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+{"text":"\\section{Introduction}\nLet ${{\\mathcal A}_g}$ be the moduli space of complex $g$-dimensional\nprincipally polarized abelian varieties (ppavs). Denote $\\pi':{\\mathcal X}_g\\rightarrow {{\\mathcal A}_g}$ the universal family of ppavs over it. Let $\\Theta_g\\subseteq {\\mathcal X}_g$, and $\\pi:\\Theta_g\\rightarrow{{\\mathcal A}_g}$ denote\nthe universal theta divisor. We shall omit the index $g$ when it is clear.\\smallskip\n\nWe can identify ${\\mathcal X}$ and ${\\mathcal A}$ respectively  with the spaces\n$$\n  \\operatorname{Sp}(g,\\Z)\\times{\\mathbb Z}^{2g}\\backslash{\\mathcal H}_g\\times{\\mathbb C}^g,\\quad \\operatorname{Sp}(g,\\Z)\\backslash{\\mathcal H}_g,\n$$\nwhere ${\\mathcal H}_g$ is the Siegel upper half-space. For $\\tau\\in{{\\mathcal H}}$,  we denote by $X_{\\tau}$, resp. $\\Theta_\\tau$, the fiber of $\\pi'$, resp. $\\pi$, over $\\tau$ (more precisely, over the image of $\\tau$ in ${\\mathcal A}$.\n\nA symmetric principal polarization $\\Theta$ is the zero set of the holomorphic function (where $^tn$ stands for the transpose of a vector $n$)\n$$\n \\vartheta(\\tau, z)=\\sum_{n\\in {\\mathbb Z}^g}\\exp(i\\pi ({}^tn\\tau n+2{}^tnz),\n$$\nand all the other principal polarization divisors symmetric under the $\\pm 1$ involution of $X_\\tau$ are obtained from this divisor by translating by points of order two $\\frac{\\tau\\epsilon+\\delta}{2}\\in X_\\tau[2]\\subset X_\\tau$, for some $\\epsilon,\\delta\\in ({\\mathbb Z}\/2{\\mathbb Z})^g$.\n\nWe denote by $\\T_{\\rm sing}\\subseteq\\Theta$  the singular  locus on $\\Theta$, i.e. the set of the classes of the points $(\\tau, z)$ defined by\n$$\n \\T_{\\rm sing}:=\\lbrace (\\tau,z)\\in{\\mathcal H}_g\\times{\\mathbb C}^g\\mid \\vartheta(\\tau, z)=\\partial_i\\vartheta(\\tau, z)=\\partial_i\\partial_j\\vartheta(\\tau, z) =0\\rbrace\n$$\n(where from here on we denote by $\\partial_i=\\frac{\\partial}{\\partial z_i}$ the partial derivative in the $z_i$ direction). Moreover we  denote by ${\\mathcal S}$  the union of the singular points of $\\Theta_{\\tau}$, i.e. the set of  the classes of the points $(\\tau, z)$ defined by\n$$\n {\\mathcal S}:=\\lbrace (\\tau,z)\\in{\\mathcal H}_g\\times{\\mathbb C}^g\\mid \\vartheta(\\tau, z)= \\partial_i\\vartheta(\\tau, z)=0\\rbrace.\n$$\nObviously we have\n$$\n  \\T_{\\rm sing}\\subset{\\mathcal S}\\subset \\Theta.\n$$\nThere has been great interest in the singularities of the theta divisor and the loci of ppavs for which the theta divisor is singular at least since the ground-breaking work of Andreotti and Mayer \\cite{am}, who defined what are now called the Andreotti-Mayer loci\n$$\n N_k:=\\pi_*({\\mathcal S})=\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\operatorname{dim}\\operatorname{Sing}\\Theta_{\\tau}\\ge k\\}.\n$$\nIt is known that $N_0\\subset{\\mathcal A}$ is a divisor, which has at most two components, cf. \\cite{mu,bea,deb}:\n$$\n \\theta_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid X_\\tau[2]^{\\rm even}\\cap\\operatorname{Sing}  \\Theta_{\\tau}\\ne\\emptyset\\}\n$$\n$$\n N_0' :={\\rm the\\ closure\\ of\\ }\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid (X_\\tau\\setminus X_\\tau[2]^{\\rm even})\\cap\\operatorname{Sing}  \\Theta_{\\tau}\\ne\\emptyset\\}\n$$\n(where $X_\\tau[2]^{\\rm even}$ denotes the even points of order two). The intersection of these two components was studied in \\cite{deb,sing}.\n\nIn general the dimensions of the loci $N_k$ are not known. They were studied in detail by Ciliberto and van der Geer in \\cite{cilvdg},\\cite{amsp}, who conjecture that within the locus of ppavs with endomorphism ring ${\\mathbb Z}$ (i.e. essentially with Picard group ${\\mathbb Z}$) the codimension of any component of $N_k$ is equal to at least $\\frac{(k+1)(k+2)}{2}$. They prove that for $g\\ge 4$ and $1\\le k\\le g-3$ one has $\\operatorname{codim} N_k\\ge k+2$.\n\nIn this paper we will be interested in the loci of ppavs for which the theta divisor has points of higher multiplicity. For points of multiplicity two the natural loci to consider are\n$$\n \\begin{aligned}G_k:=\n &\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\operatorname{dim}\\{z\\in X_\\tau : {\\rm mult}_z\\Theta_{\\tau}\\ge 2\\}\\ge k\\}\\\\\n=&\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid\n \\operatorname{dim} (X_\\tau\\cap\\T_{\\rm sing})\\ge k\\}\\end{aligned}\n$$\n(the two definitions are equivalent by the heat equation).\n\nAlready the locus $G_0$ is still a rather unknown object, and we will mostly concentrate on studying it. It has a natural subset\n$$\n (\\partial\\theta)_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid {\\mathcal X}_{\\tau}[2]^{\\rm odd}\\cap\\T_{\\rm sing}\\ne\\emptyset\\}.\n$$\nWe can further generalize this  to define\n$$\n (\\partial^k\\theta)_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\exists x\\in {\\mathcal X}_{\\tau}[2]: {\\rm mult}_x\\Theta_\\tau- k\\in 2{\\mathbb Z}_{>0}\\}\n$$\nFor some computations it will be important to keep track of which dimension we are in. In this case we will use an upper index $(g)$ and write $\\T_{\\rm sing}^{(g)},(\\partial\\theta)_{\\rm null}^{(g)},$ etc.\n\n\\smallskip\nIn this paper we give equations for  $(\\partial\\theta)_{\\rm null}$ and $(\\partial^2\\theta)_{\\rm null}$  using modular forms and so we can say that we solve Schottky's problem  for them. We describe explicitly some irreducible components of $(\\partial^k\\theta)_{\\rm null}$, and thus obtain an estimate for their codimension in ${\\mathcal A}_g$. Moreover we shall prove that these loci are reducible for $1\\leq k\\leq g-4$ (from \\cite{el} it follows that $(\\partial^{g-2}\\theta)_{\\rm null}$ is irreducible). Finally  we shall give some evidence for the expected dimensions of  $(\\partial\\theta)_{\\rm null}$,  $G_0$ and $(\\partial^2\\theta)_{\\rm null}$.  Doing this we  will relate the dimension of these three  varieties with the dimension of $\\T_{\\rm sing}$.\n\nThe methods we use consist mostly of working with theta functions and their derivatives, computing and bounding the dimensions of the tangent spaces by the ranks of explicit matrices of derivatives. We use the heat equation in many places, and compute the intersections of the loci we are interested in with the boundary of the partial compactification of ${\\mathcal A}$. In \\cite{amsp} Ciliberto and van der Geer study primarily the dimensions of the loci $N_k$, while we are mostly interested in the dimensions of $G_k$ and $(\\partial^k\\theta)_{\\rm null}$. Perhaps uniting the two approaches may yield further insights into the geometry of the theta divisor.\n\n\\section{Notations}\nWe start by recalling  some basic facts about theta functions and modular forms. We denote ${\\mathcal H}_g$ the {\\it Siegel upper\nhalf-space}, i.e. the set of symmetric complex $g\\times g$ matrices\n$\\tau$ with positive definite imaginary part. Each such $\\tau$\ndefines a complex   abelian variety   ${\\mathbb C}^g\/\\tau{\\mathbb Z}^g+{\\mathbb Z}^g$. If $\\sigma=\\left(\\begin{matrix} a&b\\\\ c&d\\end{matrix}\\right)\\in\\operatorname{Sp}(g,\\Z)$ is a symplectic matrix in a $g\\times g$ block form, then its action on $\\tau\\in{\\mathcal H}_g$ is defined by $\\sigma\\cdot\\tau:=(a\\tau+b)(c\\tau+d)^{-1}$, and the moduli space of ppavs is the quotient ${\\mathcal A}_g={\\mathcal H}_g\/\\operatorname{Sp}(g,\\Z)$. A period matrix $\\tau$ is called {\\it decomposable} if there exists $\\sigma\\in\\operatorname{Sp}(g,\\Z)$ such that\n$$\n \\sigma\\cdot\\tau=\\left(\\begin{matrix} \\tau_1&0\\\\\n 0&\\tau_2\\end{matrix}\\right),\\quad\\tau_i\\in{\\mathcal H}_{g_i},\\ g_1+g_2=g, g_i>0;\n$$\notherwise we say that $\\tau$ is indecomposable.\n\nFor $\\epsilon,\\delta\\in ({\\mathbb Z}\/2{\\mathbb Z})^g$, thought of as vectors of zeros and ones,\n$\\tau\\in{\\mathcal H}_g$ and $z\\in {\\mathbb C}^g$, the {\\it theta function with\ncharacteristic $[\\epsilon,\\delta]$} is\n$$\n \\tt\\epsilon\\delta(\\tau,z):=\\sum\\limits_{m\\in{\\mathbb Z}^g} \\exp \\pi i \\left[\n ^t(m+\\frac{\\epsilon}{2})\\tau(m+\\frac{\\epsilon}{2})+2\\ ^t(m+\\frac{\\epsilon}{2})(z+\n \\frac{\\delta}{2})\\right].\n$$\nWe write  $\\vartheta(\\tau, z)$ for the theta function with\ncharacteristic $[0,0]$. Observe that\n$$\n \\tt{0}{0}\\left(\\tau,z+\\tau\\frac\\e2+\\frac\\de2\\right)= \\exp \\pi i\n \\left(-\\frac{^t\\epsilon}{2}\\,\\tau\\frac{\\epsilon}{2}\\,\\,-\\frac{^t\\epsilon}{2}\\,(z+\n \\frac{\\delta}{2})\\right)\\tt\\epsilon\\delta(\\tau,z),\n$$\ni.e. theta functions with characteristics are, up to some non-zero\nfactor, equal to $\\vartheta(\\tau,z)$ shifted by points of order two.\n\nA {\\it characteristic} $[\\epsilon,\\delta]$ is called {\\it even} or {\\it odd}\ndepending on whether  the scalar product\n$\\epsilon\\cdot\\delta\\in{\\mathbb Z}\/2{\\mathbb Z}$ is zero or one, respectively. The function\n$\\tt\\epsilon\\delta(\\tau,z)$ is even or odd as a function\nof $z$, according to the parity of the characteristic; thus a characteristic $[\\epsilon,\\delta]$ is even (resp. odd) if the multiplicity of the theta function $\\vartheta(\\tau,z)$ at the point $z=(\\tau\\epsilon+\\delta)\/2$ is even (resp. odd). We denote by $X_\\tau[2]^{\\rm even\/odd}$ the set of even\/odd points of order two on $X_\\tau$.\n\nA {\\it theta constant} is the evaluation at $z=0$ of a theta function. All odd theta constants of course vanish identically in $\\tau$, but their first derivatives at zero\ndo not vanish identically, and in fact transform in a nice way under the $\\operatorname{Sp}(g,\\Z)$ action.\n\n\\smallskip\nFor a finite index subgroup $\\Gamma\\subset\\operatorname{Sp}(g,\\Z) $ a multiplier system of weight $r\/2$ is a map $v:\\Gamma\\to {\\mathbb C}^*$, such that the map\n$$\n  \\sigma\\mapsto v(\\sigma)\\det(c\\tau+d)^{r\/2}\n$$\nsatisfies the cocycle condition for every $\\sigma\\in\\Gamma$ and\n$\\tau\\in{\\mathcal H}_g$.\n\n\\smallskip\nGiven a pair $\\rho=(\\rho_0,r)$, where $r$ is half integral, and $\\rho_0:{\\rm GL}(g,{\\mathbb C})\\to \\operatorname{End} V$ is an irreducible rational representation with the highest weight $(k_1,k_2,\\dots,k_g)$, $k_1\\geq k_2 \\geq\\dots\\geq k_g=0$, we use the notation\n$$\n  \\rho(A)=\\rho_0(A)\\det A^{r\/2}\\ .\n$$\n\nA map $f:{\\mathcal H}_g\\to V$ is called a modular form for $\\rho$,\nor simply a {\\it vector-valued modular form}, if the choice\nof $\\rho$ is clear, {\\it with multiplier $v$}, with respect to a\nfinite index subgroup $\\Gamma\\subset\\operatorname{Sp}(g,\\Z)$ if\n\\begin{equation}\\label{transform}\n  f(\\sigma\\cdot\\tau)=v(\\sigma)\\rho(c\\tau+d)f(\\tau)\\qquad\\forall\n  \\tau\\in{\\mathcal H}_g,\\forall\\sigma\\in\\Gamma,\n\\end{equation}\nand if additionally $f$ is holomorphic at all cusps of\n${\\mathcal H}_g\/\\Gamma$.\n\nWe define the {\\it level}  and {\\it Igusa}'s subgroups of the symplectic group to be\n$$\n  \\Gamma_g(n):=\\left\\lbrace \\sigma=\\left(\\begin{matrix} a&b\\\\ c&d\\end{matrix}\\right)\n  \\in\\operatorname{Sp}(g,\\Z)\\, |\\, \\sigma\\equiv\\left(\\begin{matrix} 1&0\\\\\n  0&1\\end{matrix}\\right)\\ {\\rm mod}\\ n\\right\\rbrace\n$$\n$$\n  \\Gamma_g(n,2n):=\\left\\lbrace \\sigma\\in\\Gamma_g(n)\\, |\\, {\\rm\n  diag}(a^tb)\\equiv{\\rm diag} (c^td)\\equiv0\\ {\\rm mod}\\\n  2n\\right\\rbrace.\n$$\nWhen $n$ is even, these are finite index normal subgroups of $\\operatorname{Sp}(g,\\Z)$.\n\n\\smallskip\n\\noindent Under the action of $\\sigma\\in\\operatorname{Sp}(g,\\Z)$ the theta functions transform\nas follows:\n$$\n  \\theta\\bmatrix \\sigma\\left(\\begin{matrix} \\epsilon\\\\ \\delta\\end{matrix}\\right)\\endbmatrix\n  (\\sigma\\cdot\\tau,\\,^{t}(c\\tau+d)^{-1}z)\\qquad\\qquad\\qquad\n$$\n$$\n  \\qquad\\qquad\\qquad=\\phi(\\epsilon,\\,\\delta,\\,\\sigma,\\,\n  \\tau,\\,z)\\det(c\\tau+d)^{\\frac{1}{2}}\\tt\\epsilon\\delta(\\tau,\\,z),\n$$\nwhere\n$$\n  \\sigma\\left(\\begin{matrix} \\epsilon\\cr \\delta\\end{matrix}\\right) :=\\left(\\begin{matrix} d&-c\\cr\n  -b&a\\end{matrix}\\right)\\left(\\begin{matrix} \\epsilon\\cr \\delta\\end{matrix}\\right)+ \\left(\\begin{matrix} {\\rm\n  diag}(c \\,^t d)\\cr {\\rm diag}(a\\,^t b)\\end{matrix}\\right),\n$$\nconsidered in $({\\mathbb Z}\/2{\\mathbb Z})^g$, and $\\phi(\\epsilon,\\,\\delta,\\,\\sigma,\\,\\tau,\\,z)$ is some complicated explicit function. For more details, we refer to\n\\cite{igbook}. Thus theta constants with characteristics are (scalar) modular forms of weight $1\/2$ with multiplier with respect to $\\Gamma_g(2)$, i.e. we have\n$$\n  \\tt\\epsilon\\delta(\\sigma\\cdot\\tau,0)=v(\\sigma, \\epsilon, \\delta) \\det(c\\tau+d)^{1\/2}\\tt\\epsilon\\delta(\\tau,0)\n  \\qquad \\forall \\sigma\\in\\Gamma_g(2).\n$$\nwhere the multiplier $v$ becomes trivial if we assume  $\\sigma\\in\\Gamma_g(4,8)$.\n\n\\smallskip\nWe call the {\\it theta-null divisor} $\\t_{\\rm null}\\subset{\\mathcal A}_g$ the\nzero locus of the product of all even theta constants. We  denote  by $\\operatorname{grad}\\tt\\epsilon\\delta(\\tau)$ the gradient of the theta function of characteristic $[\\epsilon,\\,\\delta]$  with respect to $ z_1,\\dots,z_g$ and evaluating at $z=0$. This gradient is not identically zero if and only if the characteristic is odd. The gradient is a vector valued modular form  for $\\rho=(St, 1\/2)$  with multiplier $v$, cf.\\cite{gs}.\n\nWe recall that theta functions  (and their derivatives) satisfy the heat equation, i.e.\n$$\n \\frac{\\partial^2\\tt\\epsilon\\delta(\\tau,z)}{\\partial z_j\\partial z_k} =2\\pi\n i(1+\\delta_{j,k})\\frac{\\partial\\tt\\epsilon\\delta(\\tau,z)}{\\partial\\tau_{jk}},\n$$\n(where $\\delta_{j,k}$ is Kronecker's symbol).\n\n\\smallskip\nThe   symmetric matrix  associated to the second derivatives\n\\begin{equation}\\label{2z}\n \\left(\\frac{\\partial^2\\tt\\epsilon\\delta(\\tau,z)}{\\partial z_j\\partial z_k}|_{z=0}\\right)\n\\end{equation}\nis a  vector valued modular form for $\\Gamma_g(2)$ if we restrict  to the locus $\\tt\\epsilon\\delta(\\tau,0)=0$.\nThis is a general fact: the derivative of a section of a line bundle is a section of the same bundle when restricted to the zero locus of the section --- the modularity of this particular gradient is discussed in \\cite{genus4}. Similarly note that third derivatives of a theta function with an odd characteristics for a modular form,\nwhen restricted to the  locus   $\\operatorname{grad}\\tt\\epsilon\\delta(\\tau,0)=0$.\n\n\\section{Equations for the loci $(\\partial^k\\theta)_{\\rm null}$}\nSimilarly to the case of $\\t_{\\rm null}$, we give vector valued (and alternatively, scalar valued) equations for the loci\n$(\\partial\\theta)_{\\rm null}$ and  $(\\partial^2\\theta)_{\\rm null}$. We will work on the level covers ${\\mathcal A}_g(2):={\\mathcal H}_g\/\\Gamma_g(2)$ or\n${\\mathcal A}_g(4, 8):={\\mathcal H}_g\/\\Gamma_g(4, 8)$, where the $\\t_{\\rm null}$ divisor decomposes into a union of components corresponding to the individual characteristics. Note also that  ${\\mathcal A}_g(4,8)$ is a smooth manifold cover of the stack\/orbifold ${\\mathcal A}_g$, so working on the level cover takes care of the stackiness.\n\nOn ${\\mathcal A}_g(2)$ for any odd $[\\epsilon, \\delta]$ the vector valued equation\n$\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$ defines a certain set of components of $(\\partial\\theta)_{\\rm null}.$ The (possibly reducible) loci $\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$ for various $\\epsilon,\\delta$ are conjugate under the action of $\\operatorname{Sp}(g,\\Z)$.\n\nFor any $[\\epsilon_1,\\,\\delta_1],\\dots,[\\epsilon_g,\\,\\delta_g]$ we define\n$$\n D([\\epsilon_1,\\,\\delta_1],\\dots,[\\epsilon_g,\\,\\delta_g])(\\tau):=\\operatorname{grad} \\tt{\\epsilon_1}{\\delta_1}\\wedge\\operatorname{grad}\\tt{\\epsilon_2}{\\delta_2}\\wedge \\dots\\wedge \\operatorname{grad}\\tt{\\epsilon_g}{\\delta_g}(\\tau)\n$$\nwhich is a scalar modular form with multiplier of weight $\\frac{g+2}{2}$ with respect to $\\Gamma_g(2)$, cf. \\cite{sm}.\n\nIt follows from Lefschetz theorem  for abelian varieties, cf. \\cite{gs}, that the $g\\times 2^{g-1}(2^g -1)$ matrix\n$$\n \\left(\\dots\\operatorname{grad}\\tt \\epsilon\\delta(\\tau)\\dots\\right)_{{\\rm all\\, odd} [\\epsilon,\\,\\delta]}\n$$\nhas maximal rank for all $\\tau$. Thus if all its minors including a fixed characteristic $[\\epsilon,\\delta]$ vanish, the corresponding gradient must be zero, and we get\n\\begin{prop}\nThe common zero locus of the scalar modular forms\n$$\n D([\\epsilon,\\,\\delta],[\\epsilon_2,\\,\\delta_2], \\dots([\\epsilon_g,\\,\\delta_g](\\tau)\n$$\nfor $[\\epsilon,\\delta]$ fixed, and all possible odd characteristics $[\\epsilon_2,\\,\\delta_2],\\dots, [\\epsilon_g,\\,\\delta_g]$,\nis equal to the locus $\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$.\n\\end{prop}\n\\begin{prop}\nSimilarly, if $[\\epsilon, \\delta]$ is an even characteristic,\nthe common zero locus of the scalar valued equation\n\\begin{equation}\\label{theta}\n  \\tt \\epsilon\\delta(\\tau)=0\n\\end{equation}\nand the vector valued equations\n\\begin{equation}\\label{theta2}\n \\left((1+\\delta_{i, j})\\frac{\\partial^2\\tt\\epsilon\\delta}{\\partial\\tau_{i\\,j}}\\tt\\alpha\\beta- (1+\\delta_{i,j})\\frac{\\partial^2\\tt\\alpha\\beta}{\\partial\\tau_{i\\,j}}\\tt\\epsilon\\delta\\right)(\\tau)=0\n\\end{equation}\nfor all even characteristics  $[\\alpha,\\, \\beta]$, defines a union of some irreducible components of  $(\\partial^2\\theta)_{\\rm null}$ on ${\\mathcal A}_g(2)$. (To see this, one notes that all even theta constants cannot vanish simultaneously, cf. \\cite{genus4}.)\n\\end{prop}\n\nNote that of course the components of $(\\partial^2\\theta)_{\\rm null}$ on ${\\mathcal A}_g(2)$ are given by\n\\begin{equation}\\label{theta2bis}\n \\left((1+\\delta_{i,j})\\frac{\\partial^2\\tt\\epsilon\\delta}{\\partial\\tau_{i\\,j}}\\right)(\\tau)=0;\n\\end{equation}\nhowever, this expression is a modular form only along the divisor $ \\tt \\epsilon\\delta(\\tau,0)=0$, cf. \\cite{genus4}.\n\nThere is also another method to produce scalar valued modular forms  vanishing exactly on the  component of  $(\\partial^2\\theta)_{\\rm null}$. This method is  similar to the method used for  $(\\partial\\theta)_{\\rm null}$. Indeed, the $2^{g-1}(2^g+1)\\times\\left(\\frac{g(g+1)}{2}+1\\right)$ matrix\n$$\n\\left(\\begin{array}{ccc}\n  \\dots &\\tt \\epsilon\\delta&\\dots \\\\\n\\dots&\\dots&\\dots\\\\\n \\dots& {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{i\\, j}}&\\dots\n  \\end{array}\\right)_{{\\rm all\\, even} [\\epsilon,\\,\\delta]}\n$$\nhas maximal rank, cf.\\cite{igbook}. Hence, setting $N=(1\/2)g(g+1)$, the form\n$$\n D^2([\\epsilon,\\,\\delta],[\\epsilon_1,\\,\\delta_1]\\dots[\\epsilon_N,\\,\\delta_N])(\\tau):=\n$$\n$$\n {\\rm det}\n \\left(\\begin{array}{llll}\n  \\ \\ \\tt \\epsilon\\delta&\\ \\ \\tt{\\epsilon_1}{\\delta_1}&\\dots&\\ \\ \\tt{\\epsilon_N}{\\delta_N}\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{1\\, 1}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{1\\, 1}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{1\\, 1}}}\\\\\n\\dots&\\dots&\\dots&\\dots\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{i\\, j}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{i\\, j}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{i\\, j}}}\\\\\n\\dots&\\dots&\\dots&\\dots\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{g\\, g}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{g\\, g}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{g\\, g}}}\\\\\n  \\end{array}\\right)(\\tau) ,\n$$\nis a modular form with multiplier, of weight $g+1+(1\/2)(N+1)$ relatively to $\\Gamma_g(2)$, cf. \\cite{diff}, and similarly to the previous case we have\n\\begin{prop}\nThe modular forms\n$$\n D^2([\\epsilon,\\,\\delta],[\\epsilon_1,\\,\\delta_1]\\dots[\\epsilon_N,\\,\\delta_N])(\\tau)\n$$\nfor $[\\epsilon,\\delta]$ and all possible even characteristics $[\\epsilon_1,\\,\\delta_1],\\dots, [\\epsilon_N,\\,\\delta_N]$, vanish  simultaneously along the locus defined by (\\ref{theta}) and (\\ref{theta2bis}), and give scalar equations for one  component of the locus $(\\partial^2\\theta)_{\\rm null}$.\n\\end{prop}\n\\begin{rem}\nOne could also write more complicated equations for the locus $(\\partial^k\\theta)_{\\rm null}$ for $k\\geq 3$, using suitable vector-valued modular forms. Unfortunately our method for obtaining equations using scalar modular forms does not work in this case, since the jacobian matrices do not have maximal rank everywhere, cf. \\cite{diff}.\n\\end{rem}\n\n\\section{Some components of $(\\partial^k\\theta)_{\\rm null}$ within the locus of decomposable ppavs}\nIn this section we start our investigation of the irreducible components of  $(\\partial^k\\theta)_{\\rm null}$ and their possible dimensions.\n\nAs an immediate consequence of the results of \\cite{ko} or \\cite{el}, we have\n\\begin{prop}\n$$(\\partial^{k}\\theta)_{\\rm null}=\\emptyset \\quad {\\rm for}\\, k\\geq g-1$$\n$$(\\partial^{g-2}\\theta)_{\\rm null}={\\mathcal A}_1\\times\\dots\\times{\\mathcal A}_1 $$\n\\end{prop}\nFor lower values of $k$ we can describe some components:\n\\begin{thm}\\label{irred}\nFor $1\\leq k\\leq g-2$, the variety\n$\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$  is an  irreducible components of $(\\partial^k\\theta)_{\\rm null}$.\nThe codimension of this subvariety gives the bound\n$$\n \\operatorname{codim}(\\partial^k\\theta)_{\\rm null} \\leq gk+1-(1\/2)(k^2+k)\n$$\n\\end{thm}\n\\begin{proof}\nWe perform the computation on ${\\mathcal A}_g(4,8)$ for the characteristic\n$\\epsilon=(\\alpha,1, \\dots,1)$ and $\\delta=(\\beta,1,\\dots,1)$, where $[\\alpha,\\beta]$ is a $(g-k)$-dimensional characteristic. Note that $[\\alpha,\\beta]$ is necessarily even. Consider then the set\n$$\n A_k([\\epsilon,\\delta]):=\\left\\{\\frac{\\partial^h\\tt\\epsilon\\delta(\\tau, z)}{\\partial z_{i_1}\\dots\\partial z_{i_h}}\\vert_{z=0}=0\\right\\}_{{\\rm for\\ all}\n \\ h\\leq k.}\n$$\nObviously $A_k$ contains the $\\Gamma_g(2)$ conjugates of the locus\n$$\n \\left(\\tt\\alpha\\beta(\\tau')=0\\right)\\times {\\mathcal H}_1\\times\\dots\\times{\\mathcal H}_1,\n$$\n(where $\\tau'\\in{\\mathcal A}_{g-k}$),\nwhich is of codimension $ gk+1-(1\/2)(k^2+k)$ in ${\\mathcal A}_g$. The only non-zero elements of the jacobian matrix of the equations defining $A_k$ are those involving derivatives of  order $k+2$. These form a matrix with $(1\/2)(g+1)g$ rows and the columns that can be indexed in the following three ways:\\smallskip\n\n1) the $k$ derivatives involve the indices $g-k+1,g-k+2,\\dots, g$\n\n2) the $k$ derivatives involve only one of the first $g-k$ indices and all, but one among $g-k+1,g-k+2,\\dots, g$\n\n3) the $k$ derivatives involve two ( even with multiplicity) of the first $g-k$ indices and all, but two among $g-k+1,g-k+2,\\dots, g.$\n\nThen we have to consider the derivative $\\partial\/\\partial \\tau_{a,b}$,  or equivalently, by the heat equation, $\\partial^2\/\\partial z_a \\partial z_b$. In  the first of the above cases we get a column with non-zero entries being\n$$\n \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{a}\\partial z_{b} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\,1\\leq a\\leq b\\leq g-k.\n$$\nIn the second case we get $(g-k)k$ columns involving in the $(a,b)$ row\n$$\n  \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{a}\\partial z_{j} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\,1\\leq a \\leq g-k<b\\leq g.\n$$\nIn the third case we get $1\/2)k(k-1)$ columns involving in the $(a,b)$ row\n$$\n \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{l}\\partial z_{j} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\, g-k<a< b\\leq g,\n$$\nwhere we denoted $\\tt 1 1^{'}(\\lambda_i)$ the derivative of the one-dimensional theta function with odd characteristic evaluated at zero for the elliptic curve with periods 1 and $\\lambda_i=\\tau_{g-k+i,g-k+i}$.\n\nIt can be easily  checked that all these columns are linearly independent, and hence the Jacobian has maximal rank, so that the equations give an irreducible component of the locus $(\\partial^k\\theta)_{\\rm null}$, as claimed.\n\\end{proof}\n\n\n\\begin{prop}\nWhen $k\\leq g-4$, the locus $(\\partial^k\\theta)_{\\rm null}$ is reducible.\n\\end{prop}\n\\begin{proof}\nWe recall that  $\\theta_{2,\\,\\rm null} ={\\mathcal A}_1\\times {\\mathcal A}_1$ and $\\theta_{3,\\,\\rm null}$ is the hyperelliptic locus in ${\\mathcal A}_3$ (in particular a generic element of it is indecomposable). Thus the locus\n$$\n \\theta_{g-k-1,\\,\\rm null}  \\times{\\mathcal A}_2\\times{\\mathcal A}_1\\times\\dots\\times{\\mathcal A}_1 \\subset (\\partial^k\\theta)_{\\rm null},\n$$\nand must be contained in an irreducible component of $(\\partial^k\\theta)_{\\rm null}$ different from $\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$.\n\\end{proof}\n\n\\begin{rem}\nWhen $k=g-3$,  for small $g$, we have that $(\\partial^{g-3}\\theta)_{\\rm null}$  is irreducible. In general this is not known. Repeating  the argument of the proof of corollary 2 in   \\cite{el}, one can show that if a principally polarized abelian variety in  $(\\partial^{g-3}\\theta)_{\\rm null}$  is decomposable, and thus necessarily lies in  $\\theta_{g-3,\\,\\rm null}  \\times{\\mathcal A}_{1}\\times\\dots\\times{\\mathcal A}_{1}$. It is interesting to know whether there are indecomposable ppavs in $(\\partial^{g-3}\\theta)_{\\rm null}$. See also remark \\ref{gminus3}\n\\end{rem}\n\n\\begin{rem}\nFor a generic $\\tau\\in\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$ the theta divisor $\\Theta_{\\tau}$ has a unique point of order 2 having  multiplicity $k+2$. However, for a generic point in\n$\\theta_{g-k-n,\\,\\rm null}  \\times{\\mathcal A}_{i_1}\\times\\dots\\times{\\mathcal A}_{i_k}$, with $k<i_1+\\dots i_k=k+n\\leq g-2$, the theta divisor $\\Theta_{\\tau}$ has several points of order 2 and multiplicity $k+2$.\nAs a consequence we see that the latter product is contained in the    singular locus of  $(\\partial^k\\theta)_{\\rm null}\\subset{\\mathcal A}_g(2)$.\n\\end{rem}\n\nWe now investigate the reducibility of the locus $G_0$.\n\\begin{thm}\\label{reduc}\nThe loci $N_{g-1,\\,0} ' \\times{\\mathcal A}_1$ and $\\theta_{g-1,\\,\\rm null}  \\times{\\mathcal A}_1$ are irreducible components of $G_0$.\n\\end{thm}\n\\begin{proof}\nRecall that the locus $\\T_{\\rm sing}\\subset{\\mathcal X}_g$ is defined by the following equations (using the heat equation for the theta function)\n$$\n 0=\\vartheta(\\tau,z)=\\partial_i\\vartheta(\\tau,z)=\\partial_i\\partial_j\\vartheta(\\tau,z)\\quad\\forall 1\\le i,j\\le g.\n$$\nComputing the Jacobian matrix of these defining equations for $\\tau\\in{\\mathcal A}_{g-1}\\times{\\mathcal A}_1$ we get\n$$\n \\left(\\begin{array}{cccccccc}\n \\,\\partial_1\\vartheta&\\partial_1\\partial_1\\vartheta\n &\\dots&\\partial_1\\partial_g\\vartheta&\\partial_1\\partial_1\\partial_1\\vartheta&\\dots&\\partial_1\\partial_g\\partial_g \\vartheta\\\\\n \\,\\partial_2\\vartheta&\\partial_2\\partial_1\\vartheta\n &\\dots&\\partial_2\\partial_g\\vartheta&\\partial_2\\partial_1\\partial_1\\vartheta&\\dots&\\partial_2\\partial_g\\partial_g \\vartheta\\\\  \\dots&\\dots&\\dots&\\dots&\\dots&\\dots&\\dots \\\\\\\n \\,\\partial_g\\vartheta&\\partial_g\\partial_1\\vartheta&\\dots&\\partial_g\\partial_g\\vartheta& \\partial_g\\partial_1\\partial_1\\vartheta&\\dots&\\partial_g\\partial_g\\partial_g \\vartheta\\\\\n \\,\\partial_1\\partial_1\\vartheta&\\partial_1\\partial_1\\partial_1\\vartheta\n &\\dots&\\partial_1\\partial_1\\partial_g\\vartheta&\\partial_1\\partial_1\\partial_1\\partial_1\\vartheta&\\dots& \\partial_1\\partial_1\\partial_g\\partial_g\\vartheta\\\\\n   \\dots&\\dots&\\dots&\\dots&\\dots&\\dots \\\\\\\n  \\,\\partial_g\\partial_g\\vartheta&\\partial_g\\partial_g\\partial_1\\vartheta\n &\\dots&\\partial_g\\partial_g\\partial_g\\vartheta&\\partial_g\\partial_g\\partial_1\\partial_1\\vartheta&\\dots& \\partial_g\\partial_g\\partial_g\\partial_g\\vartheta\n \\end{array}\\right)\n$$\nIn both cases, let $$(X_{\\tau}, \\Theta_{\\tau})=(X_{\\tau'}, \\Theta_{\\tau'})\\times(E_{\\lambda}, \\Theta_{\\lambda})$$\nand let $x=(x', y)$ be the point of multiplicity at least 3 such that\n$x'$ is a singular point of  $\\Theta_{\\tau'}$ and $y= \\Theta_{\\lambda}$. We can assume that $x'$ is an ordinary double point, then we have that the jacobian  matrix at the point $(\\tau, x)\\in \\T_{\\rm sing}$ is of the form\n$$\n  \\left(\\begin{array}{ccc }\n  0&0&\\partial_i\\partial_j\\partial_k\\vartheta\\\\\n  0&{}^t(\\partial_i\\partial_j\\partial_k\\vartheta)&\\partial_i\\partial_j\\partial_k\\partial_l\\vartheta\n  \\end{array}\\right)\n$$\nHere the first column is in fact a column-vector, the entry (1,2) is the $g\\times g$ null matrix; $(\\partial_i(\\partial_j\\partial_k\\vartheta))$ is  the  $g\\times (1\/2)g(g+1)$ matrix of the third derivatives of $\\vartheta$, ${}^t(\\partial_i\\partial_j\\partial_k\\vartheta)$ is its transpose, and  the entry  (2,3)  is the $(1\/2)g(g+1)\\times (1\/2)g(g+1)$ matrix of the fourth derivatives of $\\vartheta$. An elementary computation gives that the rank of the matrix $(\\partial_i\\partial_j\\partial_k\\vartheta)$ is $g$, since $x'$ is an ordinary double point, hence the previous matrix has rank $2g$.\n\nHence  in a neighborhood of the point  $(\\tau, x)$ the subvariety $\\T_{\\rm sing}$ has codimension at least $2g$ in ${\\mathcal X}_g$, and thus its projection has codimension at least $g$ in ${\\mathcal A}_g$. Since ${N}_{g-1,\\,0} ' \\times{\\mathcal A}_1$ and $\\theta_{g-1,\\,\\rm null}  \\times{\\mathcal A}_1$ both have codimension $g$ in ${\\mathcal A}_g$, these are two irreducible components of $G_0$.\n\\end{proof}\n\nIn more generality, note that $(\\partial^k\\theta)_{\\rm null}\\times{\\mathcal A}_1\\subset (\\partial^{k+1}\\theta)_{\\rm null}$, and we will now consider the question of whether these give irreducible components. The result is the following generalization of theorem \\ref{irred}.\n\n\\begin{prop}\\label{codg}\nLet $X$ be an irreducible component of $(\\partial^k\\theta)_{\\rm null}^{(g)}$.\nThen the locus $X\\times {\\mathcal A}_1$  is an irreducible component of $ (\\partial^{k+1}\\theta)_{\\rm null}^{(g+1)}$ if and only if the rank of the\n$g \\times \\binom{g+k}{k+1}$ matrix\n$$\n L:=\\left(\\frac{\\partial\\tt\\epsilon\\delta (Z)}{\\partial z_{a}\\partial z_I}\\right)\n$$\nfor $1\\leq a \\leq g$ and $I\\subset\\{1,\\dots,g\\}$ with $\\#I=k+1$, is equal to $g$.\n\nMoreover, if $k=1$, the set $X\\times {\\mathcal A}_1$  is an irreducible component of $ (\\partial^{2}\\theta)_{\\rm null}$ if and only if the codimension of $X$ in ${\\mathcal A}_g$ is equal to $g$.\n\\end{prop}\n\\begin{proof}\nAs before, we will compute the Jacobian matrix of the defining equations at a smooth point $\\tau\\in X$, lifted to the level cover ${\\mathcal A}_g(4,8)$, so that we are talking about the multiplicity of the theta function with a given characteristic $[\\alpha,\\beta]$ at zero. The   $(1\/2)g(g+1)\\times \\binom{g+k-1}{k}$ Jacobian matrix is then\n$$\n  J=\\left(\\frac{\\partial^2  \\tt\\alpha\\beta(\\tau)}{\\partial z_{a}\\partial z_{b} \\partial z_I}\\right)\n$$\nwith $1\\leq a\\leq b\\leq g$, and $I\\subset\\{1,\\dots,g\\},$ with $\\#I=k$. Since this is the Jacobian matrix of the equations defining $X$, its rank is equal to the codimension of $X$ in ${\\mathcal A}_g$.\n\nThen for any $x\\in{\\mathcal A}_1$ the point $(\\tau,x)\\in {\\mathcal A}_g\\times {\\mathcal A}_1$ is a smooth point of $(\\partial^{k+1}\\theta)_{\\rm null}$, with the vanishing theta function with high multiplicity being $\\tt{\\alpha\\ 1}{\\beta\\ 1}$. The Jacobian matrix of the equations defining $\\partial^{k+1}\\theta)_{\\rm null}\\subset{\\mathcal A}_{g+1}$ is the same as $J$ together with the matrix $L$. The first statement of the proposition follows, while for the special case of $k=1$ we observe that $L$ is the transpose of $J$ and thus has the same rank.\n\\end{proof}\n\n\\section{Codimensions of $(\\partial\\theta)_{\\rm null},(\\partial^2\\theta)_{\\rm null}$, and $G_0$}\nNote that the dimension of a reducible variety means the maximum dimension of the components, and thus $\\operatorname{codim}$ means the minimal codimension of the components.\n\nIn the previous section we have computed some components of the loci $(\\partial^k\\theta)_{\\rm null}$ and thus know their dimensions. It is natural to conjecture the following.\n\\begin{conj}\\label{dthetanull}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim}_{{\\mathcal A}_g} (\\partial\\theta)_{\\rm null}=\\operatorname{codim}_{{\\mathcal A}_g} G_0=g.\n$$\n\\end{conj}\n\nWe give some evidence for this conjecture by using the degeneration techniques. We want to consider the  intersection of $ (\\partial\\theta)_{\\rm null}$ with $\\partial{\\mathcal A}_{g+1}$, the corank one boundary component of any toroidal compactification of ${\\mathcal A}_{g+1}$. A point of $\\partial{\\mathcal A}_{g+1}$ is a semiabelian variety of torus rank one, i.e. a pair $(\\overline{G}, D)$, where $\\overline{G}$ is a non-normal compactification of the extension $1\\to{\\mathbb G}_m\\to G\\to B\\to 0$ for some $B\\in{\\mathcal A}_g$, and $D$ is an ample divisor that is the limit of the theta divisors. The restriction of the map\n\\begin{equation} \\label{fib}\n \\pi_{|\\partial{\\mathcal A}_{g+1}}:\\partial{\\mathcal A}_{g+1}\\to{\\mathcal A}_{g}\n\\end{equation}\nhas $B\/Aut (B, \\Xi)$ as fiber over $(B,\\Xi)$. This means that the\nfiber over a general $(B,\\Xi)\\in{\\mathcal A}_{g}$ is the Kummer variety\n$B\/\\pm 1$.\n\nStudying the boundary behavior of the universal theta divisor and using the Fourier-Jacobi expansion similarly to \\cite{mu}, we get the following\n\\begin{prop}\n\\begin{equation}\\label {inter0}\n (\\partial\\theta)_{\\rm null}^{(g+1)}\\cap\\partial{\\mathcal A}_{g+1}=\\left(\\bigcup_{ (\\overline{G},D)}2_B({\\mathcal S}^{(g)}\\cap\\pi^{-1}((B,\\Xi))\\right)\\cup \\left(\\pi^{-1} (\\partial\\theta)_{\\rm null}^{(g)}\\cap\\Theta_g\\right)\n \\end{equation}\n\\end{prop}\n\\begin{proof}\nAnalytically   we consider the Fourier-Jacobi expansion of\n$$\n \\operatorname{grad}\\tt\\epsilon\\delta(Z,0)\\quad{\\rm with}\\quad Z=\\left(\\begin{array}{cc}\n\\tau&z\\\\\nz'&w\n \\end{array}\\right)\n$$\nalong the boundary components. Since the symplectic group acts transitively on the boundary components and on the odd characteristics, considering the  Fourier-Jacobi expansion of  a fixed $\\operatorname{grad}\\tt\\epsilon\\delta$ at all boundary components is equivalent to considering the Fourier-Jacobi expansion of $\\operatorname{grad}\\tt\\epsilon\\delta$ for all possible $[\\epsilon,\\delta]$ at the standard boundary component where the period matrix element $\\tau_{g+1,g+1}$ goes to $i\\infty$. It is easy to check that the Fourier-Jacobi expansions are of two types depending on whether the $(g+1)$'th entry  of the characteristic is 0 and 1. If it is 1  one we get\n$$\n \\partial_i\\,  \\tt \\epsilon\\delta(Z, 0) =\\partial_i\\tt {\\alpha}{\\beta}(\\tau, z\/2)exp(i\\pi w\/4)+\\dots\\quad {\\rm when}\\,i=1,\\dots, g\n$$\n$$\n \\partial_{g+1}\\,  \\tt \\epsilon\\delta(Z, 0) =\\tt {\\alpha}{\\beta}(\\tau, z\/2)exp(i\\pi w\/4)+\\dots\n$$\nIf it is 0 we get\n$$\n  \\partial_i\\,  \\tt \\epsilon\\delta(Z, 0) =\\partial_i\\tt {\\alpha}{\\beta}(\\tau, 0)+\\dots\n$$\n$$\n  \\partial_{g+1}\\,  \\tt \\epsilon\\delta(Z, 0) =\\tt {\\alpha}{\\beta}(\\tau, z)exp(i\\pi w)+\\dots\n$$\nNow the intersection  of $\\operatorname{grad}\\tt\\epsilon\\delta=0$ with the blow up of the first boundary components gives $2_{X_{\\tau}}(\\operatorname{Sing} (\\Theta_{\\tau}))$ along each fiber, and $\\Theta_{\\tau}$ along $p^{-1}((\\partial\\theta)_{\\rm null}^{(g)})$\n\\end{proof}\n\nNotice that by results of Keel and Sadun \\cite{ke}, any component of $(\\partial\\theta)_{\\rm null}^{(g)}$, which is a priori of codimension at most $g$, must intersect the boundary of any toroidal compactification of ${\\mathcal A}_g$. We can handle the degenerations only for the case when the components intersect the torus rank one boundary component, i.e. when they intersect the boundary of the partial compactification.\n\\begin{thm}\\label{boun}\nIf all irreducible components of $(\\partial\\theta)_{\\rm null}^{(g)}$  meet $\\partial{\\mathcal A}_g$ (the boundary of the partial compactification),  and $\\operatorname{codim}_{{\\mathcal A}_{g-1}} (\\partial\\theta)^{(g-1)}_{\\rm null}=g-1$, then   $\\operatorname{codim}_{{\\mathcal A}_g} (\\partial\\theta)_{\\rm null}=g$, i.e. if for any $g$ all irreducible components of $(\\partial\\theta)_{\\rm null}^{(g)}$ meet $\\partial{\\mathcal A}_g$, then conjecture \\ref{dthetanull} holds.\n\\end{thm}\n\\begin{proof}\nObviously we  know that the codimension is less or equal to $g$.\nThe first case  in the intersection above gives a subvariety of codimension $g$. By induction the second case also gives a subvariety of codimension $g$, since $(\\partial\\theta)_{\\rm null}^{(g-1)}$ has codimension $g-1$ (and thus also its preimage under $\\pi^{-1}$), and we are then intersecting with a divisor.\n\\end{proof}\n\nOne can also conjecture the codimensions of the other loci we study.\n\\begin{conj} \\label{co1}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim}_{{\\mathcal X}_g}\\T_{\\rm sing}=2g.\n$$\n\\end{conj}\n\n\\begin{conj} \\label{co2}\nFor any $g\\geq 4$\n$$\n  \\operatorname{codim}_{{\\mathcal A}_g}(\\partial^{2}\\theta)_{\\rm null}= 2g-2.\n$$\n\\end{conj}\n\nWe will give a degeneration argument relating these two conjectures.\n\\begin{thm}\\label{muthm}\n\\begin{equation}\\label {inter}\n (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}\\cap\\partial{\\mathcal A}_{g+1}=\\left(\\bigcup_{ (\\overline{G},D)}2_B(\\T_{\\rm sing}\\vert_{\\Xi})\\right)\\cup (\\pi^{-1}(\\partial^2\\theta)_{\\rm null}^{(g)}\\cap {\\mathcal S}^{(g)})\n \\end{equation}\n\\end{thm}\n\\begin{proof}\nThe proof is analogous to the proof of the identity (\\ref{inter0}), if one considers the second derivatives as well. There are no new technicalities or ideas involved, and we omit the details here.\n\\end{proof}\n\n\n\\begin{rem}\nThis intersection give us a rough estimate for the dimension of $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$. In fact the intersection  of any irreducible components of  $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$,  meeting  the first  boundary component $\\partial{\\mathcal A}_{g+1}$ have dimension less or equal to $\\operatorname{dim}\\,{\\mathcal S}^{(g)}=(1\/2)g(g+1)-1$. This implies that all\nirreducible components of  $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$  meeting  the first  boundary component have codimension at least\n$g+1$.\n\\end{rem}\n\nWe can relate the two conjectures by the following theorem\n\\begin{thm}\nIf for any $k\\in[4,g+1]$ all components of $(\\partial^2\\theta)_{\\rm null}^k$ intersect $\\partial{\\mathcal A}_{k}$, then conjecture $\\ref{co2}$ is true for all $k\\in[4,g+1]$ if and only if conjecture $\\ref{co1}$ is true for all $k\\in[3,g]$.\n\\end{thm}\n\\begin{proof}\nWe use the description of the boundary of the locus $(\\partial^2\\theta)_{\\rm null}$ from formula (\\ref{inter}). We know a priori that $\\T_{\\rm sing}^{(g)}$ has codimension at most $2g$ (since its projection to ${\\mathcal A}_g$  contains $(\\partial\\theta)_{\\rm null}$, which is of codimension at most $g$). Similarly we know a priori that $\\operatorname{codim}(\\partial^2\\theta)_{\\rm null}\\le 2g-2$, since this locus contains $\\t_{\\rm null}^{(g-2)}\\times{\\mathcal A}_1\\times{\\mathcal A}_1$.\n\nOne implication is easy: if conjecture \\ref{co2} holds for $k=g+1$, i.e. $\\operatorname{codim}_{{\\mathcal A}_{g+1}}(\\partial^2\\theta)_{\\rm null}^{(g+1)}=2g$, then $\\operatorname{codim}_{\\partial{\\mathcal A}_{g+1}}\\left((\\partial^2\\theta)_{\\rm null}^{(g+1)}\\cap \\partial{\\mathcal A}_{g+1}\\right)=2g$ (we know from (\\ref{inter}) that this intersection with an open divisor $\\partial{\\mathcal A}_{g+1}\\subset\\bar{{\\mathcal A}_{g+1}}$ is non-empty, so it has codimension one in $(\\partial^2\\theta)_{\\rm null}^{(g+1)}$), and thus the codimension of its component $2_B(\\T_{\\rm sing}^{(g)})$ cannot be less than $2g$.\n\nFor the other direction, assume that conjecture \\ref{co1} holds for $k=g$, i.e. that $\\operatorname{codim}_{{\\mathcal X}_g}(\\T_{\\rm sing})=2g$, and also inductively assume that we know that $\\operatorname{codim} (\\partial^2\\theta)_{\\rm null}^{(g)}=2g-2$. Since for any $\\tau\\in{\\mathcal A}_g$ we have $\\operatorname{codim}_{X_\\tau} {\\mathcal S}\\cap X_\\tau\\ge 2$ (a generic point of $\\Theta_{\\tau}$ is smooth). Thus we have $\\operatorname{codim}_{{\\mathcal X}_g} \\pi^{-1}(\\partial^2\\theta_{\\rm null}^{(g)}\\cap{\\mathcal S}^{(g)})\\ge 2g$, and thus $\\operatorname{codim}_{{\\mathcal A}_{g+1}}(\\partial^2\\theta)_{\\rm null}^{(g+1)}=2g$.\n\nFor the base of the induction, observe that the locus $\\T_{\\rm sing}^{(3)}$ consists of the family of odd points of order two over ${\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1 $ and so has codimension $6$ in ${\\mathcal X}_3$, while $(\\partial^2\\theta)_{\\rm null}^{(4)}={\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1=\\t_{\\rm null}^{(2)}\\times{\\mathcal A}_1\\times{\\mathcal A}_1$ also has codimension $6$ (see theorem \\ref{irred}).\n\\end{proof}\n\n\n\\begin{rem}\nIn lemma 9 of \\cite{sing} we proved and used  that $ (\\partial^{2}\\theta)_{\\rm null}$ has codimension greater than $2$ in ${\\mathcal A}_{g+1}$. Our proof quoted a result of  \\cite{amsp} the proof of theorem 8.11 in \\cite{amsp}, while it seems to us the statement in theorem 8.6(ii) of \\cite{amsp} yields this more directly, implying that $3\\le\\operatorname{codim} G_0\\le\\operatorname{codim} (\\partial^{2}\\theta)_{\\rm null}$.\n\\end{rem}\n\n\\medskip\nIn analogy with the situation for Andreotti-Mayer loci, it is natural to make the following\n\\begin{conj} \\label{co3}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim} G_k \\ge g+k\n$$\n\\end{conj}\n\\begin{rem}\nNotice that we know that the locus $(\\partial\\theta)_{\\rm null}$ is by definition a subset of $G_0$; however, $(\\partial\\theta)_{\\rm null}$ is locally given by the vanishing of a gradient of an odd theta function with characteristic at zero, i.e. is given by $g$ equations, and thus we have $g\\le\\operatorname{codim} (\\partial\\theta)_{\\rm null}\\le\\operatorname{codim} G_0$, so the above conjecture implies that $\\operatorname{codim} G_0=\\operatorname{codim} (\\partial\\theta)_{\\rm null}=g$.\n\\end{rem}\n\n\\begin{rem}\nIn fact it seems likely that the locus $(\\partial\\theta)_{\\rm null}$ is of pure codimension $g$, while it seems less likely that $G_0$ is of pure dimension: the second derivatives of theta automatically vanish at odd points of order two, but not generically on ${\\mathcal X}_g$, and thus it is natural to expect that there exist components of $G_0\\setminus(\\partial\\theta)_{\\rm null}$ are of codimension higher than $g$.\n\\end{rem}\n\\begin{rem}\\label{gminus3}\nIn the notations of \\cite{amsp} the loci $N_{g,k,3}=G_k^{(g)}$ are studied, and the following results are know for them. Theorem 8.6(ii) in \\cite{amsp} then says that $\\operatorname{codim} G_k\\ge k+3$, while theorem 7.5(ii) in \\cite{amsp} in our notations is that $$G_{g-3}^{(g)}=\\mathop{\\bigcup}\\limits_{g_1+g_2+g_3=g,\n\\ g_1g_2g_3>0}{\\mathcal A}_{g_1}\\times{\\mathcal A}_{g_2}\\times{\\mathcal A}_{g_3},$$\nwhich is not purely dimensional, but has codimension $2g-3$ --- so that conjecture \\ref{co3} is true for $k=g-3$.\n\\end{rem}\n\nThe statement  about the codimension of $G_k$ seems to be   rather convincing, but we could not find a proof.  Note that the argument to prove $\\operatorname{codim} N_1>1$ given by Mumford in \\cite{mu} uses an involved heat equations argument, while Ciliberto and van der Geer in \\cite{cilvdg}, \\cite{amsp} show, with a lot of work, that $\\operatorname{codim} N_k\\ge k+2$. The basic question seems to be whether the locus $\\T_{\\rm sing}$ is pure-dimensional or not, and the relationship of the conjecture is the following\n\\begin{prop}\nConjecture \\ref{co1}, for a fixed $g$, is equivalent to conjecture \\ref{co3}, for the same $g$, and all $k$.\n\\end{prop}\n\\begin{proof}\nIndeed recall that by definition the map $\\pi$ restricted to $\\T_{\\rm sing}\\cap\\pi^{-1}(G_k)$ has fiber dimension $k$, and thus $\\operatorname{dim}\\T_{\\rm sing}\\ge k+\\operatorname{dim} G_k$. Thus if we know that $\\operatorname{codim}_{{\\mathcal X}_g}\\T_{\\rm sing}=2g$, it follows that $\\operatorname{codim}_{{\\mathcal A}_g}G_k\\ge g+k$. In the other direction, if $\\operatorname{codim} G_k\\ge g+k$ for all $k$, we have $\\operatorname{dim}\\T_{\\rm sing}=\\max_k(k+\\operatorname{dim} G_k)\\le \\max_k(k+\\operatorname{dim}{{\\mathcal A}_g}-g-k)$, and thus $\\operatorname{codim}\\T_{\\rm sing}\\ge 2g$, in which case it must be equal to $2g$.\n\\end{proof}\n\nNow we would like to give some evidence for the validity of these conjectures. Note that if $z=(\\tau\\epsilon+\\delta)\/2\\in X_\\tau[2]$ is an odd point, then $0=\\vartheta(\\tau,z)=\\partial_i\\partial_j\\vartheta(\\tau,z)$ automatically as the value and derivatives of an odd function, and thus the locus\n$$\n Y:=\\{(\\tau,z)\\in{\\mathcal X}_g \\mid \\operatorname{grad} \\vartheta(\\tau,z)=0;\\ z=(\\tau\\epsilon+\\delta)\/2\\}\n$$\nis a subset of $\\T_{\\rm sing}\\subset{\\mathcal X}_g$, while the projection $\\pi(Y)=(\\partial\\theta)_{\\rm null}\\subset{\\mathcal A}_g$ (notice that by definition $(\\partial\\theta)_{\\rm null}$ is the union of such projections over all odd $[\\epsilon,\\delta]$, but they are all conjugate under $\\operatorname{Sp}(g,\\Z)$, and thus have the same image on ${\\mathcal A}_g$). The expected codimension of $Y$ in ${\\mathcal X}_g$ is equal to $2g$ ($g$ for fixing a point on $X_\\tau$, and $g$ for the vanishing of the gradient).\n\\begin{thm}\\label{tch}\nLet $Z$ be a reduced irreducible component of $\\T_{\\rm sing}$ that is contained in $Y$ as above. Then $Z$ has codimension $2g$ in ${\\mathcal X}_g$ and $\\pi(Z)$ has codimension $g$ in ${\\mathcal A}_g$ (and is thus an irreducible component of $(\\partial\\theta)_{\\rm null}$).\n\\end{thm}\n\\begin{proof}\nWe apply  the jacobian criterion, in a smooth point  of $Z$. As in the theorem \\ref{reduc}, the dimension of the tangent space to $\\T_{\\rm sing}$ is $(1\/2)g(g+1)-g+2k$ if $rk (\\partial_i\\partial_j\\partial_k\\vartheta)=g-k .$\n\nOn the other hand, the normal space to $Y$ at the same point is given by the matrix\n$$\n  \\left(\\begin{array}{cc }\n  0&1_g\\\\\n  (\\partial_i\\partial_j\\partial_k\\vartheta)'&M(\\epsilon)\n  \\end{array}\\right)\n$$\nwith $M(\\epsilon)$  a $(1\/2)g(g+1)\\times g$   matrix  depending on $\\epsilon$. This matrix has rank $2g-k$, hence the dimension of the tangent space is $(1\/2)g(g+1)-g+k$. Since the  two computations    give the same result , we get $k=0$.  We get that the codimension of $\\pi(Y)$ is at least $g$. Since on a suitable covering of ${\\mathcal A}$ we have that this locus is defined by the equations\n$$\n \\partial_i \\tt \\epsilon\\delta(\\tau, 0)=0,\n$$\nWe have that the codimension is at most $g$, hence it is exactly $g$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThroughout the paper, a domain is a not necessarily commutative unital ring in which the zero element is the unique zero divisor.\nIn \\cite{claborn66}, Claborn showed that every abelian group $G$ is the class group of a commutative Dedekind domain. An exposition is contained in \\cite[Chapter III \\S14]{fossum73}.\nSimilar existence results, yielding commutative Dedekind domains which are more geometric, respectively number theoretic, in nature, were obtained by Leedham-Green in \\cite{leedham-green72} and Rosen in \\cite{rosen73,rosen76}.\nRecently, Clark in \\cite{clark09} showed that every abelian group is the class group of an elliptic commutative Dedekind domain, and that this domain can be chosen to be the integral closure of a PID in a quadratic field extension.\nSee Clark's article for an overview of his and earlier results.\nIn commutative multiplicative ideal theory also the distribution of nonzero prime ideals within the ideal classes plays an important role.\nFor an overview of realization results in this direction see \\cite[Chapter 3.7c]{ghk06}.\n\nA ring $R$ is a Dedekind prime ring if every nonzero submodule of a (left or right) progenerator is a progenerator (see \\cite[Chapter 5]{mcconnell-robson01}).\nEquivalently, $R$ is a hereditary Noetherian prime ring which is also a maximal order in its simple Artinian quotient ring.\nA Dedekind domain is a Dedekind prime ring $R$ which is also a domain (equivalently, $\\udim R_R = \\udim {}_R R = 1$).\nTo a Dedekind prime ring $R$ one can associate an (abelian) class group $G(R)$ in such a way that $K_0(R) \\cong G(R) \\times \\bZ$.\nEquivalently, $G(R)$ can also be interpreted as a group of stable isomorphism classes of essential right ideals of $R$.\nSince $K_0$ is Morita invariant, the same holds for the class group.\nEvery Dedekind prime ring is Morita equivalent to a Dedekind domain.\n\nRealization questions for class groups within the class of strictly noncommutative Dedekind prime rings have an easy answer.\nIf $R$ is a commutative Dedekind domain with class group $G(R)$ and $M$ is a finitely generated projective $R$-module, then $S=\\End_R(M)$ is a Dedekind prime ring with $G(S) \\cong G(R)$.\nHowever, $S$ is a PI ring, and thus in many aspects close to being commutative.\n\nOn the other hand, there exist Dedekind prime rings (and domains) of a very different nature.\nFor instance, the first Weyl algebra $A_1(K)$ over a field $K$ of characteristic $0$ is a simple Dedekind domain with trivial class group.\nThe ring $R=\\bR[X,Y]\/(X^2+Y^2-1)$ is a commutative Dedekind domain with $G(R) \\cong \\bZ\/2\\bZ$.\nIf $\\sigma \\in \\Aut(R)$ denotes the automorphism induced by the rotation by an irrational angle, then the skew Laurent polynomial ring $T = R[x,x^{-1};\\sigma]$ is a noncommutative Dedekind domain with $G(T) \\cong \\bZ\/2\\bZ$.\nSimilar constructions exist that show that $\\bZ^n$ for $n \\in \\bN_0$ appears as class group of a noncommutative Dedekind prime ring.\n(See \\cite[\\S7.11 and \\S12.7]{mcconnell-robson01} for details.)\n\nThe mentioned rings are not Morita equivalent to commutative Dedekind domains and they are all simple rings.\nIn fact, in \\cite{goodearl-stafford05}, a striking dichotomy is established: A Dedekind domain which is finitely generated as an algebra over $\\bC$ is commutative or simple.\nMore generally, if $K$ is a field and a $K$-algebra $R$ is a Dedekind prime ring such that $\\dim_K R < \\card{K}$ and $R \\otimes_K \\overline K$ is Noetherian, then $R$ is a PI ring or simple.\n\nIn \\cite[Problem 54.7]{levy-robson11}, Levy and Robson state it as an open problem to determine which abelian groups can appear as class groups of simple Dedekind prime rings.\nThe present paper answers this question by showing that any abelian group can be realized as the class group of a simple Dedekind domain.\nThe main theorem we prove is the following.\n\n\\begin{thm} \\label{t-main}\n  Let $G$ be an abelian group, $K$ a field, and $\\kappa$ a cardinal.\n  Then there exists a $K$-algebra $T$ which is a noncommutative simple Dedekind domain, $G(T)\\cong G$, and each class of $G(T)$ contains at least $\\kappa$ maximal right ideals of $T$.\n\\end{thm}\n\nSimple noncommutative Dedekind domains are canonically obtained either as skew Laurent polynomial rings $R[x,x^{-1};\\sigma]$ or as skew polynomial rings $R[x;\\delta]$, where $R$ is a commutative Dedekind domain and $\\sigma$ is an automorphism, respectively $\\delta$ a derivation.\nThe domains we construct are skew Laurent polynomial rings.\nIt is well understood how class groups behave under this extension.\nIn this way, the problem reduces to the construction of a commutative Dedekind domain $R$ with prescribed class group and automorphism $\\sigma$ of $R$.\nThe automorphism $\\sigma$ must be such that no proper nonzero ideal $\\fa$ is $\\sigma$-stable (that is, $\\sigma(\\fa) = \\fa$), but such that the induced automorphism on the class group of $R$ is trivial.\n\nThe actual construction is very conceptual in nature and proceeds through the following steps:\n\n\\begin{enumerate}\n  \\item Construct a commutative Krull monoid with class group $G$ and a monoid automorphism $\\tau$ of $H$ such that no nonempty proper divisorial ideal of $H$ is $\\tau$-stable.\n  \\item Extend $\\tau$ to $K[H]$.\n    The semigroup algebra $K[H]$ is a commutative Krull domain with class group isomorphic to $G$.\n    The crucial step lies in establishing that no nonzero proper divisorial ideal of $K[H]$ is $\\tau$-stable.\n  \\item A suitable localization $R=S^{-1}K[H]$ is a commutative Dedekind domain, has the same class group as $K[H]$, and $\\tau$ extends to $R$.\n    This is analogous to the same step in Claborn's proof.\n  \\item The skew Laurent polynomial ring $T=R[x,x^{-1};\\tau]$ is a noncommutative simple Dedekind domain with $G(T) \\cong G$.\n\\end{enumerate}\n\nThe methods work in greater generality.\nFor instance, the field $K$ can be replaced by a commutative Krull domain with suitable automorphism.\nThe full result is stated in \\cref{t-extend}.\n\\Cref{t-main} is an immediate consequence of \\cref{t-ex-mon-aut} and \\cref{t-extend}.\nThe actual construction is mostly commutative in nature.\nBefore giving the proofs in \\cref{s-proofs}, a number of preliminary results are recalled in \\cref{s-prelim}.\n\n\\begin{remark}\n  Let $R$ be a commutative Dedekind domain which is an affine algebra over a field $K$ of characteristic $0$.\n  Then the ring of differential operators $\\cD(R)$ is a simple Dedekind domain, and the inclusion $R \\hookrightarrow \\cD(R)$ induces an isomorphism $K_0(R) \\cong K_0(\\cD(R))$ (see \\cite[Chapter 15]{mcconnell-robson01}).\n  This induces an isomorphism $G(R) \\cong G(\\cD(R))$.\n  In \\cite{rosen73}, Rosen has shown that any finitely generated abelian group is the class group of a commutative Dedekind domain which is affine over a number field.\n  This gives a different way of showing that any finitely generated abelian group is the class group of a simple Dedekind domain.\n  Using the results from \\cite{clark09}, this can be extended to groups of the form $F\/H$ where $F$ is free abelian and $H$ is a finitely generated subgroup.\n\\end{remark}\n\n\\section{Background: Krull monoids and skew Laurent polynomial rings} \\label{s-prelim}\n\nAll rings and modules are unital.\nRing homomorphisms preserve the multiplicative identity.\nIf $X$ is a subset of a domain, we set $X^\\bullet=X \\setminus \\{0\\}$.\nA \\emph{monoid} is a cancellative semigroup with a neutral element.\nMonoid homomorphisms preserve the neutral element.\nIf $H$ is a monoid, $H^\\times$ denotes its group of units.\n$H$ is \\emph{reduced} if $H^\\times = \\{1\\}$.\nA commutative monoid is \\emph{torsion-free} if its quotient group is torsion-free.\n$\\bN$ denotes the set of positive integers and $\\bN_0$ the set of all nonnegative integers.\nFor sets $A$ and $B$, inclusion is denoted by $A \\subset B$ and strict inclusion by $A \\subsetneq B$.\n\nFor a set $P$, let $\\cF(P)$ denote the multiplicatively written free abelian monoid with basis $P$.\nThe quotient group $\\quo(\\cF(P))$ of $\\cF(P)$ is the free abelian group with basis $P$.\nEach $a \\in \\quo(\\cF(P))$ has a unique (up to order) representation of the form $a = p_1^{n_1}\\cdots p_r^{n_r}$ with $r \\in \\bN_0$, pairwise distinct $p_1$,~$\\ldots\\,$,~$p_r \\in P$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\nWe define $\\supp(a) = \\{p_1,\\ldots, p_r\\}$, $\\val_{p_i}(a)=n_i$ for $i \\in [1,r]$ and $\\val_q(a) = 0$ for all $q \\in P \\setminus \\supp(a)$.\n\n\\subsection{Commutative Krull monoids and commutative Krull domains}\nWe use \\cite[Chapter 2]{ghk06} as a reference for commutative Krull monoids and \\cite{fossum73} as reference for commutative Krull domains.\nFor semigroup algebras we refer to \\cite{gilmer84}.\n\nLet $(H,\\cdot)$ be a commutative monoid and let $\\tau \\in \\Aut(H)$.\nLet $\\quo(H)$ denote the quotient group of $H$.\nThe automorphism $\\tau$ naturally extends to an automorphism of $\\quo(H)$, which we also denote by $\\tau$.\nFor subsets $X$,~$Y \\subset \\quo(H)$ we define $\\cc{Y}{X} = \\{\\, a \\in \\quo(H) \\mid aX \\subset Y \\,\\}$.\nWe set $X^{-1} = \\cc{H}{X}$ and $X_v = (X^{-1})^{-1}$.\nThen $\\tau\\big({\\cc{Y}{X}}\\big) = \\cc{\\tau(Y)}{\\tau(X)}$ and hence $\\tau(X^{-1}) = \\tau(X)^{-1}$ and $\\tau(X)_v = \\tau( X_v )$.\n\nA subset $\\fa \\subset \\quo(H)$ is a \\emph{fractional ideal} of $H$ if $H\\fa \\subset \\fa$ and there exists a $d \\in H$ such that $d\\fa \\subset H$.\nIf in addition $\\fa \\subset H$, then $\\fa$ is an \\emph{ideal} of $H$.\nA fractional ideal $\\fa$ is \\emph{divisorial} if $\\fa = \\fa_v$.\nFor all $a \\in \\quo(H)$, $(aH)_v = aH$ and hence principal fractional ideals are divisorial.\nIf $\\fa$ and $\\fb$ are divisorial fractional ideals of $H$, their \\emph{divisorial product} is $\\fa \\cdot_v \\fb = (\\fa \\cdot \\fb)_v$.\nFor principal fractional ideals, the divisorial product coincides with the usual ideal product.\n\n$H$ is a \\emph{commutative Krull monoid} if it is $v$-Noetherian (i.e., satisfies the ascending chain condition on divisorial ideals) and completely integrally closed (i.e, whenever $x \\in \\quo(H)$ is such that there exists a $c \\in H$ such that $cx^n \\in H$ for all $n \\in \\bN$, then already $x \\in H$).\nFrom now on, let $H$ be a commutative Krull monoid.\nIf $\\fa$ is a nonempty divisorial fractional ideal of $H$, then $\\fa$ is invertible with respect to the divisorial product, i.e., $\\fa \\cdot_v \\fa^{-1} = H$.\nWe denote by $\\cF_v(H)^\\times$ the group of all nonempty divisorial fractional ideals, and by $\\cI_v^*(H)$ the monoid of all nonempty divisorial ideals.\nLet $\\mathfrak X(H)$ be the set of nonempty divisorial prime ideals.\nRecall that $\\mathfrak X(H)$ consists precisely of the prime ideals of height $1$.\n\nWith respect to the divisorial product, $\\cI_v^*(H)$ is the free abelian monoid with basis $\\mathfrak X(H)$, and $\\cF_v(H)^\\times$ is the free abelian group with basis $\\mathfrak X(H)$.\nHence, every $\\fa \\in \\cF_v(H)^\\times$ has a unique representation of the form\n\\[\n\\fa = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}\n\\]\nwith $r \\in \\bN_0$, pairwise distinct $\\fp_1$,~$\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\nWe have $\\supp(\\fa) = \\{\\fp_1,\\ldots,\\fp_r\\}$ and $\\val_{\\fp_i}(\\fa)= n_i$ for $i \\in [1,r]$.\n\nThe principal fractional ideals form a subgroup of $\\cF_v(H)^\\times$. The \\emph{class group} of $H$ is the factor group\n\\[\n\\cC(H) = \\cF_v(H)^\\times \/ \\{\\, aH \\mid a \\in \\quo(H) \\,\\}.\n\\]\nWe use additive notation for $\\cC(H)$.\nIf $\\fa \\in \\cF_v(H)^\\times$, we write $[\\fa]=[\\fa]_H$ for its class in $\\cC(H)$.\nIf $\\fa$,~$\\fb \\in \\cF_v(H)^\\times$, then $[\\fa \\cdot_v \\fb] = [\\fa] + [\\fb]$.\n\nAny $\\tau \\in \\Aut(H)$ induces an automorphism $\\tau_*$ of $\\cF_v(H)^\\times$ by means of $\\tau_*(\\fa) = \\tau(\\fa)$.\nThen $\\tau_*(\\mathfrak X(H)) = \\mathfrak X(H)$, the restriction $\\tau_*=\\tau_*|_{\\cI_v^*(H)}$ is a monoid automorphism of $\\cI_v^*(H)$, and $\\tau_*(aH)=\\tau(a)H$ for all $a \\in \\quo(H)$.\nIn particular, $\\tau_*$ induces an automorphism of $\\cC(H)$, also denoted by $\\tau_*$, by means of $\\tau_*([\\fa])=[\\tau_*(\\fa)]$.\n\nA \\emph{commutative Krull domain} is a domain $D$ such that $D^\\bullet$ is a commutative Krull monoid.\nWe use similar notation for Krull domains as we have introduced for Krull monoids.\nIf $\\quo(D)$ denotes the quotient field of $D$ and $X \\subset \\quo(D)$, then $\\cc{D}{X}$ is always additively closed.\nThis implies that there exists an isomorphism\n\\[\n\\cF_v(D)^\\times \\to \\cF_v(D^\\bullet)^\\times,\\quad \\fa \\mapsto \\fa^\\bullet.\n\\]\nConcepts related to divisorial ideals on $D$ correspond to ones on $D^\\bullet$.\nWe make use of this without further mention.\nIn particular, $\\cC(D) \\cong \\cC(D^\\bullet)$ canonically, and we identify.\nA commutative domain $D$ is a Dedekind domain if and only if it is a Krull domain with $\\dim(D) \\le 1$.\nThen every nonzero fractional ideal of $D$ is invertible and hence divisorial.\nIn particular, $\\cC(D)$ is the usual ideal class group of the Dedekind domain.\n\nWe will construct Krull domains from Krull monoids using semigroup algebras.\nThe following result is essential.\n\n\\begin{prop}[{\\cite[Theorem 15.6 and Corollary 16.8]{gilmer84}}] \\label{p-krull-semigroup-domain}\n  Let $D$ be a commutative domain and $H$ a torsion-free commutative monoid.\n  The semigroup algebra $D[H]$ is a Krull domain if and only if $D$ is a Krull domain, $H$ is a Krull monoid, and $H^\\times$ satisfies the ascending chain condition on cyclic subgroups.\n  In this case $\\cC(D[H]) \\cong \\cC(D) \\times \\cC(H)$.\n\\end{prop}\n\nThe isomorphism between $\\cC(D) \\times \\cC(H)$ and $\\cC(D[H])$ is obtained naturally by extending representatives of the divisorial ideal classes in $D$, respectively $H$, to $D[H]$.\nIf $\\fa$ is a fractional ideal of $D$, let $\\fa[H]=\\fa D[H]$ be the extension of $\\fa$ to $D[H]$.\nIt consists of all elements all of whose coefficients are contained in $\\fa$.\nIf $\\fb$ is a fractional ideal of $H$, let $D[\\fb]= \\fb D[H]$ be the extension of $\\fb$ to $D[H]$.\nIt consists of all elements whose support is contained in $\\fb$.\nBy $\\fa[\\fb]$ we denote the fractional ideal whose support is contained in $\\fb$ and whose coefficients are contained in $\\fa$.\nThen $\\fa[\\fb] = \\fa[H] \\cdot D[\\fb]$.\nExplicitly, the isomorphism of class groups is given by\n\\[\n\\cC(D) \\times \\cC(H) \\to \\cC(D[H]), \\quad\n([\\fa]_D, [\\fb]_H)   \\mapsto \\big[ \\fa[\\fb] \\big]_{D[H]}.\n\\]\nLet $\\sigma \\in \\Aut(D)$, $\\tau \\in \\Aut(H)$ and let $\\varphi \\in \\Aut(D[H])$ be the extension of $\\sigma$ and $\\tau$ to $D[H]$ (i.e., $\\varphi|_D=\\sigma$ and $\\varphi|_H = \\tau$).\nUnder the stated isomorphism of the class groups, the automorphism $(\\sigma_*, \\tau_*)$ on $\\cC(D) \\times \\cC(H)$ corresponds to $\\varphi_*$ on $\\cC(D[H])$.\nFrom now on we identify $\\cC(D[H]) \\cong \\cC(D) \\times \\cC(H)$.\n\n\\begin{prop}[Nagata's Theorem, {\\cite[Corollary 7.2]{fossum73}}] \\label{p-nagata}\n  Let $D$ be a commutative Krull domain and $S \\subset D^\\bullet$ a multiplicative subset.\n  Then the localization $S^{-1}D$ is a Krull domain and the map $\\cI_v^*(D) \\to \\cI_v^*(S^{-1}D)$, $\\fa \\mapsto S^{-1}\\fa$ induces an epimorphism $\\cC(D) \\to \\cC(S^{-1}D)$ with kernel generated by those $\\fp \\in \\mathfrak X(D)$ with $\\fp \\cap S \\ne \\emptyset$.\n  In particular, if $S$ is generated by prime elements of $D$, then $\\cC(D) \\cong \\cC(S^{-1}D)$.\n\\end{prop}\n\nLet $D$ be a commutative Krull domain and let $S \\subset D^\\bullet$ be a multiplicative subset.\nThen $S^{-1}D$ is a Dedekind domain if and only if $\\dim(S^{-1}D) \\le 1$.\nThis is the case if and only if $S \\cap \\fP \\ne \\emptyset$ for all $\\fP \\in \\spec(D)$ with $\\height(\\fP) > 1$.\n\n\\subsection{Skew Laurent polynomial rings}\n\nLet $R$ be a ring and $\\sigma \\in \\Aut(R)$.\nBy $R[x,x^{-1};\\sigma]$ we denote the ring of \\emph{skew Laurent polynomials}.\n$R[x,x^{-1};\\sigma]$ consists of polynomial expressions in $x$ and $x^{-1}$ with coefficients in $R$ and subject to $ax=x\\sigma(a)$ for all $a \\in R$.\nLet $\\fa$ be an ideal of $R$.\nIf $\\sigma \\in \\Aut(R)$, then $\\fa$ is \\emph{$\\sigma$-stable} if $\\sigma(\\fa) = \\fa$.\nThe ring $R$ is \\emph{$\\sigma$-simple} if $\\mathbf 0$ and $R$ are the only $\\sigma$-stable ideals of $R$.\n\n\\begin{prop}[{\\cite[Theorem 1.8.5]{mcconnell-robson01}}]\n  Let $R$ be a ring, $\\sigma \\in \\Aut(R)$ and $T = R[x,x^{-1};\\sigma]$.\n  Then $T$ is a simple ring if and only if $R$ is $\\sigma$-simple and no power of $\\sigma$ is an inner automorphism.\n\\end{prop}\n\nIf $R$ is a commutative ring, the identity is the only inner automorphism of $R$.\nHence the second condition in the previous theorem reduces to $\\sigma$ having infinite order.\nIf $R$ is a $\\sigma$-simple commutative domain which is not a field, then $\\sigma$ has infinite order.\nFor suppose $\\sigma^n = \\id$ for some $n \\in \\bN$.\nLet $\\mathbf 0 \\ne \\fa \\subsetneq R$ be an ideal of $R$.\nThen $\\fa\\sigma(\\fa) \\cdots \\sigma^{n-1}(\\fa) \\ne \\mathbf 0$ is a proper ideal of $R$ which is $\\sigma$-stable.\n\nCombining our observations so far with \\cite[Theorem 7.11.2]{mcconnell-robson01}, we obtain the following.\n\\begin{prop}[{\\cite[Theorem 7.11.2]{mcconnell-robson01}}] \\label{p-nc-dedekind}\n  Let $R$ be a commutative Dedekind domain which is not a field, let $\\sigma \\in \\Aut(R)$, and let $T=R[x,x^{-1};\\sigma]$.\n  The following conditions are equivalent:\n  \\begin{equivenumerate}\n    \\item $T$ is simple.\n    \\item $T$ is hereditary.\n    \\item The Krull dimension of $T$ is $1$.\n    \\item $T$ is a noncommutative Dedekind domain.\n    \\item $R$ is $\\sigma$-simple.\n  \\end{equivenumerate}\n\\end{prop}\n\nThe behavior of the Grothendieck group $K_0$ under skew Laurent polynomial extensions is well understood.\nWe denote classes in $K_0$ using angle brackets.\nWe recall the result from \\cite[\\S12.5]{mcconnell-robson01}.\nLet $R$ be a ring and $\\sigma \\in \\Aut(R)$.\nLet $M$ be a right $R$-module.\nDefine a new right $R$-module $M^\\sigma$ as follows:\nAs a set, $M^\\sigma$ is in bijection with $M$, where the element of $M^\\sigma$ corresponding to $m \\in M$ is written as $m^\\sigma$.\nThe abelian group structure on $M^\\sigma$ is the one induced from $M$, i.e., $m^\\sigma + n^\\sigma = (m+n)^\\sigma$.\nThe right $R$-module structure on $M^\\sigma$ is defined by $(m^\\sigma) r = (m \\sigma^{-1}(r))^\\sigma$.\nA similar construction works for left modules: To a left module $M$ associate ${}^\\sigma M$ with $r ({}^\\sigma m) = {}^\\sigma (\\sigma(r) m)$.\nIn particular, ${}^\\sigma R$ with the usual right $R$-module structure is an $R$-bimodule, and $M \\otimes_R ({}^\\sigma R) \\cong M^\\sigma$ as right $R$-modules.\nNow, $\\sigma$ induces an automorphism $\\sigma_*$ of $K_0(R)$ by means of $\\kcls{M} \\mapsto \\kcls{M^\\sigma}$.\n\nLet $T = R[x,x^{-1};\\sigma]$.\nFor a finitely generated projective right $R$-module $M$, $M \\otimes_R T$ is a finitely generated projective right $T$-module.\nThis induces a homomorphism $\\alpha\\colon K_0(R) \\to K_0(T)$.\nA ring $R$ is right regular if each finitely generated right $R$-module has a projective resolution of finite length.\n\n\\begin{prop}[{\\cite[Theorem 12.5.6]{mcconnell-robson01}}] \\label{p-k0-ext}\n  Let $R$ be a right regular, right Noetherian ring.\n  Let $\\sigma \\in \\Aut(R)$ and $T = R[x,x^{-1};\\sigma]$.\n  Then the sequence\n  \\[\n  \\xymatrix@C=1.5cm{\n    K_0(R) \\ar[r]^{\\id-\\sigma_*} & K_0(R) \\ar[r]^{\\alpha} & K_0(T) \\ar[r] & \\mathbf 0\n  }\n  \\]\n  is exact.\n\\end{prop}\n\nLet $R$ be a Dedekind prime ring.\nEach finitely generated projective right $R$-module $P$ has a uniform dimension $\\udim_R (P) \\in \\bN_0$.\nThe uniform dimension is additive on direct sums and induces an epimorphism $\\udim_R \\colon K_0(R) \\to \\bZ$.\nThe \\emph{(ideal) class group} of $R$ is $G(R) = \\ker(\\udim_R\\colon K_0(R) \\to \\bZ)$.\nThe epimorphism $\\udim_R$ splits, hence $K_0(R) \\cong G(R) \\times \\bZ$.\nLet $G'$ denote the set of stable isomorphism classes of essential right ideals of $R$.\n$G'$ can be endowed with the structure of an abelian group by setting $[\\fa] + [\\fb] = [\\fc]$ if and only if $\\fa \\oplus \\fb \\cong R \\oplus \\fc$.\nThen $G'$ is isomorphic to $G(R)$ by means of $G' \\to G(R), [\\fa] \\mapsto \\kcls{\\fa} - \\kcls{R}$, and we identify.\nWhen we say that a class $g \\in G(R)$ contains an essential right ideal $\\fa$ of $R$, we mean $g=[\\fa]=\\langle \\fa \\rangle - \\langle R \\rangle$.\n\nIf $R$ is commutative, $G(R)$ is indeed isomorphic the usual ideal class group.\nThe isomorphism $\\cC(R) \\to G(R)$ is given by $[\\fa] \\mapsto \\kcls{\\fa} - \\kcls{R}$.\nIf $\\sigma$ is an automorphism of $R$, we note that under the stated isomorphism of $\\cC(R)$ and $G(R)$, the induced automorphism $\\sigma_* \\colon \\cC(R) \\to \\cC(R)$ corresponds to $\\sigma_* \\colon G(R) \\to G(R)$.\nThis is so, because for an ideal $\\fa \\subset R$, we have $\\fa^\\sigma \\cong \\sigma(\\fa)$ as right $R$-modules, via $a^\\sigma \\mapsto \\sigma(a)$.\n\nLet $R$ be a commutative Dedekind domain.\nSince $\\udim_T(P \\otimes_R T) = \\udim_R(P)$ for all finitely generated projective $R$-modules $P$, we obtain a commutative diagram\n\\[\n  \\xymatrix@C=1.5cm{\n    K_0(R) \\ar[r]^{\\id-\\sigma_*} \\ar[d] & K_0(R) \\ar[r]^{\\alpha} \\ar[d] & K_0(T) \\ar[r] \\ar[d] & \\mathbf 0 \\\\\n    G(R) \\times \\bZ \\ar[r]^{(\\id-\\sigma_*,\\id)} & G(R) \\times \\bZ \\ar[r]^{(\\alpha_0,\\id)} & G(T) \\times \\bZ \\ar[r] & \\mathbf 0 \\\\\n  }\n\\]\nwith the vertical arrows being isomorphisms induced by the splitting of $\\udim_R$, $\\udim_R$, and $\\udim_T$ respectively.\nHere $\\alpha_0$ is the map induced on $G(R) \\to G(T)$ by $\\alpha$.\nUsing the isomorphism $\\cC(R) \\cong G(R)$, we obtain a short exact sequence\n\\[\n  \\xymatrix@C=1.5cm{\n    \\cC(R) \\ar[r]^{\\id-\\sigma_*} & \\cC(R) \\ar[r]^{\\beta} & G(T) \\ar[r] & \\mathbf 0.\n  }\n\\]\nHere, $\\beta([\\fa]_R) = \\kcls{\\fa \\otimes_R T} - \\kcls{T} = [\\fa \\otimes_R T]_T \\in G(T)$.\n\n\\begin{remark}\n  \\begin{enumerate}\n  \\item\n    If $R$ and $S$ are Morita equivalent Dedekind prime rings, the Morita equivalence induces an isomorphism $K_0(R) \\cong K_0(S)$, which restricts to an isomorphism $G(R) \\cong G(S)$.\n  \\item\n  Let $R$ be a Dedekind prime ring.\n  If $\\fa$ and $\\fb$ are stably isomorphic essential right ideals of $R$, that is $[\\fa]=[\\fb]$ in $G(R)$, then, in general, it does not follow that $\\fa \\cong \\fb$.\n  However, if $\\udim_R R \\ge 2$, then $[\\fa]=[\\fb]$ does imply $\\fa \\cong \\fb$ (\\cite[Corollary 35.6]{levy-robson11}).\n  Note that $S=M_n(R)$, with $n \\ge 2$ is a Dedekind prime ring with $G(R) \\cong G(S)$ and $\\udim_S S =n \\ge 2$.\n\\end{enumerate}\n\\end{remark}\n\n\\section{Construction and main results} \\label{s-proofs}\n\n\\begin{lemma}\n  Let $H$ be a commutative Krull monoid, $\\tau \\in \\Aut(H)$, and $\\fa \\in \\cF_v^\\times(H)$.\n  Then $\\tau(\\fa) = \\fa$ if and only if $\\tau(\\fa) \\subset \\fa$.\n\\end{lemma}\n\n\\begin{proof}\n  Suppose that $\\tau(\\fa) \\subset \\fa$.\n  Then there exists $\\fb \\in \\cI_v^*(H)$ such that $\\tau(\\fa) = \\fa \\cdot_v \\fb$.\n  Let $\\fa = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}$ with $r \\in \\bN_0$, $\\fp_1$,~$\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Similarly, let $\\fb = \\fq_1^{m_1} \\cdot_v \\ldots \\cdot_v \\fq_s^{m_s}$ with $s \\in \\bN_0$, $\\fq_1$,~$\\ldots\\,$,~$\\fq_s \\in \\mathfrak X(H)$ and $m_1$,~$\\ldots\\,$,~$m_s \\in \\bN$.\n  Then\n  \\[\n  \\tau(\\fa) = \\tau(\\fp_1)^{n_1} \\cdot_v \\ldots \\cdot_v \\tau(\\fp_r)^{n_r} = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r} \\cdot_v \\fq_1^{m_1} \\cdot_v \\ldots \\cdot_v \\fq_s^{m_s}.\n  \\]\n  Then necessarily $n_1+\\cdots + n_r = n_1 + \\cdots + n_r + m_1+ \\cdots + m_s$.\n  Hence $s=0$ and $\\fb=H$.\n  Thus $\\tau(\\fa) = \\fa$.\n\\end{proof}\n\nOf course, the claim of the previous lemma does not hold for ideals which are not divisorial.\nFor a counterexample, let $K$ be a field, $H=K[...,X_{-1},X_0,X_1,\\ldots]^\\bullet$, $\\tau(X_i) = X_{i+1}$ with $\\tau|_K=\\id$, and $\\fa = (X_0,X_1,\\ldots)$.\n\n\\begin{lemma} \\label{l-simple}\n  Let $H$ be a commutative Krull monoid and let $\\tau \\in \\Aut(H)$.\n  The following statements are equivalent:\n  \\begin{equivenumerate}\n    \\item\\label{l-simple:fracideal} $\\tau(\\fa) \\ne \\fa$ for all $\\fa \\in \\cF_v(H)^\\times \\setminus \\{H\\}$.\n    \\item\\label{l-simple:ideal} $\\tau(\\fa) \\ne \\fa$ for all $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$.\n    \\item\\label{l-simple:sqf} $\\tau(\\fa) \\ne \\fa$ for all squarefree $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$.\n    \\item\\label{l-simple:prime} For all finite $\\emptyset \\ne X \\subset \\mathfrak X(H)$, it holds that $\\tau_*(X) = \\{\\, \\tau(\\fp) \\mid \\fp \\in X \\,\\} \\ne X$.\n    \\item\\label{l-simple:orbits} The induced permutation $\\tau_*$ of $\\mathfrak X(H)$ has no finite orbits.\n  \\end{equivenumerate}\n  If $\\cC(H)=\\mathbf 0$, then any of the above conditions is equivalent to\n  \\begin{equivenumerate}\n    \\setcounter{enumi}{5}\n    \\item\\label{l-simple:principal} For all $a \\in H\\setminus H^\\times$ and $\\varepsilon \\in H^\\times$, $\\tau(a) \\ne \\varepsilon a$.\n  \\end{equivenumerate}\n\n  In particular, if these equivalent conditions are satisfied and $\\emptyset \\ne A \\subset \\quo(H)$ is finite with $A \\not\\subset H^\\times$, then $\\tau(A) \\ne A$.\n\\end{lemma}\n\n\\begin{proof}\n  \\ref*{l-simple:fracideal}${}\\Rightarrow{}$\\ref*{l-simple:ideal}${}\\Rightarrow{}$\\ref*{l-simple:sqf}: Trivial.\n\n  \\ref*{l-simple:sqf}${}\\Rightarrow{}$\\ref*{l-simple:prime}:\n  By contradiction. Suppose that $\\emptyset \\ne X \\subset \\mathfrak X(H)$ is such that $\\tau_*(X)=X$.\n  Set $\\fa = ( \\prod_{\\fp \\in X} \\fp )_v$.\n  Then $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$, $\\fa$ is squarefree, and $\\tau(\\fa) = \\tau\\big( ( \\prod_{\\fp \\in X} \\fp )_v \\big ) = ( \\prod_{\\fp \\in X} \\tau(\\fp) )_v = \\fa$.\n  This contradicts \\ref*{l-simple:sqf}.\n\n  \\ref*{l-simple:prime}${}\\Rightarrow{}$\\ref*{l-simple:orbits}: Clear.\n\n  \\ref*{l-simple:orbits}${}\\Rightarrow{}$\\ref*{l-simple:fracideal}:\n  Let $\\fa \\in \\cF_v^*(H) \\setminus \\{H\\}$.\n  Then $\\fa = \\fp_1^{n_1}\\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}$ with $r \\in \\bN$, $\\fp_1$, $\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$, $\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Now $\\tau(\\fa) = \\tau(\\fp_1)^{n_1} \\cdot_v \\ldots \\cdot_v \\tau(\\fp_r)^{n_r}$ is the unique representation of $\\tau(\\fa)$ as divisorial product of divisorial prime ideals.\n  Suppose that $\\tau(\\fa)=\\fa$.\n  Then $\\tau^n(\\fa) = \\fa$ for all $n \\in \\bZ$.\n  Hence the $\\tau_*$-orbit of $\\fp_1$ is contained in $\\supp(\\fa) = \\{ \\fp_1,\\ldots,\\fp_r \\}$.\n  This contradicts \\ref*{l-simple:orbits}.\n\n  \\ref*{l-simple:ideal}${}\\Leftrightarrow{}$\\ref*{l-simple:principal}\n  Suppose that $\\cC(H)$ is trivial.\n  Then every divisorial ideal is principal.\n  The claim follows since $aH = bH$ for $a$,~$b \\in H$ if and only if there exists $\\varepsilon \\in H^\\times$ with $a = b \\varepsilon$.\n\n  We still have to show the final implication and do so by contradiction.\n  Let $\\emptyset \\ne A=\\{a_1, \\ldots, a_n\\} \\subset \\quo(H)$ with $A \\not\\subset H^\\times$.\n  Since $A \\not\\subset H^\\times$, the set $X = \\bigcup_{i=1}^n \\supp(a_i H) \\subset \\mathfrak X(H)$ is nonempty.\n  Thus $\\tau_*(X) \\ne X$ by \\ref*{l-simple:prime}, and hence $\\tau(A) \\ne A$.\n\\end{proof}\n\n\\begin{defi}\n  Let $H$ be a commutative Krull monoid and $\\tau \\in \\Aut(H)$.\n  $H$ is called \\emph{$\\tau$-$v$-simple} if the equivalent conditions of \\cref{l-simple} are satisfied.\n  If $D$ is a commutative Krull domain and $\\sigma \\in \\Aut(D)$, then $D$ is called \\emph{$\\sigma$-$v$-simple} if the commutative Krull monoid $D^\\bullet$ is $(\\sigma|_{D^\\bullet})$-$v$-simple.\n\\end{defi}\n\nA lemma analogous to \\cref{l-simple} holds for commutative Krull domains.\nSince there is a correspondence between divisorial ideals of $D$ and divisorial ideals of $D^\\bullet$, $D$ is $\\sigma$-$v$-simple if and only if $\\sigma(\\fa) \\ne \\fa$ for all divisorial ideals $\\fa$ of $D$, etc.\n\nWe first construct a reduced commutative Krull monoid $H$ with given class group $G$, as well as an automorphism of $H$ such that $H$ is $\\tau$-$v$-simple, and such that $\\tau_*$ acts trivially on the class group.\n\n\\begin{thm} \\label{t-ex-mon-aut}\n  Let $G$ be an abelian group and $\\kappa$ an infinite cardinal.\n  Then there exists a reduced commutative Krull monoid $H$ and an automorphism $\\tau$ of $H$ such that $\\cC(H) \\cong G$, $\\tau_*=\\id_{\\cC(H)}$, and $H$ is $\\tau$-$v$-simple.\n  Each class of $\\cC(H)$ contains $\\kappa$ nonempty divisorial prime ideals.\n\\end{thm}\n\n\\begin{proof}\nLet $(G,+)$ be an additive abelian group, and let $\\Omega$ be a set of cardinality $\\kappa$.\nLet $\\tau_0 \\colon \\Omega \\to \\Omega$ be a permutation such that $\\tau_0(X) \\ne X$ for all finite $\\emptyset \\ne X \\subset \\Omega$.\n(Such a permutation always exists. $\\Omega$ is in bijection with $\\Omega \\times \\bZ$, and the map $\\Omega \\times \\bZ \\to \\Omega \\times \\bZ$, $(x,n) \\mapsto (x,n+1)$ has the desired property.)\n\nLet $D = \\cF(\\Omega \\times G)$ be the free abelian monoid with basis $\\Omega \\times G$.\nThen $\\tau_0$ induces an automorphism $\\tau \\in \\Aut(D)$ with the property that $\\tau((x,g)) = (\\tau_0(x),g)$ for all $x \\in \\Omega$ and $g \\in G$.\nLet $\\psi\\colon D \\to G$ be the unique homomorphism such that $\\psi((x,g)) = g$ for all $x \\in \\Omega$ and $g \\in G$.\nSet $H = \\psi^{-1}(0_G)$.\nSince $\\psi(\\tau((x,g))) = g = \\psi((x,g))$, we find $\\tau(H) \\subset H$.\nHence $\\tau$ restricts to an automorphism of $H$, again denoted by $\\tau$.\n\nWe claim that $(H,\\cdot)$ is a reduced commutative Krull monoid with class group $G$, that $H$ is $\\tau$-$v$-simple, and that the induced automorphism $\\tau_*$ of $\\cC(H)$ is the identity.\nMoreover, each class of $\\cC(H)$ contains $\\card{\\Omega}$ divisorial prime ideals.\nThat $H$ is a reduced commutative Krull monoid with class group $G$ follows from \\cite[Proposition 2.5.1.4]{ghk06}.\nIt also follows that the inclusion $\\iota\\colon H \\hookrightarrow D$ is a divisor theory.\nHence, $\\mathfrak X(H) = \\{\\, (x,g)D \\cap H \\mid x \\in \\Omega,\\; g \\in G \\,\\}$.\nBy construction, $\\tau$ does not fix any finite nonempty subset of $\\mathfrak X(H)$, and hence $H$ is $\\tau$-$v$-simple.\nOn the other hand, $\\psi(\\tau(x,g)) = \\psi((x,g)) = g$, so that $\\tau_*$ acts trivially on $\\cC(H)$.\n\\end{proof}\n\n\\begin{remark}\n  Let $H$ be a reduced commutative Krull monoid.\n  We note that it is easy to determine $\\Aut(H)$.\n  Let $\\tau \\in \\Aut(H)$.\n  Then $\\tau$ induces an automorphism $\\tau_*$ of $\\cI_v^*(H)$ and further an automorphism of $\\cC(H)$, that we denote by $\\tau_*$ again.\n  For $g \\in \\cC(H)$, denote by $\\mathfrak X(H)(g) = \\{\\, \\fp \\in \\mathfrak X(H) \\mid [\\fp]=g \\,\\}$ the nonempty divisorial prime ideals in class $g$.\n  For all $\\fp \\in \\mathfrak X(H)$, it holds that $[\\tau_*(\\fp)] = \\tau_*([\\fp])$.\n  In particular, if $g$ and $h \\in \\cC(H)$ lie in the same $\\tau_*$-orbit, then $\\card{\\mathfrak X(H)(g)} = \\card{\\mathfrak X(H)(h)}$.\n  The automorphism $\\tau$ is uniquely determined by the induced $\\tau_* \\in \\Aut(\\cC(H))$ as well as the family of bijections $\\mathfrak X(H)(g) \\to \\mathfrak X(H)(\\tau_*(g))$ induced by $\\tau_*$.\n\n  Conversely, suppose that $\\alpha$ is an automorphism of $\\cC(H)$ such that for all $g \\in \\cC(H)$, $\\card{\\mathfrak X(H)(g)}=\\card{\\mathfrak X(H)(\\alpha(g))}$.\n  For each class $g \\in \\cC(H)$, let $\\beta_g \\colon \\mathfrak X(H)(g) \\to \\mathfrak X(H)(\\alpha(g))$ be a bijection.\n  Then there exists a (uniquely determined) automorphism $\\tau \\in \\Aut(H)$ with $\\tau_*(\\fp) = \\beta_g(\\fp)$ for all $g \\in \\cC(H)$ and $\\fp \\in \\mathfrak X(H)(g)$.\n\n  In particular, we obtain the following strengthening of \\cref{t-ex-mon-aut}:\n  If $H$ is a commutative Krull monoid such that each class contains either zero or infinitely many divisorial prime ideals, then there exists a $\\tau \\in \\Aut(H)$ such that $H$ is $\\tau$-$v$-simple and $\\tau_*$ is the identity on $\\cC(H)$.\n\\end{remark}\n\nLet $P$ be a set, $\\cF(P)$ the (multiplicatively written) free abelian monoid with basis $P$, and $G=\\quo(\\cF(P))$ the free abelian group with basis $P$.\nSince every element of $G\\cong \\bZ^{(P)}$ has finite support, any total order on $P$ induces a total order on $G$ by means of the lexicographical order and the natural total order on $\\bZ$.\nExplicitly, for $a \\in G \\setminus \\{1\\}$ we define $a \\ge 1$ if and only if $\\val_p(a) \\ge 0$ for $p = \\max\\supp(a)$.\nWith respect to any such order, $G$ is a totally ordered group.\nIf $(G,\\cdot,\\le)$ is a totally ordered group, we set $G_{>1} = \\{\\, a \\in G \\mid a > 1 \\,\\}$ and $G_{\\ge 1} = \\{\\, a \\in G \\mid a \\ge 1 \\,\\}$.\n\n\\begin{lemma} \\label{l-ex-order}\n  \\begin{enumerate}\n    \\item\\label{l-ex-order:set} Let $P$ be a set and let $\\tau\\colon P \\to P$ be a permutation having no finite orbits.\n      Then there exists a total order $\\le$ on $P$ such that $\\tau$ is order-preserving with respect to $\\le$.\n      Moreover, $\\tau(x) > x$ for all $x \\in P$.\n    \\item\\label{l-ex-order:group}\n      Let $(P,\\le_P)$ be a totally ordered set.\n      Let $\\tau\\colon P \\to P$ be a permutation such that $\\tau$ is order-preserving and $\\tau(x) >_P x$ for all $x \\in P$.\n      Let $G = \\quo(\\cF(P))$,  $\\overline\\tau \\in \\Aut(G)$ with $\\overline\\tau|_P=\\tau$, and let $\\le$ be the total order on $G$ induced by $\\le_P$.\n      Then $\\overline\\tau(a) > a$ for all $a \\in G_{>1}$.\n      In particular, $\\overline\\tau$ is order-preserving and $\\overline\\tau(G_{>1}) \\subset G_{>1}$.\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n  \\ref*{l-ex-order:set}\n  For $x \\in P$, let $x^\\tau = \\{\\, \\tau^n(x) \\mid n \\in \\bZ \\,\\}$ be its $\\tau$-orbit.\n  Since $x^\\tau$ is infinite, it is naturally totally ordered by $\\tau^m(x) \\le \\tau^n(x)$ if and only if $m \\le n$.\n  Fix an arbitrary total order on the set of all $\\tau$-orbits.\n  For $x$, $y \\in P$, define $x \\le y$ if and only if either $x^\\tau < y^\\tau$, or if $x^\\tau=y^\\tau$ and there exists an $n \\in \\bN_0$ such that $y = \\tau^n(x)$.\n  Then $\\le$ is a total order on $P$, and $\\tau$ is order-preserving with respect to this order.\n  Moreover, $\\tau(x) > x$ for all $x \\in P$.\n\n  \\ref*{l-ex-order:group}\n  As already observed, $(G, \\cdot, \\le)$ is a totally ordered group.\n  Let $a \\in G$ with $a > 1$.\n  We show $\\overline\\tau(a) > a$.\n  We have $a = p_1^{n_1}\\cdots p_r^{n_r}$ with $r \\in \\bN$, pairwise distinct $p_1$,~$\\ldots\\,$,~$p_r \\in P$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Using the total order on $P$, we may assume $p_1 > \\cdots > p_r$.\n  Since $a > 1$, we have $n_1 > 0$.\n  Now, $\\overline\\tau(a) = \\overline\\tau(p_1)^{n_1} \\cdots \\overline\\tau(p_r)^{n_r}$.\n  Since $\\overline\\tau$ is order-preserving with respect to $\\le_P$, we have $\\overline\\tau(p_1) > \\cdots > \\overline\\tau(p_r)$ and moreover $\\overline\\tau(p_1) > p_1$.\n  From the way we defined the total order on $G$, it follows that $\\overline\\tau(a) > a > 1$.\n  In particular, $\\overline\\tau$ is order-preserving and $\\overline\\tau(G_{>1}) \\subset G_{>1}$.\n\\end{proof}\n\nIf $\\cF(P)$ is a free abelian monoid and $\\overline\\tau$ is an automorphism of $\\cF(P)$ which has no finite orbits on $P$, then \\cref{l-ex-order} implies that the quotient group $\\quo(\\cF(P))$ admits the structure of a totally ordered group with respect to which $\\overline\\tau$ is order-preserving, etc.\nThe following is a strengthening of this result to quotient groups of reduced commutative Krull monoids.\n\n\\begin{prop} \\label{p-grp-order}\n Let $H$ be a reduced commutative Krull monoid and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n Let $G$ denote the quotient group of $H$, and denote the extension of $\\tau$ to $\\Aut(G)$ again by $\\tau$.\n Then there exists an order $\\le$ on $G$ such that $(G,\\cdot,\\le)$ is a totally ordered group, $H \\subset G_{\\ge 1}$, and $\\tau(a) > a$ for all $a \\in G_{> 1}$.\n In particular, $\\tau$ is order-preserving on $G$ and $\\tau(G_{>1}) \\subset G_{>1}$.\n\\end{prop}\n\n\\begin{proof}\n  Since $H$ is a commutative Krull monoid, it has a divisor theory.\n  Because $H$ is reduced, this divisor theory can be taken to be an inclusion.\n  Thus, explicitly, there exists a set $P$ such that $\\iota\\colon H \\hookrightarrow F=\\cF(P)$, $G \\subset \\quo(F)$, and such that the inclusion $\\iota$ induces a monoid isomorphism\n  \\[\n  \\iota^*\\colon F \\to \\cI_v^*(H),\\quad  a \\mapsto aF \\cap H.\n  \\]\n  Moreover, $(\\iota^*)^{-1}(aH) = a$ for all $a \\in H$, and $\\iota^*|_P \\colon P \\to \\mathfrak X(H)$ is a bijection.\n  (See \\cite[Theorem 2.4.7.3]{ghk06}.)\n  Recall that $\\tau$ induces a monoid automorphism $\\tau_*\\colon \\cI_v^*(H) \\to \\cI_v^*(H)$ and that $\\tau_*(aH) = \\tau(a)H$ for all $a \\in H$.\n\n  We first show that $\\tau$ extends to an automorphism of $F$.\n  Through $\\iota^*$, we obtain an automorphism $\\overline\\tau = (\\iota^*)^{-1} \\circ \\tau_* \\circ \\iota^* \\in \\Aut(F)$.\n  But we also have $H \\subset F$ via the inclusion $\\iota$.\n  We claim that in fact $\\overline\\tau|_H = \\tau$.\n  Let $a \\in H$.\n  Then\n  \\[\n  \\overline\\tau(a) = (\\iota^*)^{-1} \\circ \\tau_* \\circ \\iota^*(a)\n           =  (\\iota^*)^{-1} \\circ \\tau_*(aH) = (\\iota^*)^{-1}(\\tau(a)H) = \\tau(a).\n  \\]\n  Moreover, $\\overline\\tau$ extends to an automorphism of $\\quo(F)$, again denoted by $\\overline\\tau$, and then also $\\overline\\tau|_G=\\tau$ on $G$.\n\n  The automorphism $\\overline\\tau$ induces a permutation on $P$, and \\subref{l-simple:orbits} implies that $\\overline\\tau$ does not have any finite orbits on $P$.\n  Thus, \\subref{l-ex-order:set} implies that there exists a total order $\\le_P$ on $P$ such that $\\overline\\tau|_P \\colon P \\to P$ is order-preserving and $\\overline\\tau(p) > p$ for all $p \\in P$.\n  Let $\\le$ denote the order on $\\quo(F)$ induced by $\\le_P$.\n  Then $(\\quo(F), \\cdot, \\le)$ is a totally ordered group.\n  By \\subref{l-ex-order:group}, $\\overline\\tau(a) > a$ for all $a \\in \\quo(F)_{>1}$.\n  Denote the restriction of $\\le$ to $G$ again by $\\le$.\n  Then $(G,\\cdot,\\le)$ is a totally ordered group, and $\\tau(a) > a$ for all $a \\in G_{>1}$.\n  Clearly $H \\subset G_{\\ge 1}.$\n\\end{proof}\n\nLet $(G,\\cdot,\\le)$ be a totally ordered group and $K$  a field.\nThe group algebra $K[G]$ is naturally $G$-graded.\nUsing the total order on $G$, it is easy to check that $K[G]$ is a domain.\nEvery unit of $K[G]$ is homogeneous, that is, $K[G]^\\times = \\{\\, \\lambda g \\mid \\lambda \\in K^\\times,\\; g \\in G \\,\\}$.\nIt follows that every nonzero principal ideal $\\fa$ of $K[G]$ has a uniquely determined generator of the form $1 + f$ with $\\supp(f) \\subset G_{>1}$.\nWe call $1+f$ the \\emph{normed generator} of $\\fa$.\n\n\\begin{prop} \\label{p-kg-simple}\n  Let $H$ be a reduced commutative Krull monoid, and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n  Let $G$ denote the quotient group of $H$ and let $K$ be a field.\n  If $\\varphi \\in \\Aut(K[G])$ with $\\varphi|_H = \\tau$ and $\\varphi(K) \\subset K$, then $K[G]$ is $\\varphi$-$v$-simple.\n\\end{prop}\n\n\\begin{proof}\n  Denote the extension of $\\tau$ to $G$ again by $\\tau$.\n  Note that $\\varphi|_G = \\tau$.\n  By \\cref{p-grp-order}, there exists an order $\\le$ on $G$ such that $(G,\\cdot,\\le)$ is a totally ordered group and $\\tau(G_{> 1}) \\subset G_{> 1}$.\n  Since $G$ is a subgroup of a free abelian group, it satisfies the ascending chain condition on cyclic subgroups.\n  Hence, $K[G]$ is a commutative Krull domain with trivial class group by \\cref{p-krull-semigroup-domain}.\n  Thus, every divisorial ideal of $K[G]$ is principal.\n  To show that $K[G]$ is $\\varphi$-$v$-simple it therefore suffices to show $\\varphi(\\fa) \\ne \\fa$ for all principal ideals $\\fa$ of $K[G]$ with $\\fa \\notin \\{\\mathbf 0, K[G]\\}$.\n  Let $\\fa$ be such an ideal.\n  Let $f \\in K[G]$ with $\\supp(f) \\subset G_{>1}$ be such that $1+f$ is the normed generator of $\\fa$.\n  Since $\\fa \\ne K[G]$, we have $\\supp(f) \\ne \\emptyset$.\n  Now $\\varphi(1+f) = \\varphi(1) + \\varphi(f) = 1 + \\varphi(f)$.\n  Moreover, $\\supp(\\varphi(f)) = \\tau(\\supp(f)) \\subset G_{>1}$.\n  Hence $1+\\varphi(f)$ is the normed generator of $\\varphi(\\fa)$.\n  Since $H$ is $\\tau$-$v$-simple, $\\tau(\\supp(f)) \\ne \\supp(f)$ by \\cref{l-simple}.\n  Thus $1+\\varphi(f) \\ne 1 + f$, and $\\varphi(\\fa) \\ne \\fa$.\n\\end{proof}\n\n\\begin{thm} \\label{t-extend-simple}\n  Let $D$ be a commutative Krull domain and let $\\sigma \\in \\Aut(D)$ be such that $D$ is $\\sigma$-$v$-simple.\n  Let $H$ be a reduced commutative Krull monoid and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n  Let $\\varphi\\in \\Aut(D[H])$ denote the extension of $\\sigma$ and $\\tau$ to $D[H]$, i.e., $\\varphi|_D=\\sigma$ and $\\varphi|_H=\\tau$.\n  Then $D[H]$ is $\\varphi$-$v$-simple.\n\\end{thm}\n\n\\begin{proof}\n  Let $\\varphi_*$ denote the permutation of $\\mathfrak X(D[H])$ induced by $\\varphi$.\n  There are injective maps\n  \\begin{align*}\n    \\iota_D^*\\colon&\n    \\begin{cases} \\mathfrak X(D) &\\to \\mathfrak X(D[H]), \\\\\n      \\fp &\\mapsto \\fp[H],\n    \\end{cases}\n    &\n    \\iota_H^*\\colon&\n    \\begin{cases} \\mathfrak X(H) &\\to \\mathfrak X(D[H]), \\\\\n      \\fp &\\mapsto D[\\fp].\n    \\end{cases}\n  \\end{align*}\n  The image of $\\iota_D^*$ consists of all $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap D^\\bullet \\ne \\emptyset$, while the image of $\\iota_H^*$ consists of all $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap H \\ne \\emptyset$.\n  Let $K=\\quo(D)$ and $G = \\quo(H)$.\n  The group algebra $K[G]$ is the localization of $D[H]$ by $H$ and $D^\\bullet$.\n  There is a bijection\n  \\[\n  \\iota_{K[G]}^* \\colon\n  \\begin{cases}\n      \\{\\, \\fp \\in \\mathfrak X(D[H]) \\mid \\fp \\cap (D^\\bullet \\cup H) = \\emptyset \\,\\} &\\to \\mathfrak X(K[G]), \\\\\n      \\fp           & \\mapsto    \\fp K[G], \\\\\n      \\fP\\cap D[H]  & \\mapsfrom  \\fP.\n  \\end{cases}\n  \\]\n  In particular,\n  \\[\n  \\mathfrak X(D[H]) = \\iota_D^*\\mathfrak X(D) \\;\\cup\\; \\iota_H^*\\mathfrak X(H) \\;\\cup\\; (\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G]).\n  \\]\n  All of this follows from \\cite[Chapter III, Sections 15 and 16]{gilmer84}, together with the fact that nontrivial essential discrete valuation overmonoids (overrings) of commutative Krull monoids (domains) bijectively correspond to nonempty (nonzero) divisorial prime ideals.\n  That $\\iota_D^*$ takes the stated form follows from \\cite[Theorem 15.3]{gilmer84}, and the corresponding fact for $\\iota_H^*$ is a consequence of \\cite[Theorem 15.7]{gilmer84}.\n  The stated decomposition of $\\mathfrak X(D[H])$ follows from \\cite[Corollary 15.9]{gilmer84}.\n\n  Since $\\varphi(H) = H$ and $\\varphi(D) = D$, each of the sets $\\iota_D^*\\mathfrak X(D)$, $\\iota_H^*\\mathfrak X(H)$, and $(\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G])$ is fixed by $\\varphi_*$.\n  To show $\\varphi_*(X) \\ne X$ for all finite $\\emptyset \\ne X \\subset \\mathfrak X(D[H])$, it therefore suffices to consider subsets of each of these three sets.\n  If $X \\subset \\iota_D^*\\mathfrak X(D)$, then $\\varphi_*(X) \\ne X$, since $D$ is $\\sigma$-$v$-simple.\n  If $X \\subset \\iota_H^*\\mathfrak X(H)$, then $\\varphi_*(X) \\ne X$, since $H$ is $\\tau$-$v$-simple.\n\n  Finally, consider the case where $X \\subset (\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G])$.\n  Since $\\varphi(H) \\subset H$ and $\\varphi(D^\\bullet) \\subset D^\\bullet$, $\\varphi$ extends to an automorphism of $K[G]$, which we again denote by $\\varphi$.\n  It now suffices to show that $K[G]$ is $\\varphi$-$v$-simple.\n  However, this follows from \\cref{p-kg-simple}.\n\\end{proof}\n\nThe following \\subref{l-loc:basic} is a slight reformulation of the original localization argument of Claborn, which can be found in  \\cite[Theorem 14.2]{fossum73} and \\cite[Theorem 7]{claborn66}.\nSince we need to observe some details in the argument, we give the proof anyway.\nRecall that if $D$ is a commutative domain and $a$,~$b \\in D$ are coprime (that is, $aD \\cap bD = abD$), then $aX + b$ is a prime element of $D[X]$ (see \\cite[Lemma 14.1]{fossum73}).\n\n\\begin{lemma} \\label{l-loc}\n  Let $D$ be a commutative Krull domain, and let $H$ be a commutative Krull monoid containing a countable set $P$ of non-associated prime elements, so that $H = H_0 \\times \\cF(P)$ with a commutative Krull monoid $H_0$.\n  Suppose that $D[H]$ is a Krull domain.\n  \\begin{enumerate}\n    \\item\\label{l-loc:basic} There exists a multiplicative subset $S \\subset D[H]^\\bullet$ such that $S$ is generated by prime elements of $D[H]$, $S \\cap D[H_0] = \\emptyset$, and $S^{-1}D[H]$ is a Dedekind domain but not a field.\n    \\item\\label{l-loc:auto} If $\\varphi \\in \\Aut(D[H])$ with $\\varphi(P) \\subset P$, then $S$ can be chosen in such a way that $\\varphi(S) \\subset S$.\n    \\item\\label{l-loc:prop}\n      Let $S$ be a multiplicative subset of $D[H]^\\bullet$ such that $S^{-1}D[H]$ is a Dedekind domain.\n      Let $\\varphi \\in \\Aut(D[H])$ be such that $D[H]$ is $\\varphi$-$v$-simple and  $\\varphi(S) \\subset (S^{-1}D[H])^\\times$.\n      Then $\\varphi$ extends to an automorphism $\\varphi_S \\in \\Aut(S^{-1}D[H])$ and $S^{-1}D[H]$ is $\\varphi_S$-simple.\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\ref*{l-loc:basic}\nWe have $D[H]\\cong D[H_0][\\cF(P)] = D[H_0][\\ldots,X_{-1},X_0,X_1,\\ldots]$.\nLet $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$, and let $a_\\fP \\in \\fP^\\bullet$.\nLet $\\fp_1$, $\\ldots\\,$,~$\\fp_r$ be the divisorial prime ideals of $D[H]$ that contain $a_\\fP$.\nBy prime avoidance, there exists an element $b_\\fP \\in \\fP \\setminus (\\fp_1 \\cup \\ldots \\cup \\fp_r)$.\nLet $X_\\fP \\in \\{ \\ldots, X_{-1},X_0,X_1, \\ldots \\}$ be such that $X_\\fP$ is not contained in the support of $a_\\fP$ or $b_\\fP$.\nThen $D[H] = R_0[X_\\fP]$ with $R_0=D[H_0][\\{\\, X_i \\mid i \\in \\bZ,\\; X_i \\ne X_\\fP \\,\\}]$ and $a_\\fP$,~$b_\\fP \\in R_0$ are coprime.\nHence $f_\\fP = a_\\fP X_\\fP + b_\\fP$ is a prime element of $D[H]$, and $f_\\fP \\in \\fP$.\nBy construction, $f_\\fP \\notin D[H_0]$.\n\nLet $Q = \\{\\, f_\\fP \\mid \\fP \\in \\spec(D[H]),\\, \\height(\\fP) > 1 \\,\\}$ and let $S$ be the multiplicative set generated by $Q$.\nSince $\\spec(S^{-1}D[H])$ is in bijection with $\\{\\, \\fp \\in \\spec(D[H]) \\mid \\fp \\cap S \\ne \\emptyset \\,\\}$, it follows that $S^{-1}D[H]$ is a Krull domain of dimension at most $1$, i.e., a Dedekind domain.\nMoreover, $S \\cap D[H_0] = \\emptyset$.\n\nIf $D[H_0]$ is not a field, then neither is $S^{-1}D[H]$.\nIt only remains to consider the, degenerate, special case where $D[H_0]$ is a field, i.e., $H_0$ is the trivial monoid and $D=K$ is a field.\nThen $D[H] = K[\\ldots,X_{-1},X_0,X_1,\\ldots]$ is a polynomial ring in countably many indeterminates.\nBy construction, $Q$ only contains elements with $X_i$-degree equal to $1$ for some $i \\in \\bZ$.\nHowever, $D[H]$ contains prime elements which are not of this form (e.g., $X_1^2 + X_0^2 X_1 + X_0$).\n\n\\ref*{l-loc:auto}\nIf $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$, then also $\\height(\\varphi(\\fP))>1$.\nThus $\\varphi$ induces a permutation of prime ideals of height greater than $1$.\nDenote a set of representatives for the orbits by $\\Omega$.\nFor each $\\fP$ in $\\Omega$, choose $f_\\fP$ as in \\ref*{l-loc:basic}.\nFor all $n \\in \\bZ$, $\\varphi^n(f_\\fP)$ is a prime element contained in $\\varphi^n(\\fP)$.\nSince $\\varphi(P) \\subset P$, we have $\\varphi^n(f_\\fP) \\notin D[H_0]$.\nSet $Q = \\bigcup_{\\fP \\in \\Omega} \\bigcup_{n \\in \\bZ} \\varphi^n(f_\\fP)$, and let $S$ be the multiplicative subset of $D[H]^\\bullet$ generated by $Q$.\nSince $\\varphi(Q) \\subset Q$, also $\\varphi(S) \\subset S$.\nThus, $S$ has the stated properties.\n\n\\ref*{l-loc:prop}\nSince $\\varphi(S) \\subset (S^{-1}D[H])^\\times$, $\\varphi$ extends to an automorphism $\\varphi_S$ of $S^{-1}D[H]$.\nLocalization induces a bijection between $\\{\\, \\fp \\in \\mathfrak X(D[H]) \\mid \\fp \\cap S = \\emptyset \\,\\}$ and $\\mathfrak X(S^{-1}D[H])$.\nHence $S^{-1}D[H]$ is $\\varphi_S$-$v$-simple.\nSince $S^{-1}D[H]$ is a Dedekind domain, every ideal is divisorial.\nThus $S^{-1}D[H]$ is $\\varphi_S$-simple.\n\\end{proof}\n\n\\begin{remark}\n  Most of the technicalities in the previous proof can be avoided as long as $\\cC(H)$ is non-trivial and we are not picky about whether or not $S \\cap D[H_0] = \\emptyset$.\n  In this case, we take $S$ to be the multiplicative set generated by all prime elements of $D[H]$.\n  Claborn's argument shows that each $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$ contains some prime element, so that indeed $\\dim(S^{-1}D[H]) \\le 1$.\n  We have $\\varphi(S) \\subset S$, since prime elements are mapped to prime elements by $\\varphi$.\n  And, finally, since $\\cC(H)$ is non-trivial, there must exist a non-principal divisorial prime ideal $\\fp \\in D[H]$.\n  Then $\\fp \\cap S= \\emptyset$, hence $\\dim(S^{-1}D[H]) = 1$.\n\\end{remark}\n\n\\begin{thm} \\label{t-extend}\n  Let $D$ be a commutative Krull domain and let $\\sigma \\in \\Aut(D)$ be such that $D$ is $\\sigma$-$v$-simple.\n  Let $H$ be a reduced commutative Krull monoid containing prime elements,\n  and let $\\tau$ be an automorphism of $H$ such that $H$ is $\\tau$-$v$-simple.\n  Let $\\varphi\\colon D[H]\\to D[H]$ denote the extension of $\\sigma$ and $\\tau$ to $D[H]$, that is, $\\varphi|_D = \\sigma$ and $\\varphi|_H = \\tau$.\n  Let $p \\in H$ be a prime element and let $p^\\tau = \\{\\, \\tau^n(p) \\mid n \\in \\bZ \\,\\}$ be its $\\tau$-orbit, so that $H = H_0 \\times \\cF(p^\\tau)$ for a Krull monoid $H_0$.\n  \\begin{enumerate}\n    \\item \\label{t-extend:exist} There exists a multiplicative subset $S$ of the semigroup algebra $D[H]$ such that $S$ is generated by prime elements, $S \\cap D[H_0] = \\emptyset$, $\\varphi(S) \\subset S$, and the localization $S^{-1}D[H]$ is a Dedekind domain but not a field.\n    \\item \\label{t-extend:clsgrp}\n      Let $S \\subset D[H]^\\bullet$ be a multiplicative subset such that $R=S^{-1}D[H]$ is a Dedekind domain but not a field and $\\varphi(S) \\subset R^\\times$.\n      Let $A$ be the subgroup of $\\cC(D) \\times \\cC(H) = \\cC(D[H])$ generated by classes of $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap S \\ne \\emptyset$.\n      Then $\\varphi$ extends to an automorphism $\\varphi_S$ of $R$, the skew Laurent polynomial ring $T=R[x,x^{-1};\\varphi_S]$ is a noncommutative simple Dedekind domain, and the following sequence of abelian groups is exact:\n      \\[\n      \\xymatrix@C=0.75cm{\n        \\cC(D) \\times \\cC(H)\/A \\ar[rr]^{\\id - (\\sigma_*,\\tau_*)}\n        & & \\cC(D) \\times \\cC(H)\/A \\ar[r]^{\\beta}\n        & G\\big(R[x,x^{-1};\\varphi_S]\\big) \\ar[r]\n        & \\mathbf 0.\n      }\n      \\]\n      Here, $(\\sigma_*,\\tau_*)$ is the automorphism of $\\cC(D) \\times \\cC(H)\/A$ induced by $\\sigma$ and $\\tau$.\n      The map $\\beta$ is induced as follows: If $\\fa \\in \\cF_v(D)^\\times$, the class of $\\fa$ is mapped to $\\kcls{\\fa[H] \\otimes_R T} - \\kcls{T}$.\n      If $\\fb \\in \\cF_v(H)^\\times$, the class of $\\fb$ is mapped to $\\kcls{D[\\fb] \\otimes_R T} - \\kcls{T}$.\n  \\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n  \\ref*{t-extend:exist}\n  Since $\\tau$ does not have finite orbits on $\\mathfrak X(H)$, the orbit $p^\\tau$ of $p$ consists of countably many non-associated prime elements.\n  Let $S$ be a multiplicative subset of $D[H]$ as in \\subref{l-loc:auto}, where we take $P=p^\\sigma$.\n\n  \\ref*{t-extend:clsgrp}\n  By \\cref{t-extend-simple}, $D[H]$ is $\\varphi$-$v$-simple.\n  By \\subref{l-loc:prop}, $\\varphi$ extends to an automorphism $\\varphi_S \\in \\Aut(R)$, and $R$ is $\\varphi_S$-simple.\n  By Nagata's Theorem and the identifications we have made, $\\cC(D) \\times \\cC(H)\/A \\cong \\cC(R)$ with the isomorphism given by $([\\fa]_D,[\\fb]_H) +A \\mapsto [S^{-1}\\fa[\\fb]]$.\n  Since $R$ is $\\varphi_S$-simple, $T = R[x,x^{-1};\\sigma]$ is a noncommutative simple Dedekind domain by \\cref{p-nc-dedekind}.\n  Under the identification of $\\cC(D) \\times \\cC(H) \/ A$ with $G(R)$, the automorphism $(\\sigma_*,\\tau_*)$ corresponds to $\\varphi_*$.\n  The exact sequence of class groups follows from \\cref{p-k0-ext} and the discussion that followed it.\n\\end{proof}\n\n\\begin{remark} \\label{r-primes-techn}\n  The technical condition that $H$ contains a prime element (and hence, since $\\tau$ does not have any finite orbits on $\\mathfrak X(H)$, infinitely many non-associated ones) is necessary so that $D[H]$ has the form $D[H_0][\\ldots,X_{-1},X_0,X_1,\\ldots]$ with $\\varphi$ acting by $\\varphi(X_i) = X_{i+1}$.\n  The countably many indeterminates are used to construct the prime elements which generate $S$, see \\cref{l-loc}.\n  (See \\cite[Proposition 14]{chang11} for a refinement that only needs one indeterminate.)\n  If $H$ does not contain a prime element, we may replace $H$ by $H'=H \\times \\cF(\\ldots,p_{-1},p_0,p_1,\\ldots)$ and extend $\\tau$ by $\\tau(p_i)=p_{i+1}$.\n  Then $H'$ satisfies the conditions of the theorem, and $\\cC(H)\\cong \\cC(H')$.\n  By formulating the theorem in the slightly more technical way, we avoid the need to enlarge $H$ if it already contains prime elements.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem 1.1]\n  We assume without restriction that $\\kappa$ is infinite.\n  Let $G$ be an abelian group.\n  \\Cref{t-ex-mon-aut} implies that there exist a reduced commutative Krull monoid with $\\cC(H) \\cong G$ and an automorphism $\\tau$ of $H$ such that $H$ is $\\tau$-$v$-simple, $\\tau_* \\colon \\cC(H) \\to \\cC(H)$ is the identity, and each divisorial ideal class of $H$ contains $\\kappa$ nonempty divisorial prime ideals.\n  Let $K$ be a field.\n  Then $K$ is simple, and hence $\\id_K$-$v$-simple.\n  Let $\\varphi\\colon K[H] \\to K[H]$ be the automorphism of $K[H]$ with $\\varphi|_H = \\tau$ and $\\varphi|_K=\\id_K$.\n  Let $P \\subset H$ be a countable set of prime elements such that $H\\setminus P$ still contains $\\kappa$ prime elements.\n  Then $H = H_0 \\times \\cF(P)$ with a Krull monoid $H_0$.\n  Each class of $\\cC(H_0)$ contains $\\kappa$ divisorial prime ideals.\n  Applying \\cref{t-extend}, we find a subset $S \\subset K[H]$ such that $S \\cap K[H_0] = \\emptyset$, the localization $R=S^{-1}K[H]$ is a commutative Dedekind domain but not a field, and $T = S^{-1}K[H][x,x^{-1};\\varphi_S]$ is a noncommutative simple Dedekind domain with $G(T) \\cong G$.\n\n  If $\\fp \\in \\mathfrak X(H)$, then $K[\\fp] \\in \\mathfrak X(K[H])$.\n  If $\\fp \\notin \\{\\, (p) \\mid p \\in P \\,\\}$, then $K[\\fp] \\cap S = \\emptyset$ by construction.\n  In this case, $S^{-1}\\fp$ is a nonzero prime ideal of $R$.\n  Thus, each ideal class of $R$ contains at least $\\kappa$ nonzero prime ideals.\n  If $\\fq$ is a nonzero prime ideal of $R$, then $\\fq T$ is a maximal right ideal of $T$ by \\cite[Lemma 6.9.15]{mcconnell-robson01}.\n  Since $T$ is flat over $R$, we have $\\fq T \\cong \\fq \\otimes_R T$.\n  If $\\fp \\in \\mathfrak X(H)$, the isomorphism $\\beta\\colon \\cC(H) \\to G(T)$ maps $[\\fp]$ to $\\kcls{K[\\fp] \\otimes_R T} - \\kcls{T}$.\n  It follows that each class of $G(T)$ contains at least $\\kappa$ maximal right ideals.\n\\end{proof}\n\n\\begin{remark}\n  \\begin{enumerate}\n    \\item We can only give a lower bound on the cardinality of maximal right ideals in each class.\n      Apart from the divisorial prime ideals of the form $K[\\fp]$ with $\\fp \\in \\mathfrak X(H)$, additional divisorial prime ideals arise from prime elements of $K[\\quo(H)]$.\n      In \\cite{chang11}, Chang has shown that if $D[H]$ is a commutative Krull domain and $H$ is non-trivial, then each divisorial ideal class contains a nonzero divisorial prime ideal.\n\n    \\item\n      A domain $D$ is \\emph{half-factorial} if every element of $D^\\bullet$ can be written as a product of irreducibles and the number of irreducibles in each such factorization is uniquely determined.\n      It is conjectured that every abelian group is the class group of a half-factorial commutative Dedekind domain.\n      See \\cite[\\S5]{gilmer06} for background.\n      The conjecture is equivalent to one purely about abelian groups (\\cite[Proposition 3.7.9]{ghk06}).\n      See \\cite{geroldinger-goebel03} for progress on this question.\n  \\end{enumerate}\n\\end{remark}\n\n\\noindent\n\\textbf{Acknowledgments.}\n\\phantomsection\n\\addcontentsline{toc}{section}{Acknowledgments}\nI thank Alfred Geroldinger for feedback on a preliminary version of this paper.\n\n\\bibliographystyle{hyperalphaabbr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAccelerating beams, i.e., electromagnetic fields that propagate along curved trajectories in free space without being subject to any external force, have been the subject of a thorough investigation in the last years. The most famous representative of such class of beams is, without doubts, the Airy beam. Firstly introduced in the context of quantum mechanics by Berry and Balazs as an exotic solution of the Schr\\\"odinger equation \\cite{berryAiry}, it was then introduced in optics in 2007 by Siviloglou and co-workers \\cite{siviloglou,siviloglou2}, as an exact solution of the paraxial equation propagating along a parabolic trajectory in free space. Due to their intriguing features, Airy beams were studied in different contexts, such as nonlinear optics \\cite{airy1}, particle manipulation \\cite{airy2}, and proposed as an efficient way to generate curved plasma channels \\cite{airy3}. \n\nInspired by these results, the last years witnessed the emergence of many different types of accelerating beams in different coordinate systems, such as parabolic \\cite{acc1} and Weber \\cite{acc2} beams. Moreover, beams of light capable to propagate along curved \\cite{acc3,curved1, curved2} and arbitrary \\cite{arbitrary1,arbitrary2} trajectories, has also been proposed. Recently, two new classes of accelerating beams have been introduced, namely angular \\cite{angularAcc, vettiOE} and radially self-accelerating beams \\cite{nostroPRL, nostroAPL}. While the former acquire angular acceleration during rotation around their optical axis \\cite{angularAcc},  the latter exhibit radial acceleration, a feature which makes them propagate along spiralling trajectories around their optical axis. \n\nRadially self-accelerating beams (RSABs) can be understood in terms of superpositions of Bessel beams, where each single component is characterised by an angular velocity proportional to the amount of orbital angular momentum it carries. This, ultimately, results in an electromagnetic field, whose transverse field or intensity distribution rotates around the propagation direction with a given constant angular velocity $\\Omega$ \\cite{nostroPRL}. Among the vast zoology of RSABs, in particular, helicon beams, i.e., a subclass of RSABs consisting of rotating diffraction-free beams based on the superposition of two Bessel beams with opposite orbital angular momentum, have attracted a lot of interest in the last decades \\cite{nostroAPL,helicon1,helicon2,helicon3,helicon4,helicon5,helicon6,helicon7,helicon8,helicon9}. Beyond helicon beams, RSABs have potentially significant  applications in different areas of physics, such as sensing \\cite{airy3}, material processing \\cite{matProc1,matProc2}, and particle manipulation \\cite{partMan1,partMan2}. \n\nDespite this broad interest, RSABs have only been defined within the scalar electromagnetic theory, and their vector nature, as well as the effect of focussing on their self-accelerating character, has not been yet investigated. In this work, therefore, we introduce vector RSABs, and study their vector properties, in terms of their linear and angular momentum content. Moreover, we carefully analyse what is the impact of focussing on the self-accelerating character of RSABs, and under which conditions the focussing process does not spoil this property.\n\nThis work is organised as follows: in Sect. 2 we briefly recall the definition of RSABs, and recall some of their main properties. In Sect. 3 we use the method of Hertz potentials to construct vector RSABs, and use these solutions as a model for focussed RSABs. Then, we derive a condition on the polarisation that a scalar RSAB must possess, in order to maintain its self-accelerating character upon focussing. Section 4 is then devoted to calculate the linear and angular momentum for paraxial, intensity rotating RSABs.  Conclusions are then drawn in Sect. 5.\n\n\\section{Radially Self-Accelerating  Beams}\nWe start our analysis by considering scalar, monochromatic, free space solutions of the Helmholtz equation \n\\begin{equation}\\label{eq1}\n\\left(\\nabla^2+k_0^2\\right)\\psi(\\vett{r})=0,\n\\end{equation}\nwhere $k_0$ is the vacuum wave vector. The most general solution of the above equation in cylindrical coordinates, can be given in terms of superposition of Bessel beams, i.e.,\n\\begin{equation}\\label{eq2}\n\\psi(\\vett{r})=\\sum_{m}\\,\\int\\,d\\xi\\,C_m(\\xi)\\text{J}_m(\\rho\\sqrt{1-\\xi^2} )e^{i(m\\theta+\\xi\\zeta)},\n\\end{equation}\nwhere $\\rho=k_0R$, and $\\zeta=k_0 z$ are normalised radial and longitudinal coordinates, $\\text{J}_m(x)$ is the Bessel function of the first kind \\cite{nist}, and the integration variable $\\xi=\\cos\\vartheta_0$  plays the role of the Bessel cone angle $\\vartheta_0$ \\cite{durnin}. \n\nFrom the above solution, it is possible to extract RSABs by applying the requirements that Eq. \\eqref{eq2} must fulfil, in order to be a RSAB \\cite{nostroPRL}. First, $\\psi(\\vett{r})$ must propagate freely, and not under the action of a certain potential. Then, there should exist a suitable reference frame, in which $\\psi(\\vett{r})$ is manifestly propagation invariant, i.e., no explicit $\\zeta$-dependence must appear. Finally, an observer at rest in such reference frame should experience a fictitious force, which, ultimately, is at the core of self-accelerating character of RSABs.\n\nWhle the first requirement is automatically met by the fact that we are considering free space propagation, the second one is very useful to define RSABs properly. Once it is fulfilled, in fact, it is not hard to show that the third requirement follows accordingly. We therefore require, that, after a suitable coordinate transformation $\\vett{r}'=S\\, \\vett{r}$,  the field $\\psi(\\vett{r}')$ in the new coordinate frame is manifestly propagation invariant, i.e., \n$\\partial\\psi(\\vett{r}')\/\\partial\\zeta=0$. To this aim, we introduce the co-rotating coordinate $\\Phi=\\theta+\\Lambda\\zeta$, and choose the expansion coefficient as  $C_m(\\xi)=D_m\\delta(\\xi-(m\\Lambda+\\beta))$, where $\\Lambda=\\Omega\/k_0>0$ is the normalised angular velocity of the RSAB, and $\\beta$ is a free (dimensionless) parameter, with the physical meaning of a normalised propagation constant. Substituting this Ansatz in Eq. \\eqref{eq2} we get the following result\n\\begin{equation}\\label{eq3}\n\\psi_{RSAB}(\\rho,\\Phi)=e^{i\\beta\\zeta}\\sum_{m\\in\\mathcal{M}}D_m\\text{J}_m(\\alpha_m\\rho)e^{im\\Phi},\n\\end{equation}\nwhere $\\alpha_m=\\sqrt{1-(m\\Lambda+\\beta)^2}$, and $\\mathcal{M}=\\{m\\in\\mathbb{N}: \\alpha_m>0\\}$. For $\\beta=0$, the above field is manifestly propagation invariant, as no explicit $\\zeta$-dependence is present. Moreover, its amplitude and phase both rotate with normalised angular velocity $\\Lambda$ during propagation. For $\\beta\\neq 0$, on the other hand, the field itself is not anymore propagation invariant, due to the presence of the global phase factor $\\exp{(i\\beta\\zeta)}$. Nevertheless, the intensity $|\\psi_{RSAB}(\\vettGreek{\\rho})|^2$ is propagation invariant also for $\\beta\\neq 0$. In this case, however, while both intensity and phase propagate describing spiralling trajectories, they are not synchronised anymore. These two classes of RSABs are called field rotating, and intensity rotating, respectively \\cite{nostroPRL}. \n\nIt is worth noticing, moreover, that while for $\\beta=0$ the set $\\mathcal{M}$ contains only positive integers, for $\\beta\\neq 0$ positive and negative values of $m$ are allowed. Thus, helicon beams, for example, are a particular case of intensity rotating RSABs, where only two Bessel beams are participating in the sum in Eq. \\eqref{eq3} An Example of both classes of RSABs is given in Fig. \\ref{figure1}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure1.pdf}\n\\caption{Intensity and phase distribution for field rotating RSABs (top row) and intensity rotating RSABs (bottom row). Panels (a), (b), (e), and (f) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d), (g), and (h) to $z=0.8(2\\pi\/\\Lambda)$. Moreover, for the top row, the intensity and phase distributions have been plotted in the region $0\\leq\\rho\\leq 10$, while for the lower row in the region $0\\leq\\rho\\leq 1200$ has been chosen. The difference in the plotting range for the normalised radial coordinate $\\rho$ reflects the paraxial (bottom) and nonparaxial (top) character of the plotted RSABs. In all these figures, $\\Lambda=10^{-5}$ (corresponding to an angular velocity of $\\Omega\\simeq 75 $ rad\/m at $\\lambda=800$ nm), $m_{max}=4$, and $D_m=1$ has been used. For the top row, $\\beta$ is set to zero, while for the bottom row $\\beta=1-m_{max}\\Lambda=0.99996$ (corresponding to a value of a global propagation constant $\\beta_0\\simeq 7.8$ $\\mu m^{-1}$ for $\\lambda=800$ nm) has been used. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nOf particular interest are RSABs with $\\Lambda\\ll 1$. Since $\\Lambda=\\Omega\/k_0$, this condition corresponds to RSABs, whose actual angular velocity $\\Omega$ is much smaller than the beam's wave vector $k_0$. This, ultimately, corresponds to experimentally realisable RSABs. In the rest of this manuscript, if not specified otherwise, we will always implicilty assume that $\\Lambda\\ll 1$ holds. This assumption, moreover, has different consequences for field and intensity rotating RSABs.\n \n In the former case (i.e., for $\\beta=0$),  $\\Lambda\\ll1$ implies that the (normalised) transverse momentum of each Bessel component is given by $\\alpha_m=\\sqrt{1-m^2\\Lambda^2}\\simeq 1+\\mathcal{O}(m^2\\Lambda^2)$. If we recall, that the transverse momentum of a Bessel beam is related to the Bessel cone angle by the relation $k_{\\perp}=k_0\\sin\\vartheta_0$, a value of the normalised transverse momentum $\\alpha_m\\simeq 1$ corresponds to $\\vartheta_0\\simeq\\pi\/2$, i.e., to a highly nonparaxial Bessel beam. \n \n Despite this fact, however, the nature of the resulting RSAB can be tuned at will between paraxial and nonparaxial, by simply changing the number of Bessel beams that participate to the sum in Eq. \\eqref{eq3}. To obtain nonparaxial RSABs, it is sufficient to limit the summation in Eq. \\eqref{eq3} to $m_{max}=\\text{max}\\{\\mathcal{M}\\}<\\ceil[\\big]{\\Lambda^{-1}}$. In this case, in fact, the transverse momentum of every Bessel component will be $\\alpha_m\\simeq 1$, and the resulting RSAB will be highly nonparaxial. \n \n On the other hand, if one includes only values of $m$, that are close to $\\ceil[\\big]{\\Lambda^{-1}}$, i.e., if $m\\in[m_{max}-\\bar{m},m_{max}]$ in Eq. \\eqref{eq3}\\footnote{with $\\bar{m}$ small compared to $m_{max}$, such that $\\bar{m}^2\\Lambda^2\\ll1$ still holds}, then $m\\Lambda\\simeq 1$, and, correspondingly, $\\alpha_m\\simeq 0$. In this case, all Bessel components will be paraxial (i.e., the correspondent cone angle will be $\\vartheta_0\\ll1$), and the resulting RSAB can also be interpreted as a paraxial beam.\n \n For intensity rotating RSABs (i.e., for $\\beta\\neq 0$), instead, $\\alpha_m$ can be made arbitrarily small, independently from the value of $\\Lambda$, by suitably tuning the parameter $\\beta$. In this case, then, the paraxial limit is simply obtained by choosing $\\beta$ such that $\\alpha_m\\simeq 0$, i.e., $\\beta=1-m_{max}\\Lambda$, with \n $m_{max}=\\text{max}\\{\\mathcal{M}\\}$. Notice, that with this choice of $\\beta$, $\\alpha_{m_{max}}=0$, and therefore the sum in Eq.\\eqref{eq3} extends to $m_{max}-1$, as $\\text{J}_{m_{max}}(\\alpha_{m_{max}}\\rho)=0$. \n \nThis extra flexibility in tuning the propagation constant $\\beta$ and the angular velocity $\\Lambda$ independently makes intensity rotating RSABs easier to generate and manipulate experimentally, than their field rotating counterparts \\cite{nostroPRL, nostroAPL}.  \n\n\n\\section{Vector Radially Self-Accelerating Beams}\\label{vectorialisation}\nThe solution presented in Eq. \\eqref{eq3} describes scalar RSABs. In many situations, however, a scalar representation of the electromagnetic field is not enough to fully describe its properties. A typical example is the focussing of a beam of light by means of a thick lens. On the focal plane of the lens, in fact, the scalar approximation given by Eq. \\eqref{eq3} would fail to describe the properties of a focussed RSAB, and a full vector theory should be instead employed. A simple way to retrieve a full vector solution of Maxwell's equations from a solution to the scalar Helmholtz equation is given by the method of Hertz potentials  \\cite{stratton, joptHertz}. First, one defines the Hertz potential $\\boldsymbol\\Pi(\\vett{r},t)=\\psi(\\vett{r})\\exp{(-i\\omega t)}\\uvett{f}$, where $\\uvett{f}$ is a suitable polarisation unit vector, and $\\psi(\\vett{r})$ is a solution of Eq. \\eqref{eq1}. Then, the vector electric and magnetic fields can then be retrieved from $\\vettGreek{\\Pi}(\\vett{r},t)$ as follows:\n\\begin{subequations}\\label{eq5}\n\\begin{align}\n\\vett{E}(\\vett{r},t) & =-\\frac{\\partial\\boldsymbol\\Pi(\\vett{r},t)}{\\partial t},\\\\\n\\vett{B}(\\vett{r},t) & =\\nabla\\times\\nabla\\times\\boldsymbol\\Pi(\\vett{r},t).\n\\end{align}\n\\end{subequations}\nIn the general case, both an electric ($\\boldsymbol\\Pi_e$) and magnetic ($\\boldsymbol\\Pi_m$) Hertz potential should be introduced, each accounting for the sources of electric and magnetic field, respectively. For free space propagation, however, no sources are present, and the electric and magnetic Hertz potential coincide (up to a global constant), i.e., $\\boldsymbol\\Pi_e=\\boldsymbol\\Pi_m\\equiv\\boldsymbol\\Pi$ \\cite{stratton}. According to the convention adopted by Jackson \\cite{jackson}, the electric and magnetic fields defined by Eqs. \\eqref{eq5} correspond to TE fields. The TM fields, however, can be obtained straightforwardly from the TE ones by setting $\\vett{E}_{TM}\\rightarrow\\vett{B}_{TE}$, and $\\vett{B}_{TM}\\rightarrow-\\vett{E}_{TE}\/c^2$.\n\nTo calculate the vector electric and magnetic fields corresponding to RSABs, we first rewrite Eq. \\eqref{eq3} as $\\psi_{RSAB}(\\vett{r})=\\sum_mD_m\\phi_m(\\vett{r})$, with $\\phi_m(\\vett{r})=\\exp{[i m\\theta+i(m\\Lambda+\\beta)\\zeta]}\\text{J}_m(\\alpha_m\\rho)$ being the usual Bessel beam \\cite{durnin}. This allows us to define the Hertz potentials for RSABs in terms of the Hertz potentials for ordinary Bessel beams, i.e.,\n\\begin{equation}\\label{eq6}\n\\boldsymbol\\Pi(\\vett{r},t) =\\sum_{m\\in\\mathcal{M}}D_m\\vett{P}^{(m)}(\\vett{r},t)\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq6bis}\n\\vett{P}^{(m)}(\\vett{r},t)=\\phi_m(\\vett{r})e^{-i\\omega t}\\uvett{f}\n\\end{equation}\nis the Hertz potential corresponding to a single Bessel beam $\\phi_m(\\vett{r})$, whose polarisation is defined by the unit vector $\\uvett{f}$. Then, using Eq. \\eqref{eq5}, we can first calculate the electric and magnetic vector fields for a single Bessel beam, namely\n\\begin{subequations}\\label{eq8}\n\\begin{align}\n\\vett{E}^{(m)}(\\vett{r},t) &=-\\frac{\\partial\\vett{P}^{(m)}(\\vett{r},t)}{\\partial t},\\\\\n\\vett{B}^{(m)}(\\vett{r},t) &=\\nabla\\times\\nabla\\times\\vett{P}^{(m)}(\\vett{r},t).\n\\end{align}\n\\end{subequations}\nThen, the electric and magnetic fields of vector RSABs can be written as follows:\n\\begin{subequations}\\label{eq9}\n\\begin{align}\n\\vett{E}(\\vett{r},t) & =\\sum_{m\\in\\mathcal{M}}D_m\\vett{E}^{(m)}(\\vett{r},t),\\\\\n\\vett{B}(\\vett{r},t) & =\\sum_{m\\in\\mathcal{M}}D_m\\vett{B}^{(m)}(\\vett{r},t).\n\\end{align}\n\\end{subequations}\n\\subsection{The Role of Polarisation of Hertz Potential in Determining the Properties of Vector RSABs}\nVector beams are frequently used as a model to describe focused light. From this perspective, the method of Hertz potential offers an intuitive and insightful perspective on the process of focussing of a beam of light by a lens, or an objective, for example. In fact, one can interpret the Hertz potential \n$\\boldsymbol\\Pi$ as the electromagnetic field before the focussing system, consisting of a scalar field distribution, and a given polarisation $\\uvett{f}$. The vectorialisation procedure described in Eqs. \\eqref{eq5}, then, represents the full vector field after the focussing process (for example, in the focal plane of a lens). Because of the structure of Eqs. \\eqref{eq5}, it is not difficult to see, that the initial polarisation $\\uvett{f}$ possessed by the field will contribute in determining all the components of the focussed field. \n\nFor the case of RSABs, it is interesting to see whether the vectorialisation procedure described above (i.e., the focussing process) preserves their self-accelerating character, or, in case it does not, under which conditions the self-accelerating character of RSABs is preserved. To do so, first we introduce the polarisation vector $\\uvett{f}=f_p\\uvett{x}+f_s\\uvett{y}$ (where $f_{p,s}\\in\\mathbb{C}$, and $|f_p|^2+|f_s|^2=1$). Then, we use Eqs. \\eqref{eq6bis} and \\eqref{eq8} to calculate the vector electric and magnetic fields corresponding to arbitrary polarised Bessel beams. Because of the intrinsic cylindrical symmetry of RSABs, we also introduce a (normalised) cylindrical reference frame $\\{\\uvettGreek{\\rho},\\uvettGreek{\\theta},\\uvettGreek{\\zeta}\\}$. In this reference frame, the electric and magnetic fields of a single vector Bessel component can be written as \n\\begin{subequations}\\label{eq10}\n\\begin{align}\n\\vett{E}^{(m)}(\\vettGreek{\\rho},t) &= e^{i(\\beta\\zeta-\\omega t+m\\Phi)}\\left[E^{(m)}_{\\rho}(\\vettGreek{\\rho})\\uvettGreek{\\rho}+E^{(m)}_{\\theta}(\\vettGreek{\\rho})\\uvettGreek{\\theta}+E^{(m)}_{\\zeta}(\\vettGreek{\\rho})\\uvettGreek{\\zeta}\\right],\\\\\n\\vett{B}^{(m)}(\\vettGreek{\\rho},t) &= e^{i(\\beta\\zeta-\\omega t+m\\Phi)}\\left[B^{(m)}_{\\rho}(\\vettGreek{\\rho})\\uvettGreek{\\rho}+B^{(m)}_{\\theta}(\\vettGreek{\\rho})\\uvettGreek{\\theta}+ B^{(m)}_{\\zeta}(\\vettGreek{\\rho})\\uvettGreek{\\zeta}\\right],\\\\\n\\end{align}\n\\end{subequations}\nwhere $\\Phi=\\theta+\\Lambda\\zeta$ is the co-rotating coordinate defined in the previous section, and the field components are given by\n\\begin{subequations}\\label{eq11}\n\\begin{align}\nE^{(m)}_{\\rho}(\\vettGreek{\\rho})&=\\omega(\\beta+m\\Lambda)(f_s\\cos\\theta-f_p\\sin\\theta)\\text{J}_m(\\alpha_m\\rho),\\\\\nE^{(m)}_{\\theta}(\\vettGreek{\\rho})&=-\\omega(\\beta+m\\Lambda)(f_p\\cos\\theta+f_s\\sin\\theta)\\text{J}_m(\\alpha_m\\rho),\\\\\nE^{(m)}_{\\zeta}(\\vettGreek{\\rho})&=\\frac{m\\omega}{\\rho}\\text{J}_m(\\alpha_m\\rho)(f_p\\cos\\theta+f_s\\sin\\theta)\\nonumber\\\\\n&+i\\omega\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_{m}(\\alpha_m\\rho)\\right](f_s\\cos\\theta-f_p\\sin\\theta),\n\\end{align}\n\\end{subequations}\nfor the electric field, and\n\\begin{subequations}\\label{eq12}\n\\begin{align}\nB^{(m)}_{\\rho}(\\vettGreek{\\rho})&=-\\frac{\\alpha_m}{\\rho}\\left[\\left(f_p-imf_s\\right)\\cos\\theta+\\left(f_s+imf_p\\right)\\sin\\theta\\right]\\text{J}_{m}^{'}(\\alpha_m\\rho)\\nonumber\\\\\n&-2im\\text{J}_m(\\alpha_m\\rho)\\left[\\left(f_s+imf_p\\right)\\cos\\theta-\\left(f_p-imf_s\\right)\\sin\\theta\\right],\\\\\nB^{(m)}_{\\theta}(\\vettGreek{\\rho})&=\\frac{im}{\\rho}\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right]\\nonumber\\\\\n&+\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\alpha^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right],\\\\\nB^{(m)}_{\\zeta}(\\vettGreek{\\rho})&=-\\frac{m(\\beta+m\\Lambda)}{2}\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\text{J}_m(\\alpha_m\\rho)\\nonumber\\\\\n&+i\\alpha_m(\\beta+m\\Lambda)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\text{J}_m^{'}(\\alpha_m\\rho),\n\\end{align}\n\\end{subequations}\nfor the magnetic field. In the equations above, $\\text{J}_{m}^{'}(\\alpha_m\\rho)$, and $\\text{J}_{m}^{''}(\\alpha_m\\rho)$  are the first and second derivative of the Bessel function with respect to their argument, respectively \\cite{nist}. The electric and magnetic fields of arbitrary polarised RSABs can be then constructed by substituting the expresisons above into Eqs. \\eqref{eq11}. Their explicit expression is reported in Appendix A, for completeness.\n\\subsection{Polarisation Constraint for Vector RSABs}\nIn Sect. 2, we have described the requirements that a scalar field must fulfill, in order to be a RSAB. In particular, the most important requirement is the existence of a suitable co-rotating reference frame, in which the field appears propagation invariant. If such reference frame exists, an observer at rest in such reference frame would then experience a fictitious centrifugal force. \n\nFor scalar fields, however, this condition is independent on polarisation, as it only applies to the field distribution, and not to the constant polarisation pattern possessed by the field. For vector beams, on the other hand, this assumption may not be valid anymore, as different polarisation states are focussed in different ways, thus resulting in a mixing of the various field components \\cite{bornWolf}. In this case, then, it is necessary to investigate under which condition the polarisation coefficients $f_p$ and $f_s$ preserve the self-accelerating character of vector RSABs. A natural way to prove this, is to impose that vector RSABs fulfill the same requirements described in Sect. 2. \n\nTo do so, we first define a suitable co-rotating reference frame, in which the electric and magnetic fields of a vector RSAB appear propagation invariant. If such reference frame exists, this automatically implies that the self-accelerating character has been preserved by the vectorialisation procedure. This means, that the electric (magnetic) field described by the first (second) of Eqs. \\eqref{eq9} must be propagation invariant in a co-rotating reference frame $\\vettGreek{\\rho}'=\\mathcal{S}\\vettGreek{\\rho}$ defined by  the following coordinate transformation\n\\begin{equation}\n\\left\\{\\begin{array}{ll}\n\\rho'=\\rho,\\\\\n\\Phi=\\theta+\\Lambda\\zeta,\\\\\n\\zeta'=\\zeta.\n\\end{array}\\right.\n\\end{equation}\nIn principle, one should check that both the electric and the magnetic field are independently propagation invariant in this reference frame. However, Maxwell's equation impose that if one field fulfills the requirement, the other must fulfill it too. For this reason, we limit or analysis to the electric field only. The same condition that we will derive for the polarisation coefficients $f_p$ and $f_s$ will apply to the magnetic field as well, and can be also derived using the same approach with the magnetic, rather than electric, field. \n\nWe then start by separating the electric field into its transverse and longitudinal parts, namely  $\\vett{E}(\\vettGreek{\\rho}')=\\vett{E}_{\\perp}(\\vettGreek{\\rho}')+\\vett{E}_{\\parallel}(\\vettGreek{\\rho}')$, and require that they are both propagation invariant, i.e., $\\partial\\vett{E}_{\\perp,\\parallel}(\\vettGreek{\\rho}')\/\\partial\\zeta=0$. Instead of dealing directly with this condition, however, we can require that the transverse and longitudinal intensities, rather than amplitudes, are propagation invariant. By doing this, we are formally requiring that only intensity rotating RSABs remain propagation invariant upon focussing. However, if the intensity of a field is independent from $\\zeta$, its amplitude will be $\\zeta$-independent as well, and the $\\zeta$ dependence can be at most contained into a phase factor. Once the condition on the intensity has been met, one could then look at the phase of the corresponding field, and check, whether it remains synchronised with its corresponding intensity profile.\n\nThe transverse  $\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2=\\left|E_{\\rho}\\right|^2+\\left|E_{\\theta}\\right|^2$, and longitudinal $\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2=\\left|E_{\\zeta}\\right|^2$ intensities can be calculated using the expressions given in Appendix A, thus obtaining\n\\begin{subequations}\\label{eq14}\n\\begin{align}\n\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2 &= \\sum_{m\\in\\mathcal{M}}\\left|\\mathcal{E}_m^{(1)}(\\rho)\\right|^2+2\\sum_{n\\neq m\\in\\mathcal{M}}\\mathcal{E}_m^{(1)}(\\rho)\\left[\\mathcal{E}_n^{(1)}(\\rho)\\right]^*\\cos\\left[\\left(m-n\\right)\\Phi\\right],\\\\\n\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2 &=G_1(\\rho,\\theta)+2\\sum_{n\\neq m\\in\\mathcal{M}}\\mathcal{E}_m^{(2)}(\\rho)\\left[\\mathcal{E}_n^{(3)}(\\rho)\\right]^*\\Big\\{a_pa_s\\cos^2\\theta\\sin\\left[\\left(m-n\\right)\\Phi-\\Delta\\right]\\nonumber\\\\\n&-(a_p^2-a_s^2)\\sin\\theta\\cos\\theta\\sin\\left[\\left(m-n\\right)\\Phi\\right]-a_pa_s\\sin^2\\theta\\sin\\left[\\left(m-n\\right)\\Phi+\\Delta\\right]\\Big\\}\\label{parallel}\n\\end{align}\n\\end{subequations}\nwhere we have rewritten the polarisation coefficients as $f_p=a_p$, $f_s=a_s\\exp{(i\\Delta)}$ (with $a_p, a_s, \\Delta \\in \\mathbb{R}$), being $\\Delta$ the relative phase between the two polarisation components,  and\n\\begin{eqnarray}\\label{G1}\nG_1(\\rho,\\theta)&=&\\sum_{m\\in\\mathcal{M}}\\Big\\{a_p^2\\left[\\cos^2\\theta \\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2+\\sin^2\\theta \\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2\\right]\\nonumber\\\\\n&+&a_s^2\\left[\\cos^2\\theta \\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2+\\sin^2\\theta \\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2\\right]\\nonumber\\\\\n&+&2a_pa_s\\left(\\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2-\\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2\\right)\\sin\\theta\\cos\\theta\\cos\\Delta\\Big\\}.\n\\end{eqnarray}\nEquations \\eqref{eq14} already contain an important information. No matter the polarisastion, the transverse intensity always remains propagation invariant, as no explicit $\\zeta$-dependence appears in the expression of $\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2$. \n\nThe longitudinal part of the intensity, on the other hand, \ncontains terms that depend on $\\sin\\theta$ and $\\cos\\theta$. Once transformed in the co-rotating frame, these terms become $\\zeta$-dependent, as $\\theta=\\Phi-\\Lambda\\zeta$. To avoid this problem, the polarisation coefficients must be chosen in such a way to guarantee the propagation invariance of the longitudinal intensity as well.  The condition on $a_p$, $a_s$, and $\\Delta$ can be then found by requiring that \n\\begin{eqnarray}\\label{eq15}\n\\frac{\\partial\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2}{\\partial\\zeta} &=\\left(a_p^2-a_s^2\\right)\\Big\\{\\left[F_2(\\rho)-F_3(\\rho)\\right]\\sin2\\theta+F_4(\\rho)\\cos2\\theta\\Big\\}\\nonumber\\\\\n&+a_pa_s\\cos\\Delta\\left\\{F_4(\\rho)\\sin2\\theta-\\left[F_2(\\rho)-F_3(\\rho)\\right]\\cos2\\theta\\right\\}=0,\n\\end{eqnarray}\nwhere the functions $F_k(\\rho)$ (with $k=\\{1,2,3,4\\}$) can be determined from Eqs. \\eqref{parallel} and \\eqref{G1}. It is not difficult to see, that the above equation is satisfied if and only if $a_p=a_s$, and $\\Delta=\\pm\\pi\/2$. Moreover, since $|f_p|^2+|f_s|^2=1$, this condition implies that $f_p=1\/\\sqrt{2}$, and $f_s=\\pm i\/\\sqrt{2}$., which correspond to left-handed ($+$) and right-handed ($-$) circular polarisation, respectively\n\nThis is the main result of our work. Vector RSABs only maintain their self-accelerating character if the polarisation of the Hertz vector is chosen to be circular. In other words, when focussing polarised RSABs, only circular polarisation is allowed, in order to preserve the self-accelerating character of the focused RSABs.  \n\nA simple explanation of this result can be given by looking at the symmetry of the scalar and vector beams, respectively. In the scalar case, in fact, RSABs naturally possess cylindrical symmetry, due to their transverse profile. By virtue of this symmetry, the co-rotating coordinate can be chosen as a $\\zeta$-dependent azimuthal coordinate, namely $\\Phi=\\theta+\\Lambda\\zeta$. Upon focussing, the overall cylindrical symmetry must be preserved, in order for the vector RSAB to maintain its self-accelerating character. This, ultimately, constraints the polarisation to be chosen as circular.\n\\subsection{Vector Fields from Circularly Polarised RSABs}\nWe now apply the polarisation constraints derived above and investigate the form of the electric and magnetic fields generated by focussing circularly polarised RSABs. By substituting $f_p=1\/\\sqrt{2}$ and $f_s=i\\sigma\/\\sqrt{2}$ into Eqs. \\eqref{eq11} and \\eqref{eq12}, and using Eqs. \\eqref{eq9}, the electric and magnetic fields of a circularly polarised focussed RSAB can be written as\n\\begin{subequations}\\label{eq16}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t)&=\\sum_{m\\in\\mathcal{M}}\\vett{e}_m(\\rho)\\,e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]},\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&=\\sum_{m\\in\\mathcal{M}}\\vett{b}_m(\\rho)\\,e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]},\n\\end{align}\n\\end{subequations}\nwhere $\\sigma=\\pm 1$ is the helicity index, which distinguishes between left-handed ($+$) and right-handed ($-$) circular polarisation \\cite{mandelWolf}, $\\vett{e}_m(\\rho)$, and $\\vett{b}_m(\\rho)$ are  radially dependent vector field, whose explicit expression, is given by\n\\begin{subequations}\\label{eq16bis}\n\\begin{align}\n\\vett{e}_m(\\vettGreek{\\rho})&=\\sigma D_m\\omega\\left\\{ i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\,\\uvett{h}_{\\sigma}-\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\zeta}\\right\\},\\\\\n\\vett{b}_m(\\rho) &= \\frac{D_m}{\\sqrt{2}}\\Big\\{(1+m\\sigma)\\left[\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2m\\sigma\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+i\\left\\{\\frac{m}{\\rho}\\left[\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right]+\\sigma\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\alpha_m^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right]\\right\\}\\uvettGreek{\\theta}\\nonumber\\\\\n&+i(\\beta+m\\Lambda)\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m\\sigma}{2}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\zeta}\\Big\\},\n\\end{align}\n\\end{subequations}\nwhere $\\uvett{h}_{\\sigma}=\\left(\\uvettGreek{\\rho}+i\\sigma\\uvettGreek{\\theta}\\right)\/\\sqrt{2}=\\left(\\uvett{x}+i\\sigma\\uvett{y}\\right)\\sqrt{2}$ is the helicity basis \\cite{mandelWolf}.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure2.pdf}\n\\caption{Intensity and phase distribution for the longitudinal component $E_{\\zeta}$ of the electric field described by Eq. \\eqref{eq17a}, for $\\sigma=1$.  Panels (a) and (b) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d) to $z=0.8(2\\pi\/\\Lambda)$. These plots are made assuming $0\\leq\\rho\\leq 10$.  The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure2}\n\\end{center}\n\\end{figure}\nFor experimentally realisable RSABs, $\\Lambda\\ll 1$. Within this approximation, one should distinguish between field rotating, and intensity rotating vector RSABs. For the former, $\\beta=0$, and the radial and azimuthal components of the electric field, as well as the longitudinal component of the magnetic field, are $\\mathcal{O}(\\Lambda)$, and can therefore be neglected, leaving a purely longitudinal electric field, and a purely transverse magnetic field, namely\n\\begin{subequations}\\label{eq17}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t) &\\simeq\\left(-\\frac{\\sigma\\omega}{\\sqrt{2}}\\right)\\sum_{m\\in\\mathcal{M}}D_m\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]e^{i[(m+\\sigma)\\theta+m\\Lambda\\zeta-\\omega t]}\\uvettGreek{\\zeta},\\label{eq17a}\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&\\simeq\\frac{1}{\\sqrt{2}}\\sum_{m\\in\\mathcal{M}}D_m\\Bigg\\{(1+m\\sigma)\\left[\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2m\\sigma\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+i\\left[\\frac{m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)-\\sigma\\alpha_m^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right]\\uvettGreek{\\theta}\\Bigg\\}e^{i[(m+\\sigma)\\theta+m\\Lambda\\zeta-\\omega t]}.\n\\end{align}\n\\end{subequations}\nFor intensity rotating vector RSABs, and within the paraxial approximation, $(\\beta+m\\Lambda)\\simeq 1$, and therefore $\\alpha_m\\ll 1$. In this case, all three components  of the electric and magnetic field are nonzero, and assume the following, simplified, form:\n\\begin{subequations}\\label{eq19}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t) &\\simeq\\sum_{m\\in\\mathcal{M}}\\sigma\\omega D_m\\left\\{i\\text{J}_m(\\alpha_m\\rho)\\,\\uvett{h}_{\\sigma}-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\zeta}\\right\\}e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]}\\label{eq19a},\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&\\simeq\\sum_{m\\in\\mathcal{M}}\\frac{D_m}{\\sqrt{2}}\\Bigg\\{2m(\\sigma+m)\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\rho}+i\\left[\\frac{m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\theta}\\nonumber\\\\\n&-\\frac{i m\\sigma}{2}\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\zeta}\\Bigg\\}e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]}.\n\\end{align}\n\\end{subequations}\nFrom the expressions above, it appears clear that, upon focussing, the electric field maintains the original circular polarisation (in the transverse plane) of the focussed beam, while the polarisation of the magnetic field gets mixed. This, however, is only a result of the fact that we only considered TE fields to start with. If one would repeat the above calculations for TM fields, in fact, the result would be the same, with the magnetic field retaining the original polarisation and the electric field being mixed up. In the most general case, where both TE and TM waves are present, each field has these two components of polarisation, thus resulting in a more complex polarisation pattern. \n\nThe intensities and phases of the electric field components for field rotating and intensity rotating RSABs are reported in Figs. \\ref{figure2}, and \\ref{figure3}, respectively.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure3.pdf}\n\\caption{Intensity and phase distribution for the radial (top row) and longitudinal (bottom row) components  of the electric field described by Eq. \\eqref{eq19a}, for $\\sigma=1$.  Panels (a), (b), (e), and (f) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d), (g), and (h) to $z=0.8(2\\pi\/\\Lambda)$. These plots are made assuming $0\\leq\\rho\\leq 1200$.  The azimuthal component of the field is not shown, as its intensity profile is the same as the radial one [ see Eq. \\eqref{eq19a}]. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure3}\n\\end{center}\n\\end{figure}\nAs it can be seen from Fig. \\ref{figure2}(d), upon focussing, field rotating vector RSABs lose their property, that intensity and phase profile are synchronised in rotation during propagation. This, ultimately is due to the fact that while the field intensity contains terms of the form $\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)\\right]$, the phase contains terms that oscillate like $\\cos\\left[m\\left(\\theta+\\Lambda\\zeta\\right)+\\sigma\\theta\\right]$. The presence of the extra term $\\sigma\\theta$ (which disappears in the intensity) is then responsible for the different evolution of amplitude and phase of the field, as it corresponds to a $\\zeta$-dependent term, once transformed in the co-rotating frame.\n\\section{Linear and Angular Momentum Densities of Vector RSABs}\nIn this section, we calculate the linear and angular momentum for intensity rotating vector RSABs. We limit ourselves to the paraxial case, as within this approximation, we can separate the angular momentum in its spin and orbital parts. This gives us the possibility to distinguish between intrinsic and extrinsic orbital angular momentum of vector RSABs, and to then isolate the extrinsic contribution given by the fact that the intensity rotates with angular velocity $\\Lambda$. \n\nFollowing Jackson, the linear and angular momentum of the electromagnetic field are defined as follows \\cite{jackson}:\n\\begin{subequations}\\label{eq21}\n\\begin{align}\n\\vett{P}=\\int\\,d^2\\rho\\,\\vett{p}(\\vettGreek{\\rho}),\\\\\n\\vett{J}=\\int\\,d^2\\rho\\,\\vett{j}(\\vettGreek{\\rho}),\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{subequations}\\label{eq20}\n\\begin{align}\n\\vett{p}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\,\\operatorname{Re}\\left\\{\\vett{E}(\\vettGreek{\\rho})\\times\\vett{B}^*(\\vettGreek{\\rho})\\right\\},\\label{eq20a}\\\\\n\\vett{j}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\,\\operatorname{Re}\\left\\{\\vettGreek{\\rho}\\times\\vett{p}(\\vettGreek{\\rho})\\right\\},\\label{eq20b}\n\\end{align}\n\\end{subequations}\nare the correspondent densities, $d^2\\rho=\\rho d\\rho d\\theta$, and the integrals are extended over the whole space. \n\nAs it can be seen from Eqs. \\eqref{eq3} and \\eqref{eq16}, RSABs are defined in terms of superpositions of Bessel beams. Therefore, as Bessel beams cary infinite energy, the above integrals diverge, and linear and angular momentum (as well as energy) are not well defined quantities for RSABs. This problem, however, can be overcome in different ways, by introducing different forms of regularisation. For example, one could limit the radial integration, up to a maximum radius. Alternatively, one could insert a regularisation function, such a Gaussian function, in the radial integrals to make them finite. Physically speaking, both regularisations can be implemented. The former, in fact, corresponds to use a pupil of a fixed diameter to filter the field. The latter, on the other hand, corresponds to describe RSABs in terms of Bessel-Gauss beams, which, de facto, are the closest approximation to Bessel beams that can be realised experimentally.\n\nIn the remaining of this section, we calculate the explicit expressions for both the momentum densities, and their integrated counterpart. For the sake of simplicity, however, we will not compute the radial integrals. These, in fact, only contribute to a multiplicative constant, and do not carry any valuable information for the purpose of investigating the properties of linear and angular momentum of RSABs.\n\\subsection{Linear Momentum}\nIf we substitute the expressions of the electric and magnetic fields of a paraxial RSABs as given by Eqs. \\eqref{eq16} into Eq. \\eqref{eq20a}, the linear momentum density can be written as follows:\n\\begin{eqnarray}\\label{eq22}\n\\vett{p}(\\vettGreek{\\rho}) &=& \\sum_{m,n\\in{\\mathcal{M}}}\\frac{|D_mD_n|\\omega\\varepsilon_0}{4}\\Big\\{P_{\\rho}^{(m,n)}(\\rho)\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+&P_{\\theta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\theta}\\nonumber\\\\\n&+&P_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere $\\phi_{m,n}=\\arg[D_{m,n}]$, and\n\\begin{subequations}\\label{eq23}\n\\begin{align}\nP_{\\rho}^{(m,n)}(\\rho) &=\\sigma\\text{J}_m(\\alpha_m\\rho)\\Bigg\\{\\left[\\frac{mn}{\\rho^2}-\\frac{m\\sigma(\\beta+n\\Lambda)^2}{\\rho}-\\frac{n(\\beta+m\\Lambda)(\\beta+n\\Lambda)}{2}\\right]\\text{J}_n(\\alpha_n\\rho)\\nonumber\\\\\n&-\\frac{mn}{\\rho^2}\\text{J}_n^{'}(\\alpha_n\\rho)\\Bigg\\},\\\\\nP_{\\theta}^{(m,n)}(\\rho) &=\\sigma\\left[\\frac{2mn(n+\\sigma)}{\\rho}+\\frac{n\\sigma(\\beta+m\\Lambda)(\\beta+n\\Lambda)}{2}\\right]\\text{J}_m(\\alpha_m\\rho)\\text{J}_n(\\alpha_n\\rho),\\\\\nP_{\\zeta}^{(m,n)}(\\rho) &=(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\Bigg\\{\\left[(\\beta+n\\Lambda)^2+2n(n+\\sigma)-\\frac{n\\sigma}{\\rho}\\right]\\text{J}_n(\\alpha_n\\rho)\\nonumber\\\\\n&+\\frac{n}{\\rho}\\text{J}_n^{'}(\\alpha_n\\rho)\\Bigg\\},\n\\end{align}\n\\end{subequations}\nwhere terms of order $\\mathcal{O}(\\alpha_m)$ have been neglected, since in the paraxial regime $\\alpha_m\\ll 1$.  The components of the linear momentum density are shown in Fig. \\ref{figure5}. Notice, that the transverse part of the linear momentum presents an unusual characteristic. While it rotates clockwise along the propagation direction, as the intensity distribution of the correspondent RSAB does, the local orientation of the transverse momentum is purely azimuthal (despite $\\vett{p}(\\vettGreek{\\rho})$ has a nonzero radial component), and always directed in the opposite direction, with respect to the rotation direction of the RSAB, as it can be seen from the white arrows in Fig. \\ref{figure5}(a). This has an interesting consequence for applications such particle manipulation and material processing, where the local, rather than the global, behaviour of the momentum plays an important role. While the RSAB (and, with it, the transverse momentum density) rotates clockwise during propagation, a particle placed in the vicinity of a RSAB will experience a local momentum, that will tend to push it in the opposite direction. This effect, however, is purely local, and it disappears when considering the whole momentum. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure5.pdf}\n\\caption{Transverse (a) and longitudinal (b) components of the linear momentum density, as given by Eq. \\eqref{eq22}, in the plane $\\zeta=0$, for $\\sigma=1$. The white arrows in panel (a) represent the flow of the transverse component of the linear momentum density. As it can be seen, the transverse momentum density always points in the opposite direction with respect to the field rotation (red arrow). These plots are made assuming $0\\leq\\rho\\leq 1200$. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The red arrow in both panels show the direction of rotation of the RSAB intensity.}\n\\label{figure5}\n\\end{center}\n\\end{figure}\nTo understand this, let us integrate Eq. \\eqref{eq22} over the transverse space. The linear momentum can be then written as follows\n\\begin{equation}\\label{momentumP}\n\\vett{P}=\\sum_{m\\in\\mathcal{M}}|D_m|^2\\Bigg[\\mathcal{P}^{(m)}_{\\theta}\\uvettGreek{\\theta}+\\mathcal{P}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\Bigg],\n\\end{equation}\nwhere\n\\begin{equation}\\label{Plambda}\n\\mathcal{P}_{\\lambda}^{(m)}=\\frac{\\pi\\omega\\varepsilon_0}{2}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,P_{\\lambda}^{(m,m)}(\\rho),\n\\end{equation}\nwhere $\\lambda\\in\\{\\theta,\\zeta\\}$. Notice that the radial integrals (once regularised) amount to a positive constant. Moreover, there is no radial component of the momentum, since the radial part of $\\vett{p}(\\vett{\\rho})$ depends on $\\sin[(m-n)(\\theta+\\Lambda\\zeta)+\\phi_m-\\phi_n]$, which gives zero once integrated with respect to the azimuthal coordinate $\\theta$. \n\\subsection{Spin and Orbital Angular Momentum}\nTo calculate the spin and orbital angular momentum for intensity rotating, paraxial RSABs, we make use of the usual decomposition of the total angular momentum in its spin (SAM) and orbital (OAM) components, namely $\\vett{J}=\\vett{S}+\\vett{L}$ \\cite{libroOAM}. To do so, we first need to introduce the vector potential $\\vett{A}(\\vettGreek{\\rho})$ associated to the electric and magnetic fields defined above, since the decomposition assumes a rather simple form if expressed in terms of the vector potential. Looking at Eqs. \\eqref{eq5}, it is not difficult to see that $\\vett{A}=\\nabla\\times\\vettGreek{\\Pi}(\\vettGreek{\\rho},t)$. The explicit expression of $\\vett{A}$ for an intensity rotating RSAB is given in Appendix B. \n\nFollowing Ref. \\cite{libroOAM}, the angular momentum density then assumes the following form\n\\begin{equation}\\label{eq24}\n\\mathbf{j}(\\vettGreek{\\rho})=\\vett{s}(\\vettGreek{\\rho})+\\vett{l}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\operatorname{Re}\\left\\{-i\\vett{A}^*\\times\\vett{A}\\right\\}+\\frac{\\varepsilon_0}{2}\\operatorname{Re}\\left\\{\\vett{A}^*\\cdot\\left(-i\\vettGreek{\\rho}\\times\\nabla\\right)\\vett{A}\\right\\},\n\\end{equation}\nwhere $-i\\vettGreek{\\rho}\\times\\nabla$ is the angular momentum operator \\cite{jackson} in the normalised cylindrical reference frame $\\{\\uvettGreek{\\rho},\\uvettGreek{\\theta},\\uvettGreek{\\zeta}\\}$. Using the expression for the vector potential given in Appendix B, the SAM and OAM of a paraxial, intensity rotating vector RSAB are given as follows:\n\\begin{subequations}\\label{eq25}\n\\begin{align}\n\\vett{s}(\\vettGreek{\\rho})&=\\frac{\\varepsilon_0}{2}\\sum_{m,n\\in\\mathcal{M}}\\Bigg\\{-S_{\\rho}^{(m,n)}\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&+S_{\\theta}^{(m,n)}\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&-S_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\zeta}\\Bigg\\},\\\\\n\\vett{l}(\\vettGreek{\\rho}) &=\\frac{\\varepsilon_0}{2}\\sum_{m,n\\in\\mathcal{M}}\\Bigg\\{L_{\\rho}^{(m,n)}(\\rho)\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&-L_{\\theta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&+L_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\zeta},\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{eqnarray}\\label{SAMdensity}\nS_{\\rho}^{(m,n)}(\\rho)&=&\\frac{\\mathcal{D}_{m,n}(\\rho)\\beta(m-n)}{\\rho},\\\\\nS_{\\theta}^{(m,n)}(\\rho)&=&\\sigma S_{\\rho}(\\rho),\\\\\nS_{\\zeta}^{(m,n)}(\\rho)&=&2\\sigma\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)(\\beta+n\\Lambda),\n\\end{eqnarray}\nare the components of the spin angular momentum density, while\n\\begin{eqnarray}\nL_{\\rho}^{(m,n)}(\\rho)&=&\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)\\left[\\sigma\\frac{n(m+\\sigma)}{\\rho}+\\rho(\\beta+m\\Lambda)(\\beta+n\\Lambda)\\right],\\\\\nL_{\\theta}^{(m,n)}(\\rho)&=&\\sigma L_{\\rho}(\\rho),\\\\\nL_{\\zeta}^{(m,n)}(\\rho)&=&\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)\\left[\\frac{m(\\beta+m\\Lambda)+n(m+\\sigma)}{\\rho}\\right],\n\\end{eqnarray}\nare the components of the orbital angular momentum density, and $\\mathcal{D}_{m,n}(\\rho)=|D_mD_n|\\text{J}_m(\\alpha_m\\rho)\\text{J}_n(\\alpha_n\\rho)\/2$. In the above expressions, terms of order $\\mathcal{O}(\\alpha_m)$ have been neglected, since, for paraxial fields, $\\alpha_m\\ll 1$.  The longitudinal and transverse SAM densities are plotted in Fig. \\ref{figure6}. As it can be seen, the SAM density can become negative. This means, that locally, the helicity of the vector RSAB can change sign. However, a close comparison between the SAM density distribution in Fig. \\ref{figure6} and the transverse and longitudinal intensity distributions depicted in Fig. \\ref{figure3} reveals, that regions of negative SAM density occur where the RSAB intensity is very low, or even zero. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure6.pdf}\n\\caption{Radial (a) and longitudinal (b) components of the SAM density, as given by Eqs. \\eqref{SAMdensity}, in the plane $\\zeta=0$, for $\\sigma=1$. The blue regions in both panels indicate areas of negative SAM density, where the helicity is oriented in the opposite direction, with respect to the propagation direction. The azimuthal component of the SAM density is not reported here, as it is, up to a constant, the same as the radial one. These plots are made assuming $0\\leq\\rho\\leq 1200$. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The red arrow in both panels show the direction of rotation of the RSAB intensity.}\n\\label{figure6}\n\\end{center}\n\\end{figure}\n\nThe spin and orbital angular momenta of paraxial vector RSABs are then obtained by integrating Eqs. \\eqref{eq25} over the transverse space. By doing so we obtain\n\\begin{subequations}\\label{eq26}\n\\begin{align}\n\\vett{S}&=\\sigma\\sum_{m\\in\\mathcal{M}}|D_m|^2\\mathcal{S}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\label{eq26a},\\\\\n\\vett{L}&=\\sum_{m\\in\\mathcal{M}}|D_m|^2\\Bigg[\\mathcal{L}^{(m)}_{\\theta}\\uvettGreek{\\theta}+\\mathcal{L}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\Bigg],\\label{eq26b}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation}\n\\mathcal{S}_{\\zeta}^{(m)}=\\frac{\\varepsilon_0}{4}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,S_{\\zeta}^{(m,m)}(\\rho),\n\\end{equation}\nand\n\\begin{equation}\n\\mathcal{L}_{\\lambda}^{(m)}=\\frac{\\varepsilon_0}{4}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,L_{\\zeta}^{(m,m)}(\\rho),\n\\end{equation}\nbeing $\\lambda\\in\\{\\theta,\\zeta\\}$. The total SAM is purely longitudinal, and given as the sum of the longitudinal components of the individual Bessel components. This is not surprising, since we are dealing with paraxial fields, for which the SAM is only directed along the propagation direction \\cite{libroOAM}.\n\nThe OAM, on the other hand, can be seen as the sum of two contributions: an intrinsic component relative to the intrinsic OAM carried by Bessel beams, and an extrinsic one, connected to the fact that the beam rotates around the $\\zeta$-axis during propagation. Their explicit expression read then as follows:\n\\begin{subequations}\\label{eq27}\n\\begin{align}\n\\vett{L}^{(int)}&=\\sum_{m\\in\\mathcal{M}}|\\tilde{D}_m|^2\\mathcal{L}_{int}^{(m)}\\left(\\uvettGreek{\\theta}+\\uvettGreek{\\zeta}\\right),\\\\\n\\vett{L}^{(ext)}&=\\sum_{m\\in\\mathcal{M}}|\\tilde{D}_m|^2\\left[\\sigma\\mathcal{L}^{(m)}_{ext,\\theta}\\uvettGreek{\\theta}+\\mathcal{L}_{ext,\\zeta}^{(m)}\\,\\uvettGreek{\\zeta}\\right],\n\\end{align}\n\\end{subequations} \nwhere $\\tilde{D}_m=D_m\\sqrt{\\beta+m\\Lambda}$, and\n\\begin{subequations}\\label{eq28}\n\\begin{align}\n\\mathcal{L}_{int}^{(m)}&=\\frac{\\varepsilon_0}{4}\\, m(m+\\sigma)\\int_0^{\\infty}\\,d\\rho\\,\\text{J}_m^2(\\alpha_m\\rho),\\\\\n\\mathcal{L}_{ext,\\theta}^{(m)}&=\\frac{\\varepsilon_0}{4}\\,(\\beta+m\\Lambda)^2\\int_0^{\\infty}\\,d\\rho\\,\\rho^2\\text{J}_m^2(\\alpha_m\\rho),\\\\\n\\mathcal{L}_{ext,\\zeta}^{(m)}&=\\frac{\\varepsilon_0}{4}\\,m(\\beta+m\\Lambda)\\int_0^{\\infty}\\,d\\rho\\,\\text{J}_m^2(\\alpha_m\\rho).\n\\end{align}\n\\end{subequations}\nThe intrinsic part of the OAM has the standard spin-orbit interaction form, through the mixed term $(m+\\sigma)$ \\cite{libroOAM}. The extrinsic part, on the other hand, depends on $(\\beta+m\\Lambda)$, which is, essentially, the angular velocity of the beam along the $\\zeta$-axis. The beam rotation, moreover, also induces a longitudinal OAM, which is, as well, proportional to the angular velocity $(\\beta+m\\Lambda)$.\n\\section{Conclusions}\nIn this work, we have analysed the properties of vector RSABs, generated by focussing a scalar, polarised RSAB. Using the method of Hertz potentials as a model for the focussing process, we have demonstrated that only circularly polarised scalar RSABs, when focussed, maintain their self-accelerating character. For this case, we have given explicit expressions of the TE vector electric and magnetic fields for both field and intensity rotating RSABs. In particular, we have shown, that the vectorialisation (focussing) process does not allow anymore the amplitude and phase of field rotating RSABs to rotate synchronously during propagation. Within the paraxial approximation, moreover, we have presented the explicit expressions for the linear and angular momentum densities of intensity rotating RSABs. For SAM, in particular, we have shown, that, locally, the SAM density can be negative, thus meaning a local inversion of the helicity axis. Moreover, for the case of OAM, we have distinguished between the intrinsic and extrinsic contributions, and shown how the rotation of the RSAB around the propagation axis is connected with the extrinsic OAM.\n\nOur work represents a useful guideline for investigating experimentally focussed RSABs and their properties. Moreover, the properties highlighted in this work represent a useful toolbox for studying the interaction of RSABs with matter and dielectric particles. In particular, the fact that locally the linear momentum density flows in the opposite direction, with respect to the overall beam rotation during propagation, could open new possibilities for particle manipulation\n\n\\section*{Acknowledgements}\nThe authors wish to thank the Deutsche Forschungsgemeinschaft (grant SZ 276\/17-1) for financial support.\n\n\\section*{Appendix A: Explicit Form of RSAB Electric and Magnetic Fields}\nThe vector electric and magnetic fields for single Bessel beams defined in Eqs. \\eqref{eq11} and \\eqref{eq12} can be used to write the expressions for the RSAB vector electric and magnetic fields explicitly. Substituting these expressions into  Eqs. \\eqref{eq9} we then get\n\\begin{eqnarray}\\label{electric}\n\\vett{E}(\\vettGreek{\\rho}) &=&e^{i(\\beta\\zeta-\\omega t)} \\sum_{m\\in\\mathcal{M}}e^{im\\Phi}\\Big\\{\\mathcal{E}_m^{(1)}(\\rho)\\Big[\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\uvettGreek{\\rho}\\nonumber\\\\\n&-&\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\uvettGreek{\\theta}\\Big]+\\Big[\\mathcal{E}_m^{(2)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{E}_m^{(3)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\mathcal{E}_m^{(1)}(\\rho)&=&D_m\\omega(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{E}_m^{(2)}(\\rho)&=&D_m(m\\omega\/\\rho)\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{E}_m^{(3)}(\\rho)&=&iD_m\\omega\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right],\n\\end{eqnarray}\nare the radially dependent expansion coefficients for the electric field\n\\begin{eqnarray}\\label{magnetic}\n\\vett{B}(\\vettGreek{\\rho}) &=&e^{i(\\beta\\zeta-\\omega t)}\\sum_{m\\in\\mathcal{M}}e^{im\\Phi}\\Big\\{\\Big[\\mathcal{B}_m^{(1)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(2)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\rho}+\\Big[\\mathcal{B}_m^{(3)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(4)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\theta}+\\Big[\\mathcal{B}_m^{(5)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(6)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\mathcal{B}_m^{(1)}(\\rho) &=&D_m\\left[-\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho) +2m^2\\text{J}_m(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(2)}(\\rho) &=&D_m\\left[\\frac{im\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2im\\text{J}_m(\\alpha_m\\rho)\\right],\\\\  \n\\mathcal{B}_m^{(3)}(\\rho) &=&\\frac{imD_m}{\\rho}\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(4)}(\\rho) &=& D_m\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\text{J}_m^{''}(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(5)}(\\rho) &=&-\\frac{mD_m(\\beta+m\\Lambda)}{2}\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{B}_m^{(6)}(\\rho) &=&iD_m\\alpha_m(\\beta+m\\Lambda)\\text{J}_m^{'}(\\alpha_m\\rho),\n\\end{eqnarray}\nare the radially dependent expansion coefficients for the magnetic field.\n\\section*{Appendix B: Explicit Expression for the Vector Potential for RSABs}\nThe vector potential can be defined from the Hertz potential as $\\vett{A}(\\vettGreek{\\rho},t)=\\nabla\\times\\vettGreek{\\Pi}(\\vettGreek{\\rho},t)$ \\cite{stratton}. Using Eqs. \\eqref{eq6} and \\eqref{eq6bis} the vector potential for an arbitrary polarised vector RSAB is given, in cylindrical coordinates, as follows:\n\\begin{eqnarray}\\label{eqB1}\n\\vett{A}(\\vettGreek{\\rho},t) &=& \\sum_{m\\in\\mathcal{M}}D_me^{i[m(\\theta+\\Lambda\\zeta)+\\beta\\zeta-\\omega t]}\\Bigg\\{-i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\Bigg[(f_s\\cos\\theta-f_p\\sin\\theta)\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&+&i(f_p\\cos\\theta+f_s\\sin\\theta)\\,\\uvettGreek{\\theta}\\Bigg]\\nonumber\\\\\n&+&\\Bigg[\\alpha_m\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{im}{\\rho}\\Big(f_p\\cos\\theta\\nonumber\\\\\n&+&f_s\\sin\\theta\\Big)\\text{J}_m(\\alpha_m\\rho)\\Bigg]\\,\\uvettGreek{\\zeta}\\Bigg\\}.\n\\end{eqnarray}\nFor the case of circular polarisation, the above expression simplifies to\n\\begin{eqnarray}\\label{eqB2}\n\\vett{A}(\\vettGreek{\\rho},t) &=& \\sum_{m\\in\\mathcal{M}}\\frac{D_m}{\\sqrt{2}}e^{i[m(\\theta+\\Lambda\\zeta)+\\sigma\\theta+\\beta\\zeta-\\omega t]}\\Bigg[A_{\\rho}^{(m)}(\\rho)\\,\\uvettGreek{\\rho}+A_{\\theta}^{(m)}(\\rho)\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&+&A_{\\rho}^{(m)}(\\zeta)\\,\\uvettGreek{\\zeta}\\Bigg],\n\\end{eqnarray}\nwhere\n\\begin{subequations}\\label{eqB3}\n\\begin{align}\nA_{\\rho}^{(m)}(\\rho) &=\\sigma(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\nA_{\\theta}^{(m)}(\\rho) &=i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\nA_{\\zeta}^{(m)}(\\rho) &=i\\Bigg[\\sigma\\alpha_m\\,\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\Bigg].\n\\end{align}\n\\end{subequations}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGavitational lensing is the phenomenon of deflection of light rays in a\ngravitational field, which has been successfully employed to explain the astronomical observations in the weak field approximation \\cite{Schneider_Ehlers_Falco_1992,Petters_Levine_Wambsganss_2001,Schneider_Kochanek_Wambsganss_2006, Bartelmann:2010fz} when deflection angle is small. \nWhen the light rays approach towards the photon sphere of black hole where the gravitational field is extremely strong, the deflection angle becomes so large that the weak field method is no longer valid. \nIt was first noticed by Darwin \\cite{Darwin} in 1959 that the light rays passing very close to a black hole would make complete one or more loops around it before falling into the event horizon, hence an infinite series of exotic images were produced.\nLater, strong gravitational lensing was regained wide attention \\cite{Atkinson, Luminet:1979nyg, Ohanian, Nemiroff:1993he}. \nThe  exact lensing equation with arbitary large value of deflection angle is obtained in 2000 \\cite{Frittelli:1999yf, Virbhadra:1999nm}. In 2001 Bozza et al. \\cite{Bozza:2001xd} developed a reliable and analytical method to obtain the deflection angle of Schwarzschild black hole in strong field region and they found the logarithmic divergence of the deflection angle in strong field limit. Later Bozza \\cite{Bozza:2002zj} extended the conclusion to a general asymptotically flat, static, and spherically symmetric spacetime.\nWith the help of strong gravitational lensing it is possible to compare alternative theories of gravity \\cite{Claudel:2000yi, Hasse:2001by, Iyer:2006cn, Virbhadra:2007kw, Bozza:2008ev, Bozza:2009yw, Ghosh:2010uw, Wei:2011nj, Chen:2009eu} and pick up information from different compact objects \\cite{Tsukamoto:2016qro, Bozza:2002af, Vazquez:2003zm, Bozza:2005tg, Bozza:2006nm, Chen:2010yx, Chen:2011ef, Cunha:2015yba, Cavalcanti:2016mbe, Gyulchev:2008ff, Sahu:2012er, Sahu:2013uya, Kuhfittig:2013hva, Nandi:2006ds, Tsukamoto:2012xs, Tsukamoto:2016jzh}.\nLast year, the first image of the supermassive black hole M87$^{*}$ at the center of the galaxy M87 has been captured by the Event Horizon Telescope (EHT)\\cite{Akiyama:2019cqa, Akiyama:2019fyp, Akiyama:2019eap}, which provides us the deeper understanding of the strong gravitational physics.\n\nOne of the simplest natural extension of Einstein's gravity by higher curvature correction is the Einstein-Gauss-Bonnet (EGB) gravity, the action of which in $D$-dimensional spacetime is given by\n\\begin{equation}\nS\n= \\frac{1}{16\\pi}\\int d^{D}x \\sqrt{-g}\n\\left[\\frac{M_{\\rm P}^2}{2}R+\\alpha\\mathcal{G}\\right],\n\\end{equation}\nwhere $\\alpha$ is the coupling constant of the Gauss-Bonnet (GB) term\n\\begin{equation}\n\\mathcal{G}= {R^{\\mu\\nu}}_{\\rho\\sigma} {R^{\\rho\\sigma}}_{\\mu\\nu}-\n4 {R^\\mu}_\\nu {R^\\nu}_\\mu + R^2 = \n6 {R^{\\mu\\nu}}_{[\\mu\\nu} {R^{\\rho\\sigma}}_{\\rho\\sigma]},\n\\end{equation}\nwith $R_{\\mu\\nu\\rho\\sigma}$ the Riemann tensor, $R_{\\mu\\nu}$ the Ricci tensor and $R$ the Ricci scalar.\nIn the  $4$-dimensional spacetime, GB term is a total derivative \\cite{Lanczos:1938sf}, so it has no contribution to the gravitational dynamics.\nHowever, the role of the GB term in $4$-dimensional gravity, in particular, holographic implications to the addition of it to the gravity action was studied in Ref. \\cite{Miskovic:2009bm}.\nNotice that standard thermodynamics for AdS black holes is recovered in this way.\nRecently, Glavan and Lin \\cite{Glavan:2019inb} reformulate the $D$-dimensional EGB gravity by rescaling the coupling $\\alpha\\rightarrow\\alpha\/(D-4)$. They obtain a novel $4$-dimensional EGB gravity theory in the limit $D\\rightarrow4$, where the GB term can give the nontrivial contribution of gravitational dynamics. They also have shown that it can bypass the Lovelock's theorem \\cite{Lovelock:1971yv,Lovelock:1972vz} and prevent Ostrogradsky instability \\cite{Woodard:2015zca}.\nThis idea of regularization can be traced back to Refs \\cite{Tomozawa:2011gp, Cognola:2013fva}, which gives the quantum corrections of Einstein's gravity.\nIn addition, a novel static spherically symmetric black hole solution  was obtained within this theory.\nNote that the black hole solution was found earlier in the gravity theories with conformal anomaly \\cite{Cai:2009ua} and quantum corrections \\cite{Tomozawa:2011gp, Cognola:2013fva}, and recently in regularized Lovelock gravity \\cite{Casalino:2020kbt}, respectively.\n\nThe novel $4$-dimensional EGB black holes are free from singularity problem. Their photon sphere and shadow, as well as the innermost stable circular orbit (ISCO) of a spinless test particle \\cite{Guo:2020zmf} and spinning test particle \\cite{Zhang:2020qew} around them, have been calculated. \nQuasinormal modes of bosonic fields \\cite{Konoplya:2020bxa} and fermionic fields \\cite{Churilova:2020aca} of these black holes have been investigated, and it is found that for the bosonic fields the damping rate is more sensitive than the real part of quasinormal modes by changing of the GB coupling constant $\\alpha$,\nwhile for the fermionic fields the damping rate usually decreases and the real part of the quasinormal modes increases with the increase of $\\alpha$. Konoplya and Zhidenko discussed the stability \\cite{Konoplya:2020juj} of spherically symmetric black holes in the novel EGB gravity. Moreover, other topics in this new theory including the charged black holes in AdS spaces \\cite{Fernandes:2020rpa}, the shadow of dS black holes \\cite{Roy:2020dyy}, the bending of light in dS black holes \\cite{Heydari-Fard:2020sib}, the rotating black holes \\cite{Wei:2020ght,Kumar:2020owy}, radiating black holes \\cite{Ghosh:2020vpc},  the structure of relativistic stars \\cite{Doneva:2020ped}, the thermodynamics of the black holes \\cite{Hegde:2020xlv, Singh:2020xju, Zhang:2020qam, HosseiniMansoori:2020yfj} and the accretion disk around the black hole \\cite{Liu:2020vkh} have also been studied.\nHowever, several problems, such as completeness, about the regularization procedure have been put foward in Refs \\cite{Ai:2020peo, Gurses:2020ofy, Shu:2020cjw, Hennigar:2020lsl, Mahapatra:2020rds, Tian:2020nzb,Ge:2020tid}, in the meantime some prescriptions have been suggested \\cite{Casalino:2020kbt, Hennigar:2020lsl, Lu:2020iav,  Kobayashi:2020wqy}.\tL\\\"u and Pang \\cite{Lu:2020iav} proposed a more rigorous way to regularize the EGB gravity by compactifying the $D$ dimensional EGB gravity on the $(D-4)$ dimensional maximally symmetric space and redefining the coupling constant as $\\alpha\/(D-4)$. \nIn accordance with the results of Ref. \\cite{Kobayashi:2020wqy}, a special scalar-tensor theory that belongs to the family of Horndeski gravity is obtained by this method.\nRef. \\cite{Hennigar:2020lsl} extends the method for obtaining the $D\\rightarrow2$ limit of general relativity \\cite{Mann:1992ar} to the $D\\rightarrow4$ limit of EGB gravity.\nAnyhow, in these regularised theories \\cite{Casalino:2020kbt, Hennigar:2020lsl, Lu:2020iav} the spherically symmetric 4D black hole solution obtained in Refs \\cite{Glavan:2019inb, Cognola:2013fva}  is still valid.\n\nOn the other hand, it is believed that there exists plasma fluid surrounding black holes and other compact objects. When the light moves towards the compact objects through the plasma, the trajectory of light is different from the vacuum case. The theory of the light propagation in a curved spacetime in the presence of an isotropic dispersive medium was considered in the classical book of Synge \\cite{Synge}. Synge used the general relativistic Hamiltonian approach to deal with the geometrical optics in a dispersive medium.\nFurthermore, the influence of a spherically symmetric and time-independent plasma on the light defection in Schwarzschild spacetime and Kerr spacetime was discussed in the book of Perlick \\cite{Perlick1}.\nThe effect of plasma on the shadows of black holes and wormholes has been investigated in \\cite{Bisnovatyi-Kogan:2017kii, Abdujabbarov:2015pqp, Perlick:2017fio, Abdujabbarov:2016efm, Huang:2018rfn}. \nGravitational lensing by the compact object in homogeneous and inhomogeneous plasma was considered in \\cite{BisnovatyiKogan:2008yg, BisnovatyiKogan:2010ar, Morozova, Er:2013efa, Atamurotov:2015nra, Rogers, Perlick:2015vta, Tsupko:2013cqa}.\n\nIn this work, we shall study the strong gravitational lensing by this novel $4$-dimensional EGB black hole in an unmagnetized homogeneous plasma medium. The rest of the paper is organized as follows.\nIn Sec.~II, we study the photon sphere radius and the critical value of impact parameter of this novel black hole in the presence of plasma and derive the expression for the deflection angle of light in Sec.~III.\nIn Sec.~IV, we investigate the effects of  plasma on the deflection angle, the coefficients and the observable quantities for gravitational lensing in the strong field limit. Finally, We end the paper with a summary in Sec.~V.\nThroughout this paper we use the units in which $G=c=1$.\n\n\\section{Photon sphere of an Einstein-Gauss-Bonnet black hole in the presence of plasma}\n\nLet us start from the line element of the EGB black hole spacetime \\cite{Glavan:2019inb}, which is given by\n\\begin{eqnarray}\\label{metric}\nds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)(d\\theta^{2}+\\sin^{2}\\theta d\\varphi^{2}),\n\\end{eqnarray}\nwhere the functions $A(r)$, $B(r)$ and $C(r)$ have respectively the following form,\n\\begin{eqnarray}\n&&A(r)= 1 + \\frac{r^2}{2\\alpha}\n\\Biggl( 1- \\sqrt{1+\\frac{8\\alpha M}{r^3}}\\Biggr),\\label{A}\\\\\n&&B(r)=\\Biggl[1 + \\frac{r^2}{2 \\alpha}\n\\Biggl( 1- \\sqrt{1+\\frac{8\\alpha M}{r^3}}\\Biggr)\\Biggr]^{-1},\\label{B}\\\\\n&&C(r)=r^2\\label{C}.\n\\end{eqnarray}\nIt has been shown that the metric is asymptotic flat by the expansion at large $r$.\nHere $M$ is the mass of the EGB black hole and the GB coupling constant $\\alpha$  is constrained in the range $-8\\le {\\alpha}\/{M^2}\\le 1$ \\cite{Guo:2020zmf}. For the case $0<{\\alpha}\/{M^2}\\le 1$, there are two horizons \n\\begin{equation}\nr_{\\pm}=M\\pm \\sqrt{M^2-\\alpha}.\n\\end{equation}\nWhile for the case $-8\\le{\\alpha}\/{M^2}< 0$, there is only one horizon $r_{+}$,\nwhere the singular short radial distances $r<\\sqrt[3]{-8\\alpha M}$ are concealed inside this outer horizon. We will take the region $-8\\le {\\alpha}\/{M^2}\\le 1$ for the coupling constant in this paper.\n \nWe assume that the spacetime is filled with a spherically symmetric distribution of plasma with electron plasma frequency\n\\begin{equation}\n\\omega_p(r)^2 = \\frac{4\\pi e^2}{m} N(r),\n\\end{equation}\nwhere $e$ and $m$ are the charge of the electron and the mass of the electron respectively. The number density of the electrons $N(r)$ is the function of the radius coordinate only.\nThe relation between the refraction index $n$ and the photon frequency $\\omega$ is given as\n\\begin{equation}\nn ^2 = 1 - \\frac{\\omega_p^2(r)}{\\omega^2}.\n\\end{equation}\nIt is found that when $\\omega > \\omega_p$, the photon can propagate through the plasma. On the other hand, when $\\omega < \\omega_p$, the photon motion is forbidden \\cite{BisnovatyiKogan:2010ar, Rogers}.\nNote that one has $n=1$ in the vacuum case. \n\nWe start to calculate the strong gravitational lensing of the EGB black hole surrounded by plasma. The trajectories of photons in a curved space-time with plasma mediums, were obtained by Synge \\cite{Synge}. The Hamiltonian for\nthe light rays around the black hole surrounded by plasma has the\nfollowing form \\cite{Kulsrud:1991jt}\n\\begin{equation}\\label{H}\nH(x,p) = \\frac{1}{2} \\left[ g^{\\mu\\nu} p_{\\mu} p_{\\nu} +\\omega_p^2(r)\n\\right] = 0,\n\\end{equation}\nwhere $p_{\\mu}$ is the four-momentum of the photon and \n$g^{\\mu\\nu}$ is the contravariant metric tensor.\nSubstituing (\\ref{metric}) into (\\ref{H}), we get the equation\n\\begin{equation}\\label{H1}\n0=-\\frac{p_t^2}{A(r)}+\\frac{p_r^2}{B(r)}+\\frac{p_{\\varphi}^2}{C(r)}+\\omega_p^2(r).\n\\end{equation}\nUsing the Hamiltonian (\\ref{H}) for the photon around\nthe EGB black hole, the paths of light rays are then described in terms of the\naffine parameter $\\lambda$ by\n\\begin{equation}\\label{dxp} \n\\frac{dx^{\\mu}}{d \\lambda} = \\frac{\\partial H}{\\partial\n\tp_{\\mu}},~ \\frac{dp_{\\mu}}{d \\lambda} = - \\frac{\\partial\n\tH}{\\partial x^{\\mu}}.\n\\end{equation}\nBecause of the spherical symmetry, we can confine the photon orbits in the equatorial plane by taking $\\theta=\\pi\/2$ without the loss of generality.\nThe coordinates $t$ and $\\varphi$ are cyclic, leading two costants of motions which are the energy $E$ and the angular\nmomentum $L$ of the photon \n\\begin{equation}\nE=-p_t=\\omega_{\\infty},~ L=p_\\varphi,\n\\end{equation}\nwhere $\\omega_{\\infty}$ is the photon frequency at infinity. From Eqs. (\\ref{metric}) and (\\ref{dxp}), the expression for \n${dr}\/{d\\lambda}, ~{d\\varphi}\/{d\\lambda}$ is obtained in terms of $p_r$ and $p_\\varphi$ \n\\begin{eqnarray}\n\\label{dr}&&\\frac{dr}{d\\lambda} =  \\frac{\\partial H}{\\partial p_r}= \n\\frac{p_r}{B(r)},\\\\\n\\label{dphi}&&\\frac{d\\varphi}{d\\lambda} =  \\frac{\\partial H}{\\partial p_{\\varphi}} =  \n\\frac{p_{\\varphi}}{C(r)}.\n\\end{eqnarray}\nUsing Eqs. (\\ref{H1}), (\\ref{dr}) and (\\ref{dphi}), we obtain the equation of trajectory for a photon which is similar to the formalism in Ref. \\cite{Tsukamoto:2016qro}\n\\begin{equation}\\label{drphi}\n\\left(\\frac{dr}{d\\varphi}\\right)^{2}=\\frac{R_p(r)C(r)}{B(r)}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n&&R_p=\\frac{E^2}{L^2}\\frac{C(r)}{A(r)}W(r)-1,\\label{Rp}\\\\\n&&W(r)=1 -\\frac{\\omega _p (r) ^2}{E^2}A(r).\\label{W}\n\\end{eqnarray}\nIn the case $\\omega_p(r)=0$ or equivalently, $W(r)=1$, Eq. (\\ref{drphi}) gives the motion of light ray in vacuum.\n\nWe are interested in a photon with a given enegy $E$ that comes in from infinity, reaches a closest distance $r=r_{0}$, and goes out to infinity.\nAs $r_0$ corresponds to the turning point of the path, $dr\/d\\varphi$ vanishes and $R_p(r_0)=0$. Hereafter subscript $0$ indicates the quantity at the closest distance $r=r_{0}$. \nFor a light ray initially in the asymptotically flat spacetime, the impact parameter can be represented as \n\\begin{equation}\\label{impact}\nb(r_{0})=\\frac{L}{E}=\\sqrt{\\frac{C_{0}W_{0}}{A_{0}}}. \n\\end{equation}\nWith the help of Eq. (\\ref{impact}), $R_p(r)$ can be rewritten as\n\\begin{equation}\\label{R2}\nR_p(r)= \\frac{A_{0}CW}{AC_{0}W_{0}}-1.\n\\end{equation}\nTo find the radius of photon sphere, which is the unstable circular photon orbit of static, spherically symmetric compact objects, one can introduce a function $h(r)$ given by Perlick \\cite{Perlick:2015vta}\n\\begin{equation}\\label{h}\nh(r)^2=\\frac{C(r)}{A(r)}W(r)=\\frac{C(r)}{A(r)}\\left[1 -\\frac{\\omega _p (r) ^2}{E^2}A(r)\\right]. \n\\end{equation}\nThe photon sphere radius $r_m$ is the biggest real root of the equation\n\\begin{equation}\\label{dh}\n\\frac{d}{dr}h(r)^2=0.\n\\end{equation}\nFrom Eq. (\\ref{dh}), we obtain\n\\begin{equation}\\label{Dp}\n\\frac{C'}{C}+\\frac{W'}{W}-\\frac{A'}{A}=0,\n\\end{equation}\nwhere prime denotes the differentiantion with respect to the radical coordinate $r$.\n\nNow we consider the EGB black hole surrounded by homogeous plasma, which has the following form\n\\begin{equation}\\label{beta0}\n\\frac{\\omega_p(r)}{E}=\\beta_0,\n\\end{equation}\nwhere $\\beta_0$ is a positive dimensionless constant. Then we rewrite Eq. (\\ref{Dp}) as\n\\begin{equation}\\label{eqrm}\nr\\left[\\beta _0 \\left(2 \\alpha ^2+r^4+2 \\alpha  r^2+4 \\alpha\nMr\\right)-2 \\alpha ^2 \\right]\\sqrt{\\frac{8 \\alpha M}{r^3}+1}=\\beta _0 \\left(16 \\alpha^2M+r^5+2 \\alpha  r^3+8 \\alpha M\nr^2\\right)-6 \\alpha ^2M.\n\\end{equation}\nWe can solve this equation numerically to get the radius of the photon sphere  which is plotted in Fig. \\ref{rm}.\nIn the left panels of Fig. \\ref{rm} we\nshow the function $r_m\/M$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, \nand we demonstrate that the radius of the photon sphere of the EGB black hole decreases with the increase of $\\alpha\/M^2$ for fixed $\\beta_0$. \nIn the right panels of Fig. \\ref{rm} we\nshow the function $r_m\/M$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, \nand we find that the radius of the photon sphere of the EGB black hole increases with the increase of $\\beta_0$ for fixed $\\alpha\/M^2$.\nIt is clear that the presence of coupling constant $\\alpha$ and the plasma parameter $\\beta_0$, affects the photon sphere radius significantly.\n In the absence of $\\beta_0$, from Eq. (\\ref{eqrm}), the largest real root has a form\n\\begin{equation}\nr_{m}=2 \\sqrt{3}M \\cos \\left[\\frac{1}{3} \\cos ^{-1}\\left(-\\frac{4 \\alpha }{3 \\sqrt{3}M^2}\\right)\\right],\n\\end{equation}\nwhich is the photon radius of the EGB black hole in vacuum \\cite{Guo:2020zmf}.\nOn the other hand, in the case $\\alpha= 0$, we can get the photon radius of Schwarzschild black hole with homogeneous plasma   \n\\begin{equation}\nr_{m}=\\frac{3-4 \\beta_0 +\\sqrt{9-8 \\beta _0}}{2\\left(1-\\beta _0\\right)}M,\n\\end{equation}\nwhich has been obtained in Ref. \\cite{Tsupko:2013cqa}.\n   \n\\begin{figure}\n\t\\includegraphics[width=80mm,angle=0]{rmEGB.eps}\\,\n\t\\includegraphics[width=80mm,angle=0]{rm1EGB.eps}\n\\caption{Left panel: The plot of the radius of the photon sphere $r_m\/M$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the radius of the photon sphere $r_m\/M$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{rm}\n\\end{figure}\n\nWe define the critical value of the impact parameter $b_{c}$ for the light ray as \n\\begin{equation}\\label{cip}\nb_{c}\n\\equiv \\lim_{r_{0}\\rightarrow r_{m}} \\sqrt{\\frac{C_{0}W_{0}}{A_{0}}}.\n\\end{equation}\nThe strong deflection limit corresponds to the limit $r_0\\rightarrow r_m$ or $b\\rightarrow b_c$. From Eqs. (\\ref{A}), (\\ref{C}) and (\\ref{W}), the critical impact parameter is given by\n\\begin{eqnarray}\nb_{c}(r_{m})=\\sqrt{\\frac{\\beta _0 \\left[r_m^4 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha r_m^2\\right]+2 \\alpha  r_m^2}{2 \\alpha +r_m^2 \\left(1-\\sqrt{\\frac{8 \\alpha\tM}{r_m^3}+1}\\right)}}.\n\\end{eqnarray}\nThe dependence of the critical impact parameter from  the coupling constant and the plasma parameters is shown in Fig. \\ref{bc}.\nThe left panels of Fig. \\ref{bc} presents the function $b_c\/M$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, \nand it shows that the critical impact parameter of the EGB black hole decreases with the increase of $\\alpha\/M^2$ for fixed $\\beta_0$. \nThe right panels of Fig. \\ref{bc} presents the function $b_c\/M$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, and it shows that the critical impact parameter of the EGB black hole decreases with the increase of $\\beta_0$ for fixed $\\alpha\/M^2$.\nWe found that both the coupling constant and the presence of plasma have remarkable influences on the critical impact parameter.\n\\begin{figure}\n\\includegraphics[width=80mm,angle=0]{bcEGB.eps}\\,\n\\includegraphics[width=80mm,angle=0]{bc1EGB.eps}\n\\caption{Left panel: The plot of the critical impact parameter $b_c\/M$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the critical impact parameter $b_c\/M$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{bc}\n\\end{figure}\n\n\\section{Strong gravitational lensing of EGB black hole in homogeneous plasma}\n\nIn this section, we will calculate the deflection angle of a light ray in the strong deflction limit in the EGB black hole spacetime with plasma medium.\nFrom Eq. (\\ref{drphi}), the deflection angle $\\hat{\\alpha}_p(r_{0})$ for the photon coming from infinite to the EGB black hole in homogeneous plasma is given by\n\\begin{equation}\\label{alpha}\n\\hat{\\alpha}_p(r_{0})=I_p(r_{0})-\\pi,\n\\end{equation}\nwhere $I_p(r_{0})$ is defined as\n\\begin{equation}\\label{angle2}\nI_p(r_{0})\\equiv 2\\int^{\\infty}_{r_{0}}\\frac{1}{\\sqrt{\\frac{R_p(r)C(r)}{B(r)}}}dr.\n\\end{equation}\nIt is found that the deflection angle increases when the closest distance $r_0$ decreases, and for a special point, the deflection angle will arrive at $2\\pi$ which means the photon winds a complete loop around the black hole. Furthermore, when $r_{0}$ approach the radius of the photon sphere $r_m$ the  deflection angle will diverge \\cite{Virbhadra:1999nm}.\nTo discuss the divergence, following Ref.\\cite{Chen:2009eu}, we  introduce a new variable $z$ \n\\begin{equation}\\label{z1}\nz\\equiv 1-\\frac{r_{0}}{r}.\n\\end{equation}\nUsing Eqs. (\\ref{A})-(\\ref{C}), (\\ref{Rp}) and (\\ref{W}), we can rewrite $I_p(r_{0})$ as\n\\begin{equation}\nI_p(r_{0})=\\int^{1}_{0}f_p(z,r_{0})dz\n=\\int^{1}_{0}\\frac{2r_0}{\\sqrt{G_p(z,r_{0})}}dz,\n\\end{equation}\nwhere the function $G_p(z,r_{0})$ in the EGB spacetime is given by\n\\begin{eqnarray}\nG_p(z,r_{0})=&&\\frac{R_p(z,r_0)C(z,r_0)}{B(z,r_0)}(1-z)^4=r_0^2\\left((1-z)^2-\\frac{r_0^2 \\left(\\sqrt{1-\\frac{8 \\alpha  M (z-1)^3}{r_0^3}}-1\\right)}{2 \\alpha}\\right)\n\\\\\\nonumber\n&&\\left(\\frac{\\left(r_0^2\n\t\t\t\\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-1\\right)-2 \\alpha \\right) \\left(-\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{r_0^3-8 \\alpha  M\n\t\t\t\t\t(z-1)^3}{r_0^3}}-1\\right)-2 \\alpha  (z-1)^2\\right)-2 \\alpha  (z-1)^2\\right)}{(z-1)^2 \\left(2 \\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha\n\t\t\t\t\tM}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right) \\left(2 \\alpha  (z-1)^2-r_0^2 \\left(\\sqrt{\\frac{r_0^3-8 \\alpha  M\n\t\t\t\t\t(z-1)^3}{r_0^3}}-1\\right)\\right)}-1\\right).\n\\end{eqnarray}\nWe can expand the above expression into a power series of z in the following form\n\\begin{equation}\nG_p(z,r_{0})=\\sum_{n=1}^{\\infty}c_n(r_0)z^n,\n\\end{equation}\nwhere $c_{1}(r_{0})$ and $c_{2}(r_{0})$ are given by\n\\begin{eqnarray}\nc_{1}(r_{0})=&&\\frac{\\beta _0 \\left(-4 \\alpha ^2 r_0^2 \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}+32 \\alpha ^2 M r_0+\\alpha  r_0^4 \\left(4-4 \\sqrt{\\frac{8 \\alpha \n\t\t\tM}{r_0^3}+1}\\right)-8 \\alpha  M r_0^3 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-2\\right)\\right)}{\\alpha  \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1} \\left(2\n\t\\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right)}\n\\\\\\nonumber\n&&+\\frac{\\beta _0 r_0^6 \\left(2-2 \\sqrt{\\frac{8 \\alpha M}{r_0^3}+1}\\right)+4 \\alpha ^2 r_0 \\left(r_0 \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-3 M\\right)}{\\alpha  \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1} \\left(2 \\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha\tM}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right)},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\\label{c2}\nc_{2}(r_{0})=&&\\frac{2 \\alpha ^2 {r_0}^2 \\left(96 \\alpha ^2 M^3 {r_0}+{r_0}^8 \\left(-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+{r_0}^8+2 \\alpha \n\t{r_0}^6\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8\\alpha M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber\n&&+\\frac{2 \\alpha ^2 M^2 {r_0}^2 \\left(128 \\alpha ^3+\\alpha  {r_0}^4 \\left(60-12 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+\\alpha ^2 {r_0}^2\n\t\\left(64-88 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber\n&&+\\frac{2 \\alpha ^2 M {r_0}^2 \\left({r_0}^7 \\left(6-6 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+\\alpha  {r_0}^5 \\left(16-28 \\sqrt{\\frac{8\n\t\t\t\\alpha  M}{{r_0}^3}+1}\\right)+32 \\alpha ^2 {r_0}^3\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left(2 \\alpha ^2 {r_0}^3 \\left(\\alpha -24 M^2\\right)+32 \\alpha ^3 M^2 {r_0}+16\n\t\\alpha ^4 M+\\alpha ^2 {r_0}^5 \\left(\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-1\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left(6 \\alpha  M {r_0}^6 \\left(3 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-5\\right)-8\n\t\\alpha ^3 M {r_0}^2 \\left(2 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}+1\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left({r_0}^9 \\left(3 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-3\\right)+\\alpha \n\t{r_0}^7 \\left(5 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-5\\right)+4 \\alpha ^2 M {r_0}^4 \\left(4 \\sqrt{\\frac{8 \\alpha \n\t\t\tM}{{r_0}^3}+1}-9\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}.\n\\end{eqnarray}\nIt is easy to get $c_{1}(r_{m})= 0$ in the limit $r_{0}\\rightarrow r_{m}$,  while $c_2(r_m)$ is complex in this limit. Furthermore, when $\\beta_0= 0$, i.e., in vacuum, the $c_2(r_m)$ term in the limit $r_{0}\\rightarrow r_{m}$ becomes\n\\begin{eqnarray}\nc_{2}(r_{m})=&&\\frac{96 \\alpha ^2 M^3 r_m^3+\\alpha  M^2 r_m^2 \\left(128 \\alpha ^2-12 r_m^4 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-5\\right)+8 \\alpha  r_m^2 \\left(8-11 \\sqrt{\\frac{8\n\t\t\t\\alpha  M}{r_m^3}+1}\\right)\\right)}{\\left(r_m^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha \\right) \\left(r_m^3+8 \\alpha  M\\right)^2}\n\\\\\\nonumber\n&&+\\frac{M \\left(-6 r_m^9 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-4 \\alpha  r_m^7 \\left(7 \\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-4\\right)+32 \\alpha ^2\n\tr_m^5\\right)+r_m^{10} \\left(-\\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)\\right)+2 \\alpha  r_m^8}{\\left(r_m^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha \\right) \\left(r_m^3+8 \\alpha  M\\right)^2}.\n\\end{eqnarray}\nSince this expression is still intricate, for the sake of clarity, let's continue to look at the form under the limit $\\alpha\\rightarrow 0$.\nIn the case $\\beta_0=0$ and $\\alpha=0$, $r_m=3M$, and Eq. (\\ref{c2}) has a form\n\\begin{equation}\nc_{2}(r_{m})=(6 M-r_m) r_m=9M^2,\n\\end{equation}\nwhere the vacuum Schwarzschild solution is recovered.\nBy the discussion above, we can find that the leading term of the divergence in $f_p(z,r_{0})$ is $z^{-1}$ in the strong deflection limit, which implies $I_p(r_{0})$ diverges logarithmically.\n\nOne can separate $I_p(r_{0})$ into two parts which are the divergent\npart $I_{D}(r_{0})$ and the regular part $I_{R}(r_{0})$\n\\begin{equation}\nI_p(r_{0})=I_{D}(r_{0})+I_{R}(r_{0}).\n\\end{equation}\nThe divergent part $I_{D}(r_{0})$ is defined as\n\\begin{equation}\nI_{D}(r_{0})\\equiv \\int^{1}_{0}f_{D}(z,r_{0})dz,\n\\end{equation}\nwhere\n\\begin{equation}\nf_{D}(z,r_{0})\\equiv \\frac{2r_{0}}{\\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}}}.\n\\end{equation}\n$I_{D}(r_{0})$ can be integrated and the result is\n\\begin{equation}\\label{ID}\nI_{D}(r_{0})=\\frac{4r_{0}}{\\sqrt{c_{2}(r_{0})}}\\log \\frac{\\sqrt{c_{2}(r_{0})}+\\sqrt{c_{1}(r_{0})+c_{2}(r_{0})}}{\\sqrt{c_{1}(r_{0})}}.\n\\end{equation}\nThe regular part $I_{R}(r_{0})$ is defined as\n\\begin{equation}\nI_{R}(r_{0})\\equiv \\int^{1}_{0}f_{R}(z,r_{0})dz,\n\\end{equation}\nwhere\n\\begin{equation}\nf_{R}(z,r_{0}) \\equiv f(z,r_{0})-f_{D}(z,r_{0}).\n\\end{equation}\n\nUsing a similar derivation as in Ref. \\cite{Tsukamoto:2016jzh}, we obtain the deflection angle $\\hat{\\alpha}_p(b)$ in the strong deflection limit $r_{0}\\rightarrow r_{m}$ or $b \\rightarrow b_{c}$ in the EGB black hole with homogeneous plasma\n\\begin{equation}\\label{alpha1}\n\\hat{\\alpha}_p(b)= -\\bar{a}\\log \\left( \\frac{b}{b_{c}}-1 \\right) +\\bar{b}+O((b-b_{c})\\log (b-b_{c})).\n\\end{equation}\nThe coefficients $\\bar{a}$ and $\\bar{b}$ are obtained as\n\\begin{eqnarray}\\label{abar}\n&&\\bar{a}=\\sqrt{\\frac{2B_{m}}{C_{m}\\left[\\frac{(CW)_{m}^{''}}{(CW)_{m}}-\\frac{A_{m}^{''}}{A_{m}}\\right]}},\\\\\n&&\\bar{b}=\\bar{a}\\log \\left\\{r^{2}_{m}\\left[\\frac{(CW)_{m}^{''}}{(CW)_{m}}-\\frac{A_{m}^{''}}{A_{m}}\\right]\\right\\} +I_{Rp}(r_{m})-\\pi,\\label{bbar}\n\\end{eqnarray}\nwhere the subscript $m$ denotes the quantities at $r=r_m$.\nIn the vacuum case, i.e., $\\beta_0=0$, and the coupling constant $\\alpha= 0$,  $\\bar{a}$ and $\\bar{b}$ will reduce to the formalism in Ref. \\cite{Tsukamoto:2016jzh}, which are the cofficients of a Schwarzschild black hole without plasma.\n\nThe numerical results of the strong field limit coefficients $\\bar{a}$ and $\\bar{b}$ are shown in  Fig. \\ref{ab1} and Fig. \\ref{bb1}. \nThe left panels of Fig. \\ref{ab1} show the strong field limit coefficient $\\bar{a}$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nWe find that the strong field limit coefficient $\\bar{a}$ increases with the increase of the coupling constant $\\alpha\/M^2$ for fixed $\\beta_0$.\nFrom the right panels of Fig. \\ref{ab1}, which refers to the strong field limit coefficient $\\bar{a}$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively,\nwe find that the strong field limit coefficient $\\bar{a}$ increases with the increase of the plasma parameter $\\beta_0$ for fixed  $\\alpha\/M^2$.\nIn the left panels of Fig. \\ref{bb1}, we illustrate the strong field limit coefficient $\\bar{b}$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, and show\nthat the strong field limit coefficient $\\bar{b}$ decreases with the increase of the coupling parameter $\\alpha\/M^2$ for fixed $\\beta_0$.\nFrom the right panels of Fig. \\ref{bb1}, which refers to the strong field limit coefficient $\\bar{b}$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, we find that the strong field limit coefficient $\\bar{b}$ increases with the increase of the plasma parameter $\\beta_0$ for fixed  $\\alpha\/M^2$.\nObviously, the strong field limit coefficients $\\bar{a}$ and $\\bar{b}$ are influenced by the choice of coupling constant $\\alpha$ and plasma parameter $\\beta_0$.\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{aEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{a1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the strong field limit coefficients $\\bar{a}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the strong field limit coefficients $\\bar{a}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{ab1}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{bEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{b1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the strong field limit coefficients $\\bar{b}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the strong field limit coefficients $\\bar{b}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{bb1}\n\\end{figure}\n\n\\section{Observables in the strong deflection limit}\n\nIn this section we consider the observables of the strong gravitational lensing by the EGB black hole in the presence of a uniform plasma.\nWe are interested in the case where the observer, the lens and the source are nearly in alignment, and the source and the observer are very far from the lens. \nThe distance between the lens and the source, and the distance between the lens and the observer are represented by $D_{LS}$ and $D_{OL}$ respectively. \n$D_{OS}$ is the distance between the observer and the source, and $D_{OS}=D_{LS}+D_{OL}$.\n$\\beta$ denotes the angular position with respect to the optical axis of the source.\n$\\theta$ is the angular position with respect to the optical axis of the image and can be expressed as $\\theta=b\/D_{OL}$. \nThus the lens eqaution can be written as \\cite{Bozza:2002zj}\n\\begin{equation}\\label{beta}\n\\beta=\\theta-\\frac{D_{LS}}{D_{OS}}\\triangle\\alpha_{n},\n\\end{equation}\nwhere $\\triangle\\alpha_{n}=\\alpha-2n\\pi$ is the offset of deflection angle, and $n$ denotes the loop numbers of the light ray around the light sphere.\n\nThe angular position $\\theta_{n}$ between the lens and the $n$-th relativistic image and the magnification of the $n$-th relativistic image $\\mu_{n}$ can be obtained approximately as\n\\begin{eqnarray}\\label{theta}\n&&\\theta_{n}=\\theta^{0}_{n}+\\frac{b_{c}(\\beta-\\theta_{n}^{0})D_{OS}}{\\bar{a}D_{LS}D_{OL}}\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right),\\\\\n&&\\mu_{n}=\\frac{b_{c}^{2}D_{OS}}{\\bar{a}\\beta D^{2}_{OL}D_{LS}}\n\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\left[1+\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\right].\\label{mun}\n\\end{eqnarray}\nHere $\\theta_{n}^{0}$ is the angular position corresponding to the case that the light ray winds completely $2n\\pi$ and can be expressed as \n\\begin{equation}\\label{theta1}\n\\theta_{n}^{0}=\\frac{b_{c}}{D_{OL}}\\left[1+\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\right].\n\\end{equation}\nIn the limit $n\\rightarrow\\infty$, we can find the relation between \nthe critical impact parameter $b_{c}$ and the asymptotic position $\\theta_{\\infty}$  approached by a set of images\n\\begin{equation}\\label{theta2}\n\\theta_{\\infty}=\\frac{b_{c}}{D_{OL}}.\n\\end{equation}\nSince the outermost relativistic image is the brightest, one can use the observable $s$ to describe the separation between this first image $\\theta_{1}$ and all the others packed images at $\\theta_{\\infty}$ \\cite{Bozza:2002zj}. The other observable $\\mathcal{R}$ represents the ratio of the received flux beteween this first image and all the others images \\cite{Bozza:2002zj}. \nUsing Eqs. (\\ref{theta}) and (\\ref{mun}), the angular separation $s$ and the ratio of the flux $\\mathcal{R}$ can be obtained as\n\\begin{eqnarray}\\label{sr}\n&&s=\\theta_{1}-\\theta_{\\infty}=\\theta_{\\infty} \\exp\\left({\\frac{\\bar{b}-2\\pi}{\\bar{a}}}\\right),\\\\\n&&\\mathcal{R}=\\frac{\\mu_{1}}{\\sum\\limits_{n=2}^{\\infty}\\mu_{n}}=\\exp\\left({\\frac{2\\pi}{\\bar{a}}}\\right).\n\\end{eqnarray}\nIf the observables  $s$, $\\theta_{\\infty}$ and $\\mathcal{R}$ are available, the coefficients $\\bar{a}$ and $\\bar{b}$ in the strong deflction limit and the critical impact parameter $b_{c}$ can be obtained easily by\n\\begin{eqnarray}\\label{ab2}\n&& \\bar{a}=\\frac{2\\pi}{\\log \\mathcal{R}},\\\\\n&& \\bar{b}=\\bar{a}\\log(\\frac{\\mathcal{R}s}{\\theta_{\\infty}}),\\\\\n&& b_{c}=\\theta_{\\infty}D_{OL}.\n\\end{eqnarray}\nThen one can numerically compute the above value by measuring  the observables  $s$, $\\theta_{\\infty}$ and $\\mathcal{R}$  and study their difference with the corresponding theoretical coefficients.\n\nLet's take the supermassive black hole M87$^{*}$ as an example.  The results from the EHT show that the angular diameter of the shadow of M87$^{*}$ is $(42\\pm 3) \\mu$as, and the observed shadow is almost circular which is supported by the fact that the axis ratio is smaller than $4\/3$ and the deviation from circularity is less than 10\\% \\cite{Akiyama:2019cqa, Akiyama:2019fyp, Akiyama:2019eap}.\nTherefore, the image of M87$^{*}$ is nearly circular due to the relatively small value of spin and low inclination angle $\\sim 17 ^\\circ$ of the source \\cite{Walker:2018muw}. So it is reasonable to choose spherically symmetric metric as  an approximation to discuss the strong gravitational lensing of M87$^{*}$.\nMeanwhile, the mass of M87$^{*}$ is estimated by the EHT collaboration as  $M=(6.5\\pm 0.7)\\times 10^{9}~M_{\\odot}$ \\cite{Akiyama:2019eap}. Note that the mass of M87$^{*}$ is also estimated to be $6.2^{+1.1}_{-0.5}\\times10^9$ $M_{\\odot}$  and $3.5^{+0.9}_{-0.3}\\times10^9$ $M_{\\odot}$   by the stellar dynamics \\cite{Gebhardt:2011yw} and gas dynamics measurements \\cite{Walsh:2013uua}, respectively.\nIn addition, the distance $D_{OL}$ of M87$^{*}$ from us is estimated to be $D_{OL}$=(16.8$\\pm$0.8) Mpc from stellar population measurements \\cite{Blakeslee:2009tc, Bird:2010rd, Cantiello:2018ffy}.\n\n\nIn the following the lens is supposed to be M87$^{*}$ which is described by the EGB black hole. For simplicity, we use the following data from M87$^{*}$, $D_{OL}$=16.8 Mpc and $M=$6.5$\\times10^9$ $M_{\\odot}$. With these data we can estimate the values of the  angular image position $\\theta_{\\infty}$, the angular image separation $s$ and the relative magnifications $r$ of the relativistic images which is defined as $r=2.5\\log_{10}\\mathcal{R}$.\nFigs. \\ref{th}-\\ref{dm} show the behaviors of these observables and the \ninfluences on them by the choince of coupling constant $\\alpha$ and plasma parameter $\\beta_0$.\nThe left panels of Fig. \\ref{th} show the value of $\\theta_{\\infty}$ as function of\n$\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases,\nthe angular image position decreases for fixed $\\beta_0$. \nThe right panels of Fig. \\ref{th} show the value of $\\theta_{\\infty}$ as function of\n$\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases,\nthe angular image position decreases for fixed $\\alpha\/M^2$.\n \nAs shown in the left panels of Fig. \\ref{s}, the value of $s$ is expressed as a function of $\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases, the angular image separation increases for fixed $\\beta_0$.\nIt is shown that in the right panels of Fig. \\ref{s}, the value of $s$ is expressed as function of $\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases, the angular image separation increases for fixed $\\alpha\/M^2$.\nInterestingly, as $\\alpha\/M^2\\rightarrow-8$ for different plasma parameter $\\beta_0$, the angular image separation converges to the value at $\\alpha\/M^2=-8$ in the vacuum case,\nwhich means plasma has little effect on the angular image separation. \n\nIn the left panels of Fig. \\ref{dm}, we show the value of relative magnifications as function of\n$\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases,\nthe relative magnifications  decreases for fixed $\\beta_0$. \nIn the right panels of Fig. \\ref{dm}, we show the value of relative magnifications as function of\n$\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases,\nthe relative magnifications decreases for fixed $\\alpha\/M^2$.\nWe find that as $\\alpha\/M^2\\rightarrow 1$ for different plasma parameter $\\beta_0$, the relative magnifications converges to the value at $\\alpha\/M^2=1$ in the vacuum case,\nwhich means plasma has little influence on the relative magnifications.\n\nIn Table \\ref{tab1}, we list the numerical estimates of the observables as well as strong field limit coefficients of a EGB black hole in uniform plasma. \nThe parameter $\\beta_0=0$  corresponds to the case of the EGB black hole in vacuum and $\\alpha=0$ means the case of Schwarzschild\nblack hole in homogeneous plasma.\nFrom Table \\ref{tab1}, we can easily obtain the differences between the Schwarzschild black hole and the EGB  black hole, as well as the EGB  black hole with various plasma parameter.\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{thetaEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{theta1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the angular image position $\\theta_{\\infty}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the angular image position $\\theta_{\\infty}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{th}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{sEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{s1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the angular image separation $s$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the angular image separation $s$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{s}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{dmEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{dm1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the relative magnifications $r$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the relative magnifications $r$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{dm}\n\\end{figure}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{cccccccc}\n\\hline\\hline\n$\\beta_0$ & $\\alpha\/M^2$ & $\\theta_{\\infty}$($\\mu\\text{as}$) & $s$($\\mu \\text{as}$) & $r(\\text{mag})$  & $b_{c}\/R_{s}$ & $\\bar{a}$ & $\\bar{b}$ \\\\\n\\hline\n & -8 & 52.10 & 0.000511 & 12.30 & 3.41 & 0.555 & -0.1151 \\\\\n & -4 & 47.39 & 0.00272  & 10.35 & 3.10 & 0.659 & -0.1558 \\\\\n & -2 & 44.21 & 0.00867  & 8.97  & 2.89 & 0.760 & -0.2087 \\\\\n0& 0  & 39.69 & 0.0497   & 6.82  & 2.60 & 1     & -0.4003 \\\\\n & 0.4& 38.41 & 0.0838   & 6.14  & 2.51 & 1.111 & -0.5264 \\\\\n & 1  & 35.85 & 0.253    & 4.51  & 2.35 & 1.511 & -1.2017 \\\\\\hline\n   & -8 & 50.82 & 0.000644 & 12.03 & 3.33 & 0.567 & -0.1088 \\\\\n   & -4 & 46.33 & 0.00320  & 10.16 & 3.03 & 0.671 & -0.1490 \\\\\n   & -2 & 43.31 & 0.00980  & 8.83  & 2.83 & 0.773 & -0.2024 \\\\\n0.1& 0  & 39.01 & 0.0530   & 6.74  & 2.55 & 1.012 & -0.3971 \\\\\n   & 0.4& 37.80 & 0.0878   & 6.08  & 2.47 & 1.123 & -0.5249 \\\\\n   & 1  & 35.38 & 0.254    & 4.50  & 2.32 & 1.515 & -1.1947 \\\\\\hline\n   & -8 & 48.08 & 0.000111 & 11.43 & 3.15 & 0.597 & -0.08858 \\\\\n   & -4 & 44.08 & 0.00473  & 9.73  & 2.89 & 0.701 & -0.1274 \\\\\n   & -2 & 41.39 & 0.00132  & 8.50  & 2.71 & 0.803 & -0.1815 \\\\\n0.3& 0  & 37.57 & 0.0622   & 6.55  & 2.46 & 1.041 & -0.3833 \\\\\n   & 0.4& 36.50 & 0.0991   & 5.93  & 2.39 & 1.150 & -0.5144 \\\\\n   & 1  & 34.38 & 0.261    & 4.47  & 2.25 & 1.527 & -1.1715 \\\\\\hline   \n   & -8 & 45.03 & 0.00220  & 10.70 & 2.95 & 0.638 & -0.04854 \\\\\n   & -4 & 41.58 & 0.00788  & 9.18  & 2.72 & 0.743 & -0.08515 \\\\\n   & -2 & 39.26 & 0.00196  & 8.08  & 2.57 & 0.845 & -0.1400 \\\\\n0.5& 0  & 35.97 & 0.0783   & 6.30  & 2.35 & 1.082 & -0.3494 \\\\\n   & 0.4& 35.05 & 0.119    & 5.73  & 2.29 & 1.190 & -0.4843 \\\\\n   & 1  & 33.25 & 0.278    & 4.41  & 2.18 & 1.548 & -1.1255 \\\\\\hline   \n   & -8 & 41.54 & 0.00574  & 9.72  & 2.72 & 0.702 &  0.04408 \\\\\n   & -4 & 38.73 & 0.0165   & 8.44  & 2.54 & 0.808 &  0.01158 \\\\\n   & -2 & 36.83 & 0.0352   & 7.50  & 2.41 & 0.910 & -0.04313 \\\\\n0.7& 0  & 34.14 & 0.113    & 5.95  & 2.24 & 1.146 & -0.2606 \\\\\n   & 0.4& 33.40 & 0.160    & 5.45  & 2.19 & 1.251 & -0.3993 \\\\\n   & 1  & 31.95 & 0.322    & 4.30  & 2.09 & 1.588 & -1.0194 \\\\\\hline\n\\hline\n\\end{tabular}\n\\caption{Numerical estimation for the observables and the strong deflection limit coefficients for EGB black holes supposed to describe the supermassive black hole M87$^{*}$. $R_S=2GM\/c^2$ is the Schwarzschild radius.}\\label{tab1}\n\t\\end{center}\n\\end{table}\n\n\\section{Conclusions}\n\nIn this work, we have investigated the strong gravitational lensing generated by a 4-dimensional Einstein-Gauss-Bonnet black hole in a plasma. \nIn the presence of plasma around the black hole, the trajectory of a photon differs from the null geodesic in vacuum, resulting in the changes of the deflection angle of light.\nUsing Hamilton's equation of the light ray in plasma with a frequency dependent refraction index, we have derived the equation of motion for light rays in the novel $4$-dimensional EGB black hole. \nFurthermore, we numerically obtained the theoretical strong field limit parameters for the lensing by the black hole in a uniform plasma. Among these parameters we found that the radius of the photon sphere $r_{m}$,\nthe critical impact parameter $b_c$ and the strong field limit coefficient $\\bar{b}$ \ndecrease monotonically, while the strong field limit coefficient $\\bar{a}$ \nincreases, with the increase of the coupling constant $\\alpha\/M^2$ for fixed value of plasma parameter $\\beta_0$.\nOn the other hand, for a fixed value of the coupling constant $\\alpha\/M^2$,   with the increase of the plasma parameter $\\beta_0$, $r_{m}$, $\\bar{a}$ and  $\\bar{b}$ increase, but  $b_c$ decreases monotonically.\nModelling the supermassive $\\mathrm{M}87^{\\ast}$ with this EGB black hole,\nwe have estimated the observables including the angular image position $\\theta_{\\infty}$, the angular image separation $s$ and the relative magnifications $r$ of the relativistic images in the uniform plasma.\nWe have shown that among these observables, when the coupling parameter $\\alpha\/M^2$ increases for fixed plasma parameter $\\beta_0$, the angular image position $\\theta_{\\infty}$ and the relative magnifications $r$ decrease,\nwhile the angular image separation $s$ increases. \nWhen the  plasma parameter $\\beta_0$ increases for fixed coupling constant $\\alpha\/M^2$, the angular image position $\\theta_{\\infty}$ and the relative magnifications $r$ decrease, but the angular image separation $s$ increases.\nAbove all, both the coupling constant $\\alpha$ and plasma parameter $\\beta_0$ have significant effects on the parameters and observables in strong gravitational lensing.\nInterestingly, it is found that  plasma has little effect on the angular image separation as $\\alpha\/M^2\\to -8$ and  the relative magnifications as $\\alpha\/M^2\\to 1$, respectively. \n\nTheoretically we can use the observations on strong gravitational lensing to test this modified gravity, although the relativistic images of strong gravitational lensing are so faint that it is hard to detect. However, with the improvement of technology, we wish observations in the future may provide the opportunity to distinguish the EGB black hole from those in general relativity. Finally, it is worth noting that in this paper we ignore the rotation of M87$^{*}$ and use the spherically symmetric metric to  provide some hints about its gravitational lensing signals. The gravitational lensing effect of rotating EGB black hole deserves a new work in the future.\n\n \n\\begin{acknowledgments}\n\t{{This work is supported in part by Science and Technology Commission of Shanghai Municipality under Grant No. 12ZR1421700 and Shanghai Normal University  KF201813.}}\n\\end{acknowledgments}\n\n\\textbf{Note added:} After this work is completed, we are  aware of a similar work by \nIslam et al. \\cite{Islam:2020xmy}, which appeared in arXiv a couple of days before.\nThey focus on the strong gravitational lensing in vacuum, in which the variation range of the coupling constant is $0\\le {\\alpha}\/{M^2}\\le 0.019$, while we discuss the strong gravitational lensing in homogeneous plasma, in which the variation range of coupling constant is $-8\\le {\\alpha}\/{M^2}\\le 1$.\nOur results agree with theirs where we overlap.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $f$ be a cusp form of positive real weight with multiplier system, and let $a(n)$ be its $n$-th Fourier coefficient. In \\cite{KKP,Pr} Knopp, Kohnen and Pribitkin, proved that the sequence $\\{a(n)\\}_{n\\in\\mathbb{N}}$, is oscillatory i.e., for each real number $\\phi\\in [0,\\pi)$, either the sequence $\\{{\\Re e\\,} (a(n)e^{-i\\phi})\\}_{n\\in\\mathbb{N}}$ changes sign infinitely often or is trivial. Geometrically speaking, this means that no matter how we slice the plane with a straight line going through the origin, there will always be infinitely many terms of $\\{a(n)\\}_{n\\in\\mathbb{N}}$ on either side of the line, unless all the terms are on the line itself, this fact motivates the following questions:\n\\begin{itemize}\n\\item[$\\bullet$] What is the proportion of integers for which the $a(n)$ lies in the same half-plane? \n\\item[$\\bullet$] If all the $a(n)$ are on a line, what is the proportion of integers for which the $a(n)$ lies in either side of the origin?\n\\end{itemize}\n\nThe latter question was asked in the particular case when $a(n)$ are real in \\cite[Section 6]{KKP}. In the case when $f$ is a newform of integral weight without complex multiplication (CM), the celebrate Sato-Tate conjecture suggests that no matter how we slice the plane with a straight line going through the origin, the proportion of primes for which the $a(p)$ lies in the same half-plane is equal to the half of the proportion of primes for which the $a(p)$ are not on the line. The questions that then naturally arise are: whether this is still true for newforms with CM? Can we infer similar results for $a(n)$ when $n$ runs through natural integers? Numerical calculations seem to suggest that the answer is positive. \n\nGoing further in this direction, in the case when $f$ is a cusp form of half-integral weight with real Fourier coefficients contained in the plus space, Kohnen and Bruinier \\cite{bruinier} gave the conjecture  \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; a(n)\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; a(n)\\neq 0\\}}=\\frac{1}{2}.\n$$\nIn \\cite{IW2} Inam and Wiese partially verified this conjecture, more precisely they proved that for a fixed square-free integer $t$, the proportion of integers from $\\{tn^2\\}_{n\\in\\mathbb{N}}$ on which the $a(tn^2)$ are of the same sign is equal to the half of the proportion of integers from $\\{tn^2\\}_{n\\in\\mathbb{N}}$ on which the $a(tn^2)$ are non-zero.\n\nThis work was intended as an attempt to answer the questions mentioned above. However, it seems quite difficult to prove any general theorem here and we can only prove results that seem to point into the right direction. One of the motivations of this paper is an earlier work \\cite{Amri} of the author, in which he proved that the Fourier coefficients of a newform supported on prime powers have infinitely many ``angular changes\". Moreover, he established the ``angular changes\" of some subfamilies of Fourier coefficients of holomorphic cusp forms of half-integral weight reachable via the Shimura correspondence, using a robust analytic tool.\n\nThe outline of the paper is as follows. In Section \\ref{sec:1} we prove a Sato-Tate theorem for CM newforms with non-trivial nebentypus, which is  presumably well-known to experts. But, it seems that this result has not previously appeared in the literature. We also recall the Sato-Tate theorem for newforms without CM due to its importance in the sequel. In Section \\ref{sec:2}, we shall prove that for a given newform, the sub-sequences of its Fourier coefficients $\\{a(p^\\nu)\\}_{p,\\text{primes}}$ (for a fixed $\\nu\\in\\mathbb{N}$), and $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ (for a fixed prime $p$) are oscillatory. Moreover, we calculate the proportion of primes and integers, respectively from $\\{p^\\nu\\}_{p,\\text{primes}}$, and $\\{p^\\nu\\}_{\\nu\\in\\mathbb{N}}$, on which $a(p^\\nu)$ lies in the same half-plane. In Section \\ref{sec:3} we study the oscillatory behavior results for the Fourier coefficients of eigenforms of half-integral weight which are accessible via the Shimura correspondence. Indeed, we prove that the sequences  $\\{a(tp^2)\\}_{p,\\text{primes}}$ and $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$, are oscillatory. Furthermore, we calculate the proportion of primes from $\\{tp^2\\}_{p,\\text{primes}}$ on which the $a(tp^2)$ lies in the same half-plane. Some conclusions are drawn in the final section.\n\n\n\n\\paragraph{\\textbf{Notations}}\nThroughout the paper, for any $k\\ge 2$, $N\\ge 1$ and any Dirichlet character $\\varepsilon\\pmod N$, we\ndenote by $r_\\varepsilon$ the order of $\\varepsilon$. We write $S_k^{\\mathrm{new}}(N,\\varepsilon)$ for the space of newforms of weight $k$ for the group $\\Gamma_0(N)$, with nebentypus $\\varepsilon$. If $4\\mid N$ we write $S_{k+1\/2}(N,\\varepsilon)$ for the space of half-integral weight cusp forms, when $k=1$, we shall work only with the orthogonal complement (with respect to the Petersson inner product) of the subspace of $S_{k+1\/2}(N,\\varepsilon)$ spanned by single-variable unary theta functions. The  letter $\\mathcal{H}$ stands for the upper half-plane, for $z\\in\\mathcal{H}$ we set $q:=e^{2\\pi iz}$.\n\nLet $\\mathbb{P}$ denote the set of all prime numbers. If $S$ is a subset of $\\mathbb{P}$,  we denote by  $\\delta(S)$ its natural density (if it exists), and we shall denote by $\\pi(x)$ the prime-counting function.\n\n\n\\section{Preliminaries}\\label{sec:1}\nIn this section, we mention some results about the equidistribution of eigenvalues of newforms with non-trivial nebentypus, which are crucial for our purpose. In the CM-case (newforms with CM), we provide a full proof, as we are not aware of any appropriate reference.\n\nLet  $f(z)=\\sum_{n\\ge1}a(n)q^n\\in S_k^{\\text{new}}(N,\\varepsilon)$ be a normalized newform. Fixing a root of unity $\\zeta$ such that $\\zeta^2\\in\\text{Im}(\\varepsilon)$, let $p$ be a prime number satisfying $\\varepsilon(p)=\\zeta^2$, it is clear that $\\frac{a(p)}{\\zeta}$ is real, hence in view of the Ramanujan-Petersson bound, we have \n\\begin{equation}\n\\frac{a(p)}{2 p^{(k-1)\/2}\\zeta}\\in[-1,1].\\label{eq1}\n\\end{equation} \n\nThe distribution of the sequence $\\left(\\frac{a(p)}{2 p^{(k-1)\/2}\\zeta}\\right)_p$ on $[-1,1]$ as $p$ varies over primes such that $\\varepsilon(p)=\\zeta^2$, depends on whether $f$ has complex multiplication (of CM-type) or not (not of CM-type), then we have to consider two cases. Let us first look at the CM-case. We shall prove.\n\\begin{thm}\\label{thmCMST}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of CM-type, write\n$$\nf(z)=\\sum_{n\\ge1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Assume that the order of $\\varepsilon$ and the discriminant of the imaginary quadratic field by which $f$ has complex multiplication are coprime. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Then the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2,$ with respect to the measure $\\mu_{\\text{CM}}:=\\frac{1}{2\\pi}\\frac{dt}{\\sqrt{1-t^2}}+\\frac{1}{2}\\delta_{0}$, where $\\delta_0$ denotes the Dirac measure concentrated at zero. In particular, for any sub-interval $I\\subset [-1,1]$, we have \n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\le x: \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\in I\\}}{\\#\\{p\\le x\\;:\\; \\varepsilon(p)=\\zeta^2\\}}=\\mu_{\\text{CM}}(I)=\\dfrac{1}{2\\pi}\\int_{I}\\dfrac{dt}{\\sqrt{1-t^2}}+\\dfrac{1}{2}\\delta_{0}(I).\n$$\n\\end{thm}\n\nNotice that if $\\varepsilon$ is trivial we find Deuring's equidistribution theorem \\cite{deuring}. Before we proceed to prove the theorem, we first recall some basic facts concerning newforms with complex multiplication, following the exposition given in \\cite[Section 2, pp.8-9]{Fite2015}.\n\nAssume that $f$ is of CM-type, by definition (cf. \\cite[Definition, pp. 34]{Ribet}) there exists a Dirichlet character $\\chi$ such that \n\\begin{equation}\\label{eq2}\na(p)=\\chi(p)a(p),\n\\end{equation}\nfor a set of primes $p$ of density $1$. We see that $\\chi$ here has to be a quadratic character and then it corresponds to a quadratic imaginary field say $F$, let $d_F$ denote its discriminant. \n\nAccording to \\cite[Corollary 3.5, Theorem 4.5]{Ribet} the newform $f$ should arise from an algebraic Hecke character $\\xi_f$ of $F$ of modulus $\\mathfrak{m}$ (integral ideal of $F$), in the sense that the Fourier expansion of $f$ at $\\infty$ can be written as\n$$\nf(z)=\\sum_{\\mathfrak{a}}\\xi_f(\\mathfrak{a})q^{\\mathcal{N}(\\mathfrak{a})},\n$$\nwhere $\\mathfrak{a}$ runs through all integral ideals of $F$ and $\\mathcal{N}=\\mathcal{N}_{F\/\\mathbb{Q}}$ is the norm relative to the extension $F\/\\mathbb{Q}$. \n\nLet $K_f$ denote the number field obtained by adjoining to $\\mathbb{Q}$ the Fourier coefficients of $f$.  Let $\\ell$ be prime.  For each prime ideal $\\lambda$  of $K_f$ lying above $\\ell$, there is an irreducible 2-dimensional Galois representation\n$$\n\\rho_{f,\\lambda} : \\mathrm{Gal}(\\overline{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\mathrm{GL}_2(K_{f,\\lambda})\n$$\nwhich is unramified outside $N\\ell$, and satisfies\n\\begin{equation}\\label{eq3}\n\\det(\\rho_{f,\\lambda})=\\varepsilon\\chi_{\\ell}^{k-1},\n\\end{equation}\nwhere $K_{f,\\lambda}$ is the completion of $K_{f}$ at $\\lambda$ and $\\chi_{\\ell}$ is the $\\ell$-adic cyclotomic character. \n\nLet $\\mathfrak{p}$ be a prime of $F$ lying above a prime $p$ not dividing $N\\ell$, and let $\\overline{\\mathfrak{p}}$ be its conjugate, then $\\rho_{f,\\lambda}(\\mathrm{Frob}_{\\mathfrak{p}})$ has a characteristic polynomial \n$$\nP_{\\mathfrak{p},\\rho_{f,\\lambda}}(T):=T^2-a(p)T+p^{k-1}\\varepsilon(p),\n$$ \nwhich can be factored into\n\\begin{equation}\\label{eq4}\nP_{\\mathfrak{p},\\rho_{f,\\lambda}}(T)=(T-\\xi_f(\\mathfrak{p}))(T-\\xi_f(\\mathfrak{\\overline{p}})).\n\\end{equation}\nTherefore, in view of \\eqref{eq3} we have\n\n\\begin{equation}\\label{eq5}\n\\xi_f(\\mathfrak{\\overline{p}})=\\varepsilon(\\mathcal{N}(\\mathfrak{p}))\\overline{\\xi_f(\\mathfrak{p})}.\n\\end{equation}\n\nNext, consider the commutative group\n$$\nG=\\left\\{\\left(\\begin{array}{cc}\nu & 0 \\\\ \n0 & \\zeta \\bar{u}\n\\end{array}\\right)\\; | \\; \\zeta\\in\\text{Im}(\\varepsilon), \\; u\\in\\mathbb{C}^*\\;\\;|u|=1 \\right\\}.$$\nLet $\\mu$ denote the Haar measure of $G$ and $X:=\\mathrm{conj}(G)$ the set of\nits conjugacy classes. Let $S$ be the set of primes of $F$ lying over $N\\ell.$ For every\n$\\mathfrak{p}\\notin S$, define the sequence\n$$x_{\\mathfrak{p}}:=\\left(\\begin{array}{cc}\n\\xi_f(\\mathfrak{p})\/\\mathcal{N}(\\mathfrak{p})^{(k-1)\/2} & 0 \\\\ \n0 & \\varepsilon(\\mathcal{N}(\\mathfrak{p}))\\overline{\\xi_f(\\mathfrak{p})}\/\\mathcal{N}(\\mathfrak{p})^{(k-1)\/2}\n\\end{array}\\right)\\in X,$$\nwith $\\mathfrak{p}$ runs over prime ideals of $F$ such that $\\text{Frob}_{\\mathfrak{p}}=\\mathcal{C}$, where $\\mathcal{C}$ is a certain conjugacy class of $\\mathrm{Gal}(F_{\\varepsilon}\/F)$, and $\\mathrm{Frob}_{\\mathfrak{p}}$ denotes the Frobenius element at $\\mathfrak{p}$ in the cyclic extension $F_{\\varepsilon}\/F$, where $F_{\\varepsilon}$ denotes the compositum of the field $\\mathbb{Q}(\\varepsilon)$ generated by the values of the character $\\varepsilon$ and $F$. \n\nAt this point we state the following proposition which can be proved by a similar argument to the one in \\cite[Proof of Proposition 3.6]{Fite2014}.\n\\begin{pro}\\label{pro1}\nFor any conjugacy class $\\mathcal{C}$ of $\\mathrm{Gal}(F_{\\varepsilon}\/F)$  the sequence $\\{x_{\\mathfrak{p}}\\}_{\\mathfrak{p}}$ is $\\mu$-equidistributed on $X$.\n\\end{pro}\n\nHaving disposed of this preliminary setup, we can now return to prove Theorem \\ref{thmCMST}.\n\\begin{proof}[Proof of Theorem \\ref{thmCMST}]\nLet $\\zeta$ be a fixed root of unity, such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Let $p$ be a prime satisfying $\\varepsilon(p)=\\zeta^2$. We distinguish two cases.\n\nIf $p=\\mathfrak{p}\\overline{\\mathfrak{p}}$ splits in $F$, then from \\eqref{eq4} we have\n\\begin{equation}\\label{eq6}\na(p)=\\xi_{f}(\\mathfrak{p})+\\xi_{f}(\\overline{\\mathfrak{p}}).\n\\end{equation}\nConsider the map $\\vartheta : X \\to[-1,1]$, got by associating a given element of $X$ to its trace divided by $2\\zeta$. Altogether from \\eqref{eq5} and \\eqref{eq6} we see that the conjugacy class of $x_{\\mathfrak{p}}$ is mapped to $\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}$ by $\\vartheta$. From Proposition \\ref{pro1} and taking into account the isomorphism $\\mathrm{Gal}(F_{\\varepsilon}\/F)\\cong \\mathrm{Im}(\\varepsilon)$, it follows that the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed on $[-1,1]$ as $p$ varies over primes that split in $F$ and satisfying $\\varepsilon(p)=\\zeta^2$ with respect to the measure $\\frac{1}{\\pi}\\frac{dt}{\\sqrt{1-t^2}}$, which is the push-forward measure with respect to $\\vartheta$ of the Haar measure $\\mu$ on the Sato-Tate group $G$.\n\nIf $p$ remains inert in $F$, from \\eqref{eq2} we have $a(p)=0$, since $(r_{\\varepsilon},d_F)=1$ then the fields $F$ and $\\mathbb{Q}(\\varepsilon)$ are linearly disjoint over $\\mathbb{Q}$. Hence, by Chebotarev's density theorem, half of the primes $p$ for which $\\varepsilon(p)=\\zeta^2$ split in $F$ and the other half are inert in $F$. Consequently the sequence $\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2$, with respect to the measure $\\mu_{\\text{CM}}=\\frac{1}{2\\pi}\\frac{dt}{\\sqrt{1-t^2}}+\\frac{1}{2}\\delta_{0}$, where we use the Dirac measure $\\delta_0$ to put half the mass at $0$ to account for the inert primes.\n\\end{proof} \n\nIn the non-CM situation the equidistribution of the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ in $[-1,1]$ as $p$ varies over primes such that $\\varepsilon(p)=\\zeta^2$, is given by case 3 of \\cite[Theorem B]{ST}.\n\\begin{thm}(Barnet-Lamb, Geraghty, Harris, Taylor)\\label{thmST}\nLet  $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform not of CM-type, write\n$$\nf(z)=\\sum_{n\\ge1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Then the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2,$ with respect to the Sato-Tate measure $\\mu_{\\text{ST}}:=\\frac{2}{\\pi}\\sqrt{1-t^2}dt$. In particular, for any sub-interval $I\\subset [-1,1]$ we have \n$$\\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\le x: \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\in I\\}}{\\#\\{p\\le x\\;:\\; \\varepsilon(p)=\\zeta^2\\}}=\\mu_{\\text{ST}}(I)=\\frac{2}{\\pi}\\int_{I}\\sqrt{1-t^2}dt.$$\n\\end{thm}\n\\section{Equidistribution of sign results for integral weight Newforms}\\label{sec:2}\nIn this section, we shall state two of our main results and shall give a proof of them. Throughout this section, we shall stick to the following notations. \n\n\\begin{hyp}\\label{hyp}\nLet \n$$\nf(z)=\\sum_{n\\ge1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$. If $f$ has CM by a quadratic imaginary field $F$, with discriminant $d_F$, we suppose that $(d_F,r_\\varepsilon)=1$.\n\\end{hyp}\n\nConsider $\\zeta$ a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, if $p$ is a prime number satisfying $\\varepsilon(p)=\\zeta^2$, in view of \\eqref{eq1}, we may write the $p$-th Fourier coefficient of $f$ as follows\n\\begin{equation}\\label{eq7}\na(p)=2\\zeta \\cos\\theta_p,\n\\end{equation}\nfor a uniquely defined angle $\\theta_p\\in [0,\\pi]$. Notice that Theorem \\ref{thmCMST} and \\ref{thmST} are equivalent to say that the sequence $\\{\\theta_p\\}_p$  is equidistributed in $[0,\\pi]$, when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta^2$ with respect to the measure $\\mu$, where $\\mu=\\frac{1}{2\\pi}d\\theta+\\frac{1}{2}\\delta_{\\pi\/2}$ if $f$ has CM, and $\\mu=\\frac{2}{\\pi}\\sin^2\\theta d\\theta$ otherwise.  \n\nFor an integer $\\nu\\ge 1$, and a real $\\phi$ belonging to $[0,\\pi)$, we let\n$$P_{> 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})> 0\\},$$\n\n$$P_{< 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})< 0\\},$$\nand  \n$$P_{\\neq 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\neq 0\\}.$$ \n\nHere is our first main theorem.\n\\begin{thm}\\label{thm:1}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$, satisfying Hypothesis \\ref{hyp}. Let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe its Fourier expansion at $\\infty$. Let $\\nu$ be a positive odd integer. Then the sequence $\\{a(p^\\nu)\\}_{p\\in\\mathbb{P}}$ is oscillatory, and for each $\\phi\\in[0,\\pi)$ the sets $P_{>0}(\\phi,\\nu)$ and $P_{<0}(\\phi,\\nu)$ have equal positive natural density, that is, both are precisely half of the natural density of the set $P_{\\ne 0}(\\phi,\\nu)$.\n\\end{thm}\n\nBefore proving this theorem, we need the following preliminary lemmas.\n\\begin{lem}\\label{lem:1}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$, let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe its Fourier expansion at $\\infty$. Let $p$ be a prime number, and $\\zeta$ be a root of unity such that $\\zeta^2\\in \\mathrm{Im}(\\varepsilon)$. If $\\varepsilon(p)=\\zeta^2$ then for any positive integer $\\nu$, the $p^\\nu$--th Fourier coefficient of $f$ is expressible by the trigonometric identity\n$$\na(p^\\nu)=\\frac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}\\zeta^\\nu,\n$$\nfor some $\\theta_p\\in (0,\\pi)$ and in the limiting cases when $\\theta_p=0$ and $\\theta_p=\\pi$ respectively we have $a(p^{\\nu})=(\\nu+1)\\zeta^\\nu$ and $a(p^{\\nu})=(-1)^\\nu(\\nu+1)\\zeta^\\nu$.\n\\end{lem}\n\\begin{rem}\nIt is worth pointing out that if the weight $k$ is even, then the cases when $a(p^{\\nu})=(\\nu+1)\\zeta^\\nu$ and $a(p^{\\nu})=(-1)^\\nu(\\nu+1)\\zeta^\\nu$ can happen for at most finitely many primes $p$ only. In fact, if we denote by $K_f$ the field generated by all the Fourier coefficients of $f$, and pick a prime $p$ satisfying $\\varepsilon(p)=\\zeta^2$, so that $\\theta_p=0$, or $\\pi$, then we should have $\\sqrt{p}\\in K_f$, which can happen for only finitely many primes, because $K_f$ is a number field.   \n\\end{rem}\n\\begin{proof}\nSince $f$ is a normalized newform, we have the following power series expansion\n$$\n\\sum_{\\nu\\ge0}a(p^\\nu)X^\\nu=\\dfrac{1}{1-a(p)X+p^{k-1}\\varepsilon(p)X^2}.\n$$\nSetting $X=x\\zeta^{-1}$, and write \n$$\n1-a(p)\\zeta^{-1}x+p^{k-1}x^2=(1-\\alpha_px)(1-\\beta_px),\n$$ \none sees\n$$\n\\sum_{\\nu\\ge0}a(p^\\nu)\\zeta^{-\\nu} x^\\nu=\\dfrac{1}{(\\alpha_p-\\beta_p)x}\\left(\\dfrac{1}{1-\\alpha_px}-\\dfrac{1}{1-\\beta_px}\\right).\n$$\nNow, expanding both geometric series, we deduce that the $p^\\nu$-th Fourier coefficient of $f$ is\n\\begin{equation}\na(p^\\nu)=\\dfrac{\\alpha^{\\nu+1}_p-\\beta^{\\nu+1}_p}{\\alpha_p-\\beta_p}\\zeta^{\\nu}.\\label{eq:7}\n\\end{equation}\nOn the other hand, since $\\frac{a(p)}{\\zeta}\\in\\mathbb{R}$, then $\\beta_p=\\overline{\\alpha_p}$ and by Deligne's theorem \\cite[Theorem 8.2]{Deligne} we have $|\\alpha_p|=|\\beta_p|=1$. Thus, we may write $\\alpha_p=e^{i\\theta_p}$ and $\\beta_p=e^{-i\\theta_p}$ for some $\\theta_p\\in[0,\\pi]$. Inserting this in \\eqref{eq:7} we obtain the desired identities. \n\\end{proof}\n\\begin{lem}\\label{lem:2}\nWe make the same assumptions as in Theorem \\ref{thm:1}, and let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, then $\\frac{a(p^\\nu)}{\\zeta^\\nu}$ is real if $\\varepsilon(p)=\\zeta^2$. With the notation\n$$\\mathbb{P}_{\\gtrless0}(\\zeta,\\nu):=\\left\\{p\\in\\mathbb{P}\\;:\\; \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p^\\nu)}{\\zeta^\\nu}\\gtrless0\\right\\},$$ \nwe have\n$$\n\\delta(\\mathbb{P}_{>0}(\\zeta,\\nu))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type},\n    \\end{array}\n\\right.\n$$\nand \n$$\n\\delta(\\mathbb{P}_{<0}(\\zeta,\\nu))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type}.\n    \\end{array}\n\\right.\n$$\n\\end{lem}\n\\begin{proof}\nLet $p$ be a prime number such that $\\varepsilon(p)=\\zeta^2$. By the previous lemma, the $p^\\nu$--th Fourier coefficient $a(p^\\nu)$ of $f$ is expressible by the trigonometric identity\n$$\na(p^\\nu)=\\frac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}\\zeta^\\nu,\n$$\nwhere $\\theta_p\\in (0,\\pi)$. Since the set $\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta^2, \\theta_p=0\\;\\text{or}\\;\\pi\\}$ has density zero, we may assume that $\\theta_p$ is different from $0$ and $\\pi$. \n\nTherefore, the sign of $\\frac{a(p^\\nu)}{\\zeta^\\nu}$ is the same as the sign of $\\sin((\\nu+1)\\theta_p)$, it follows that\n$$\np\\in \\mathbb{P}_{>0}(\\zeta,\\nu)\\Longleftrightarrow\\varepsilon(p)=\\zeta^2,\\;\\theta_p\\in A_{>0}:=\\bigcup_{j=1}^{\\frac{\\nu+1}{2}}\\left(\\frac{(2j-2)\\pi}{\\nu+1},\\frac{(2j-1)\\pi}{\\nu+1}\\right),\n$$\nand\n$$\np\\in \\mathbb{P}_{<0}(\\zeta,\\nu)\\Longleftrightarrow \\varepsilon(p)=\\zeta^2,\\;\\theta_p\\in A_{<0}:=\\bigcup_{j=1}^{\\frac{\\nu+1}{2}}\\left(\\frac{(2j-1)\\pi}{\\nu+1},\\frac{2j\\pi}{\\nu+1}\\right).\n$$\n\nOn the other hand from \\cite[Proof of Theorem 1.1, odd case]{Meher2017}, we have\n$$\n\\mu(A_{>0})=\\mu(A_{<0})=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type}.\n    \\end{array}\n\\right.\n$$\nTaking into account that\n$$\n\\delta\\left(\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta^2\\}\\right)=\\dfrac{1}{r_{\\varepsilon}},\n$$\nthe desired conclusion can be derived easily from Theorem \\ref{thmCMST} and \\ref{thmST}.\n\\end{proof}\n\nNow we are in the position to prove Theorem \\ref{thm:1}.\n\\begin{proof}[Proof of Theorem \\ref{thm:1}]\n\\sloppy Fix $\\phi\\in[0,\\pi)$. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. For the notational convenience throughout the proof we let $\\pi_{>0}(x,\\zeta):=\\#\\{p\\le x : p\\in\\mathbb{P}_{>0}(\\zeta,\\nu)\\}$ and similarly $\\pi_{<0}(x,\\zeta)$, where $\\mathbb{P}_{>0}(\\zeta,\\nu)$ and $\\mathbb{P}_{<0}(\\zeta,\\nu)$ be as in Lemma \\ref{lem:2}. \n\nLet us first examine the oscillatory behavior of the sequence $\\{a(p^\\nu)\\}_{p\\in\\mathbb{P}}$. We need to consider the following two cases.\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\mathrm{arg}(\\zeta^\\nu)\\not\\equiv\\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.]\nThe sequence $\\{{\\Re e\\,}(a(p^\\nu) e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ is not trivial. Moreover, by Lemma \\ref{lem:1} we have the trigonometric identity\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})=\\dfrac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})\\quad\\text{with}\\quad\\varepsilon(p)=\\zeta^2,\n$$\nfor some $\\theta_p\\in(0,\\pi)$. There is no loss of generality in assuming ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0$. Then the sign of the sequence ${\\Re e\\,}(a(p^\\nu)e^{-i\\phi})$ is the same as the sign of $\\sin((\\nu+1)\\theta_p)$, when $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2$. Thus we have\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})>0\\quad \\text{if and only if}\\quad \\theta_p\\in A_{>0},\n$$\nand\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})<0 \\quad \\text{if and only if}\\quad \\theta_p\\in A_{<0},\n$$\nwhere $A_{\\lessgtr 0}$ be defined as in the proof of Lemma \\ref{lem:2}. From Theorem \\ref{thmCMST} and \\ref{thmST}, we know that the sequence $\\{\\theta_p\\}_{p,\\varepsilon(p)=\\zeta^2}$ is equidistributed in $[0,\\pi]$ with respect to the measure $\\mu$. Thereby, there are infinitely many primes $p$ satisfying $\\varepsilon(p)=\\zeta^2$ such that $\\theta_p\\in A_{>0}$, and infinitely many primes $p$ satisfying $\\varepsilon(p)=\\zeta^2$ such that $\\theta_p\\in A_{<0}$. Hence the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ changes sign infinitely often.\n\n\\item[\\textbf{Case 2}: $\\mathrm{arg}(\\zeta^\\nu)\\equiv \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(p^\\nu) e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ is trivial.\n\\end{description}\n\nSummarizing, we have thus proved that for any root of unity $\\zeta$ such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p}$ when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta^2$, either changes sign infinitely often or is trivial. Accordingly, for each $\\phi\\in [0,\\pi)$ either the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p\\in\\mathbb{P}}$ is trivial or changes sign infinitely often.\n\nIt remains to calculate the natural density of the sets $P_{>0}(\\phi,\\nu)$ and $P_{<0}(\\phi,\\nu)$. Here we restrict ourselves to the case when $f$ is not of CM-type, as the argument is entirely similar to the CM situation. The key point here is to see that \n$$P_{>0}(\\phi,\\nu)=\\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\mathbb{P}_{>0}(\\zeta,\\nu)\\bigsqcup \\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\mathbb{P}_{<0}(\\zeta,\\nu),$$\nand\n$$P_{<0}(\\phi,\\nu)=\\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\mathbb{P}_{>0}(\\zeta,\\nu)\\bigsqcup \\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\mathbb{P}_{<0}(\\zeta,\\nu),$$\nup to finitely many primes, where $\\zeta=e^{\\frac{\\pi i j}{r_{\\varepsilon}}}$, is so chosen that $1\\le j \\le r_{\\varepsilon}$, when $r_{\\varepsilon}$ is even, (note that there is a unique choice of $\\zeta$ when $r_{\\varepsilon}$ is odd). \n\nThe above displayed formula combined with Lemma \\ref{lem:2}, gives\n\\begin{eqnarray*}\n\\delta\\left(P_{>0}(\\phi,\\nu)\\right) &=& \\!\\!\\lim_{x\\to\\infty}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{ \\xi=\\zeta^2 ,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\!\\!\\!\\!\\!\\dfrac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}+\\lim_{x\\to\\infty}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2 ,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\!\\!\\!\\!\\!\\!\\!\\dfrac{\\pi_{<0}(x,\\zeta)}{\\pi(x)},\\\\\n  &=& \\!\\!\\frac{1}{2}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{ \\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})\\neq 0}}\\frac{1}{r_{\\varepsilon}},\\\\\n  &=&\\!\\!\\frac{\\delta(P_{\\neq 0}(\\phi,\\nu))}{2}.\n\\end{eqnarray*}\nIn the same manner we can see that $\\delta\\left(P_{<0}(\\phi,\\nu)\\right) =\\frac{\\delta(P_{\\neq 0}(\\phi,\\nu))}{2}$, which concludes the proof.\n\\end{proof}\n\n\\sloppy Our next concern will be the oscillatory behavior of the sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$, and the equidistribution of signs of $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$, for a fixed prime number $p$. \n\\begin{thm}\\label{thm:2}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform, and let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H}\n$$ \nbe its Fourier expansion at $\\infty$. Then there exists a set $S$ of primes of density zero, such that the following holds: For every prime $p\\notin S$, the sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory, and for any $\\phi\\in [0,\\pi)$ we have \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x\\; :\\; {\\Re e\\,}\\{a(p^\\nu)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{\\nu\\le x \\; :\\; {\\Re e\\,}\\{a(p^\\nu)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2}.\n$$\n\\end{thm}\n\\begin{rem}\n\\begin{enumerate}\n\\item The sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory means that the sequence escape infinitely often from any half-plane. Hence, it improves the result of the author in \\cite[Theorem 2.1]{Amri}.\n\\item It is worth pointing out that the theorem holds for prime in $S$, under some further restrictions, which can be easily deduced from the techniques of our proof. \n\\end{enumerate}\n\\end{rem}\n\\begin{proof}[Proof of Theorem \\ref{thm:2}] Set\n$$\nS :=\\coprod_{\\xi\\in\\mathrm{Im}(\\varepsilon)}\\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\xi, \\theta_p=0\\:\\text{or}\\:\\pi\\right\\},\n$$\nwhere $\\theta_p$ is defined as in \\eqref{eq7}. By Theorem \\ref{thmCMST} and \\ref{thmST} we see that the set $S$ has density zero. Let $p$ be a prime outside $S$, then there exists a root of unity $\\zeta$ satisfying $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, such that $\\varepsilon(p)=\\zeta^2$. From Lemma \\ref{lem:1}, one can write\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})=\\dfrac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi}),\n$$ \nfor some $\\theta_p\\in(0,\\pi)$. We derive the oscillatory behavior of the sequence $\\{a(p^\\nu))\\}_{\\nu\\in\\mathbb{N}}$, from the well-known behavior of the sequence $\\left\\{\\sin((\\nu+1)\\theta_p)\\right\\}_{\\nu\\in\\mathbb{N}}$.\n\nWe are left with the task of studying the equidistribution of signs of $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$, to this end, write ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})=\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)$, if $r_{\\varepsilon}$ is even, and ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})=\\cos\\left(\\frac{2\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)$, if  $r_{\\varepsilon}$ is odd, for some $1\\le j\\le r_{\\varepsilon}$.  We treat only the former case, the second one being completely similar. We need to distinguish the following two cases.\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\frac{\\theta_p}{2\\pi}$ is irrational.] Write\n$$\n{\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})=(\\sin\\theta_p)^{-1}\\sin\\left(2\\pi \\left<\\!\\frac{(\\nu+1)\\theta_p}{2\\pi}\\!\\right>\\right)\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right),\n$$ \nwhere $\\left<\\!\\frac{(\\nu+1)\\theta_p}{2\\pi}\\!\\right>$ denotes the fractional part of $\\frac{(\\nu+1)\\theta_p}{2\\pi}$. Note that the sequence $\\left\\{\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)\\right\\}_{\\nu\\in\\mathbb{N}}$ is $t_{\\varepsilon}$-periodic and takes only finitely many different values, with $t_{\\varepsilon}=2r'_{\\varepsilon}$ if $\\frac{j}{(r_{\\varepsilon},j)}$ is odd, and $t_{\\varepsilon}=r'_{\\varepsilon}$ if $\\frac{j}{(r_{\\varepsilon},j)}$ is even, where $r'_\\varepsilon=\\frac{r_{\\varepsilon}}{(r_{\\varepsilon},j)}$. So, one may split the total range for $\\nu$ into different arithmetic progressions $d\\pmod{ t_{\\varepsilon}}$ where $1\\le d \\le t_{\\varepsilon}$, so that when $\\nu$ runs through each of these arithmetic progressions the $\\cos$-factor becomes constant and takes always the same value, say $c_d$. Accordingly we may write\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})>0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1,\n$$\nand\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})<0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1.\n$$ \nOn the other hand we have\n\\begin{eqnarray*}\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1 &=& \\lim_{x\\to\\infty}\\frac{1}{x}\\displaystyle \\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)\\in [0,1]}} 1,\\\\\n &=&  \\lim_{x\\to\\infty}\\frac{1}{x}\\displaystyle \\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\in \\left[0,\\pi\\right]}} 1,\\\\\n  &=& \\frac{1}{2t_\\varepsilon},\n \\end{eqnarray*}\n\\sloppy where we used the fact that the sequence $\\left\\{\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right\\}_{\\nu\\in\\mathbb{N}}$ is uniformly distributed $\\pmod 1$ in $[0,1]$ in the last step, (Weyl's equidistribution theorem, see, e.g., \\cite{Kuipers}[Example 2.1, pp.8]). Similarly we have\n$$\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1=\\frac{1}{2t_{\\varepsilon}}.\n$$\nIt follows\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{x}=\\sum_{\\mycom{d=1}{c_d\\ne 0}}^{t_{\\varepsilon}}\\frac{1}{2 t_{\\varepsilon}},\n$$\nand therefore \n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\ne 0\\}}=\\frac{1}{2}.\n$$\n\\item[\\textbf{Case 2}: $\\frac{\\theta_p}{2\\pi}=\\frac{n}{m}\\in (0,\\frac{1}{2})$ is rational, where $m$ and $n$ are coprime.] Write\n$$\n{\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})=(\\sin\\theta_p)^{-1}\\sin\\left(\\frac{2\\pi(\\nu+1) n}{m}\\right)\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right).\n$$\nBy similar considerations as in the previous case, we have\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})>0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)>0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)<0}}1,\n$$\nand\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})<0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)<0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)>0}}1.\n$$ \nThus, the study of the distribution of signs of ${\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})$, turned out to study the distribution of signs of the sequence $\\left\\{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)\\right\\}_{\\nu}$ when $\\nu$ runs through an arithmetic progression $d\\pmod{t_\\varepsilon}$. Note that this sequence is $t_{\\varepsilon}m'$-periodic and takes only finitely many different values,  where $m'=\\frac{m}{(m,t_\\varepsilon)}$. Thus, we can split this arithmetic progression into $m'$ different sub-arithmetic progressions $d+t_{\\varepsilon}\\ell\\pmod{m't_\\varepsilon}$ where $1\\le\\ell\\le m'$, such that when $\\nu$ runs through each of these sub-arithmetic progressions the sequence becomes constant and has always the same value, say $s_{d,\\ell}$. Consequently\n\\begin{eqnarray*}\n\\nonumber \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi \\nu n}{m}\\right)>0}}1 &=& \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\frac{\\nu-d}{t_\\varepsilon}\\equiv \\ell \\!\\!\\!\\!\\!\\pmod{m'}}}1, \\\\  \n&=& \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\sum_{\\mycom{\\nu\\le x}{\\nu\\equiv d+t_\\varepsilon\\ell \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}m'}}}1,\\\\ \n&=& \\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\frac{1}{t_{\\varepsilon}m'}.\n\\end{eqnarray*}\nSimilarly, we obtain \n$$\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi \\nu n}{m}\\right)<0}}1=\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}<0}}^{m'}\\frac{1}{t_{\\varepsilon}m'}.\n$$\nTherefore\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{x}=\\sum_{\\mycom{d=1}{c_d\\ne 0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}\\ne 0}}^{m'}\\frac{1}{2t_{\\varepsilon}m'},\n$$\nand hence\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\ne 0\\}}=\\frac{1}{2},\n$$\nas desired.\n\\end{description}\nThis completes the proof of Theorem \\ref{thm:2}.\n\\end{proof}\n\n\\section{Equidistribution of sign results of half-integral weight cuspidals eigenforms}\\label{sec:3}\nIn this section, we shall present and prove our results concerning the oscillatory behavior and signs equidistribution of Fourier coefficients of cuspidal Hecke eigenforms following a similar philosophy to that in the previous section. In order to state our results, we need to develop some notations and make some assumptions for this section. \n\n\\begin{hyp}\\label{hyp:2}\nLet $N\\ge 4$ be divisible by $4$ and $k\\ge 1$ be a natural number. Fix any Dirichlet character $\\varepsilon\\pmod N$. Let $$f(z)=\\sum_{n=1}^{\\infty}a(n)q^n\\quad z\\in\\mathcal{H},$$ be a non-zero cuspidal Hecke eigenform of half-integral weight $k+1\/2$ and level $N$ with Dirichlet character $\\varepsilon\\pmod N$, and let  $t$ be a square-free integer such that $a(t)\\ne 0$.  The Shimura correspondence \\cite{Shi} lifts $f$ to a Hecke eigenform $\\mathrm{Sh}_t(f)$ of weight $2k$ for the group $\\Gamma_0(N\/2)$ with character $\\varepsilon^2$. Let us write\n$$\n\\mathrm{Sh}_t(f)=\\sum_{n\\ge 1} A_t(n)q^n,\n$$\nfor its expansion at $\\infty$. For simplicity we assume that $a(t)=1.$ According to \\cite{Shi}, the $n$-th Fourier coefficient of $\\mathrm{Sh}_t(f)$ is given by \n\\begin{equation}\nA_t(n)=\\sum_{d|n}\\varepsilon_{t,N}(d)d^{k-1}a\\left(\\frac{n^2}{d^2}t\\right),\\label{eq8}\n\\end{equation}\nwhere $\\varepsilon_{t,N}$ denotes the character $\\varepsilon_{t,N}(d):=\\varepsilon(d)\\left(\\frac{(-1)^{k}N^{2}t}{d}\\right)$, we let $\\chi_0(d):=\\left(\\frac{(-1)^{k}N^{2}t}{d}\\right)$.  If $\\mathrm{Sh}_t(f)$ has complex multiplication by an imaginary quadratic field $F$ denote by $d_F$ its fundamental discriminant. We suppose that $(r_{\\varepsilon},d_F)=1$ and the fields $F_\\varepsilon$ and $\\mathbb{Q}(\\sqrt{(-1)^k t})$ are linearly disjoint over $\\mathbb{Q}$, where $F_{\\varepsilon}$ is the field obtained by adjoining to $F$ the values of $\\varepsilon$. We let $\\chi_F$ be the quadratic character associated to $F$.\n\\end{hyp}\n\nLet $\\zeta$ be a root of unity such that $\\zeta\\in\\mathrm{Im}(\\varepsilon)$, if $p$ is a prime number satisfying $\\varepsilon(p)=\\zeta,$ then we have\n$$B_\\zeta(p):=\\frac{A_t(p)}{2p^{k-1\/2}\\zeta}\\in [-1,1].$$\nBy \\eqref{eq8} we have $a(tp^2)=A_t(p)-\\varepsilon_{t,N}(p)p^{k-1}$, and hence \n\\begin{equation}\\label{eq9}\n\\frac{a(tp^2)}{2p^{k-1\/2}\\zeta}=B_{\\zeta}(p)-\\frac{\\chi_0(p)}{2\\sqrt{p}}.\n\\end{equation}\n\nFor abbreviation  we let $A_\\zeta(p)$ stand for $\\frac{a(tp^2)}{2p^{k-1\/2}\\zeta}$.\nSet\n$$P_{> 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})> 0\\},$$\n\n$$P_{< 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})< 0\\},$$\nand \n$$P_{\\neq 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})\\neq 0\\},$$\nwhere $\\phi$ is a real number belonging to $[0,\\pi)$.\n\\begin{thm}\\label{thm:3}\nLet $f\\in S_{k+1\/2}(N,\\varepsilon)$ be a cuspidal Hecke eigenform  satisfying Hypothesis \\ref{hyp:2}. Let us write \n$$\nf(z)=\\sum_{n\\ge 1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Then the sequence $\\{a(tp^2)\\}_{p\\in\\mathbb{P}}$ is oscillatory, and if moreover $\\mathrm{Sh}_t(f)$ is not of CM-type, or of CM-type and $\\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F$, then for each $\\phi\\in[0,\\pi)$ the sets $P_{>0}(\\phi)$ and $P_{<0}(\\phi)$ have equal positive natural density, that is, both are precisely half of the natural density of the set $P_{\\ne 0}(\\phi)$. In the remaining cases we have\n$$\n\\delta(P_{>0}(\\phi))=\\left\\{\n    \\begin{array}{llll}\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+\\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})>0}} \\frac{1}{r_{\\varepsilon}} &\\mbox{if}\\;\\;\\chi_0 =\\chi_F, \\\\\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+ \\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})<0}} \\frac{1}{r_{\\varepsilon}} & \\mbox{if}\\;\\;  \\chi_0 =\\chi_{\\mathrm{triv}},\n    \\end{array}\n\\right.\n$$\nand\n$$\n\\delta(P_{<0}(\\phi))=\\left\\{\n    \\begin{array}{llll}\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+\\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})<0}} \\frac{1}{r_{\\varepsilon}} &\\mbox{if}\\;\\;\\chi_0 =\\chi_F, \\\\\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+ \\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})>0}} \\frac{1}{r_{\\varepsilon}} & \\mbox{if}\\;\\;  \\chi_0 =\\chi_{\\mathrm{triv}}.\n    \\end{array}\n\\right.\n$$\n\\end{thm}\n\nWe shall need the following lemma.\n\\begin{lem}\\label{lem:3}\nWe make the same assumptions as in Theorem \\ref{thm:3}, and let $\\zeta$ be a root of unity belonging to $\\mathrm{Im}(\\varepsilon)$. If we denote\n \n$$\\mathbb{P'}_{\\gtrless0}(\\zeta):=\\left\\{p\\in\\mathbb{P}\\;:\\; \\varepsilon(p)=\\zeta,\\;A_\\zeta(p)\\gtrless0\\right\\}.$$\nThen we have\n$$\n\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F,\\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_{\\mathrm{triv}},\\\\\n        \\frac{3}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_F,\n    \\end{array}\n\\right.\n$$\nand\n$$\n\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F,\\\\\n        \\frac{3}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_{\\mathrm{triv}},\\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_F.\n    \\end{array}\n\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nDenote by $\\pi_{>0}(x,\\zeta):=\\#\\{p\\le x\\; :\\;p\\in\\mathbb{P'}_{>0}(\\zeta)\\}$ and similarly $\\pi_{\\ge 0}(x,\\zeta)$, $\\pi_{<0}(x,\\zeta)$ and $\\pi_{\\le0}(x,\\zeta)$. \n\nFirst assume that $\\mathrm{Sh}_t(f)$ is not of CM-type, we follow closely the method of \\cite{IW}. Let $p$ be a prime satisfying $\\varepsilon(p)=\\zeta$, from \\eqref{eq9} we have \n$$\nA_\\zeta(p)>0 \\Longleftrightarrow B_\\zeta(p)>\\frac{\\chi_0(p)}{2\\sqrt{p}}.\n$$\nIt follows that for any fixed $\\epsilon >0$, we have the following inclusion of sets\n$$\\{p\\leq x : \\varepsilon(p)=\\zeta, B_\\zeta(p)>\\epsilon\\}\\!\\subset\\!\\{p\\in\\mathbb P : p\\leq\\frac{1}{4\\epsilon^2}, \\varepsilon(p)=\\zeta\\}\\cup\\{p\\leq x : p\\in\\mathbb{P'}_{>0}(\\zeta)\\}.$$\nTherefore \n\\begin{equation}\\label{eq10}\n\\pi_{>0}(x,\\zeta)+\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)\\geq \\#\\{p\\leq x :\\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\},\n\\end{equation}\nwhere $\\pi_{\\zeta}(x):=\\#\\{p\\in\\mathbb{P}: p\\le x,\\;\\; \\varepsilon(p)=\\zeta\\}$. Now dividing \\eqref{eq10} by $\\pi_{\\zeta}(x)$ we obtain \n\\begin{equation}\\label{eq:10}\n\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}+\\dfrac{\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)}{\\pi_{\\zeta}(x)}\\geq \\dfrac{\\#\\{p\\leq x :\\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\}}{\\pi_{\\zeta}(x)}.\n\\end{equation}\nSince $\\pi_{\\zeta}(x)\\underset{x\\to\\infty}{\\sim}\\frac{x}{r_{\\varepsilon}\\log x}$, and the term $\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)$ is finite, it follows\n\\begin{equation}\\label{eq11}\n\\lim_{x\\to\\infty}\\dfrac{\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)}{\\pi_{\\zeta}(x)}=0.\n\\end{equation}\nOn the other hand, by Theorem \\ref{thmST} we have \n\\begin{equation}\\label{eq12}\n \\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\leq x : \\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\}}{\\pi_{\\zeta}(x)}=\\mu_{\\mathrm{ST}}([\\epsilon,1]).\n \\end{equation}\n\nTaking into account \\eqref{eq11} and \\eqref{eq12} a passage to the limit in \\eqref{eq:10} implies that\n\\begin{equation}\\label{eq13}\n\\liminf_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\mu_{\\mathrm{ST}}([\\epsilon,1]).\n\\end{equation}\nAs the inequality \\eqref{eq13} holds for all $\\epsilon>0$, we have\n$$\\liminf_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\frac{1}{2}.$$\nSimilarly we get $\\liminf\\limits_{x\\to\\infty}\\frac{\\pi_{\\le 0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\frac{1}{2}$ and in view of $\\pi_{\\le 0}(x,\\zeta)=\\pi_\\zeta(x)-\\pi_{>0}(x,\\zeta)$, one sees $\\limsup\\limits_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\le\\frac{1}{2}$. Consequently\n$$\n\\lim\\limits_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}=\\frac{1}{2}.\n$$\nSince $\\delta(\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta\\})=\\frac{1}{r_{\\varepsilon}}$, it follows\n $$\\delta(\\mathbb{P}'_{>0}(\\zeta))=\\lim_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}=\\frac{1}{2r_{\\varepsilon}}.$$\nSimilarly we have $\\delta(\\mathbb{P}'_{<0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$.\n\nNow, let us examine the CM situation, assume that $\\mathrm{Sh}_t(f)$ has CM by an imaginary quadratic field $F$. Set $I=(0,1]$, $J=[-1,0)$,\n$$\nT_{I}(\\zeta):=\\{p\\in\\mathbb{P}: \\varepsilon(p)=\n\\zeta, A_{\\zeta}(p)\\in I, B_\\zeta(p)\\ne 0\\}, S_{I}(\\zeta):=\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, B_{\\zeta}(p)\\in I\\}\n$$\nand \n$$\n T_{J}(\\zeta):=\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta, A_{\\zeta}(p)\\in J, B_\\zeta(p)\\ne 0\\}, S_{J}(\\zeta):=\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, B_{\\zeta}(p)\\in J\\}\n$$\n\nFrom Theorem \\ref{thmCMST} we have $\\delta(S_{I}(\\zeta))=\\delta(S_{J}(\\zeta))=\\frac{1}{4 r_\\varepsilon}$. Thus in view of \\cite[Theorem 4.2.1]{Arias} and Theorem \\ref{thmCMST} it follows that $$\\delta(T_{I}(\\zeta))=\\delta(T_{J}(\\zeta))=\\frac{1}{4 r_\\varepsilon}.$$\nNow by \\cite[Remark 4.2.2]{Arias} we may write\n\n$$\n\\mathbb{P'}_{>0}(\\zeta)= T_{I}(\\zeta)\\sqcup \n$$\n\\begin{equation}\\label{eq15}\n\\left(\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta,B_\\zeta(p)=0 \\}\\cap \\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, \\frac{-\\chi_0(p)}{2\\sqrt{p}}\\in I\\right\\}\\right),\n\\end{equation}\nand \n$$\n\\mathbb{P'}_{<0}(\\zeta)=T_{J}(\\zeta)\\sqcup\n$$\n\\begin{equation}\\label{eq16}\n\\left(\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta,B_\\zeta(p)=0 \\}\\cap \\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, \\frac{-\\chi_0(p)}{2\\sqrt{p}}\\in J\\right\\}\\right).\n\\end{equation}\n\nIn order to calculate $\\delta(\\mathbb{P'}_{<0}(\\zeta))$ and $\\delta(\\mathbb{P'}_{>0}(\\zeta))$, we distinguish the following three cases\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\chi_0\\ne \\chi_{\\mathrm{triv}}$ and $\\chi_F$.] By our hypothesis we have $(r_{\\varepsilon},d_F)=1$, and the fields $\\mathbb{Q}(\\sqrt{(-1)^k t})$ and $F_\\varepsilon$ are linearly disjoint over $\\mathbb{Q}$. Hence by Chebotarev's theorem the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $\\frac{1}{4r_{\\varepsilon}}$ and $\\frac{1}{4r_{\\varepsilon}}$ respectively. Consequently $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$ and $\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$.\n\n\\item[\\textbf{Case 2}: $\\chi_0=\\chi_{\\mathrm{triv}}$.] Since $(r_{\\varepsilon},d_F)=1$ the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $0$ and $\\frac{1}{2r_{\\varepsilon}}$ respectively, it follows that $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{1}{4r_{\\varepsilon}}$ and \n$\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{3}{4r_{\\varepsilon}}$.\n\n\\item[\\textbf{Case 3}: $\\chi_0=\\chi_F$.] By Hypothesis \\ref{hyp:2} we have $(r_{\\varepsilon},d_F)=1$. Thus, the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $\\frac{1}{2r_{\\varepsilon}}$ and $0$ respectively. It follows that $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{3}{4r_{\\varepsilon}}$ and $\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{1}{4r_{\\varepsilon}}$.\n\\end{description}\nwhich finishes the proof of Lemma \\ref{lem:3}.\n\\end{proof}\n\nWe proceed now to prove Theorem \\ref{thm:3}.\n\\begin{proof}[Proof of Theorem \\ref{thm:3}]\nFix $\\phi\\in[0,\\pi)$, and pick $\\zeta$ a root of unity belonging to $\\mathrm{Im}(\\varepsilon)$. We first study the oscillatory behavior of the sequence $\\{a(tp^2)\\}_{p\\in\\mathbb{P}}$. To this end, two cases we shall need to consider.\n\\begin{description}\n\n\\item[\\textbf{Case 1}: $\\mathrm{arg}(\\zeta)\\not\\equiv  \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta}$ is not trivial, and we may write\n$$\n{\\Re e\\,}(a(tp^2)e^{-i\\phi})=\\frac{a(tp^2)}{\\zeta}{\\Re e\\,}(\\zeta e^{-i\\phi}).\n$$\nBy Theorem \\ref{thmCMST} and \\ref{thmST}, it follows that the sequence $\\left(\\frac{a(tp^2)}{\\zeta}\\right)_{p,\\varepsilon(p)=\\zeta}$\nchanges sign infinitely often. Hence the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p}$ changes sign infinitely often as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta$.\n\\item[\\textbf{Case 2}: $\\mathrm{arg}(\\zeta)\\equiv  \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_p$ is trivial when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta$.\n\\end{description}\n\nWhat we have just proved is that for any root of unity $\\zeta$ such that $\\zeta\\in\\mathrm{Im}(\\varepsilon)$, the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p}$ when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta$  either changes sign infinitely often or is trivial. Consequently, for each $\\phi\\in [0,\\pi)$ either the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p\\in\\mathbb{P}}$\nis trivial or changes sign infinitely often. \n\nFor the purpose to calculate the density of the sets $P_{>0}(\\phi)$ and $P_{<0}(\\phi)$, we shall restrict ourselves to the case when $\\mathrm{Sh}_t(f)$ is not of CM-type, as the argument in the CM situation is entirely analogous. First note that\n$$P_{> 0}(\\phi)=\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{ \\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\mathbb{P'}_{>0}(\\zeta)\\bigsqcup\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{ \\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\mathbb{P'}_{<0}(\\zeta),$$\nand\n$$P_{< 0}(\\phi)=\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\mathbb{P'}_{>0}(\\zeta)\\bigsqcup\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\mathbb{P'}_{<0}(\\zeta),$$\nup to finitely many primes. \n\nThese, together with Lemma \\ref{lem:3} yields\n\\begin{eqnarray*}\n\\delta\\left(P_{>0}(\\phi)\\right) &=& \\lim_{x\\to\\infty}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\dfrac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}+\\lim_{x\\to\\infty}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\dfrac{\\pi_{<0}(x,\\zeta)}{\\pi(x)},\\\\\n  &=& \\frac{1}{2}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\mathrm{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})\\neq 0}}\\frac{1}{r_{\\varepsilon}},\\\\\n  &=&\\frac{\\delta(P_{\\neq 0}(\\phi))}{2}.\n\\end{eqnarray*}\nIn a similar way, it can be shown that $\\delta\\left(P_{<0}(\\phi)\\right) =\\frac{\\delta(P_{\\neq 0}(\\phi))}{2}$, which concludes the proof.\n\\end{proof}\n\nOur next objective is to investigate the oscillatory behavior of the sequence $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$. We shall prove the following.\n\\begin{thm}\\label{thm:4}\nLet $f\\in S_{k+1\/2}(N,\\varepsilon)$ be a cuspidal Hecke eigenform of half integral weight, and \n$$\nf(z)=\\sum_{n\\ge 1}a(n)q^n\\quad z\\in \\mathcal{H}\n$$\nits expansion at $\\infty$. Let $t$ be a square-free integer such that $a(t)\\ne 0$. For all but finitely many  primes $p$ the sequence $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory.\n\\end{thm}\n\\begin{proof}\nApplying the M\\\"obius inversion formula to \\eqref{eq8}, we derive that\n$$\na(tn^2)=\\sum_{d |n} \\mu(d)\\varepsilon_{t,N}(d)d^{k-1} A_t\\left(\\frac{n}{d}\\right).\n$$\nFor $n=p^{\\nu}$, with $\\nu\\in\\mathbb{N}$ ($p\\nmid N$ a prime), it follows that\n\\begin{equation}\\label{eq18}\na(tp^{2\\nu})=A_t(p^{\\nu})-p^{k-1}\\varepsilon_{t,N}(p) A_t(p^{\\nu-1})\n\\end{equation}\nDividing \\eqref{eq18} by $\\varepsilon(p)^\\nu$, we obtain\n$$\n\\dfrac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}=\\dfrac{A_t(p^{\\nu})}{\\varepsilon(p)^\\nu}-\\chi_0(p)p^{k-1}\\dfrac{A_t(p^{\\nu-1})}{\\varepsilon(p)^{\\nu-1}},\n$$\nhence $\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\in\\mathbb{R}$. Thus, we may write\n$$\n{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi})=\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}{\\Re e\\,}(\\varepsilon(p)^{\\nu}e^{-i\\phi}),\n$$\nfor each $\\phi\\in[0,\\pi)$. We shall have established the theorem if we prove that  for all but finitely many primes $p$ the sequence $\\left(\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\right)_{\\nu\\in\\mathbb{N}}$, changes sign infinitely often. To this end, we shall follow \\cite[Proof of Theorem 2.2]{bruinier}.\n\n\nAssume, for the sake of contradiction that there exist infinitely many primes $p$ such that the sequence $\\left(\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\right)_{\\nu\\in\\mathbb{N}}$ does not changes sign infinitely often. Let $\\lambda_p$ denote the $p$-th Hecke eigenvalue of $f$. Since \n$$\nT(p)\\mathrm{Sh}_t(f)=\\mathrm{Sh}_t(T(p^2)f),\n$$\nit follows that the $p$-th Hecke eigenvalue of $\\mathrm{Sh}_t(f)$ is $\\lambda_p$, where $T(p^2)$ is the Hecke operator on $S_{k+1\/2}(N,\\varepsilon)$ and $T(p)$ is the Hecke operator\non $S_{2k}(N\/2,\\varepsilon^2)$. By \\cite[Corolary 1.8]{Shi} we have \n\\begin{equation}\\label{eq19}\n\\sum_{\\nu\\ge 0}a(tp^{2\\nu})X^\\nu=a(t)\\dfrac{1-\\varepsilon_{N,t}(p)p^{k-1}X}{1-\\lambda_pX+\\varepsilon(p)^2p^{2k-1}X^2},\n\\end{equation}\nwrite \n$$1-\\lambda_pX+\\varepsilon(p)^2p^{2k-1}X^2=(1-\\alpha_p X)(1-\\beta_p X).$$ \nReplacing $X=\\varepsilon(p)^{-1}p^{-s}$ ($s\\in\\mathbb{C}$) in \\eqref{eq19} we get\n\\begin{equation}\\label{eq20}\n\\sum_{\\nu\\ge0} a(tp^{2\\nu})\\varepsilon(p)^{-\\nu}p^{-s\\nu}=a(t)\\dfrac{1-\\chi_{0}(p)p^{k-1-s}}{(1-\\alpha_{p}' p^{-s})(1-\\beta_{p}'p^{-s})},\n\\end{equation}\nwhere  $\\alpha_{p}'=\\alpha_p \\varepsilon(p)^{-1}$ and $\\beta_{p}'=\\beta_p \\varepsilon(p)^{-1}$. \n\n Let $p$ be a prime such that $\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\ge 0$ for all but finitely many $\\nu \\ge 0$. Thus, by Landau's theorem \\cite[ pp. 697--699]{Landau}, the series in the left-hand side of \\eqref{eq20} either (a) converges for all $s\\in\\mathbb{C}$ or (b) has a singularity at the real point of its line of convergence. It is clear that the alternative (a) cannot occur, since the right-hand side of \\eqref{eq20} has a pole for $p^{s}=\\alpha'_p$ or $p^{s}=\\beta'_p.$\nThus the alternative (b) must hold, therefore $\\alpha'_p$ or $\\beta'_p$ must be real. On the other hand since\n $\\frac{\\lambda_p}{\\varepsilon(p)}\\in\\mathbb{R}$, it follows $\\alpha_p'=\\overline{\\beta_p'}$. Moreover, by Deligne's theorem \\cite[Theorem 8.2]{Deligne}\nwe have $|\\alpha'_p|=|\\beta'_p|=p^{k-1\/2}$. Consequently\n$$\\lambda_p=\\pm 2 p^{k-1\/2}\\varepsilon(p),$$\nhence $\\mathbb{Q}(\\sqrt{p})\\subset K_f$, where $K_f$ denotes the field generated   over $\\mathbb{Q}$ by $\\lambda_p$ ($p$ runs over primes numbers) and all the values of $\\varepsilon$. Therefore, by our hypothesis $K_f$ has infinitely many quadratic subfields, this is in contradiction with the fact that $K_f$ is a number field. Consequently, for all but finitely many primes $p$ and each $\\phi\\in [0,\\pi)$ the sequence $\\{{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi}\\}_{\\nu\\in\\mathbb{N}}$ , changes sign infinitely often. \n\\end{proof}\n\\begin{rem}\nIt seems likely that we can prove similar results to \\cite[Theorem 3]{Kohnen} for the sequence $\\{{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$. However, we have not checked this as yet.\n\\end{rem}\n\\section{Concluding Remarks}\nLet $k,N$ be natural numbers and $\\varepsilon$ be a Dirichlet character modulo $N$ assume that $N$ be an odd and square-free integer. We write $S^{+}_{k+1\/2}(4N,\\varepsilon)$ for the Kohnen's plus space (cf. \\cite{Kohnen1982}). \n\nLet $f$ be a cusp form of half integral weight belonging to $S^{+}_{k+1\/2}(4N,\\varepsilon)$, we let $a(n)$ to denote its $n$-th Fourier coefficient. Motivated by Theorem \\ref{thm:3} and numerical calculations, it seems reasonable to conjecture that, for each $\\phi\\in [0,\\pi)$ \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; {\\Im m\\,}\\{a(n)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; {\\Im m\\,}\\{a(n)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2},\n$$\n\n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; {\\Re e\\,}\\{a(n)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; {\\Re e\\,}\\{a(n)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2}.\n$$\n\nIt may be noted that one can get similar statements on the imaginary part in Theorem \\ref{thm:1}, \\ref{thm:2} and \\ref{thm:3} by a rotation around $\\pi\/2$.\n\nWe believe that these results (Theorem \\ref{thm:1}, \\ref{thm:2}, \\ref{thm:3} and the above conjecture) should extend to any totally real number field by the approach taken in the present paper.\n\n\n\\section*{Acknowledgments}\nThe author is greatly grateful to Francesc Fit\\'e for a helpful conversation. He also wishes to thank Gabor Wiese for his valuable comments on the first draft of this work as well as Ilker Inam for providing him with some data for numerical experiments. Thanks are also due to the referee for his careful reading and their helpful comments which improve the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}