results}\n\\centering\n\\addtolength{\\tabcolsep}{-3pt}   \n\\scriptsize\n\\begin{tabular}{lcccc}\n\\toprule\nDataset        & LiMa Forest         & Noisy Forest        & Menon Forest         & Mithal Forest       \\\\ \\midrule\nsatimage       & 0.563 $\\pm$ 0.012   & \\textbf{0.602 $\\pm$ 0.010} & 0.549 $\\pm$ 0.0055  & 0.563 $\\pm$ 0.061 \\\\\nproteinhomo    & 0.788 $\\pm$ 0.002   & \\textbf{0.790 $\\pm$ 0.002} & 0.339 $\\pm$ 0.0076  & 0.786 $\\pm$ 0.002 \\\\\nopticaldigits  & 0.813 $\\pm$ 0.008   & \\textbf{0.859 $\\pm$ 0.008} & 0.843 $\\pm$ 0.0051  & 0.826 $\\pm$ 0.007 \\\\\npendigits      & \\textbf{0.961 $\\pm$ 0.006}   & 0.955 $\\pm$ 0.002 & 0.945 $\\pm$ 0.003  & 0.961 $\\pm$ 0.003 \\\\\nletterimg      & 0.793 $\\pm$ 0.013   & 0.809 $\\pm$ 0.007 & 0.809 $\\pm$ 0.005  & \\textbf{0.810 $\\pm$ 0.007}\\\\\ncoil2000       & 0.115 $\\pm$ 0.003   & 0.146 $\\pm$ 0.004 & \\textbf{0.160 $\\pm$ 0.002}  & 0.114 $\\pm$ 0.003 \\\\\nthyroidsick    & 0.769 $\\pm$ 0.015   & \\textbf{0.786 $\\pm$ 0.008} & 0.775 $\\pm$ 0.008  & 0.400 $\\pm$ 0.176 \\\\\nsickeuthyroid  & \\textbf{0.836 $\\pm$ 0.006}   & 0.772 $\\pm$ 0.011 & 0.701 $\\pm$ 0.009  & 0.678 $\\pm$ 0.136 \\\\\nmamography     & 0.509 $\\pm$ 0.023   & \\textbf{0.569 $\\pm$ 0.012} & 0.484 $\\pm$ 0.011  & 0.103 $\\pm$ 0.037 \\\\\nabalone        & 0.370 $\\pm$ 0.012   & \\textbf{0.384 $\\pm$ 0.006} & 0.359 $\\pm$ 0.004  & 0.300 $\\pm$ 0.038 \\\\[1pt]\navg. rank      & 2.500               & \\textbf{1.600}      & 2.800                & 3.100              \\\\\navg. f1        &  0.658              & \\textbf{0.667}      & 0.596                & 0.554              \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}[!b]\n  \\centering\n  \\include{img\/ccd}\n  \\caption{%\n    Critical difference diagram \\cite{demsar2006statistical}, based on Table~\\ref{table:classification_results}, at a confidence level of 0.9. Methods are connected with horizontal bars if and only if a Holm-corrected Wilcoxon signed-rank test cannot significantly distinguish their pairwise performances.\n  }\n  \\label{fig:ccd}\n\\end{figure}\n\n\n\\section{Related Work}\n\\label{sec:related}\nIn this work, we have addressed class-conditional label noise (CCN) for weakly supervised binary classification. Since the background of this particular learning task is already detailed in Sec.~\\ref{sec:background}, we focus on connections to other strands of weak supervision in the following.\n\nBeyond the CCN noise model, other types of label noise have been discussed in the scientific literature. For instance, the label noise of a distribution is called \\emph{uniform} \\cite{ghosh2015making} if each label has the same chance of being flipped, independent of the class and of the instance. Otherwise, if the chance of being flipped does not depend on the true class, but on the features, we are speaking of \\emph{purely instance-dependent} label noise \\cite{menon2018learning}. Generally speaking, learning is feasible under each of these noise models. However, learning under each noise model requires dedicated assumptions about the data and the learning method, with CCN combining the most appropriate assumptions in the context of gamma hadron classification.\nSince the distinction of gamma rays and hadronic particles naturally poses a \\emph{binary} classification task, we have decided to leave the difficulties that are otherwise imposed by multi-class CCN \\cite{patrini2017making} for future work on learning through the significance of detection.\n\nThe \\emph{On} and \\emph{Off} noisy labels considered here depict only a single source of weak supervision, i.e. a single noisy label is assigned to each training instance. If, however, multiple sources of weak labels were available, we could combine them \\cite{ratner2020snorkel} to learn classifiers from less data \\cite{zamani2018theory}. To this end, we see a potential for future work in gamma ray astronomy: domain experts might be able to define additional noisy labels, e.g. through heuristics that express physical background knowledge, that can be incorporated into the learning process.\n\n\n\n\n\\subsection{A label-free interpretation of noisy label learning}\nOur \\emph{On} and \\emph{Off} annotations can alternatively be understood as a special group attribute rather than being understood as noisy labels. Indeed, these annotations are not only available for the training instances, but also for any test instance that is to be predicted.\nIn particle physics, this label-free interpretation of \\emph{On} and \\emph{Off} information is actually the prevalent one \\cite{metodiev2017classification,cohen2018machine,desimone2019guiding}.\nIt allows us to draw additional connections of our work to multiple-instance (MI) learning \\cite{foulds2010review,dietterich1997solving} and learning from label proportions (LLP) \\cite{Stolpe\/Morik\/2011a}.\n\nThese strands of weak supervision challenge the traditional one-to-one correspondence between labels and instances. Namely, they assume that labels are only given for bags of instances but not for each instance individually. These bags can vary in size, which leads to a one-to-many correspondence between labels and instances. Similarly, equivalence relations \\cite{wagstaff2001constrained,kumar2007classification} exhibit a one-to-two correspondence; they assume labels only for pairs of instances. In all of these settings, the goal is to learn a classifier that predicts the labels of individual instances, just like in standard supervised learning, despite the fact that no individual labels are available for training.\n\nIn the label-free interpretation of our approach,\nall we know for supervision is that the significance of detection must be maximized. This maximization amounts to a one-to-all correspondence between supervision and instances which goes beyond MI learning and LLP: we are not learning from labels, but by maximizing a single domain-specific criterion over the \\emph{full} training set.\n\nStill, we have focused on the noisy label interpretation of \\emph{On} and \\emph{Off} annotations throughout this work. This focus is due to two reasons: to shed light on the theoretical properties of learning with \\emph{On} and \\emph{Off} information and to generalize our proposals to other CCN applications.\n\n\n\\subsection{Related applications from particle physics}\n\\emph{On} and \\emph{Off} annotations are often available in data from particle detectors. Similar to our use case, the goal of these detectors is to distinguish some kind of signal (like our gamma rays) from some kind of background instances (like our hadronic particles). Traditionally, this distinction is learned with full supervision provided by simulations. A more recent proposal, however, is to learn directly from \\emph{On} and \\emph{Off} regions, instead  \\cite{metodiev2017classification,cohen2018machine,desimone2019guiding}.\nSince this proposal essentially learns under CCN, it is facing the difficulty of choosing an optimal decision threshold for a scoring function that already is optimal.\n\nCohen et al. \\cite{cohen2018machine} suggest to choose the decision threshold from known label noise rates. The authors discuss the implications of noise rate uncertainties from a physics perspective, concluding that small inaccuracies of the assumed noise rates do not harm the classifier training. This domain-specific result falls in line with the general CCN theory by Natarajan et al. \\cite{natarajan2013learning}. If the noise rates of the full training set are unknown, they might at least be available for a small calibration set from which a decision threshold can be determined \\cite{metodiev2017classification}.\n\nThe work by de Simone et al. \\cite{desimone2019guiding} is mostly concerned with a two-sample hypothesis test. This test is used to decide whether there is \\emph{some} signal in the data, with the classification of individual instances being only proposed as a future extension of their method. However, this extension would require a meaningful decision threshold to be set, just like in any other CCN task.\n\nTwo-sample hypothesis testing, in general, is concerned with the detection of differences between the distributions of two data samples \\cite{gretton2012kernel}. Our learning criterion, the significance of detection, is motivated by one particular test which only checks whether the two \\emph{On} and \\emph{Off} samples differ in their numbers of positive instances. Other types of differences between the two data samples, which could be detected with other two-sample hypothesis tests, are not assumed here.\n\n\n\n\\section{Conclusion and Outlook}\n\\label{sec:conclusion}\n\nWe have proposed the significance of detection as a learning criterion for binary classification under class-conditional label noise. This criterion assumes one of the class-wise noise rates to be known, but does not need to estimate the other noise rate explicitly. We can use the criterion either to tune the decision threshold of any given classifier, or to learn a classification model from scratch. Most importantly, the significance of detection is interpretable in the sense that it indicates whether learning is actually feasible with the given noisy labels---a property that is established by an integrated hypothesis test. All of these qualities go beyond the state of the art in learning under class-conditional label noise.\n\nOur proposals stem from the gamma hadron classification task in gamma ray astronomy, where the significance of detection is already employed to tune the decision thresholds of strongly supervised models. These models are trained with simulated data, which is costly in terms of computation time and energy. Constrastingly, our weakly supervised algorithms learn directly from the real telescope, which produces an abundance of cheap but noisy-labeled data. Still, they match the performance of the strongly supervised state of the art in the gamma ray domain. We hope that this result in particular---and our discussion of gamma hadron classification as noisy label learning in general---will pave the way towards an even more data-driven gamma ray astronomy.\n\nFuture work should look into adapting the significance of detection for other data representations. A lot of engineering around strongly supervised classification went into the design of the numeric features \\cite{Bockermann\/etal\/2015a} that we have continued to use here. Learning from the underlying sensor data of the telescope, however, would require some form of representation learning, e.g. using deep neural networks \\cite{Buschjaeger\/etal\/2020a}. Looking into deep clustering approaches \\cite{Caron\/etal\/2018 ,Asano\/etal\/2020a}, for instance, might yield performances similar to the ones presented here, but without the need for manual feature engineering.\n\n \\section*{Acknowledgements}\n %\nThis work has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876, \"Providing Information by Resource-Constrained Data Analysis\", project A1 and C3. \\url{http:\/\/sfb876.tu-dortmund.de}. Parts of this work have been funded by the Federal Ministry of Education and Research of Germany as part of the competence center for machine learning ML2R (01|S18038A).\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\n\nA precise measurement of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{ub}|$ is important to understand the nature of weak interactions and CP violation in the Standard Model~\\cite{Petrella}. This value can be extracted from charmless semileptonic $B$ decays, where the final state hadron carries a $u$ quark and the lepton refers either to an electron or a muon. The advantage of these decays is that the leptonic and hadronic part of the final state do not interact strongly, and are thus easier to calculate theoretically. \n\nThis presentation will summarize the status of measurements of charmless semileptonic decays and $V_{ub}$ from Belle and BaBaR experiments.   \n\n\n\\section{Reconstruction of $B$ Mesons}\n\nThe study of $B$ mesons at $B$-factories are possible thanks to the tunning of the beam energies to the energy of the $\\Upsilon(4S)$ resonance. The decay products of this resonance are mostly $B\\bar{B}$ pairs, approximately 96\\% of the cases, which allows to perform  precision measurements of $B$ decays. For this purpose, one $B$ meson is reconstructed in the decay mode of interest, the signal, by combining a lepton, a charmless meson and a neutrino. Since the latter is not visible to the detector, it is inferred from the missing four-momentum, \n\\begin{equation}\n P_{\\rm miss} = P_{\\rm beam}-\\sum_{i}P_{i},\n\\end{equation}\nwhich is basically the difference between the beam four-momentum and the sum of all four-momenta of the reconstructed particles. \n\nNowadays, there are three methods for reconstructing $B$ mesons~\\cite{Dingfelder}, namely untagged, semileptonic tag and hadronic tag. In the untagged method, one only reconstruct the signal $B$, which offers a big statistical sample of signal candidates, but also incurs in a lot much bigger amount of background. Therefore, the signal reconstruction efficiency is high but the signal purity is poor. One of the challenges is the reduction of the background due to charmed semileptonic decays that is about 50 times more abundant than charmless semileptonic decays and the kinematic of the processes are very similar. Consequently, one has to apply very harsh selection on kinematic variables to suppress this background. The selection of a $B$ meson candidate is evaluated with two variables, the beam constrained mass $M_{bc}$ and the energy difference $\\Delta E$ given by,\n\\begin{equation}\n M_{\\rm bc} = \\sqrt{E_{\\rm beam}^{* 2 } - |\\vec{p}_{B}^{*}|^{2}}, \n\\end{equation}\n\\begin{equation}\n \\Delta E = E_{\\rm beam}^{*}- E_{B}^{*2}, \n\\end{equation}\nwhere $E_{\\rm beam}^{*}$ is the energy of the beam and $P_{B}^{*}$ and $E_{\\rm B}^{*}$ are the momentum and energy of the $B$ meson in the $\\Upsilon(4S)$ rest frame. \n\nThe other two techniques use information about the other $B$ meson or $B_{\\rm tag}$, this has a direct effect on the signal sample, the signal purity is increased. However, there is a price to pay, the reconstruction efficiency is dramatically lowered. In the semileptonic tag technique~\\cite{Hokuue}, the $B_{\\rm tag}$ is reconstructed in charmed semileptonic decays of the form $B\\to D^{(*)}\\ell\\nu$, with the $D^{(*)}$ meson reconstructed in hadronic modes. Since there are two semileptonic decays involved, additional requirements have to be considered. Foremostly, one identifies two leptons of opposite signs and then consider the presence of a neutrino on each side, following the massless neutrino hypothesis in the $\\Upsilon(4S)$ rest frame, i.e., $P_{\\nu}^{*2}=0=(P_{B}^{*}-P_{Y}^{*})^2$, where $P_{B}^{*}$ and $P_{Y}^{*}$ are the four-momentum of the $B$ meson and a virtual particle $Y$ respectively and all asterisked quantities are boosted to the $\\Upsilon(4S)$ resonance.  The $Y$ particle is formed by the combination of a charged lepton and the reconstructed meson. From this condition, one can infere the angle between the $B$ meson and the $Y$ system through $\\cos \\theta_{BY} = \\frac{2E_{B}^{*}E_{Y}^{*}-m_{B}^{2}m_{Y}^{2}}{2|\\vec{p}_{B}^{*}||\\vec{p}_{Y}^{*}|}$. This quantity can be used for the tag side, $\\cos\\theta_{B1}\n$, the signal side, $\\cos\\theta_{B2}$, together with the cosine of the angle between the two $Y$ particles on each side, $\\cos\\theta_{12}$, to form a discriminating variable to separate signal events from background events $x_{B}^{2}$ ( as is called within the Belle Collaboration or $\\cos^2\\phi_{B}$ for the case of BaBaR) given by\n\\begin{equation}\nx_{B}^{2}=1-\\frac{1}{\\sin^2 \\theta_{12}}\\left( \\cos^2\\theta_{B1} + \\cos^2\\theta_{B2}-2\\cos\\theta_{B1}cos\\theta_{B2}\\cos\\theta_{12}\\right), \n\\end{equation}\nthat for signal events this quantity has values in the interval $[0,1]$.\n\nFinally, in the hadronic tag method the $B_{\\rm tag}$  is fully reconstructed in hadronic decays leading to the knowledge of the momentum, flavor and charge of the signal $B$ and also offers a very high signal purity. Discriminating variables for this technique include the missing mass squared $M_{\\rm miss}^2$ that is basically the squared magnitud of the missing four-momentum, $M_{\\rm miss}^{2}=|P_{\\rm miss}|^{2}$, the hadronic invariant mass $m_{X}$ and the squared of the momentum transfer to the lepton $q^{2}=(P_{\\ell}+P_{\\nu})^2$. The significant high purity achieved with this technique leads to a high reduction of systematic uncertainties, but it also requires a large dataset to reduce statistical uncertainties.  \n\n\n\\section{Inclusive Charmless Semileptonic $B$ Decays}\n\nIn the inclusive approach the meson $X_{u}$ in the semileptonic decay $B\\to X_{u}\\ell\\nu$ is modeled as a sum over all hadronic final states that carry an $u$ quark~\\cite{Jin}. At parton level, the decay rate can be computed from a free quark decay as $\\Gamma(b\\to u\\ell\\nu)=\\frac{ G_{F}^{2}m_{b}^{5} }{ 192\\pi^{3}} |V_{ub}|^{2} $, where $G_{F}$ is the Fermi constant, $m_{b}$ the mass of the $b$ quark and $V_{ub}$ is the CKM matrix element for $b\\to u$ quark transition. However, the $b$ quark is not a free particle, instead it forms bound states inside the $B$ mesons, for which other considerations need to be taken into account, such as strong interaction effects or the movement of $b$ quarks inside B mesons. The Operator Product Expansion (OPE) and the Heavy Quark Expansion (HQE) are used to calculate the total decay rate for $B\\to X_{u}\\ell\\nu$ in powers of $1\/m_{b}$ with uncertainties at the $5\\%$ level. The additional factors to the free quark decay correspond to electroweak, perturbative and non-perturbative QCD corrections. Hence the decay rate can be written as~\\cite{AndersenGardi}:\n\\begin{equation}\n\\Gamma(B\\to X_{u}\\ell\\nu) = \\frac{ G_{F}^{2}m_{b}^{5} }{ 192\\pi^{3} } |V_{ub}|^{2} \\left[ 1 + a_{0}\\frac{ \\alpha_{s}(m_{b}) }{ \\pi }+a_{1}\\left( \\frac{ \\alpha_{s}(m_{b}) }{ \\pi }\\right)^2 +\\dots + {\\cal O}\\left(\\frac{ \\Lambda^{2} }{ m_{b}^{2}}\\right) \\right],\n\\end{equation}\nwhere $\\alpha_{s}$ and $\\Lambda$ are the strong and QCD coupling constants. \n\nThe application of very harsh selection on kinematic variables for reducing the large background from $B\\to X_{c}\\ell\\nu$, reduce the available phase space region. This represents a challenge on the theory side since the HQE convergence is spoiled and non-perturbative Shape Functions (SFs) need to be introduced~\\cite{Luth}. The configuration of the SFs cannot be determined from first principles, instead they rely on global fits to moments in inclusive $B\\to X_{c}\\ell\\nu$ and $B\\to X_{s}\\gamma$. From these fits heavy quark parameters can be determined, such as the mass of the $b$ quark $m_{b}$, the kinetic energy squared of the $b$ quark in the $B$ meson $\\mu_{b}^{2}$ and the chromomagnetic moment $\\mu_{G}^{2}$. Due to confinement and non-perturbative effects, the quantitative values of these parameters rely upon the theoretical framework in which they are defined. Therefore the results of the global fits need to be translated to other schemes depending on the QCD calculation used to extract $|V_{ub}|$.\n\nSome of the QCD calculations available to date are based on either OPE such as BLNP~\\cite{BLNP} and GGOU~\\cite{GGOU}, or non-perturbative QCD models like Dressed Gluon Exponentiation (DGE)~\\cite{AndersenGardi} and ADFR~\\cite{ADFR}. The main uncertainty in these calculations is coming from the uncertainty in the $m_{b}$. In the case of the ADFR calculation the dominant uncertainty is due to the uncertainty in the mass of the charm quark $m_{c}$.\n\n \\begin{figure}[htb]\n \\centering\n \\includegraphics[height=1.5in]{InclusiveTagged}\n \\caption{Fit Projections for $M_{X}$ and $q^{2}$. Left: results from Belle. Right: results from BaBaR.}\n \\label{fig:inclusive}\n \\end{figure}\n\nThe most recent results on inclusive measurements from Belle~\\cite{InclusiveBelle} and BaBaR~\\cite{InclusiveBabar} use their complete data sample, $467\\times 10^{6}$ $B\\bar{B}$ pairs for BaBaR and $657\\times 10^{6}$ $B\\bar{B}$ pairs for Belle, and a hadronic tag technique to reconstruct the $B_{\\rm tag}$. The two analyses differ in the treatment of the background, Belle~\\cite{InclusiveBelle} implements a multivariate discriminant while BaBaR~\\cite{InclusiveBabar} uses a cut based method for background suppression. BaBaR also reports measurements of $|V_{ub}|$ in several regions of phase space. The most precise results are extracted from a two dimensional fit to $(M_{X},q^{2})$ with no restriction on phase space other than $p_{\\ell}^{*}>1.0$ GeV, this selection allows to access approximately $90\\%$ of the total phase space. The distributions of the projections of the fit for $M_{X}$ and $q^{2}$ are shown in Fig.~\\ref{fig:inclusive}. The upper row corresponds to the projections as measured by the Belle Collaboration, for which $1032\\pm 91$ signal events are extracted leading to a partial branching fraction of $\\Delta {\\cal B} (B\\to X_{u}\\ell\\nu)=(1.96 \\pm 0.17_{\\rm stat} \\pm 0.16_{\\rm syst})\\times 10^{-3}$. The second row shows the projections from the BaBaR collaboration with $1430\\pm 130$ signal events and the corresponding partial branching ratio of $\\Delta {\\cal B} (B\\to X_{u}\\ell\\nu)=(1.80 \\pm 0.13_{\\rm stat} \\pm 0.15_{\\rm syst})\\times 10^{-3}$.\n\nThe values of $|V_{ub}|$ can be extracted using the relation~\\cite{InclusiveBabar}\n\\begin{equation}\n\\label{vubeq}  \n|V_{ub}|=\\sqrt{ \\frac{ \\Delta {\\cal B} (B\\to X_{u}\\ell\\nu )}{ \\tau_{B} \\Gamma_{\\rm theory}} },\n\\end{equation}\nwhere $\\tau_{B}$ is the $B$ lifetime and $\\Delta {\\cal B} (B\\to X_{u}\\ell\\nu )$ and $\\Gamma_{\\rm theory}$  are the measured partial branching ratio and the predicted decay rate in a given phase space region, respectively. The latter depends on the QCD calculation implemented. The inclusive $|V_{ub}|$ extracted with four different QCD calculations for the Belle and BaBaR collaborations together with the world average from HFAG  are shown in Table~\\ref{tab:inclusive}.  \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cccc}  \\hline\n             &  BLNP~\\cite{BLNP}  &  DGE~\\cite{AndersenGardi} & GGOU~\\cite{GGOU} & ADFR~\\cite{ADFR} \\\\ \\hline\nBelle~\\cite{InclusiveBelle}        &$4.47\\pm 0.27^{+0.19}_{-0.21}$&$4.60\\pm 0.27^{+0.11}_{-0.13}$&$4.54\\pm 0.27^{+0.10}_{-0.11}$&$4.48\\pm 0.30^{+0.19}_{-0.19}$  \\\\\nBaBaR~\\cite{InclusiveBabar}        &$4.28\\pm 0.24^{+0.18}_{-0.20}$&$4.40\\pm 0.24^{+0.12}_{-0.13}$&$4.35\\pm 0.24^{+0.09}_{-0.10}$&$4.29\\pm 0.24^{+0.18}_{-0.19}$  \\\\ \nWorld average~\\cite{HFAG}&$4.40\\pm 0.15^{+0.19}_{-0.21}$&$4.45\\pm 0.15^{+0.15}_{-0.16}$&$4.39\\pm 0.15^{+0.12}_{-0.14}$&$4.03\\pm 0.13^{+0.18}_{-0.12}$  \\\\ \\hline\n\\end{tabular}\n\\caption{Values of $|V_{ub}|$ reported by the Belle and BaBaR collaboration compared to the world average, where the first error is statistical and the second due to systematic uncertainties. The values are extracted using four different QCD calculation with the requirement $p_{\\ell}^{*}>1.0$ GeV.}\n\\label{tab:inclusive}\n\\end{center}\n\\end{table}\n\n\\section{Exclusive Charmless Semileptonic $B$ Decays}\nIn the exclusive approach the hadronic final state is reconstructed in a particular channel such as $B\\to\\pi\\ell\\nu$ or $B\\to\\rho\\ell\\nu$. In this case the matrix element necessary for calculating the decay rate depends not only on the CKM element $V_{ub}$ but also on non-perturbative hadronic physics contained in so called form factors. These form factors depend on the final state hadron, in particular on whether the particle is a vector or a pseudoscalar meson. For vector mesons, such as $\\rho$ and $\\omega$, the decay rate can be written as~\\cite{BabarUntaggedW}\n\\begin{equation}\n\\frac{ d\\Gamma(B\\to V\\ell\\nu)  }{ dq^{2} } = \\frac{ G_{F}^{2} p_{V} q^{2} }{ 96\\pi^{3}m_{B} C_{V}^{2} } |V_{ub}|^{2} (|H_{0}|^2+|H_{+}|^2+|H_{-}|^2)\n\\end{equation}\nwhere $G_{F}$ is the Fermi constant, $p_{V}$ is the magnitude of the vector meson momentum in the $B $ rest frame, $m_{B}$ is the $B$ mass, $C_{V}$ is the isospin factor and $H_{\\pm}$ and $H_{0}$ are the helicity functions. These helicity functions can be written in terms of two axial vector form factors $A_{1}$ and $A_{2}$, and one vector form factor $V$ as follows\n\\begin{equation}\nH_{\\pm}(q^2)=(m_{B}+m_{V})\\left[ A_{1}(q^2) \\mp \\frac{ 2m_{B}p_{V} }{ (m_{B}+m_{V})^2 }V(q^{2}) \\right],\n\\end{equation}\n\\begin{equation}\nH_{0}(q^2)= \\frac{ m_{B}+m_{V} }{ 2m_{V}\\sqrt{q^{2}} }\\left[ (m_{B}^{2}-m_{V}^{2}-q^{2})A_{1}(q^{2}) -\\frac{ 4m_{B}^{2}p_{V}^{2} }{ (m_{B} +m_{V})^{2} }A_{2}(q^{2}) \\right].\n\\end{equation}\nFor pseudoscalar mesons the decay rate only depends on one form factor, because in the limit of small lepton masses the term proportional to the second form factor $f_{0}(q^{2})$ can be neglected, hence~\\cite{Luth,BelleHad}\n\\begin{equation}\n\\frac{ d\\Gamma(B\\to P\\ell\\nu)  }{ dq^{2} } = \\frac{ G_{F}^{2} |p_{P}|^{3}  }{ 24\\pi^{3} } |V_{ub}|^{2} |f_{+}(q^{2})|^{2}, \n\\end{equation}\nwith $f_{+}(q^{2})$ a form factor. These form factors are provided by theory using different calculations valid for certain regions of $q^{2}$. These calculations include quark models like ISGW2, Lattice QCD (LQCD) that is valid for high values of $q^{2}$ and Light Cone Sum Rules (LCSR) valid for low values of $q^{2}$. \n\n\\subsection{Results for $B\\to\\pi\\ell\\nu$}\nAmong the exclusive channels studied to date, the $B\\to\\pi\\ell\\nu$ is the most promising channel to extract the CKM matrix element $|V_{ub}|$, both theorically and experimentally. Different analyses have come out using different reconstruction techniques of $B$ mesons, taking advantage of the large signal sample, with consistent results on the values on $|V_{ub}|$.\n\\begin{figure}[htb]\n\\centering\n\\subfigure[Belle]{\n\\includegraphics[width=0.45\\textwidth ]{BelleUntagged1.pdf}\n\\label{untaggedbelle}\n}\n\\subfigure[BaBaR]{\n\\includegraphics[width=0.45\\textwidth ]{BabarUntagged.pdf}\n\\label{untaggedbabar}\n}\n\\caption{Fit projections in two bins of $q^{2}$ for the $\\Delta E$ and $M_{\\rm bc(ES)}$ from the untagged measurements of $B\\to\\pi\\ell\\nu$ by the Belle and BaBaR collaborations.}\n\\end{figure}\nThe most recent untagged results from Belle~\\cite{UntaggedExclusiveBelle} and BaBaR~\\cite{UntaggedExclusiveBabar} perform a two dimensional binned extended maximum likelihood fit to $(M_{bc(ES)},\\Delta E)$, in bins of $q^{2}$ and consider basically three sources of background: charmless semileptonic $B$ decays other than the signal $(B\\to X_{u}\\ell\\nu)$, other $B$ decays specially $B\\to X_{c}\\ell\\nu$ and continuum events.  BaBaR splits further the $B$ decays into  `same $B$ category' and `both $B$ category' depending wether or not the pion and the lepton come from the same $B$ meson. In both analyses, the selection criteria are optimized separately in each bin of $q^{2}$ by maximizing the figure of merit $S\/\\sqrt{S+B}$, where $S(B)$ is the expected number of signal (background) events. In addition, an unfolding technique is performed to correct for finite resolution and acceptance, by applying the inverse detector response matrix to the measured yields.  The Belle~\\cite{UntaggedExclusiveBelle} analysis uses a data sample corresponding to $605$ fb$^{-1}$  of integrated luminosity, for which $21486\\pm 548$ signal events are obtained, leading to a branching ratio of ${\\cal B}(B^{0}\\to \\pi^{-}\\ell^{+}\\nu) = (1.49\\pm 0.04_{\\rm stat}\\pm 0.07_{\\rm syst} )\\times 10^{-4}$.   The BaBaR~\\cite{UntaggedExclusiveBabar} analysis reconstruct explicitly the $B^{0}\\to \\pi^{-}\\ell^{+}\\nu$ and $B^{+}\\to \\pi^{0}\\ell^{+}\\nu$ decay channels and combine the results using isospin symmetry, for which they report $12448\\pm 361$ signal events leading to a branching ratio of ${\\cal B} (B\\to\\pi\\ell\\nu)=(1.45\\pm 0.04_{\\rm stat}\\pm 0.06_{\\rm syst})\\times 10^{-4}$. The projections of the fit result in $\\Delta E$ and $M_{\\rm bc(ES)}$ are shown in Fig.~\\ref{untaggedbelle} for the Belle analysis and in Fig.~\\ref{untaggedbabar} for the BaBaR analyisis. The main sources of systematic uncertainties are due to detector effects. \n\nThe lattest results from Belle and BaBaR using semileptonic tag reconstruction differ in the method use to extract the signal yields, while BaBaR~\\cite{SLBabar} uses an unbinned maximum likelihood to $cos^{2}\\phi_{B}$ distribution, Belle~\\cite{Hokuue} implements a two dimensional binned maximum likelihood to the $(x_{B},m_{X})$ distributions. The fit results for $B^{0}\\to\\pi^{-}\\ell^{+}\\nu$ are shown in Fig.~\\ref{fig:SLtag} and the measurements are unfolded. The analysis by Belle reports $156\\pm 20$ $B^{0}\\to\\pi^{-}\\ell^{+}\\nu $ events using a data set of $253$ fb$^{-1}$, which leads to a branching ratio of  ${\\cal B}(B^{0}\\to \\pi^{-}\\ell^{+}\\nu) = (1.38\\pm 0.19_{\\rm stat}\\pm 0.14_{\\rm syst} )\\times 10^{-4}$. For the same decay channel, BaBaR reports $150\\pm 22$ signal events leading to a branching ratio of  ${\\cal B}(B^{0}\\to \\pi^{-}\\ell^{+}\\nu) = (1.38\\pm 0.21_{\\rm stat}\\pm 0.07_{\\rm syst} )\\times 10^{-4}$ using a data sample of 348 fb$^{-1}$ of integrated luminosity.    \n\\begin{figure}[htb]\n\\centering\n\\subfigure[BaBaR]{\n\\includegraphics[width=0.25\\textwidth ]{babarSLtag.pdf}\n}\n\\subfigure[Belle]{\n\\includegraphics[width=0.45\\textwidth ]{belleSLtag.pdf}\n}\n\\label{fig:SLtag}\n\\caption{Fit results from the semileptonic tag measurements of $B\\to\\pi\\ell\\nu$ by the Belle and BaBaR collaborations.}\n\\end{figure}\n\n\nA new analysis using hadronic reconstruction by Belle~\\cite{BelleHad} obtains $463\\pm 28$ $B^{0}\\to\\pi^{-}\\ell^{+}\\nu$ events from a extended binned maximum likelihood fit to the $M_{\\rm miss}^{2}$ distribution, using a data set of 711 fb$^{-1}$. This leads to a branching ratio of $ {\\cal B}(B^{0}\\to \\pi^{-}\\ell^{+}\\nu) = (1.49\\pm 0.09_{\\rm stat}\\pm 0.07_{\\rm syst} )\\times 10^{-4}$, which is competive with the more precise results from untagged measurements. Fig.~\\ref{fig:hadBelle} shows the fitted $M_{\\rm miss}^{2}$ distribution for $B^{0}\\to\\pi^{-}\\ell\\nu$ and $B^{+}\\to\\pi^{0}\\ell\\nu$, it can be noted a clear peak around $M_{\\rm miss}^2 = 0 $ GeV$^{2}$ with a small contribution from the background underneath this peak. The branching ratio for $B^{+}\\to\\pi^{0}\\ell^{+}\\nu$ channel is measured to be $ {\\cal B}(B^{+}\\to \\pi^{0}\\ell^{+}\\nu) = (0.80\\pm 0.08_{\\rm stat}\\pm 0.04_{\\rm syst} )\\times 10^{-4}$, which is in good agreement with the predictions using isospin symmetry, $2\\times \\frac{ {\\cal B}(B^{+}\\to\\pi^{0}\\ell^{+}\\nu) }{ {\\cal B}(B^{0}\\to\\pi^{-}\\ell^{+}\\nu) } \\frac{ \\tau_{ B^{0} } }{ \\tau_{ B^{+} } }=1.00\\pm 0.13_{\\rm tot}$, using $ \\frac{ \\tau_{ B^{+} } }{ \\tau_{ B^{} } }=1.079\\pm 0.007$~\\cite{pdg} .  The reduced contribution of background events in the hadronic tag technique is a direct consequence of the kinematic reconstruction of the full event and also contributes to the reduction of systematic uncertainties.\n\n\\begin{figure}[htb]\n\\centering\n\\subfigure[$B^{0}\\to\\pi^{-}\\ell^{+}\\nu$]{\n\\includegraphics[width=0.45\\textwidth ]{pub_mm_pic_col.pdf}\n}\n\\subfigure[$B^{+}\\to\\pi^{0}\\ell^{+}\\nu$]{\n\\includegraphics[width=0.45\\textwidth ]{pub_mm_pi0_col.pdf}\n}\n\\label{fig:hadBelle}\n\\caption{Fit results from the hadronic tag measurements of $B\\to\\pi\\ell\\nu$ from Belle.}\n\\end{figure}\n\nTo extract a value for $|V_{ub}|$ from the measured differential decay rates, the traditional method described by Eq.~\\ref{vubeq} is used with a slight modification. $\\Delta {\\cal B}$ is replaced by $C_{V}\\Delta {\\cal B}$, where the isospin factor $C_{V}$ is equal to 2 for $B^{+}$ decay modes and is equal to 1 for $B^{0}$ decays modes. A total of four form factor calculations are used to determined $V_{ub}$, two from unquenchend LQCD valid for $q^{2}>$ 16 GeV$^{2}$ provided by the FNAL~\\cite{FNAL} and HPQCD~\\cite{HPQCD} collaborations, and two from LCSR provided by Ball and Zwicky~\\cite{Ball} ($q^{2}<16$ GeV$^{2}$) and KMOW~\\cite{KMOW} ($q^{2}<12$ GeV$^2{2}$). I only present the most recent $V_{ub}$ results from the untagged BaBaR and the tagged Belle analysis and compare with the world averages according to HFAG~\\cite{HFAG} (end of 2011), which are shown in Table~\\ref{tab:exclusive}.   \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|ccc}  \\hline\n             &  Belle tagged~\\cite{BelleHad}  &  BaBaR Untagged~\\cite{UntaggedExclusiveBabar} & World Average~\\cite{HFAG} \\\\ \\hline\nKMOW~\\cite{KMOW}   &$3.40\\pm 0.13\\pm0.09^{+0.37}_{-0.32}$&$3.46\\pm 0.06\\pm 0.08^{+0.37}_{-0.32}$&$3.40\\pm 0.07^{+0.37}_{-0.32}$  \\\\\nBall-Zwicky~\\cite{Ball} &$3.58\\pm 0.12\\pm 0.09^{+0.59}_{-0.39}$&---&$3.57\\pm 0.06^{+0.59}_{-0.39}$  \\\\ \nHPQCD~\\cite{HPQCD} &$3.81\\pm 0.22\\pm 0.10^{+0.66}_{-0.43}$&$3.47\\pm 0.10\\pm 0.08^{+0.60}_{-0.39}$&$3.45\\pm 0.09^{+0.60}_{-0.39}$  \\\\ \nFNAL~\\cite{FNAL}&$3.64\\pm 0.21\\pm 0.09^{+0.40}_{-0.33}$&$3.31\\pm 0.09\\pm 0.07^{+0.37}_{-0.30}$&$3.30\\pm 0.09^{+0.37}_{-0.30}$  \\\\ \\hline\n\\end{tabular}\n\\caption{Values of $|V_{ub}|$ from $B^{0}\\to\\pi^{-}\\ell^{+}\\nu$ reported by the Belle and BaBaR collaboration compared to the world average, where the first error is statistical, the second due to systematic uncertainties and the third is the theoretical uncertainty. }\n\\label{tab:exclusive}\n\\end{center}\n\\end{table}\n\nA second method to extract $V_{ub}$ consists in applying a simultaneous fit using a model independent description of the $f_{+}(q^{2})$ hadronic form factor and the measured $q^{2}$ spectrum. The BaBaR~\\cite{UntaggedExclusiveBabar} collaboration performs a simultaneous fit of the BGL parametrization~\\cite{BGL} to their experimental data and to four points of the  FNAL predictions for the $B^{0}\\to\\pi^{-}\\ell\\nu$ to obtain $|V_{ub}|=(3.25\\pm 0.31)\\times 10^{-3}$ (see Fig.~\\ref{fig:q2BRbabar}). The Belle~\\cite{BelleHad} collaboration reports a value of $|V_{ub}|=(3.52\\pm 0.29)\\times 10^{-3}$ using the BCL parametrization~\\cite{BCL} with their measured partial branching fractions, a recent LCSR calculation and LQCD points (see Fig.~\\ref{fig:q2BRbelle}). Belle also applies the same procedure using data from previous untagged measurements by Belle~\\cite{UntaggedExclusiveBelle} and BaBaR~\\cite{UntaggedExclusiveBabar} with their tagged~\\cite{BelleHad} results and obtain a value of $|V_{ub}|=(3.41\\pm 0.22)\\times 10^{-3}$. \n\\begin{figure}[htb]\n\\centering\n\\subfigure[BaBaR]{\n\\includegraphics[width=0.35\\textwidth ]{FplusCombined.pdf}\n\\label{fig:q2BRbabar}\n}\n\\subfigure[Belle]{\n\\includegraphics[width=0.55\\textwidth ]{pub_br_pic_vs_q2_fit.pdf}\n\\label{fig:q2BRbelle}\n}\n\\caption{Simultaneous fit of form factor parametrization to $B^0\\to\\pi^-\\ell^+\\nu$ data, (a) using the BGL parametrization~\\cite{BGL} to the BaBaR data~\\cite{UntaggedExclusiveBabar} and four points of the FNAL~\\cite{FNAL} prediction and (b) using the BCL parametrization~\\cite{BCL} to the Belle data~\\cite{BelleHad} using LQCD points and LCSR prediction at $q^2 = 0$. }\n\\end{figure}\n\n\n\\subsection{Results from other exclusive channels}\n\\subsubsection{$B\\to\\rho\\ell\\nu$}\n \\begin{figure}[htb]\n \\centering\n \\includegraphics[height=2in]{hrho_rho0_after_2d_fit_col.pdf}\n \\caption{Invariant mass distribution for $\\pi^{+}\\pi^{-}$ using the hadronic full reconstruction by Belle.}\n \\label{fig:massrho}\n \\end{figure}\nThe most recent results for this channel are coming from a Belle tagged analysis~\\cite{BelleHad}, whose main results are presented in Table~\\ref{tab:rho}. When comparing the branching ratio results with previous measurements (see ref.~\\cite{HFAG}), it can be noted that the smallest systematic uncertainty is hold by hadronic tagged measurements, while, contrary to the $\\pi$ channel, the untagged measurements have the biggest systematic errors. This behaviour can be explained using the fact that the $\\rho$ meson is a wide resonance whose main background is due to other charmless semileptonic $B$ decays. This background is very similar to the signal, for which very harsh kinematic selection are required leading to an increase in the systematic uncertainties. The invariant mass of the $\\pi^{+}\\pi^{-} $ system is shown if Fig.~\\ref{fig:massrho}, for which a peak around $0.8$ GeV corresponds to the $\\rho^{0}$ resonance. It can also be noted that the current Monte Carlo scheme overestimates the $\\pi^{+}\\pi^{-}$ non-resonant contribution, for which more detailed theoretical studies are required. For the first time, Belle has an evidence of a broad resonance around 1.3 GeV dominated by  the $B^{+}\\to f_{2}\\ell\\nu$ decay.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cccc}  \\hline\nTheory &  $q^{2}$ [GeV$^{2}$]  &  $N^{\\rm fit}$ & $\\Delta {\\cal B} 10^{-4}$ & $|V_{ub}| 10^{-3}$ \\\\ \\hline\nLCSR~\\cite{LCSR2}   & $<16$                 & $477\\pm 31$    &$1.431\\pm 0.091$           &$3.56\\pm 0.11\\pm 0.09^{+0.54}_{-0.37}$  \\\\\nUKQCD~\\cite{UKQCD}  & full range            & $622\\pm 35$    &$1.834\\pm 0.103$           &$3.68\\pm 0.10\\pm 0.10^{+0.29}_{-0.34}$  \\\\  \\hline\n\\end{tabular}\n\\caption{Results for  $B^{+}\\to\\rho^{0}\\ell^{+}\\nu$ reported by the Belle collaboration, the first error is statistical, the second due to systematic uncertainties and the third is the theoretical uncertainty. }\n\\label{tab:rho}\n\\end{center}\n\\end{table}\n\\subsubsection{$B\\to\\omega\\ell\\nu$}\nIn this section I present two untagged results by BaBaR~\\cite{BabarUntaggedW,UntaggedExclusiveBabar} and one tagged measurement by Belle~\\cite{BelleHad}. The two untagged measurements by BaBaR differ in the signal selection and background suppression approach. One~\\cite{BabarUntaggedW} uses a  neural network discriminator to reduce background and subtract the combinatoric background using a fit the mass sideband data. The other analysis~\\cite{UntaggedExclusiveBabar} uses a cut based technique to reduce background and considers the combinatoric background as fit parameter. The tagged measurement by Belle follows the same procedure described earlier for the $B\\to\\pi\\ell\\nu$ decay. The results for these measurements are shown in Table~\\ref{tab:omega}, where the $\\omega $ meson has been reconstructed in the $\\omega\\to\\pi^{+}\\pi^{-}\\pi^{0}$ decay channel. \n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cccc}  \\hline\nExperiment &  $q^{2}$ [GeV$^{2}$]  &  $N^{\\rm fit}$ & $\\Delta {\\cal B} 10^{-4}$ & $|V_{ub}| 10^{-3}$ \\\\ \\hline\nBaBaR~\\cite{BabarUntaggedW}   & full range               & $1125\\pm 131$  &$1.21\\pm 0.14\\pm 0.08$     &$3.23\\pm 0.22_{\\rm exp}\\pm 0.38_{\\rm theo}$  \\\\\n        & $0-12$                   & ---            &---                        &$3.37\\pm 0.23_{\\rm exp}\\pm 0.38_{\\rm theo}$  \\\\\nBaBaR~\\cite{UntaggedExclusiveBabar}   & full range               & $1861\\pm 233$  &$1.19\\pm 0.16\\pm 0.09$     &$3.20\\pm 0.10\\pm 0.05^{+0.45}_{-0.32}$  \\\\\nBelle~\\cite{BelleHad}  & $0-12$                    & $61\\pm 11$     &$0.611\\pm 0.113$           &$3.08\\pm 0.29\\pm 0.11^{+0.44}_{-0.31}$  \\\\  \\hline\n\\end{tabular}\n\\caption{Results for  $B^{+}\\to\\omega\\ell^{+}\\nu$  using the Ball-Zwicky~\\cite{LCSR2} calculation for the two untagged BaBaR measurements and the tagged Belle measurement. The first error is statistical, the second due to systematic uncertainties and the third is the theoretical uncertainty. }\n\\label{tab:omega}\n\\end{center}\n\\end{table}\n\n\\subsubsection{$B\\to\\eta\\ell\\nu$ and $B\\to\\eta^{\\prime}\\ell\\nu$}\nThe $\\eta$ meson is reconstructed in two decay modes, $\\eta\\to\\gamma\\gamma$ and $\\eta\\to\\pi^{+}\\pi^{-}\\pi^{0}$ and then combine to quote the results. The $\\eta^{\\prime}$ meson is reconstructed in $\\eta^{\\prime}\\to\\eta(\\gamma\\gamma)\\pi^{+}\\pi^{-}$. In Table~\\ref{tab:eta}, I show the latest results from one untagged measurement form BaBaR and one tagged measurement by Belle.\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cc|cc}  \\hline\n           &\\multicolumn{2}{|c|}{$\\eta$}&\\multicolumn{2}{|c}{$\\eta^{\\prime}$}\\\\\nExperiment & $N^{\\rm fit}$ & $\\Delta {\\cal B} \\;  10^{-4}$ &$N^{\\rm fit}$ & $\\Delta {\\cal B} \\;  10^{-4}$ \\\\ \\hline\nBaBaR~\\cite{UntaggedExclusiveBabar}      & $867\\pm 101$  &$0.38\\pm 0.05\\pm 0.05$     &$141\\pm 49$   &$0.24\\pm 0.08\\pm 0.03$  \\\\\nBelle      & $39\\pm 11$    &$0.42\\pm 0.12\\pm 0.05$     &$6.1\\pm 4.7$  &$<0.57$ at $90\\% $ CL  \\\\  \\hline\n\\end{tabular}\n\\caption{Results for  $B^{+}\\to\\eta\\ell^{+}\\nu$ for one untagged BaBaR measurement and the tagged Belle measurement. The first error is statistical, the second due to systematic uncertainties and the third is the theoretical uncertainty. }\n\\label{tab:eta}\n\\end{center}\n\\end{table}\n\\section{Search for $B^{-}\\to p \\bar{p}\\ell^{-}\\nu$}\nA phenomenological calculation~\\cite{PhenoCalc} suggests that the branching ratio of exclusive semileptonic $B$ decays to a baryon-antibaryon pair is about $10^{-5}-10^{-6}$, making this kind of analysis very challenging with the current data set accumulated by the $B-$factories. However, a recent paper~\\cite{PhenoCalc2} estimates this branching ratio to be $(1.04\\pm 0.38)\\times 10^{-4}$ for $B^{-}\\to p \\bar{p}\\ell^{-}\\nu$, which is at the same level of most semileptonic decays with charmless mesons such as $B\\to\\pi\\ell\\nu$. This fact is the main motivation for study this decay for the first time in the Belle collaboration. This analysis~\\cite{pplnu} uses the complete Belle data set, 711 fb$^{-1}$ of integrated luminosity, with a hadronic full reconstruction of the other $B$ meson. They obtain $18^{+11}_{-9}$ signal events from an ubninned maximum likelihood fit to the $M_{\\rm miss}^{2}$, leading to a branching ratio of ${\\cal B} (B\\to p\\bar{p}\\ell\\nu)=(5.8^{+2.4}_{-2.1}(stat)\\pm 0.9(syst)) \\times 10^{-6}$ with a significance of $3.2\\sigma$. The fitted $M_{\\rm miss}^{2}$ distribution is shown in Fig.~\\ref{fig:ppbar}. In addition, an upper limit of $9.6\\times 10^{-6}$ is estimated at 90$\\%$ confidence level. The main source of systematic uncertainties is due to the signal decay model. \n\n \\begin{figure}[htb]\n \\centering\n \\includegraphics[height=2.5in]{ppbar_mmiss.pdf}\n \\caption{Fitted missing mass squared for the combined channels (electrons and muons) of the $B^{-}\\to p \\bar{p}\\ell^{-}\\nu$ decay.}\n \\label{fig:ppbar}\n \\end{figure}\n\n\\section{Summary}\nNew results from Belle and BaBaR using the inclusive and exclusive approach have been presented. Although the results agree between the two collaborations, the tension between the values of $|V_{ub}|$ from inclusive and exclusive measurements still persists. In addition to this, Belle and BaBaR report measurements of $|V_{ub}|$ in other exclusive channels ($B\\to\\rho,\\omega\\ell\\nu$) different to the traditional $B\\to\\pi\\ell\\nu$, which are dominated by theoretical uncertainties. In the near future more detailed analyses of exclusive charmless semileptonic decays with hadron masses above 1 GeV are expected to come out, and thus extend the range of the exclusive measurements and help to reduce systematic uncertainties due to these decays. Finally, new results from the Belle collaboration show evidence of semileptonic decays involving bound states of baryons.     \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}