%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % shacomp_v12.tex % % by William Stein and Amod Agashe % % was@math.harvard.edu % % February 3, 2004 % % Revised in response to referee's and editors remarks. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{mcom-l} \bibliographystyle{amsalpha} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \newcommand{\nf}[1]{\mbox{\bf #1}} \newcommand{\sM}{\mathfrak{M}} \newcommand{\sS}{\mathfrak{S}} \newcommand{\sB}{\mathfrak{B}} \newcommand{\Adual}{A^{\vee}} \newcommand{\Bdual}{B^{\vee}} \newcommand{\defn}[1]{{\em #1}} \DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts \newcommand{\textcyr}[1]{% {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}% \selectfont #1}} \newcommand{\Sha}{{\mbox{\textcyr{Sh}}}} \newcommand{\Shaan}{\#\Sha_?} \newcommand{\Gam}{\Gamma} \newcommand{\cA}{\mathcal{A}} \newcommand{\cAdual}{\mathcal{A}^{\vee}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\U}{\mathcal{U}} \newcommand{\Kbar}{\overline{K}} \newcommand{\q}{\mathbf{q}} \newcommand{\union}{\cup} \newcommand{\intersect}{\cap} \newcommand{\cross}{\times} \newcommand{\hra}{\hookrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\lra}{\longrightarrow} \newcommand{\la}{\leftarrow} \newcommand{\alp}{\alpha} \newcommand{\tensor}{\otimes} \newcommand{\vphi}{\varphi} \newcommand{\comment}[1]{} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Qp}{\Q_p} \newcommand{\R}{\mathbf{R}} \newcommand{\D}{{\mathbb D}} \newcommand{\cC}{\mathcal{C}} \newcommand{\K}{{\mathbb K}} \newcommand{\C}{\mathbf{C}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\T}{\mathbf{T}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\F}{\mathbf{F}} \newcommand{\Fbar}{\overline{\F}} \newcommand{\Fp}{\F_p} \newcommand{\Fpbar}{\Fbar_p} \newcommand{\Ad}{Ad} \newcommand{\A}{\mathcal{A}} \newcommand{\isom}{\cong} \newcommand{\ncisom}{\approx} \newcommand{\cB}{\mathcal{B}} \newcommand{\h}{\mathfrak{h}} \newcommand{\p}{\mathfrak{p}} \newcommand{\m}{\mathfrak{m}} \newcommand{\magma}{{\sc Magma}} \newcommand{\marginalfootnote}[1]{% \footnote{#1} \marginpar{\hfill {\sf\thefootnote}}} \renewcommand{\Re}{\mbox{\rm Re}} \renewcommand{\L}{\mathcal{L}} \renewcommand{\l}{\ell} \renewcommand{\t}{\tau} \renewcommand{\P}{\mathbf{P}} \renewcommand{\O}{\mathcal{O}} \renewcommand{\a}{\mathfrak{a}} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\ch}{char} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\moddeg}{moddeg} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\numer}{numer} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Ver}{Ver} \DeclareMathOperator{\Vis}{Vis} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\new}{new} \DeclareMathOperator{\op}{odd} \DeclareMathOperator{\tor}{tor} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Spf}{Spf} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\ur}{ur} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Hom}{Hom} %%%% Theoremstyles \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}[theorem]{Claim} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{sublemma}[theorem]{Sublemma} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{goal}[theorem]{Goal} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \usepackage[all]{xy} \DeclareMathOperator{\Princ}{Princ} \font\cyrbig=wncyr10 scaled \magstep 1 \newcommand{\BigSha}{\mbox{\cyrbig X}} \title[Modular Abelian Varieties]{Visible Evidence for the Birch and Swinnerton-Dyer Conjecture for Modular Abelian Varieties of Analytic Rank Zero\\ (With an Appendix by J.~Cremona and B.~Mazur)} \author{Amod Agashe} \address{Department of Mathematics, U niversity of Texas, Austin, Texas 78712} \email{agashe@math.utexas.edu} %\thanks{The first author was supported in part by ...} \author{William Stein} \address{Department of Mathematics, Harvard University, Cambridge, Massachussetts 02138} \email{was@math.harvard.edu} \subjclass{Primary 11G40; Secondary 11F11, 11G10, 14K15, 14H25, 14H40.} \date{\today} \keywords{Birch and Swinnerton-Dyer conjecture, modular abelian variety, visibility, Shafarevich-Tate groups} \begin{document} \UseTips \begin{abstract} This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank~$0$ abelian varieties~$A_f$ that are optimal quotients of $J_0(N)$ attached to newforms. We prove theorems about the ratio $L(A_f,1)/\Omega_{A_f}$, develop tools for computing with~$A_f$, and gather data about certain arithmetic invariants of the nearly $20000$ %all $19608$ abelian varieties $A_f$ of level $\leq 2333$. Over half %$10360$ of these $A_f$ have analytic rank~$0$, and for these we compute upper and lower bounds on the conjectural order of~$\Sha(A_f)$. We find that there are at least $168$ such that the Birch and Swinnerton-Dyer Conjecture implies that $\Sha(A_f)$ is divisible by an odd prime, and we prove for $39$ of these that the odd part of the conjectural order of $\Sha(A_f)$ really divides $\#\Sha(A_f)$ by constructing nontrivial elements of $\Sha(A_f)$ using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves. \end{abstract} \maketitle \tableofcontents \section{Introduction}\label{sec:intro} Let~$N$ be a positive integer, and~$f$ be a newform of weight~$2$ on~$\Gamma_0(N)$. A construction due to Shimura associates to~$f$ an abelian variety quotient~$A_f$ of~$J_0(N)$. We say that $A_f$ has {\it analytic rank zero} if its $L$-function~$L(A_f,s)$ is nonzero at $s=1$. In this paper we give evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank~$0$ abelian varieties~$A_f$ of arbitrary dimension. %that are optimal %quotients of $J_0(N)$ attached to newforms. For such abelian varieties, the conjecture asserts that $A_f(\Q)$ is finite, and gives a formula for the order of the Shafarevich-Tate group~$\Sha(A_f)$. Kolyvagin and Logachev proved in \cite{kolyvagin-logachev:finiteness, kolyvagin-logachev:totallyreal} that if $L(A_f,1)\neq 0$, then $A_f(\Q)$ and $\Sha(A_f)$ are both finite. To the best of our knowledge, Birch and Swinnerton-Dyer's formula for $\#\Sha(A_f)$ has not been completely verified for a single abelian variety $A_f$ of dimension greater than one. In \cite[\S1.6]{kolyvagin-logachev:totallyreal} Kolyvagin and Logachev remark that if one were able to compute the height of a certain Heegner point, their methods could be used to find an upper bound on $\#\Sha(A_f)$, but we have not done this. Instead, in this paper we focus on computing {\em nonzero subgroups} of $\Sha(A_f)$ when the conjecture predicts that $\Sha(A_f)$ is nonzero. Inspired by work of Cremona and Mazur (see \cite{cremona-mazur}), we had the idea to reverse their methods and prove, in some cases, that $\#\Sha(A_f)$ is at least as big as predicted by the Birch and Swinnerton-Dyer conjecture. Instead of assuming that $\Sha(A_f)$ is as predicted by the conjecture and trying to understand whether or not it is visible in $J_0(N)$, we prove a theorem (see~\cite{agashe-stein:visibility}) that allows us to sometimes construct the odd part of $\Sha(A_f)$ without assuming any conjectures. After developing algorithms that allow us to compute the conjectural order of $\Sha(A_f)$ in most cases, we analyzed the $19608$ abelian varieties $A_f$ of level $\leq 2333$, and constructed the tables of Section~\ref{sec:tables}. This resulted in the first systematic experimental evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of dimension greater than~$2$ (see \cite{empirical} for dimension~$2$). This paper is organized as follows. In Section~\ref{sec:background} we review background about modular abelian varieties and state the Birch and Swinnerton-Dyer conjecture. Section~\ref{sec:computing} explains the basic facts about quotients $A_f$ of $J_0(N)$ that one needs to know in order to compute with them. In Section~\ref{sec:quotient} we discuss a generalization of the Manin constant, derive a formula for the ratio $L(A_f,1)/\Omega_{A_f}$, and bound the denominator of this ratio, thus giving some theoretical evidence towards the Birch and Swinnerton-Dyer conjecture. Section~\ref{sec:tables} reports on our construction of a table of $168$ rank~$0$ abelian varieties $A_f$ of level $\leq 2333$ such that the Birch and Swinnerton-Dyer conjecture predicts that $\#\Sha(A_f)$ is divisible by an odd prime, and discusses what we computed to show that for $39$ of the $A_f$ there are {\em at least} as many elements of the odd part of $\#\Sha(A_f)$ as predicted. The part of~$\# \Sha(A_f)$ that is coprime to the modular degree of~$A_f$ (which we define below) is a perfect square, and in the several cases where we could compute the odd part of the conjectured value of~$\# \Sha(A_f)$, we found the odd part to be a perfect square, which gives computational evidence for the conjecture. The appendix, written by Cremona and Mazur, fills in some gaps in the theoretical discussion in \cite{cremona-mazur}. {\bf Acknowledgment.} It is a pleasure to thank Bryan Birch, Robert Coleman, Benedict Gross, Hednrik Lenstra, Dino Lorenzini, Lo\"\i{}c Merel, Bjorn Poonen, Ken Ribet, and John Tate for many helpful comments and discussions. Special thanks go to Barry Mazur for guiding our ideas on visibility and purchasing the second author a powerful computer, and to Allan Steel and David Kohel at {\sc Magma} for their crucial computational support. \section{Background and Definitions} \label{sec:background} \subsection{Modular Forms} Fix a positive integer~$N$. The group $$\Gamma_0(N) = \left\{ \left(\begin{matrix}a&b\\c&d\end{matrix}\right) \in \SL_2(\Z) : N \mid c \right\}$$ acts by linear fractional transformations on the extended complex upper halfplane $\h^*$. As a Riemann surface, $X_0(N)(\C)$ is the quotient $\Gamma_0(N)\backslash{}\h^*$. There is a standard model for $X_0(N)$ over~$\Q$ (see \cite[Ch.~6]{shimura:intro}), and the Jacobian $J_0(N)$ of $X_0(N)$ is an abelian variety over~$\Q$ of dimension equal to the genus~$g$ of $X_0(N)$, which is equipped with an action of the Hecke algebra $\T = \Z[\ldots T_n \ldots]$. The space $S_2(\Gamma_0(N))$ of cuspforms of weight~$2$ on $\Gamma_0(N)$ is a module over~$\T$ and $S_2(\Gamma_0(N))\isom H^0(X_0(N),\Omega_{X_0(N)})$ as $\T$-modules. \subsection{Abelian Varieties Attached to Newforms} \label{sec:abvars} A {\em newform} $$ f= \sum_{n\geq 1} a_n q^n \in S_2(\Gamma_0(N)) $$ is an eigenvector for $\T$ that is normalized so that $a_1 = 1$ and which lies in the orthogonal complement of the old subspace of $S_2(\Gamma_0(N))$. Let $I_f$ denote the annihilator $\Ann_\T(f)$ of~$f$ in~$\T$. Following Shimura \cite{shimura:factors}, attach to $I_f$ the quotient $$ A_f = J_0(N)/I_f J_0(N), $$ which is an abelian variety over~$\Q$ of dimension $[\Q(\ldots, a_n, \ldots):\Q]$, which is equipped with a faithful action of $\T/I_f$. Moreover, $A_f$ is an {\em optimal quotient} of $J_0(N)$, in the sense that $A_f^{\vee}\ra J_0(N)$ is a closed immersion, or equivalently that the kernel of $J_0(N)\to A_f$ is connected (see \cite[Prop.~3.3]{conrad-stein:compgrp}). Also, the complex torus $A_f(\C)$ fits into the exact sequence $$ H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N))[I_f],\C) \ra A_f(\C) \ra 0. $$ \subsection{The Birch and Swinnerton-Dyer Conjecture} \label{sec:bsd} The conjecture of Birch and Swinnerton-Dyer makes sense for abelian varieties over fairly general global fields, but we only state a special case. This conjecture involves the $L$-function attached to $A=A_f$: $$L(A,s) = \prod_{i=1}^d L(f^{(i)},s) = \prod_{i=1}^d \left(\sum_{n\geq 1} \frac{a_n^{(i)}}{n^{s}}\right), $$ where $f^{(i)}$ is the $i$th Galois conjugate of~$f$ and $a_n^{(i)}$ is the $i$th Galois conjugate of~$a_n$. It follows from work of Hecke that $L(A,s)$ has an analytic continuation to the whole complex plane and satisfies a functional equation. Birch and Swinnerton-Dyer made the following conjecture, which relates the rank of~$A$ to the order of vanishing of $L(A,s)$ at $s=1$. \begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsdrank} The Mordell-Weil rank of~$A$ is equal to the order of vanishing of $L(A,s)$ at $s=1$, i.e., $$\dim (A(\Q)\tensor \Q) = \ord_{s=1} L(A,s).$$ \end{conjecture} Birch and Swinnerton-Dyer also furnished a conjectural formula for the order of the Shafarevich-Tate group $$ \Sha(A) := \ker\left(H^1(\Q,A) \lra \prod_{\text{all places $v$}} H^1(\Q_v,A)\right). $$ (They only made their conjecture for elliptic curves, but Tate~\cite{tate:bsd} reformulated it a functorial way which makes sense for abelian varieties. See also \cite[\S{}III.5]{lang:nt3} for another formulation.) We now state their conjecture in the special case when $L(A,1)\neq 0$, where \cite{kolyvagin-logachev:finiteness,kolyvagin-logachev:totallyreal} implies that $\Sha(A)$ is finite. The conjecture involves the Tamagawa numbers $c_p$ of~$A$ (see Section~\ref{sec:tamagawa}), and the canonical volume $\Omega_A$ of $A(\R)$ (see Section~\ref{sec:lratio}). \begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsd} Suppose $L(A,1)\neq 0$. Then $$\frac{L(A,1)}{\Omega_A} = \frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p} {\# A(\Q)_{\tor}\cdot\#\Adual(\Q)_{\tor}},$$ where $\Adual$ is the abelian variety dual of~$A$. \end{conjecture} \begin{remark} Since $L(A,1)\neq 0$, finiteness of $\Sha(A)$ and the existence of the Cassels-Tate pairing implies that $\#\Sha(A) = \#\Sha(\Adual)$, so Conjecture~\ref{conj:bsd} can also be viewed as a formula for $\#\Sha(\Adual)$. \end{remark} The algorithms outlined in this paper take advantage of the fact that~$A$ is attached to a newform in order to compute the conjectural order of $\Sha(A)$ away from certain bad primes. \section{Explicit Approaches to Modular Abelian Varieties} \label{sec:computing} We use the algorithms of this section to enumerate the~$A_f$, compute information about the invariants of~$A_f$ that appear in Conjecture~\ref{conj:bsd}, and to verify the hypothesis of Theorem~\ref{thm:shaexists} in order to construct nontrivial subgroups of $\Sha(A_f)$. The second author has implemented the algorithms discussed in this paper, and made many of them part of the {\sc Magma} computer algebra system \cite{magma}. In Section~\ref{modsym}, we discuss modular symbols, which are the basic tool we use in many of the computations, and in Section~\ref{sec:enumerate} we discuss how we systematically enumerate modular abelian varieties. There is an analogue for $A_f$ of the usual elliptic-curve modular degree, which we discuss in Section~\ref{sec:moddegree}, and which we use to rule out the existence of visible elements of $\Sha(A_f)$ of a certain order. In Section~\ref{sec:intersect} we describe how to compute the intersection of two abelian varieties, which will be needed to verify the hypothesis of Theorem~\ref{thm:shaexists}. In Sections~\ref{sec:torsion_above} and \ref{sec:torsion_below}, we describe standard methods for bounding the torsion subgroup of an abelian variety above and below. Section~\ref{sec:tamagawa} reviews an algorithm for computing the odd part of the Tamagawa number $c_p$ when $p\mid\mid N$, and discusses the Lenstra-Oort bound in the case when $p^2\mid N$. Unless otherwise stated,~$f$ is a newform, $I_f$ its annihilator, and $A=A_f$ is the corresponding optimal quotient of $J_0(N)$. \subsection{Modular Symbols}\label{modsym} Modular symbols are crucial to many algorithms for computing with modular abelian varieties, because they can be used to construct a {\em finite} presentation for $H_1(X_0(N),\Z)$ in terms of paths between elements of $\P^1(\Q) = \Q\union \{\infty\}$. They were introduced by Birch \cite{birch:bsd} and studied by Manin, Mazur, Merel, Cremona, and others. Let~$\sM_2$ be the free abelian group with basis the set of all symbols $\{\alpha,\beta\}$, with $\alpha, \beta\in\P^1(\Q)$, modulo the three-term relations $$ \{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0, $$ and modulo any torsion. The group $\GL_2(\Q)$ acts on the left on $\sM_2$ by $$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$ where~$g$ acts on~$\alpha$ and~$\beta$ by a linear fractional transformation. The space $\sM_2(\Gamma_0(N))$ of \defn{modular symbols} for $\Gamma_0(N)$ is the quotient of $\sM_2$ by the subgroup generated by all elements of the form $x - g(x)$, for~$x \in \sM_2$ and~$g$ in $\Gamma_0(N)$, modulo any torsion. A \defn{modular symbol} for $\Gamma_0(N)$ is an element of this space, and we frequently denote the equivalence class that defines a modular symbol by giving a representative element. Let $\sB_2(\Gamma_0(N))$ be the free abelian group with basis the finite set $\Gamma_0(N)\backslash \P^1(\Q)$. The boundary map $\delta: \sM_2(\Gamma_0(N))\ra \sB_2(\Gamma_0(N))$ sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$ denotes the basis element of $\sB_2(\Gamma_0(N))$ corresponding to $\beta\in\P^1(\Q)$. The \defn{cuspidal modular symbols} are the kernel $\sS_2(\Gamma_0(N))$ of~$\delta$, and the integral homology $H_1(X_0(N),\Z)$ is canonically isomorphic to $\sS_2(\Gamma_0(N))$. Cremona's book \cite[\S2.2]{cremona:algs} contains a concrete description of how to compute $\sM_2(\Gamma_0(N))\tensor\Q$ using Manin symbols, which are a finite set of generators for $\sM_2(\Gamma_0(N))$. In general, the easiest way we have found to compute $\sM_2(\Gamma_0(N))$ is to compute $\sM_2(\Gamma_0(N))\tensor\Q$, then compute the $\Z$-submodule of $\sM_2(\Gamma_0(N))\tensor\Q$ generated by the Manin symbols. \subsection{Enumerating Newforms} \label{sec:enumerate} Since $X_0(N)$ is defined over $\Q$ it is defined over~$\R$, so complex conjugation acts on $X_0(N)(\C)$ hence on the homology $H_1(X_0(N),\Z)$. In terms of modular symbols, complex conjugation acts by sending $\{\alpha,\beta\}$ to $\{-\alpha,-\beta\}$. Let $H_1(X_0(N),\Z)^+$ denote the $+1$-eigenspace for the action of the involution induced by complex conjugation, which we can compute using modular symbols. We list all newforms of a given level~$N$ by decomposing the new subspace of $H_1(X_0(N),\Q)^+$ under the action of the the Hecke operators and listing the corresponding systems of Hecke eigenvalues (see~\cite{stein:modsyms}). First we compute the characteristic polynomial of~$T_2$, and use it to break up the new space. We apply this process recursively with $T_3, T_5, \ldots$ until either we have exceeded the bound coming from~\cite{sturm:cong} (see \cite{agashe-stein:schoof-appendix}), or we have found a Hecke operator~$T_n$ whose characteristic polynomial is irreducible. We {\em order the newforms} in a way that extends the ordering in~\cite{cremona:algs}: First sort by dimension, with smallest dimension first; within each dimension, sort in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$, $++-$, $+-+$, $+--$, $-++$, etc. When two forms have the same Atkin-Lehner sign sequence, order by $|\Tr(a_p)|$ with ties broken by taking the positive trace first. We denote a Galois-conjugacy class of newforms by a bold symbol such as $\mathbf{389E}$, which consists of a level and isogeny class, where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second, $\mathbf{E}$ the fifth, $\mathbf{BB}$ the $28$th, etc. As discussed in \cite[pg.~5]{cremona:algs}, for certain small levels the above ordering, when restricted to elliptic curves, does not agree with the ordering used in the tables of \cite{cremona:algs}. For example, our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$. \subsection{The Modular Degree}\label{sec:moddegree} Since $A_f$ is an optimal quotient, the dual map $A_f^{\vee}\ra J_0(N)$ is injective and the composite $\theta_f:A_f^{\vee} \ra A_f$ has finite degree. The map $\theta_f$ is a polarization, so $\deg(\theta_f)$ is a perfect square (see Lemma~\ref{lem:kerpoll}). The {\em modular degree} of $A_f$ is the square root of the degree of $\theta_f$: $$ \moddeg(A_f) = \sqrt{\deg(\theta_f)}. $$ When $\dim A_f=1$, $\moddeg(A_f)$ is the usual modular degree, i.e., the degree of $X_0(N)\ra A_f$. If~$M$ is an abelian group, let $M^* = \Hom_\Z(M,\Z)$. The Hecke algebra acts in a natural way on $H_1(X_0(N),\Z)$ and $H_1(X_0(N),\Z)^*$, and we have a natural restriction map $$ r_f: H_1(X_0(N),\Z)^*[I_f] \rightarrow (H_1(X_0(N),\Z)[I_f])^*. $$ The following proposition leads to an algorithm for computing the modular degree. \begin{proposition} $\coker(r_f)\isom \ker(\theta_f)$, so $\moddeg(A_f) = \sqrt{\#\coker(r_f)}$. \end{proposition} The proposition is proved in~\cite{kohel-stein:ants4}. The proof makes use of the {\em Abel-Jacobi theorem}, which realizes the Jacobian $J_0(N)(\C)$ as a complex torus: $$ 0\ra H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N)),\C) \ra J_0(N)(\C) \ra 0, $$ where $H_1(X_0(N),\Z)$ is embedded as a lattice of full rank in the complex vector space $\Hom(S_2(\Gamma_0(N)),\C)$ using the integration pairing, and this description of $J_0(N)(\C)$ is compatible with the action of Hecke operators. \subsection{Intersecting Complex Tori} \label{sec:intersect} Let $V$ be a finite dimensional complex vector space and let~$\Lambda$ be a lattice in~$V$, so that $T=V/\Lambda$ is a complex torus. Suppose that $V_A$ and $V_B$ are subspaces of~$V$ such that $\Lambda_A = V_A\intersect \Lambda$ and $\Lambda_B = V_B\intersect \Lambda$ are lattices in $V_A$ and $V_B$, respectively. \begin{proposition}\label{prop:intersection} If $A\intersect B$ is finite, then there is an isomorphism $$A\intersect B \isom \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$ \end{proposition} \begin{proof} Extend the exact sequence $$0\ra A\intersect B \ra A \oplus B \xrightarrow{(x,y)\mapsto x-y} T$$ to the following diagram: $$\xymatrix{ & {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]& {\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\ 0\ar[r]\ar[d] & {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\ {A\intersect B}\ar[r] & A\oplus B\ar[r] & T \ar[r] & T/(A+ B).}$$ The middle row is exact because $A\intersect B$ is finite so $V_A\intersect V_B = 0$. Using the snake lemma, which connects the kernel $A\intersect B$ of $A\oplus B \ra T$ to the cokernel of $\Lambda_A \oplus \Lambda_B \ra \Lambda$, we obtain an exact sequence $$0 \ra A\intersect B \ra \Lambda/(\Lambda_A + \Lambda_B) \ra V/(V_A+V_B).$$ Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$. No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could map to~$0$, because if it did then $A\intersect B$ would not be finite. The proposition follows. \end{proof} For abelian subvarieties of $J_0(N)$ attached to newforms, we use the proposition above as follows. The complex vector space $V=\Hom(S_2(\Gamma_0(N)),\C)$ is the tangent space of~$J_0(N)(\C)$ at the identity. Setting $\Lambda = H_1(X_0(N),\Z)$ and considering $\Lambda$ as a lattice in~$V$ via the integration pairing, we have $J_0(N)(\C)\isom V/\Lambda$. Suppose~$f$ and~$g$ are non-conjugate newforms, and let $I_f$ and $I_g$ be their annihilators in the Hecke algebra $\T$, and let $A=A_f^{\vee}$ and $B=A_g^{\vee}$. Then $V_A=V[I_f]$ and $V_B=V[I_g]$ are the tangent spaces to~$A$ and~$B$ at the identity, respectively. The above proposition shows that the group $A\intersect B$ is canonically isomorphic to $({\Lambda}/(\Lambda_A + \Lambda_B))_{\tor}$. Here $\Lambda_A=\Lambda[I_f]$ and $\Lambda_B=\Lambda[I_g]$, because $A_f$ and $A_g$ are optimal quotients. The following formula for the intersection of~$n$ subtori is obtained in a similar way to that of Proposition~\ref{prop:intersection}. \begin{proposition}\label{prop:intersect} For $i=1,\ldots,n$, with $n\geq 2$, let $A_i = V_i/\Lambda_i$ be a subtorus of $T=V/\Lambda$, and assume that each pairwise intersection $A_i \intersect A_j$ is finite. Define a linear map $$ f : V_1 \cross \cdots \cross V_n \lra V^{\oplus (n-1)}. $$ by $f(x_1,\ldots,x_n)=(x_1-x_2,\,x_2-x_3,\,x_3-x_4,\,\ldots,\,x_{n-1}-x_n)$. Then $$A_1\intersect \cdots \intersect A_n \isom \left(\frac{\Lambda^{\oplus (n-1)}} {f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right)_{\tor}.$$ \end{proposition} \subsection{Bounding the Torsion From Above} \label{sec:torsion_above} In this section we recall the standard upper bound on the order of $\#A(\Q)_{\tor}$, and illustrate its usefulness. Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$ with Nebentypus character $\varepsilon:(\Z/N\Z)^*\ra \C^*$ (recall that $f$ is a form on $\Gamma_0(N)$ if and only if $\varepsilon=1$), and let $A=A_f$ be the corresponding optimal quotient of $J_1(N)$, as in \cite{shimura:factors}. Shimura proved in \cite[Ch.~7]{shimura:intro} that the local Euler factor of $A_f$ at~$p$ is $$L_p(A_f,s) \,\,\,=\, \prod_{\sigma : K_f \hra \Qbar} \,\frac{1} {1 - \sigma(a_p) p^{-s} + \sigma(\varepsilon(p)) p^{1-2s}} $$ by showing that the characteristic polynomial $F_p$ of Frobenius on any $\ell$-adic Tate module of $A_{\F_p}$ (for $\ell\nmid pN$) is $$ F_p(X) = \prod_{\sigma : K_f \hra \Qbar} X^2 - \sigma(a_p) X + \sigma(\varepsilon(p))p, $$ where $K_f=\Q(\ldots, a_n, \ldots)$. Let $\Q(\varepsilon)$ be the field generated by the values of~$\varepsilon$ (note that $\Q(\varepsilon)\subset K_f$), and for any $p\nmid N$ let $G_p(X)\in\Q(\varepsilon)[X]$ be the characteristic polynomial of left multiplication by $a_p$ on the $\Q(\varepsilon)$-vector space $K_f$, which is a polynomial of degree $d'=[K_f: \Q(\varepsilon)]$. Then $$ F_p(X) = \Norm_{\Q(\varepsilon)/\Q}\left( X^{d'} \cdot G_p\left(X + \frac{\varepsilon(p)p}{X}\right)\right), $$ so \begin{align*} \#A_{\Fp}(\Fp) &= \deg(1-\Frob_p) = |\det(1-\Frob_p)| \\ &= |F_p(1)| = |\Norm_{\Q(\varepsilon)/\Q}(G_p(1 + \varepsilon(p)p))|. \end{align*} If $p\nmid N$ is odd, standard facts about formal groups imply that the reduction map $A(\Q)_{\tor}\ra A_{\Fp}(\Fp)$ is injective, so $$ \#A(\Q)_{\tor} \mid \gcd\, \left\{ \#A_{\Fp}(\Fp) \,:\, \text{\rm primes $p\nmid 2N$}\right\}. $$ Likewise, since $A^{\vee}$ is isogenous to~$A$, the same bound applies to $A^{\vee}(\Q)_{\tor}$, since $A^{\vee}$ and~$A$ have the same $L$-series. The upper bound is the same for every abelian variety isogenous to~$A$, so it is not surprising that it is not sharp in general. For example, let~$E$ (resp., $F$) be the elliptic curve labeled \nf{30A1} (resp. \nf{30A2}) in Cremona's tables \cite{cremona:algs}. Then~$E$ and~$F$ are isogenous, $E(\Q)\ncisom \Z/6\Z$, and $F(\Q)\ncisom \Z/12\Z$, so $$ 12 \mid \gcd\, \left\{ \#E_{\Fp}(\Fp) \,:\, \text{\rm primes $p\nmid 2N$}\right\}. $$ (Incidentally, since $\#E(\F_5) = 12$, the gcd is $12$.) For answers to some related deep questions about this gcd, see \cite{katz:torsion}. \begin{example}\label{ex:torsion_upper_bound} Let $$ f = q + (-1+\sqrt{2})q^2 + q^3 + (-2\sqrt{2} + 1)q^4 - 2\sqrt{2}q^5 +\cdots \in S_2(\Gamma_0(39)) $$ be the form \nf{39B}. Then $G_5(X) = X^2 - 8$, so $$ \#A_f(\Q)_{\tor} \mid G_5(1+5) = 28. $$ We find in \cite{empirical} that $A_f$ is isogenous to the Jacobian~$J$ of $y^2 + (x^3 + 1)y = -5x^4 - 2x^3 + 16x^2 -12x +2$ and that $\#J(\Q) = 28$. However $A_f$ is not isomorphic to~$J$ since, as reported in Table~2 of \cite{empirical}, the Tamagawa numbers of~$J$ are $c_3=28$, $c_{13}=1$, whereas the methods of Section~\ref{sec:tamagawa} below show that the Tamagawa numbers of $A_f$ are $c_3=14$, $c_{13}=2$. The authors do not know for sure whether $\#A_f(\Q) = 28$, but in Example~\ref{ex:torsion_lower_bound} below we show that $14\mid \#A_f(\Q)$. (Also, using the computational techniques of this paper one sees that the Birch and Swinnerton-Dyer conjecture implies that $\#A_f(\Q) = 28$.) \end{example} \begin{example} Let $$ f= q + (-\zeta_6 - 1)q^2 + (2\zeta_6 - 2)q^3 + \zeta_6q^4 + (-2\zeta_6 + 1)q^5 + \cdots $$ be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$. This form has character $\varepsilon:(\Z/13\Z)^* \ra \C^*$ of order~$6$, and $A_f = J_1(13)$. We have $G_3(X) = X - 2\zeta_6 + 2$ and $\varepsilon(3) = -\zeta_6$, so \begin{align*} \#J_1(13)(\Q)_{\tor} \mid \#J_1(13)(\F_3) &= |\Norm(G_3(1-3\zeta_6))|\\ & = |\Norm(-5\zeta_6 + 3)| = 19. \end{align*} In fact Ogg proved that $J_1(13)(\Q)_{\tor}\ncisom \Z/19\Z$ (see \cite{ogg:rational13} and \cite{mazurtate:13}). \end{example} \comment{ Consider the abelian variety quotient $A_f=A_{20}$ of $J_0(389)$, as in Section~\ref{example:first_vis_sha}. Each $G_p(x)$ is a polynomial of degree $20$. We find that $G_3(3+1) = 2^2\cdot 97 \cdot 2586967$ and $G_5(5+1) = 5^2\cdot 97 \cdot 183880478851$. Thus $97$ is an upper bound on $\#A(\Q)_{\tor}$. We will see in Example~\ref{ex:torsion_lower_bound} that in fact $\#A(\Q)_{\tor}=97$. \end{example} } \subsection{Bounding the Torsion From Below} \label{sec:torsion_below} A cusp $\alpha \in{} \Gamma_0(N)\backslash\P^1(\Q) \subset X_0(N)$ defines a point $(\alpha)-(\infty) \in J_0(N)(\Qbar)_{\tor}$. The rational cuspidal subgroup~$C$ of $J_0(N)(\Q)_{\tor}$ generated by $\Q$-rational cusps is of interest because the order of the image of~$C$ in $A_f(\Q)_{\tor}$ provides a lower bound on $\#A_f(\Q)_{\tor}$. Stevens \cite[\S1.3]{stevens:thesis} computed the action of $\Gal(\Qbar/\Q)$ on the subgroup of $J_0(N)(\Qbar)$ generated by all cusps (and for other congruence subgroups besides $\Gamma_0(N)$). He found that $\Gal(\Qbar/\Q)$ acts on the cusps through $\Gal(\Q(\zeta_N)/\Q)\isom (\Z/N\Z)^*$, and that $d\in(\Z/N\Z)^*$ acts by $x/y\mapsto x/(d'y)$, where $dd'\equiv 1\pmod{N}$. Thus, e.g., $(0)-(\infty)\in J_0(N)(\Q)_{\tor}$, and if~$N$ is square-free then all cusps are rational. To compute the image of~$C$ in $A_f(\Q)_{\tor}$, first make a list of inequivalent cusps using, e.g., the method described in \cite[\S2.2, pg.~17]{cremona:algs}. Keep only the $\Q$-rational cusps, which can be determined using the result of Stevens above and \cite[Prop.~2.2.3]{cremona:algs} (when~$N$ is squarefree all cusps are rational). Next compute the subgroup~$\mathcal{C}$ of $\sM_2(\Gamma_0(N))$ generated by modular symbols $\{\alpha,\infty\}$, where~$\alpha$ is a $\Q$-rational cusp. The image of~$C$ in $A_f(\Q)_{\tor}$ is isomorphic to the image of~$\mathcal{C}$ in $$ P = \Phi_f(\sM_2(\Gamma_0(N)))/\Phi_f(\sS_2(\Gamma_0(N))), $$ where $\Phi_f : \sM_2(\Gamma_0(N))\ra \Hom(S_2(\Gamma_0(N))[I_f],\C)$ is defined by the integration pairing. To keep everything rational, note that~$P$ can be computed using any map with the same kernel as $\Phi_f$; for example, such a map can be constructed by finding a basis for $\Hom(\sM_2(\Gamma_0(N)),\Q)[I_f]$ as described at the end of Section~\ref{sec:lratio}). \begin{example}\label{ex:torsion_lower_bound} Let the notation be as in Example~\ref{ex:torsion_upper_bound}. The cusps on $X_0(39)$ are represented by $0$, $\infty$, $-1/9$, and $-4/13$, and since $N=39$ is squarefree, these cusps are all rational. Using {\sc Magma} we find that the image of~$C$ in $A_f(\Q)_{\tor}$ is isomorphic to $\Z/14\Z$. Thus $A_f(\Q)_{\tor}$ is isomorphic to one of $\Z/14\Z$, $\Z/28\Z$, or $\Z/14\Z\cross \Z/2\Z$, but we do not know which. \end{example} \begin{example}\label{ex:below175} Let $$ f= q + \frac{1+\sqrt{5}}{2}q^2 + \frac{1-\sqrt{5}}{2}q^3 + \frac{5+\sqrt{5}}{2}q^4 + \cdots \in S_2(\Gamma_0(175)) $$ be the form \nf{175D}. The cusps of $X_0(175)$ are represented by $$0,\, \infty,\, \frac{1}{25},\, \frac{1}{28},\, \frac{1}{30},\, \frac{1}{35},\, \frac{1}{45},\, \frac{1}{60},\, \frac{1}{65},\, \frac{1}{70},\, \frac{1}{105},\, \frac{1}{140}.$$ The $\Q$-rational cusps in this list are $0, \infty, \frac{1}{25}, \frac{1}{28}$, and these generate a subgroup of $A_f(\Q)_{\tor}$ of order~$2$. (Incidentally, the group generated by all cusps, both rational and not, is isomorphic to $\Z/32\Z$.) Using $a_p$ for $p\leq 17$ and the method of the previous section, we see that $\#A_f(\Q)_{\tor} \mid 4$. The authors do not know if the cardinality is~$2$ or~$4$. \end{example} \begin{example} The form \nf{209C} is $$ f = q + \alpha{} q^2 + (1/2\alpha^4 - \alpha^3 - 5/2\alpha^2 + 4\alpha + 1)q^3 + (\alpha^2 - 2)q^4 + \cdots, $$ where $\alpha^5 - 2\alpha^4 - 6\alpha^3 + 10\alpha^2 + 5\alpha - 4=0$. As above, we find that $\#A_f(\Q)_{\tor}$ divides~$5$. The image of the (rational) cuspidal subgroup in $A_f(\Qbar)_{\tor}$ is isomorphic to $\Z/5\Z$, so $A_f(\Q)_{\tor}\ncisom\Z/5\Z$. \end{example} \subsection{Tamagawa Numbers} \label{sec:tamagawa} Suppose $p\mid N$ and let $\Phi_{A,p}$ denote the component group of~$A$ at~$p$, which is defined by the following exact sequence: $$0\ra \cA_{\Fp}^0\ra \cA_{\Fp} \ra \Phi_{A,p}\ra 0,$$ where $\cA_{\Fp}$ is the closed fiber of the N\'eron model of~$A$ over $\Z_p$ and $\cA_{\Fp}^0$ is the component of $\cA_{\Fp}$ that contains the identity. \begin{definition} The {\em Tamagawa number} of $A$ at $p$ is $$c_p = c_{A,p} = \#\Phi_{A,p}(\Fp).$$ \end{definition} When $p\mid\mid N$, the second author found a computable formula for $\#\Phi_{A,p}(\Fbar_p)$ and (sometimes only up to a power of $2$) for $\#\Phi_{A,p}(\F_p)$. There is a discussion about how to compute this number in \cite{kohel-stein:ants4} and \cite{conrad-stein:compgrp} contains a proof of the formula. Note also that in this case the Tamagawa number of~$A$ at~$p$ is the same as the Tamagawa number of $\Adual$ at~$p$. When $p^2\mid{}N$ the authors do not know an algorithm to compute $c_p$. However, in this case Lenstra and Oort (see \cite{lenstra-oort:additive}) proved that $$ \sum_{\ell\neq p}(\ell-1)\ord_\ell(\#\Phi_{A,p}(\Fpbar)) \leq 2\dim(A_f), $$ so if $\ell\mid \#\Phi_{A,p}(\Fpbar)$ then $\ell \leq 2\cdot \dim(A_f)+1$ or $\ell=p$. (Here $\ord_\ell(x)$ denotes the exponent of the largest power of~$\ell$ that divides~$x$.) \begin{example} Let~$f$ be \nf{39B} as in Example~\ref{ex:torsion_upper_bound}. Running the algorithm of \cite{kohel-stein:ants4}, we find that $c_3 = 14$ and $c_{13}=2$. \end{example} \begin{example} Let~$f$ be \nf{175D} as in Example~\ref{ex:below175}. Running the algorithm of \cite{kohel-stein:ants4}, we find that $c_7 = 1$, and the Lenstra-Oort bound implies that the only possible prime divisors of $c_{5}$ are~$2$,~$3$, and~$5$. \end{example} \subsection{Visibility Theory} We briefly recall visibility theory, which we will use to construct elements of Shafarevich-Tate groups. Section~\ref{sec:appdx} contains another approach to the results reported in this section, but in the special case of elliptic curves. \begin{definition} Let $\iota:A\hookrightarrow J$ be an embedding of abelian varities over~$\Q$. The \defn{visible subgroup of $\Sha(A)$ with respect to the embedding~$\iota$} is $$\Vis_J(\Sha(A)) = \Ker(\Sha(A)\ra{}\Sha(J)).$$ \end{definition} The following is a special case of Theorem~3.1 of \cite{agashe-stein:visibility}. \begin{theorem}\label{thm:shaexists} Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ over~$\Q$ such that $A(\Qbar)\intersect B(\Qbar)$ is finite. Let~$N$ be an integer divisible by the residue characteristics of primes of bad reduction for~$J$ (e.g., the conductor of~$J$). Suppose~$p$ is a prime such that $$p\nmid 2\cdot N \cdot \#(J/B)(\Q)_{\tor}\cdot\#B(\Q)_{\tor}\cdot \prod_{\ell} c_{A,\ell}\cdot c_{B,\ell},$$ where $c_{A,\ell} = \#\Phi_{A,\ell}(\F_\ell)$ (resp., $c_{B,\ell}$) is the Tamagawa number of~$A$ (resp., $B$) at~$\ell$. Suppose furthermore that $B[p](\Qbar) \subset A(\Qbar)$ as subgroups of $J(\Qbar)$. Then there is a natural map $$ \vphi:B(\Q)/pB(\Q)\ra \Vis_J(\Sha(A)) $$ such that $\dim_{\F_p} \ker(\vphi) \leq \dim_{\Q} A(\Q)\tensor \Q$. \end{theorem} We return to the situation where $A=A_f$ is an optimal quotient of $J_0(N)$ attached to a newform. In Proposition~\ref{prop:invis} below we show that $\Vis_{J_0(N)}(\Sha(\Adual))$ is annihilated by multiplication by $\moddeg(A)$ (see also \cite[p.19]{cremona-mazur}). We first state a lemma; the outline of the proof was indicated to us by B.~Poonen. \begin{lemma} \label{lem:kerpoll} Let $A$ be an abelian variety over~$k$, where $k$ is a field, and let $\lambda:A\ra A^{\vee}$ be a polarization. Suppose either that $k$ has characteristic~$0$ or that its characteristic does not divide the degree of~$\lambda$. Then there is a finite abelian group~$H$ such that $\ker(\lambda) \ncisom H \times H$ as groups. \end{lemma} \begin{proof} We work in the setting of Section~16 of \cite{milne:abvars}, using the notation used there. Consider the pairing $$e^{\lambda} : {\rm Ker}(\lambda) \times {\rm Ker}(\lambda) \ra \mu_m \subseteq \overline{k}^*,$$ as in \cite[p.~135]{milne:abvars}, where $m$~is an integer that kills ${\rm Ker}(\lambda)$. We will show that this pairing is nondegenerate. Suppose $a \in {\rm Ker}(\lambda)$ is such that $e^{\lambda}(a,a') = 1$ for all $a' \in {\rm Ker}(\lambda)$. Let $a'' \in A^{\vee}[m]$. There exists an isogeny~$\lambda': \Adual \ra A$ such that $\lambda' \circ \lambda$ is multiplication by~$m$ on~$A$ and $\lambda \circ \lambda'$ is multiplication by~$m$ on~$\Adual$ (to construct $\lambda'$, note that $\lambda'$ is the quotient map $\Adual \to \Adual/\lambda(A[m])$). Pick an element $b \in A(\overline{k})$ such that $\lambda b = a''$. Then $mb = \lambda' (\lambda b) = \lambda'(a'')$. So $\overline{e}_m(a,a'') = \overline{e}_m(a, \lambda b) = e^{\lambda}(a, \lambda'a'') = 0$ (note that $\lambda (\lambda'a'') = ma'' = 0$, so that $\lambda'a'' \in {\rm Ker}(\lambda)$). This is true for all $a'' \in A^{\vee}[m]$, so the non-degeneracy of $\overline{e}_m$ (\cite[p.~131]{milne:abvars}) implies that $a = 0$. Similarly, suppose $a' \in {\rm Ker}(\lambda)$ is such that $e^{\lambda}(a,a') = 1$ for all $a \in {\rm Ker}(\lambda)$. Since $e^{\lambda}$ is skew-symmetric (\cite[p.~135]{milne:abvars}), this implies that $e^{\lambda}(a',a) = 1$ for all $a \in {\rm Ker}(\lambda)$. Then by the previous paragraph, $a' = 0$. This finishes the proof of non-degeneracy. As mentioned before, the pairing~$e^{\lambda}$ is skew-symmetric. It is alternating because it extends to pairings on Tate modules (denoted by $e^{\lambda}_{\ell}$ in \cite[p.~132]{milne:abvars}), and the latter take values in a torsion-free group, so there is no distinction between skew-symmetric and alternating. Now the lemma follows from the fact that if $G$ is a finite abelian group with an alternating nondegenerate pairing, then there is a finite abelian group~$H$ such that $G \ncisom H \times H$ as groups (e.g., see~\cite[Prop.~2]{delaunay}). \end{proof} \begin{proposition} \label{prop:invis} Let $m_A = \moddeg(A)$. We have $$\Vis_{J_0(N)}(\Sha(\Adual))\subset \Sha(\Adual)[m_A].$$ \end{proposition} \begin{proof} The polarization $\theta_f$ (from Section~\ref{sec:moddegree}) is the composite map $\Adual \ra J_0(N)\ra A$. Let $e_A$ be the exponent of the finite group $\ker(\theta_f)$. By Lemma~\ref{lem:kerpoll}, multiplication by~$m_A$ kills $\ker(\theta_f)$, so $e_A \mid m_A$. Also $\theta_f$ factors through multiplication by $e_A$, so there is a map $\theta_f':A \ra \Adual$ such that $\theta_f'\circ\theta_f$ is multiplication by $e_A$. If $\phi$ is a map of abelian varieties (over~$\Q$), let $\phi_*$ denote the corresponding map on Shafarevich-Tate groups. Since $\Vis_{J_0(N)}(\Sha(\Adual))$ is contained in $\ker((\theta_f)_*)$, it is also contained in $$\ker((\delta'\circ \delta)_*) = \Sha(\Adual)[e_A] \subset \Sha(\Adual)[m_A].$$ \end{proof} Since $\Sha(\Adual)[n]$ is finite for any~$n$, we obtain the following corollary. \begin{corollary} $\Vis_{J_0(N)}(\Sha(\Adual))$ is finite. \end{corollary} \comment{ Note that if $L(A,1) \neq 0$, then by \cite{kolyvagin-logachev:finiteness}, $\Sha(A)$ is finite. The following corollary is useful in computational examples: \begin{corollary} \label{cor:invis} Let $f$ be a modular form of weight~$2$ on~$\Gamma_0(N)$ such that $L(A_f,1) \neq 0$. Let $\ell$ be a prime such that $\ell \mid \# \Sha(A_f)$, but $\ell \nmid m_{\scriptscriptstyle{A_f^{\vee}}}$. Then $\Sha(A_f^{\vee})$ has an element of order~$\ell$ that is not visible in~$J_0(N)$. \end{corollary} \begin{proof} By the remark above, $\Sha(A_f)$ is finite and so by the Cassels-Tate pairing, $\Sha(A_f^{\vee})$ is also finite, and $\# \Sha(A_f^{\vee}) = \# \Sha(A_f)$. Thus $\Sha(A_f^{\vee})$ has an element of order~$\ell$, and since $\ell \nmid m_{\scriptscriptstyle{A_f^{\vee}}}$, by Proposition~\ref{prop:invis}, this element is not visible in~$J_0(N)$. \end{proof} } \section{The Quotient $L(A,1)/\Omega_{A}$} \label{sec:quotient} Fix a newform $f\in S_2(\Gamma_0(N))$, let $I_f$ be the annihilator of $f$ in $\T$, and $A=A_f=J_0(N)/I_f J_0(N)$ the corresponding optimal quotient. Suppose for the rest of this section that $L(A,1)\neq 0$. \subsection{The Manin Constant} \label{sec:manin} When trying to compute the conjectural order of $\Sha(A)$, we try to compute the quotient $L(A,1)/\Omega_A$, but find that it is easier to compute $c_A\cdot L(A,1)/\Omega_A$ where $c_A$ is the Manin constant of~$A$, which is defined as follows: \begin{definition}[Manin constant] The \defn{Manin constant} of~$A$ is $$ c_A = \#\left(\frac{S_2(\Gamma_0(N),\Z)[I_f]} {H^0(\cA,\Omega_{\cA/\Z})}\right)\in\Z, $$ where we consider $H^0(\cA,\Omega_{\cA/\Z})$ as a submodule of $S_2(\Gamma_0(N),\Q)$ using $$ H^0(\cA,\Omega_{\cA/\Z}) \ra H^0(\cJ,\Omega_{\cJ/\Z})[I_f] \ra H^0(J,\Omega_{J/\Q})[I_f] \ra S_2(\Gamma_0(N),\Q)[I_f], $$ where~$\cA$ and $\cJ$ are the N\'eron models of~$A$ and~$J$, respectively. (See \cite{agashe-stein:manin} for a discussion of why the image of $H^0(\cA,\Omega_{\cA/\Z})$ is contained in $S_2(\Gamma_0(N),\Z)$.) \end{definition} \begin{theorem}\label{thm:manin_bound} If $\ell\mid c_A$ is a prime then $\ell^2 \mid 4N$. \end{theorem} \begin{proof} Mazur proved this when $\dim A=1$ in \cite[\S4]{mazur:rational}, and we generalized his proof in \cite{agashe-stein:manin}. \end{proof} When $\dim A=1$, Edixhoven \cite{edixhoven:manin} obtained strong results towards the folklore conjecture that $c_A=1$, and when~$A$ has arbitrary dimension the authors have made the following conjecture (see \cite{agashe-stein:manin} for evidence): \begin{conjecture}\label{conj:manin_constant} $c_A=1$. \end{conjecture} \subsection{A Formula for $L(A,1)/\Omega_A$} \label{sec:lratio} If~$L$ and~$M$ are lattices in a real vector space~$V$, then the {\em lattice index} $[L:M]$ is the absolute value of the determinant of a linear transformation of~$V$ taking~$L$ onto~$M$. The lattice index satisfies the usual properties suggested by the notation, e.g., $[L:M]\cdot [M:N] = [L:N]$. The {\em real volume $\Omega_{A}$} is defined as follows. If $L^*$ is a lattice in the cotangent space $$ T^*=H^0(A_\R,\Omega_{A_\R})=S_2(\Gamma_0(N),\R)[I_f] $$ of $A_\R$, then~$L^*$ determines a lattice $L=\Hom(L^*,\Z)$ in the tangent space $T=\Hom(T^*,\R)$, and hence a measure on~$T$ by declaring that the quotient $T/L$ has measure~$1$. Let $A(\R)^0$ denote the identity component of $A(\R)$. Then $A(\R)^0$ inherits a measure by virtue of being viewed as $T/H_1(A(\R)^0,\Z)$, and we have $$ \mu_L(A(\R)^0) = [L : H_1(A(\R)^0,\Z)]. $$ We also set $$ \mu_L(A(\R)) = \mu_L(A(\R)^0)\cdot c_{\infty}, $$ where $c_{\infty} = \#(A(\R)/A(\R)^0)$. Let $\cA$ be the N\'eron model of~$A$ (see \cite{neronmodels}). The N\'eron differentials $H^0(\cA,\Omega_{\cA/\Z})$ define a lattice $\Lambda^*$ in $T^*$, and we define $\Omega_A = \mu_\Lambda(A(\R))$. \begin{lemma} \label{lem:plus} $H_1(A(\R)^0,\Z)\isom H_1(A(\C),\Z)^+$. \end{lemma} \begin{proof} This lemma is well known, but we give a proof for the reader's convenience (which was suggested by H.~Lenstra and B.~Poonen). We have the commutative diagram $$\xymatrix@=1.3pc{ 0 \ar[r] & H_1(A(\R)^0,\Z) \ar[r] \ar[d]^{\psi} & H_1(A(\R)^0,\R) \ar[r] \ar[d]^{\isom} & A(\R)^0 \ar[r] \ar@{^(->}[d]^{i}& 0 \\ 0 \ar[r] & H_1(A(\C),\Z)^+ \ar[r] & H_1(A(\C),\R)^+ \ar[r]^{\quad\,\,\pi} & A(\C)^+ }$$ where the upper horizontal sequences is exact (we view the real torus $A(\R)^0$ as the quotient of the tangent space at the identity by the first integral homology), and the lower horizontal sequence is exact because it is the beginning of the long exact sequence of $\Gal(\R/\C)$-cohomology that arises from $$ 0\ra H_1(A(\C),\Z) \ra H_1(A(\C),\R) \ra A(\C) \ra 0. $$ The middle vertical map is an isomorphism because if it were not then its kernel would be an uncountable set that maps to~$0$ in $A(\R)^0$. The snake lemma then yields an exact sequence $$ 0 \to \ker(\psi) \to 0 \to 0 \to \coker(\psi) \to 0, $$ which implies that~$\psi$ is an isomorphism. \end{proof} Let $$ \Phi : H_1(X_0(N),\Q) \ra \Hom(S_2(\Gamma_0(N))[I_f],\C) $$ be the map induced by integration, scaled so that $$ \Phi(\{0,\infty\})(f) = L(f,1) $$ (that $\{0,\infty\}\in H_1(X_0(N),\Q)$ is the Manin-Drinfeld theorem, and that $\int_0^{\infty} f$ is a multiple of $L(f,1)$ follows from the definition of $L(f,s)$ as a Mellin transform). \begin{theorem}\label{thm:lratio} Recall that $A$ is an abelian variety attached to a newform $f\in S_2(\Gamma_0(N))$, that $c_\infty$ is the number of connected components of $A(\R)$, that $c_A$ is the Manin constant of~$A$, that $\Omega_A$ is the N\'eron canonical volume of $A(\R)$, and that $\Phi$ is the period mapping on homology induced by integrating homology classes on $X_0(N)$ against the $\C$-vector space spanned by the $\Gal(\Qbar/\Q)$-conjugates of~$f$. Then we have the following equation: $$ c_\infty \cdot c_A\cdot \frac{L(A,1)}{\Omega_A} = [ \Phi(H_1(X_0(N),\Z))^+ : \Phi(\T\{0,\infty\})]\in\Q, $$ where the lattice index on the right hand side should be interpreted as~$0$ if $\Phi(\T\{0,\infty\})$ has rank less than the dimension of~$A$. \end{theorem} \begin{proof} It is easier to compute with $\tilde{\Lambda}^*=S_2(\Gamma_0(N),\Z)[I_f]$ than with $\Lambda^*$, so let $\tilde{\Omega}_A = \mu_{\tilde{\Lambda}}(A(\R))$. Note that $\tilde{\Omega}_A \cdot c_A = \Omega_A$, where $c_A$ is the Manin constant. By Lemma~\ref{lem:plus} and Section~\ref{sec:abvars}, \begin{align*} \tilde{\Omega}_A &= c_\infty\cdot [\tilde{\Lambda} : H_1(A(\R)^0,\Z)]\\ &= c_\infty \cdot [\Hom(S_2(\Gamma_0(N),\Z)[I_f],\Z) : \Phi(H_1(X_0(N),\Z))^+]. \end{align*} For any ring~$R$ the pairing $$\T_R \cross S_2(\Gamma_0(N),R) \ra R$$ given by $\langle T_n, f \rangle = a_1(T_n f)$ is perfect, so $ (\T/I_f)\tensor R \isom \Hom(S_2(\Gamma_0(N),R)[I_f],R). $ Using this pairing, we may view $\Phi$ as a map $$ \Phi : H_1(X_0(N),\Q) \ra (\T/I_f)\tensor \C, $$ so that $$ \tilde{\Omega}_A = c_\infty \cdot [\T/I_f : \Phi(H_1(X_0(N),\Z))^+]. $$ Note that $(\T/I_f)\tensor\C$ is isomorphic as a ring to a product of copies of $\C$, with one copy corresponding to each Galois conjugate $f^{(i)}$ of~$f$. Let $\pi_i\in (\T/I_f)\tensor\C$ be the projector onto the subspace of $(\T/I_f)\tensor\C$ corresponding to $f^{(i)}$. Then $\Phi(\{0,\infty\})\cdot \pi_i = L(f^{(i)},1)\cdot \pi_i.$ Since the $\pi_i$ form a basis for the complex vector space $(\T/I_f)\tensor\C$, we see that $$ \det(\Phi(\{0,\infty\})) = \prod_i L(f^{(i)},1) = L(A,1). $$ Letting $H = H_1(X_0(N),\Z)$, we have \begin{align*} [\Phi(H)^+ : \Phi(\T \{0,\infty\})] &= [\Phi(H)^+ : (\T/I_f) \cdot \Phi(\{0,\infty\}) ] \\ &= [\Phi(H)^+ : \T/I_f] \cdot [\T/I_f : \T/I_f\cdot \Phi(\{0,\infty\})]\\ &= \frac{c_\infty}{\tilde{\Omega}_A} \cdot \det(\Phi(\{0,\infty\}))\\ &= \frac{c_\infty c_A}{\Omega_A} \cdot L(A,1), \end{align*} which proves the theorem. \end{proof} Theorem~\ref{thm:lratio} was inspired by the case when~$A$ is an elliptic curve (see~\cite[\S{}II.2.8]{cremona:algs}) or the winding quotient of $J_0(p)$ (see~\cite{agashe}), and it generalizes to forms of weight $>2$ (see \cite{stein:phd}). Theorem~\ref{thm:lratio} is true with~$\Phi$ replaced by any linear map with the same kernel as~$\Phi$. One way to find such a linear map with image in a $\Q$-vector space is to compute a basis $\vphi_1, \ldots \vphi_d$ for $\Hom(H_1(X_0(N),\Q),\Q)[I_f]$ and let $\Phi=\vphi_1\cross \cdots \cross \vphi_d$. Also, since $H_1(X_0(N),\Z)^+$ and $\T \{0,\infty\}$ are contained in $H_1(X_0(N),\Q)^+$, Theorem~\ref{thm:lratio} implies that $L(A,1)/\Omega_A\in\Q$, a fact well known to the experts (see \cite[Prop.~2.7]{gross:central} for the statement, but without proof). \subsection{The Denominator of $L(A,1)/\Omega_A$} In this section, we prove a result about the denominator of the rational number $L(A,1)/\Omega_A$ and compare it to what is predicted by the Birch and Swinnerton-Dyer conjecture. \begin{proposition} \label{prop:denr} Let $z$ be the point in $J_0(N)(\Q)$ defined by the degree~$0$ divisor $(0)-(\infty)$ on $X_0(N)$, and let $n = n_f$ be the order of the image of~$z$ in $A(\Q)$. Then the denominator of $c_\infty\cdot c_A\cdot L(A,1)/\Omega_A $ divides~$n$. \end{proposition} \begin{proof} Let~$x$ be the image of~$z$ in $A(\Q)$, and let $I=\Ann_\T(x)$ be the ideal of elements of $\T$ that annihilate~$x$. Since~$f$ is a newform, the Hecke operators $T_p$, for $p\mid{}N$, act as~$0$ or $\pm 1$ on $A(\Q)$ (see, e.g., \cite[\S6]{diamond-im}). If $p\nmid N$, then a standard calculation (see, e.g., \cite[\S2.8]{cremona:algs}) shows that $T_p(x) = (p+1)x$. Let~$C$ be the cyclic subgroup of $A(\Q)$ of order~$n$ generated by~$x$. Consider the map $\T \rightarrow C$ given by $T_p\mapsto T_p(x)$. The kernel of this map is~$I$, and the map is surjective because its image is an additive group that contains~$x$, and~$C$ is the smallest such group. Thus the map induces an isomorphism $\T/I \xrightarrow{\isom} C$. \end{proof} Conjecture~\ref{conj:bsd} predicts that $$ \#A(\Q)\cdot \#A^{\vee}(\Q) \cdot \frac{L(A,1)}{\Omega_A}= \#\Sha(A)\cdot \prod c_p \in \Z, $$ and since $n\mid \#A(\Q)$, Proposition~\ref{prop:denr} implies that $$ c_\infty \cdot c_A \cdot \#A(\Q) \cdot \frac{L(A,1)}{\Omega_A} \in \Z. $$ Since $c_\infty$ is a power of~$2$, and $c_A$ is conjecturally~$1$ (if~$N$ is prime, then by Theorem~\ref{thm:manin_bound} it is a power of~$2$), Proposition~\ref{prop:denr} provides theoretical evidence for Conjecture~\ref{conj:bsd}, and also reflects a surprising amount of cancellation between $\prod c_p$ and $\#A^{\vee}(\Q)$. \section{Results and Conclusions} \label{sec:tables} We computed all $19608$ abelian varieties $A=A_f$ attached to newforms of level $N\leq 2333$. Interesting data about some of these abelian varieties is summarized in Tables~\ref{t1}--\ref{t4}, which use the notation described in this section. Suppose that~$A$ is one of the $10360$ of these for which $L(A,1)\neq 0$, so Conjecture~\ref{conj:bsd} asserts that $\Sha(A)$ has order $$ \#\Sha_? = \frac{L(A,1)}{\tilde{\Omega}_{A}\cdot c_A}\cdot \frac{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}}{\prod_{p\mid N} c_p}. $$ (See Section~\ref{sec:lratio} for the definition of $\tilde{\Omega}_A$ and $c_A$.) For any rational number~$x$, let $x^{\op}$ be the {\em odd part} of~$x$. If~$a$ and~$b$ are rational numbers with $a\neq 0$, we say that $a\mid b$ if $b/a$ is an integer. Define integers $S_l$ and $S_u$ such that $$ S_l\mid \numer(\#\Sha_?^{\op}) \mid S_u $$ as follows: \begin{itemize} \item[$\mathbf{S}_u$] The {\em upper bound} $S_u$ is the odd part of the numerator of $$ \frac{L(A,1)}{\tilde{\Omega}_A} \cdot \frac{T^2}{\prod_{p\mid\mid N} c_p}, $$ where~$T$ is the upper bound on $\#A(\Q)$ and $\#A^{\vee}(\Q)$ computed using Section~\ref{sec:torsion_above} using $a_p$ for $p\leq 17$. Since the Manin constant and the Tamagawa numbers are integers, $S_u$ is an upper bound on the odd part of $\#\Sha_?$. \vspace{1.3ex} \item[$\mathbf{S}_l$] The {\em lower bound} $S_l$ is defined as follows: Let $S_{l,1}$ be the odd part of the rational number $$ \frac{L(A,1)}{\tilde{\Omega}_A} \cdot \frac{\#C\cdot \#D}{\prod_{p\mid\mid N} c_p}, $$ where $C\subset A(\Q)_{\tor}$ and~$D$ is the part of~$C$ coprime to the modular degree of~$A$. Usually~$C$ is the group generated by the image of $(0)-(\infty)$, and in all cases it contains this subgroup. More precisely, when~$A$ is an elliptic curve, we instead let~$C$ and~$D$ be the full torsion subgroup $A(\Q)_{\tor}$, because it is easy to calculate. When~$A$ is not an elliptic curve it would be better to let~$C$ be the subgroup generated by all rational cusps, but the authors only realized this after completing the calculations, so we did not do this. If~$N$ is square free, we let $S_l=S_{l,1}$. Otherwise, let $S_{l,2}$ be the largest part of $S_{l,1}$ coprime to all primes whose square divides~$N$. This takes care of the Manin constant, which only involves primes whose square divides~$N$. To take care of Tamagawa numbers, remove all primes $p\leq 2\dim(A)+1$ from $S_{l,2}$ to obtain $S_l$. \end{itemize} \begin{remark} When~$N$ is square free we have $$ S_l\mid \#\Sha_?^{\op} \mid S_u $$ since $c_A$ is a power of~$2$ and no Tamagawa numbers have been omitted from the formulas for $S_l$ and $S_u$. For every $N\leq 2333$ we found that $S_l$ is an integer, so when $N\leq 2333$ is squarefree, $\#\Sha_?^{\op}$ is an integer. Since Conjecture~\ref{conj:bsd} asserts that $\#\Sha_?$ is the order of a group, hence an integer, our data gives evidence for Conjecture~\ref{conj:bsd}. \end{remark} Tables~\ref{t1}--\ref{t4} list every~$A$ of level $N\leq 2333$ such that $S_l>1$. The~$A$ column contains the label of~$A$ (see Section~\ref{sec:enumerate}), and the next column (labeled dim) contains $\dim A$. A star next to the label for~$A$ indicates that we have proved that the odd part of $\#\Sha(A)$ is at least as large as conjectured by the Birch and Swinnerton-Dyer conjecture. {\em This is the case for $39$ of the $168$ examples.} The columns labeled $S_l$ contains the number $S_l$ defined above. If $S_l=S_u$ then the column labeled $S_u$ contains an $=$ sign, and otherwise, it contains $S_u$ (there are only $13$ cases in which $S_u\neq S_l$). The column labeled $\moddeg(A)^{\op}$ contains the odd part~$m$ of the modular degree of~$A$, written as a product $\gcd(m,S_u)\cdot \mbox{\tiny $m/\gcd(S_u,m)$}$, where {\tiny $m/\gcd(S_u,m)$} is shrunk to save space. The only non-square-free levels of $A_f$ for which $S_l>1$ are $1058$, $1664$, $2224$, and $2264$. The column labeled~$B$ contains all~$B$ such that $L(B,1)=0$ and $$ \gcd(S_l,\#(A^{\vee}\intersect \tilde{B}^{\vee}))>1. $$ (In retrospect, it would probably have been more interesting to list those~$B$ such that $\gcd(S_u,\#(A^{\vee}\intersect \tilde{B}^{\vee}))>1$.) Here if $B=A_g$ for some newform~$g$ of level dividing~$N$, and $\tilde{B}^{\vee}$ is the abelian subvariety of $J_0(N)$ generated by all images of $B^{\vee}$ under the degeneracy maps. Thus, e.g., when~$B^{\vee}$ is of level~$N$, $\tilde{B}^{\vee}=B^{\vee}$. The next column, labeled $\dim$, contains the dimension of~$B$. The final two columns contain information about the relationship between~$A$ and~$B$. The one labeled $A^{\vee}\intersect \tilde{B}^{\vee}$ contains the abelian group structure of the indicated abelian group, where e.g., $[a^bc^d]$ means the abelian group $(\Z/a\Z)^b\cross (\Z/c\Z)^d$. The column labeled~$\Vis$ contains a divisor of the order of $\Vis_{C}(\Sha(A^{\vee}))$, where $C=A^{\vee} + \tilde{B}^{\vee}$ (note that $\Vis_{C}(\Sha(A^{\vee}))\subset \Vis_{J_0(N)}(\Sha(A^{\vee}))$). The table is divided into three vertical regions, where the columns in the first region are about~$A$ only, the columns of the second region are about~$B$ only, and the third column is about the relationship between~$A$ and~$B$. \subsection{Example: Level 389}\label{sec:389} We illustrate what is involved in computing the first line of Table~\ref{t1}. Using the method sketched in Section~\ref{sec:enumerate}, we find that $S_2(\Gamma_0(389))$ contains exactly five Galois-conjugacy classes of newforms, and these are defined over extensions of~$\Q$ of degrees $1$, $2$, $3$, $6$, and $20$. Thus $J=J_0(389)$ decomposes, up to isogeny, as a product $A_1\cross A_2 \cross A_3\cross A_6 \cross A_{20}$ of abelian varieties, where $\dim A_d=d$ and $A_d$ is the optimal quotient corresponding to the appropriate Galois-conjugacy class of newforms. Next we consider the arithmetic of the $A_d$. Using Theorem~\ref{thm:lratio} we find that $$L(A_1,1)=L(A_2,1)=L(A_3,1)=L(A_6,1)=0,$$ and $$\frac{L(A_{20},1)}{\Omega_{A_{20}}} = \frac{5^2\cdot 2^{11}}{97\cdot c_A},$$ where~$c_A$ is the Manin constant attached to $A_{20}$, which, by Theorem~\ref{thm:manin_bound}, is of the form $2^n$ with $n\geq 0$. Using the algorithms of Sections~\ref{sec:torsion_above}, \ref{sec:torsion_below}, \ref{sec:tamagawa}, we find that $\#A_{20}(\Q) = c_{389} = 97$. Thus Conjecture~\ref{conj:bsd} predicts that $\#\Sha(A_{20}) = 5^2\cdot 2^{11}/c_A$. The following proposition provides support for this conjecture. \begin{proposition}\label{prop:389} There is a natural inclusion $$(\Z/5\Z)^2 \isom A_1(\Q)/5 A_1(\Q) \hookrightarrow \Vis_{J_{0}(389)}(\Sha(A_{20}^{\vee})).$$ \end{proposition} \begin{proof} Let $A=A_{20}^{\vee}$, $B=A_{1}^{\vee}$ and $J=A+B\subset J_0(389)$. Using Proposition~\ref{prop:intersect}, we find that $A\intersect B \isom (\Z/4)^2\cross (\Z/5\Z)^2$, so $B[5] \subset A$. Since~$5$ does not divide the numerator of $(389-1)/12$, it does not divide the Tamagawa numbers or the orders of the torsion groups, so Theorem~\ref{thm:shaexists} yields the asserted injection. To see that $(\Z/5\Z)^2 \isom A_1(\Q)/5 A_1(\Q)$ use the standard elliptic curves algorithms \cite{cremona:algs}. \end{proof} \subsection{Invisible Elements of $\Sha(A)$} \label{sec:existinvis} Tables~\ref{t1}--\ref{t4} suggest that much of $\Sha(\Adual)$ is invisible in $J_0(N)$. This is because Proposition~\ref{prop:invis} implies that if a prime divides $\#\Sha(A^{\vee})$ but not $\moddeg(A^{\vee})$ then $\Sha(A^{\vee})$ contains an element of order~$p$ that is invisible. We find many examples in the table where~$p$ divides the conjectural order of $\Sha(A^{\vee})$, but $p\nmid \moddeg(A^{\vee})$. Invisible elements might become visible at higher level (see \cite[\S4.3]{agashe-stein:visibility} for a discussion and example). \subsection{The Part of $\Sha(A)$ That Must be a Perfect Square} When $\dim A=1$, properties of the Cassels-Tate pairing imply that if $\Sha(A)$ is finite then $\#\Sha(A)$ is a perfect square, and the fact that one finds in examples (see~\cite{cremona:algs}) that $\Shaan$ is a perfect square is computational evidence for Conjecture~\ref{conj:bsd}. In contrast, when the dimension is greater than one, Poonen and Stoll~\cite{poonen-stoll} discovered Jacobians~$J$ such that $\Sha(J)$ has order twice a square, and the second author found for each prime $p<25000$ an abelian variety $A$ of dimension $p-1$ such that $\#\Sha(A)=pn^2$ for some integer~$n$ (see \cite{stein:nonsquare}). \begin{proposition} Let~$A=A_f$ be a quotient of~$J_0(N)$ and $\ell$ be a prime that does not divide the modular degree of~$A$. Suppose that $\Sha(A)[\ell^\infty]$ is finite. Then $\#\Sha(A)[\ell^{\infty}]$ is a perfect square. \end{proposition} \begin{proof} The Cassels-Tate pairing (see \cite[\S3]{tate:duality}) induces a pairing $$\phi: \Sha(A)[\ell^\infty] \times \Sha(\Adual)[\ell^\infty] \ra \Q/\Z.$$ Since $\Sha(A)[\ell^\infty]$ is finite, it follows from~\cite[Thm.~3.2]{tate:duality} that $\Sha(\Adual)[\ell^\infty]$ is also finite and $\phi$ is non-degenerate. In particular, $\# \Sha(\Adual)[\ell^\infty] = \# \Sha(A)[\ell^\infty]$. Since $J_0(N)$ is a Jacobian, it possesses a canonical polarization arising from the theta divisor; this divisor is rational over~$\Q$, since $X_0(N)$ always has a point over~$\Q$ (the cusp~$\infty$ is rational). This polarization induces a polarization $\theta: \Adual\ra A$, which also comes from a divisor that is rational over~$\Q$. Hence, by \cite[Thm.~3.3]{tate:duality} (see also \cite[Thm.~5]{poonen-stoll}), the pairing $$\phi': \Sha(\Adual)[\ell^\infty] \times \Sha(\Adual)[\ell^\infty] \ra \Q/\Z$$ obtained by composing~$\theta$ with the pairing~$\phi$ above is alternating. Since $\ell$ does not divide the modular degree of~$A$, it does not divide the degree of the isogeny~$\theta$. Hence $\theta$ induces an isomorphism $\Sha(\Adual)[\ell^\infty] \stackrel{\isom}{\ra} \Sha(A)[\ell^\infty]$. Thus by the non-degeneracy of the pairing~$\phi$, the pairing~$\phi'$ is also non-degenerate. Since $\phi'$ is also alternating, it follows from arguments similar to those in~\cite[p.~260]{cassels:arithmeticIII} that $\# \Sha(\Adual)[\ell^\infty]$ is a perfect square. Since $\# \Sha(A)[\ell^\infty] = \# \Sha(\Adual)[\ell^\infty]$, we see that $\# \Sha(A)[\ell^\infty]$ is also a perfect square. \end{proof} For the entries in Tables~\ref{t1}--\ref{t4}, $\Sha(A)$ is finite, so if $\ell \nmid \moddeg(A)$ then the $\ell$-power part of $\#\Sha(A)$ must be a perfect square. When $S_l=S_u$ and the level is square free, then $S_l$ is the odd part of the conjectural order of $\Sha(A)$. We found that $S_l$ is a perfect square whenever $S_l=S_u$, which provides evidence for Conjecture~\ref{conj:bsd}. \begin{table} \caption{Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups\label{t1} } \begin{center} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{389E}*&20&5^{2}&=&5&\mathbf{389A}&1&[20^{2}]&5^{2} \\ \mathbf{433D}*&16&7^{2}&=&7\!\cdot\!\mbox{\tiny $111$}&\mathbf{433A}&1&[14^{2}]&7^{2} \\ \mathbf{446F}*&8&11^{2}&=&11\!\cdot\!\mbox{\tiny $359353$}&\mathbf{446B}&1&[11^{2}]&11^{2} \\ \mathbf{551H}&18&3^{2}&=&\mbox{\tiny $169$}&\text{NONE} & & & \\ \hline \mathbf{563E}*&31&13^{2}&=&13&\mathbf{563A}&1&[26^{2}]&13^{2} \\ \mathbf{571D}*&2&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $127$}&\mathbf{571B}&1&[3^{2}]&3^{2} \\ \mathbf{655D}*&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $9799079$}&\mathbf{655A}&1&[36^{2}]&3^{4} \\ \mathbf{681B}&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $125$}&\mathbf{681C}&1&[3^{2}]&- \\ \hline \mathbf{707G}*&15&13^{2}&=&13\!\cdot\!\mbox{\tiny $800077$}&\mathbf{707A}&1&[13^{2}]&13^{2} \\ \mathbf{709C}*&30&11^{2}&=&11&\mathbf{709A}&1&[22^{2}]&11^{2} \\ \mathbf{718F}*&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $5371523$}&\mathbf{718B}&1&[7^{2}]&7^{2} \\ \mathbf{767F}&23&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \hline \mathbf{794G}*&12&11^{2}&=&11\!\cdot\!\mbox{\tiny $34986189$}&\mathbf{794A}&1&[11^{2}]&- \\ \mathbf{817E}*&15&7^{2}&=&7\!\cdot\!\mbox{\tiny $79$}&\mathbf{817A}&1&[7^{2}]&- \\ \mathbf{959D}&24&3^{2}&=&\mbox{\tiny $583673$}&\text{NONE} & & & \\ \mathbf{997H}*&42&3^{4}&=&3^{2}&\mathbf{997B}&1&[12^{2}]&3^{2} \\ \hline &&& && \mathbf{997C}&1&[24^{2}]&3^{2} \\ \mathbf{1001F}&3&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $1269$}&\mathbf{1001C}&1&[3^{2}]&- \\ &&& && \mathbf{91A}&1&[3^{2}]&- \\ \mathbf{1001L}&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $2029789$}&\mathbf{1001C}&1&[7^{2}]&- \\ \hline \mathbf{1041E}&4&5^{2}&=&5^{2}\!\cdot\!\mbox{\tiny $13589$}&\mathbf{1041B}&2&[5^{2}]&- \\ \mathbf{1041J}&13&5^{4}&=&5^{3}\!\cdot\!\mbox{\tiny $21120929983$}&\mathbf{1041B}&2&[5^{4}]&- \\ \mathbf{1058D}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $483$}&\mathbf{1058C}&1&[5^{2}]&- \\ \mathbf{1061D}&46&151^{2}&=&151\!\cdot\!\mbox{\tiny $10919$}&\mathbf{1061B}&2&[2^{2}302^{2}]&- \\ \hline \mathbf{1070M}&7&3 \!\cdot\! 5^{2}&3^{2} \!\cdot\! 5^{2}&3 \!\cdot\! 5\!\cdot\!\mbox{\tiny $1720261$}&\mathbf{1070A}&1&[15^{2}]&- \\ \mathbf{1077J}&15&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $1227767047943$}&\mathbf{1077A}&1&[9^{2}]&- \\ \mathbf{1091C}&62&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1094F}*&13&11^{2}&=&11^{2}\!\cdot\!\mbox{\tiny $172446773$}&\mathbf{1094A}&1&[11^{2}]&11^{2} \\ \hline \mathbf{1102K}&4&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $31009$}&\mathbf{1102A}&1&[3^{2}]&- \\ \mathbf{1126F}*&11&11^{2}&=&11\!\cdot\!\mbox{\tiny $13990352759$}&\mathbf{1126A}&1&[11^{2}]&11^{2} \\ \mathbf{1137C}&14&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $64082807$}&\mathbf{1137A}&1&[9^{2}]&- \\ \mathbf{1141I}&22&7^{2}&=&7\!\cdot\!\mbox{\tiny $528921$}&\mathbf{1141A}&1&[14^{2}]&- \\ \hline \mathbf{1147H}&23&5^{2}&=&5\!\cdot\!\mbox{\tiny $729$}&\mathbf{1147A}&1&[10^{2}]&- \\ \mathbf{1171D}*&53&11^{2}&=&11\!\cdot\!\mbox{\tiny $81$}&\mathbf{1171A}&1&[44^{2}]&11^{2} \\ \mathbf{1246B}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $81$}&\mathbf{1246C}&1&[5^{2}]&- \\ \mathbf{1247D}&32&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $2399$}&\mathbf{43A}&1&[36^{2}]&- \\ \hline \mathbf{1283C}&62&5^{2}&=&5\!\cdot\!\mbox{\tiny $2419$}&\text{NONE} & & & \\ \mathbf{1337E}&33&3^{2}&=&\mbox{\tiny $71$}&\text{NONE} & & & \\ \mathbf{1339G}&30&3^{2}&=&\mbox{\tiny $5776049$}&\text{NONE} & & & \\ \mathbf{1355E}&28&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $2224523985405$}&\text{NONE} & & & \\ \hline \mathbf{1363F}&25&31^{2}&=&31\!\cdot\!\mbox{\tiny $34889$}&\mathbf{1363B}&2&[2^{2}62^{2}]&- \\ \mathbf{1429B}&64&5^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1443G}&5&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $18525$}&\mathbf{1443C}&1&[7^{1}14^{1}]&- \\ \mathbf{1446N}&7&3^{2}&=&3\!\cdot\!\mbox{\tiny $17459029$}&\mathbf{1446A}&1&[12^{2}]&- \\ \hline \end{array}$ \end{center} \end{table} \begin{table} \caption{Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups\label{t2} } \begin{center} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{1466H}*&23&13^{2}&=&13\!\cdot\!\mbox{\tiny $25631993723$}&\mathbf{1466B}&1&[26^{2}]&13^{2} \\ \mathbf{1477C}*&24&13^{2}&=&13\!\cdot\!\mbox{\tiny $57037637$}&\mathbf{1477A}&1&[13^{2}]&13^{2} \\ \mathbf{1481C}&71&13^{2}&=&\mbox{\tiny $70825$}&\text{NONE} & & & \\ \mathbf{1483D}*&67&3^{2} \!\cdot\! 5^{2}&=&3 \!\cdot\! 5&\mathbf{1483A}&1&[60^{2}]&3^{2} \!\cdot\! 5^{2} \\ \hline \mathbf{1513F}&31&3&3^{4}&3\!\cdot\!\mbox{\tiny $759709$}&\text{NONE} & & & \\ \mathbf{1529D}&36&5^{2}&=&\mbox{\tiny $535641763$}&\text{NONE} & & & \\ \mathbf{1531D}&73&3&3^{2}&3&\mathbf{1531A}&1&[48^{2}]&- \\ \mathbf{1534J}&6&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $635931$}&\mathbf{1534B}&1&[6^{2}]&- \\ \hline \mathbf{1551G}&13&3^{2}&=&3\!\cdot\!\mbox{\tiny $110659885$}&\mathbf{141A}&1&[15^{2}]&- \\ \mathbf{1559B}&90&11^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1567D}&69&7^{2} \!\cdot\! 41^{2}&=&7 \!\cdot\! 41&\mathbf{1567B}&3&[4^{4}1148^{2}]&- \\ \mathbf{1570J}*&6&11^{2}&=&11\!\cdot\!\mbox{\tiny $228651397$}&\mathbf{1570B}&1&[11^{2}]&11^{2} \\ \hline \mathbf{1577E}&36&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $15$}&\mathbf{83A}&1&[6^{2}]&- \\ \mathbf{1589D}&35&3^{2}&=&\mbox{\tiny $6005292627343$}&\text{NONE} & & & \\ \mathbf{1591F}*&35&31^{2}&=&31\!\cdot\!\mbox{\tiny $2401$}&\mathbf{1591A}&1&[31^{2}]&31^{2} \\ \mathbf{1594J}&17&3^{2}&=&3\!\cdot\!\mbox{\tiny $259338050025131$}&\mathbf{1594A}&1&[12^{2}]&- \\ \hline \mathbf{1613D}*&75&5^{2}&=&5\!\cdot\!\mbox{\tiny $19$}&\mathbf{1613A}&1&[20^{2}]&5^{2} \\ \mathbf{1615J}&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $13317421$}&\mathbf{1615A}&1&[9^{1}18^{1}]&- \\ \mathbf{1621C}*&70&17^{2}&=&17&\mathbf{1621A}&1&[34^{2}]&17^{2} \\ \mathbf{1627C}*&73&3^{4}&=&3^{2}&\mathbf{1627A}&1&[36^{2}]&3^{4} \\ \hline \mathbf{1631C}&37&5^{2}&=&\mbox{\tiny $6354841131$}&\text{NONE} & & & \\ \mathbf{1633D}&27&3^{6} \!\cdot\! 7^{2}&=&3^{5} \!\cdot\! 7\!\cdot\!\mbox{\tiny $31375$}&\mathbf{1633A}&3&[6^{4}42^{2}]&- \\ \mathbf{1634K}&12&3^{2}&=&3\!\cdot\!\mbox{\tiny $3311565989$}&\mathbf{817A}&1&[3^{2}]&- \\ \mathbf{1639G}*&34&17^{2}&=&17\!\cdot\!\mbox{\tiny $82355$}&\mathbf{1639B}&1&[34^{2}]&17^{2} \\ \hline \mathbf{1641J}*&24&23^{2}&=&23\!\cdot\!\mbox{\tiny $1491344147471$}&\mathbf{1641B}&1&[23^{2}]&23^{2} \\ \mathbf{1642D}*&14&7^{2}&=&7\!\cdot\!\mbox{\tiny $123398360851$}&\mathbf{1642A}&1&[7^{2}]&7^{2} \\ \mathbf{1662K}&7&11^{2}&=&11\!\cdot\!\mbox{\tiny $16610917393$}&\mathbf{1662A}&1&[11^{2}]&- \\ \mathbf{1664K}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $7$}&\mathbf{1664N}&1&[5^{2}]&- \\ \hline \mathbf{1679C}&45&11^{2}&=&\mbox{\tiny $6489$}&\text{NONE} & & & \\ \mathbf{1689E}&28&3^{2}&=&3\!\cdot\!\mbox{\tiny $172707180029157365$}&\mathbf{563A}&1&[3^{2}]&- \\ \mathbf{1693C}&72&1301^{2}&=&1301&\mathbf{1693A}&3&[2^{4}2602^{2}]&- \\ \mathbf{1717H}*&34&13^{2}&=&13\!\cdot\!\mbox{\tiny $345$}&\mathbf{1717B}&1&[26^{2}]&13^{2} \\ \hline \mathbf{1727E}&39&3^{2}&=&\mbox{\tiny $118242943$}&\text{NONE} & & & \\ \mathbf{1739F}&43&659^{2}&=&659\!\cdot\!\mbox{\tiny $151291281$}&\mathbf{1739C}&2&[2^{2}1318^{2}]&- \\ \mathbf{1745K}&33&5^{2}&=&5\!\cdot\!\mbox{\tiny $1971380677489$}&\mathbf{1745D}&1&[20^{2}]&- \\ \mathbf{1751C}&45&5^{2}&=&5\!\cdot\!\mbox{\tiny $707$}&\mathbf{103A}&2&[505^{2}]&- \\ \hline \mathbf{1781D}&44&3^{2}&=&\mbox{\tiny $61541$}&\text{NONE} & & & \\ \mathbf{1793G}*&36&23^{2}&=&23\!\cdot\!\mbox{\tiny $8846589$}&\mathbf{1793B}&1&[23^{2}]&23^{2} \\ \mathbf{1799D}&44&5^{2}&=&\mbox{\tiny $201449$}&\text{NONE} & & & \\ \mathbf{1811D}&98&31^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \hline \mathbf{1829E}&44&13^{2}&=&\mbox{\tiny $3595$}&\text{NONE} & & & \\ \mathbf{1843F}&40&3^{2}&=&\mbox{\tiny $8389$}&\text{NONE} & & & \\ \mathbf{1847B}&98&3^{6}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1871C}&98&19^{2}&=&\mbox{\tiny $14699$}&\text{NONE} & & & \\ \hline \end{array}$ \end{center} \end{table} \begin{table} \caption{Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups\label{t3}} \begin{center}\vspace{-1em} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{1877B}&86&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1887J}&12&5^{2}&=&5\!\cdot\!\mbox{\tiny $10825598693$}&\mathbf{1887A}&1&[20^{2}]&- \\ \mathbf{1891H}&40&7^{4}&=&7^{2}\!\cdot\!\mbox{\tiny $44082137$}&\mathbf{1891C}&2&[4^{2}196^{2}]&- \\ \mathbf{1907D}*&90&7^{2}&=&7\!\cdot\!\mbox{\tiny $165$}&\mathbf{1907A}&1&[56^{2}]&7^{2} \\ \hline \mathbf{1909D}*&38&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $9317$}&\mathbf{1909A}&1&[18^{2}]&3^{4} \\ \mathbf{1913B}*&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $103$}&\mathbf{1913A}&1&[3^{2}]&3^{2} \\ \mathbf{1913E}&84&5^{4} \!\cdot\! 61^{2}&=&5^{2} \!\cdot\! 61\!\cdot\!\mbox{\tiny $103$}&\mathbf{1913A}&1&[10^{2}]&- \\ &&& && \mathbf{1913C}&2&[2^{2}610^{2}]&- \\ \hline \mathbf{1919D}&52&23^{2}&=&\mbox{\tiny $675$}&\text{NONE} & & & \\ \mathbf{1927E}&45&3^{2}&3^{4}&\mbox{\tiny $52667$}&\text{NONE} & & & \\ \mathbf{1933C}&83&3^{2} \!\cdot\! 7&3^{2} \!\cdot\! 7^{2}&3 \!\cdot\! 7&\mathbf{1933A}&1&[42^{2}]&3^{2} \\ \mathbf{1943E}&46&13^{2}&=&\mbox{\tiny $62931125$}&\text{NONE} & & & \\ \hline \mathbf{1945E}*&34&3^{2}&=&3\!\cdot\!\mbox{\tiny $571255479184807$}&\mathbf{389A}&1&[3^{2}]&3^{2} \\ \mathbf{1957E}*&37&7^{2} \!\cdot\! 11^{2}&=&7 \!\cdot\! 11\!\cdot\!\mbox{\tiny $3481$}&\mathbf{1957A}&1&[22^{2}]&11^{2} \\ &&& && \mathbf{1957B}&1&[14^{2}]&7^{2} \\ \mathbf{1979C}&104&19^{2}&=&\mbox{\tiny $55$}&\text{NONE} & & & \\ \hline \mathbf{1991C}&49&7^{2}&=&\mbox{\tiny $1634403663$}&\text{NONE} & & & \\ \mathbf{1994D}&26&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $46197281414642501$}&\mathbf{997B}&1&[3^{2}]&- \\ \mathbf{1997C}&93&17^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2001L}&11&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $44513447$}&\text{NONE} & & & \\ \hline \mathbf{2006E}&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $805$}&\mathbf{2006D}&1&[3^{2}]&- \\ \mathbf{2014L}&12&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $126381129003$}&\mathbf{106A}&1&[9^{2}]&- \\ \mathbf{2021E}&50&5^{6}&=&5^{2}\!\cdot\!\mbox{\tiny $729$}&\mathbf{2021A}&1&[100^{2}]&5^{4} \\ \mathbf{2027C}*&94&29^{2}&=&29&\mathbf{2027A}&1&[58^{2}]&29^{2} \\ \hline \mathbf{2029C}&90&5^{2} \!\cdot\! 269^{2}&=&5 \!\cdot\! 269&\mathbf{2029A}&2&[2^{2}2690^{2}]&- \\ \mathbf{2031H}*&36&11^{2}&=&11\!\cdot\!\mbox{\tiny $1014875952355$}&\mathbf{2031C}&1&[44^{2}]&11^{2} \\ \mathbf{2035K}&16&11^{2}&=&11\!\cdot\!\mbox{\tiny $218702421$}&\mathbf{2035C}&1&[11^{1}22^{1}]&- \\ \mathbf{2038F}&25&5&5^{2}&5^{2}\!\cdot\!\mbox{\tiny $92198576587$}&\mathbf{2038A}&1&[20^{2}]&- \\ \hline &&& && \mathbf{1019B}&1&[5^{2}]&- \\ \mathbf{2039F}&99&3^{4} \!\cdot\! 5^{2}&=&\mbox{\tiny $13741381043009$}&\text{NONE} & & & \\ \mathbf{2041C}&43&3^{4}&=&\mbox{\tiny $61889617$}&\text{NONE} & & & \\ \mathbf{2045I}&39&3^{4}&=&3^{3}\!\cdot\!\mbox{\tiny $3123399893$}&\mathbf{2045C}&1&[18^{2}]&- \\ \hline &&& && \mathbf{409A}&13&[9370199679^{2}]&- \\ \mathbf{2049D}&31&3^{2}&=&\mbox{\tiny $29174705448000469937$}&\text{NONE} & & & \\ \mathbf{2051D}&45&7^{2}&=&7\!\cdot\!\mbox{\tiny $674652424406369$}&\mathbf{2051A}&1&[56^{2}]&- \\ \mathbf{2059E}&45&5 \!\cdot\! 7^{2}&5^{2} \!\cdot\! 7^{2}&5^{2} \!\cdot\! 7\!\cdot\!\mbox{\tiny $167359757$}&\mathbf{2059A}&1&[70^{2}]&- \\ \hline \mathbf{2063C}&106&13^{2}&=&\mbox{\tiny $8479$}&\text{NONE} & & & \\ \mathbf{2071F}&48&13^{2}&=&\mbox{\tiny $36348745$}&\text{NONE} & & & \\ \mathbf{2099B}&106&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2101F}&46&5^{2}&=&5\!\cdot\!\mbox{\tiny $11521429$}&\mathbf{191A}&2&[155^{2}]&- \\ \hline \mathbf{2103E}&37&3^{2} \!\cdot\! 11^{2}&=&3^{2} \!\cdot\! 11\!\cdot\!\mbox{\tiny $874412923071571792611$}&\mathbf{2103B}&1&[33^{2}]&11^{2} \\ \mathbf{2111B}&112&211^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2113B}&91&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2117E}*&45&19^{2}&=&19\!\cdot\!\mbox{\tiny $1078389$}&\mathbf{2117A}&1&[38^{2}]&19^{2} \\ \hline \end{array}$ \end{center} \end{table} \begin{table} \caption{Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups\label{t4} } \begin{center}\vspace{-1em} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{2119C}&48&7^{2}&=&\mbox{\tiny $89746579$}&\text{NONE} & & & \\ \mathbf{2127D}&34&3^{2}&=&3\!\cdot\!\mbox{\tiny $18740561792121901$}&\mathbf{709A}&1&[3^{2}]&- \\ \mathbf{2129B}&102&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2130Y}&4&7^{2}&=&7\!\cdot\!\mbox{\tiny $83927$}&\mathbf{2130B}&1&[14^{2}]&- \\ \hline \mathbf{2131B}&101&17^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2134J}&11&3^{2}&=&\mbox{\tiny $1710248025389$}&\text{NONE} & & & \\ \mathbf{2146J}&10&7^{2}&=&7\!\cdot\!\mbox{\tiny $1672443$}&\mathbf{2146A}&1&[7^{2}]&- \\ \mathbf{2159E}&57&13^{2}&=&\mbox{\tiny $31154538351$}&\text{NONE} & & & \\ \hline \mathbf{2159D}&56&3^{4}&=&\mbox{\tiny $233801$}&\text{NONE} & & & \\ \mathbf{2161C}&98&23^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2162H}&14&3&3^{2}&3\!\cdot\!\mbox{\tiny $6578391763$}&\text{NONE} & & & \\ \mathbf{2171E}&54&13^{2}&=&\mbox{\tiny $271$}&\text{NONE} & & & \\ \hline \mathbf{2173H}&44&199^{2}&=&199\!\cdot\!\mbox{\tiny $3581$}&\mathbf{2173D}&2&[398^{2}]&- \\ \mathbf{2173F}&43&19^{2}&3^{2} \!\cdot\! 19^{2}&3^{2} \!\cdot\! 19\!\cdot\!\mbox{\tiny $229341$}&\mathbf{2173A}&1&[38^{2}]&19^{2} \\ \mathbf{2174F}&31&5^{2}&=&5\!\cdot\!\mbox{\tiny $21555702093188316107$}&\text{NONE} & & & \\ \mathbf{2181E}&27&7^{2}&=&7\!\cdot\!\mbox{\tiny $7217996450474835$}&\mathbf{2181A}&1&[28^{2}]&- \\ \hline \mathbf{2193K}&17&3^{2}&=&3\!\cdot\!\mbox{\tiny $15096035814223$}&\mathbf{129A}&1&[21^{2}]&- \\ \mathbf{2199C}&36&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $13033437060276603$}&\text{NONE} & & & \\ \mathbf{2213C}&101&3^{4}&=&\mbox{\tiny $19$}&\text{NONE} & & & \\ \mathbf{2215F}&46&13^{2}&=&13\!\cdot\!\mbox{\tiny $1182141633$}&\mathbf{2215A}&1&[52^{2}]&- \\ \hline \mathbf{2224R}&11&79^{2}&=&79&\mathbf{2224G}&2&[79^{2}]&- \\ \mathbf{2227E}&51&11^{2}&=&\mbox{\tiny $259$}&\text{NONE} & & & \\ \mathbf{2231D}&60&47^{2}&=&\mbox{\tiny $91109$}&\text{NONE} & & & \\ \mathbf{2239B}&110&11^{4}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \hline \mathbf{2251E}*&99&37^{2}&=&37&\mathbf{2251A}&1&[74^{2}]&37^{2} \\ \mathbf{2253C}*&27&13^{2}&=&13\!\cdot\!\mbox{\tiny $14987929400988647$}&\mathbf{2253A}&1&[26^{2}]&13^{2} \\ \mathbf{2255J}&23&7^{2}&=&\mbox{\tiny $15666366543129$}&\text{NONE} & & & \\ \mathbf{2257H}&46&3^{6} \!\cdot\! 29^{2}&=&3^{3} \!\cdot\! 29\!\cdot\!\mbox{\tiny $175$}&\mathbf{2257A}&1&[9^{2}]&- \\ \hline &&& && \mathbf{2257D}&2&[2^{2}174^{2}]&- \\ \mathbf{2264J}&22&73^{2}&=&73&\mathbf{2264B}&2&[146^{2}]&- \\ \mathbf{2265U}&14&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $73023816368925$}&\mathbf{2265B}&1&[7^{2}]&- \\ \mathbf{2271I}*&43&23^{2}&=&23\!\cdot\!\mbox{\tiny $392918345997771783$}&\mathbf{2271C}&1&[46^{2}]&23^{2} \\ \hline \mathbf{2273C}&105&7^{2}&=&7^{2}&\text{NONE} & & & \\ \mathbf{2279D}&61&13^{2}&=&\mbox{\tiny $96991$}&\text{NONE} & & & \\ \mathbf{2279C}&58&5^{2}&=&\mbox{\tiny $1777847$}&\text{NONE} & & & \\ \mathbf{2285E}&45&151^{2}&=&151\!\cdot\!\mbox{\tiny $138908751161$}&\mathbf{2285A}&2&[2^{2}302^{2}]&- \\ \hline \mathbf{2287B}&109&71^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2291C}&52&3^{2}&=&\mbox{\tiny $427943$}&\text{NONE} & & & \\ \mathbf{2293C}&96&479^{2}&=&479&\mathbf{2293A}&2&[2^{2}958^{2}]&- \\ \mathbf{2294F}&15&3^{2}&=&3\!\cdot\!\mbox{\tiny $6289390462793$}&\mathbf{1147A}&1&[3^{2}]&- \\ \hline \mathbf{2311B}&110&5^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{2315I}&51&3^{2}&=&3\!\cdot\!\mbox{\tiny $4475437589723$}&\mathbf{463A}&16&[13426312769169^{2}]&- \\ \mathbf{2333C}&101&83341^{2}&=&83341&\mathbf{2333A}&4&[2^{6}166682^{2}]&- \\ \hline \end{array}$ \end{center} \end{table} \newpage \section{Appendix by J.~Cremona and B.~Mazur:\\ ``Explaining'' Shafarevich-Tate via Mordell-Weil}\label{sec:appdx} \newcommand{\calC}{\mathcal{C}} \newcommand{\calE}{\mathcal{E}} \noindent{\bf Introduction.} In our article \cite{cremona-mazur} we discussed the notion of visibility and offered some tables of examples of that phenomenon. We gave, however, very little theoretical discussion in that article. Here we wish to take the opportunity to correct some gaps in our commentary on our tables and to offer the details of the proof of a general criterion that is sometimes useful to test visibility. Regarding Table~1 of \cite{cremona-mazur} we said that for each pair $(E,p)$ that occurs there and for which there is a corresponding ``$F$" on the table of the same conductor of $E$, the Shafarevich-Tate group of $E$ is {\it explained} by the Mordell-Weil group of $F$, in the technical sense that we gave to the word {\it explained} in that article. Now this is indeed the case for all entries of our table such that $E$ has semistable reduction at $p$ and it is also the case for those entries where the conductor of $F$ properly divides the conductor of $E$. We will review why this is so, below. It is also true that for each of the remaining $7$ entries ($E=$ \nf{2601H}, \nf{2718D}, \nf{2900D}, \nf{3555E}, \nf{3879E}, \nf{3933A}, \nf{5499E}) a nontrivial subgroup of the Shafarevich-Tate group of~$E$ is {\em explained} by the Mordell-Weil group of the corresponding~$F$, but we wish to notify our readers that we have not yet checked whether or not {\em all} of the ``$\Sha$" of these~$7$ elliptic curves is so explained. These~$7$ cases deserve to be looked at (the issue being local at the prime~$3$ for all but \nf{2900D}, where it is local at the prime~$5$). Regarding Table~2 of \cite{cremona-mazur}, although our commentary in \cite{cremona-mazur} does not say this clearly, for all the entries~$E$ of that table for which there is a corresponding~$F$ of the same conductor we only have checked that $E[2]= F[2]$ in $J_0(N)$ and nothing more, except, of course, for those entries we particularly signal to have shown something less; namely, in the language of our article, that they ``seem to satisfy a $2$-congruence.'' In these latter cases where we signal that we have shown something less, W.~Stein has checked that in fact $E[2]\neq F[2]$ in $J_0(N)$. Let~$p$ be an odd prime number. If~$E$ is an (optimal) elliptic curve over ${\bf Q}$ of conductor $N$ then $E$ may be unambiguously identified (up to sign) with a sub-abelian variety of the modular jacobian $J_0(N)$ (over ${\bf Q}$). If $(E,F,p)$ is an entry of Table~1 of \cite{cremona-mazur} such that~$E$ and~$F$ are of the same conductor~$N$ we checked that we have equality of the finite group schemes $E[p]_{/{\bf Q}} = F[p]_{/{\bf Q}}$ in $J_0(N)_{/{\bf Q}}$. For the remaining three entries we checked that there is an isomorphism of finite group schemes $\iota: E[p]_{/{\bf Q}}\cong F[p]_{/{\bf Q}}$. In both cases, identifying the two finite group schemes let $H$ denote the common cohomology group, $$H:= H^1(G_{\Q}, E[p]_{/{\Q}}) = H^1(G_{\Q}, F[p]_{/{\Q}}),$$ and $S_E \subset H$, and $S_F \subset H$ the $p$-Selmer groups of, respectively,~$E$ and~$F$. What we will show is that \begin{proposition} For each of the entries $(E,p)$ in Table 1 of \cite{cremona-mazur} such that~$p$ is a prime of semistable reduction for~$E$ and for which there is a ``corresponding"~$F$, we have $$S_E=S_F \subset H.$$ \end{proposition} To discuss this, we need some notation. Let $X := \Spec(\Z)$, $Y:= \Spec(\Z[1/p]) = X -\Spec(\Fp),$ and $\eta:= \Spec(\Q)$. Let $E_{\eta}:=E$ be our elliptic curve over $\Q$ of conductor~$N$, $E_{/X}$ the N\'eron model over~$X$ of $E_{\eta}$ and $E^o_{/X} \subset E_{/X}$ the ``connected component'' of N\'eron (meaning the open subgroup scheme every fiber of which is connected). We have, of course, similar notation for the corresponding elliptic curve~$F$. Let $E[p]_{/X}$ denote the closed subgroup scheme given as the kernel of multiplication by~$p$ in the N\'eron model: $E[p]_{/X} \subset E_{/X}$. We have, in general, that the restriction $E[p]_{/Y}$ of $E[p]_{/X}$ to the base~$Y$ is an \'etale quasi-finite flat group scheme; and if $p^2$ doesn't divide~$N$ we have that the group scheme $E[p]_{/X}$ is a quasi-finite flat group scheme \cite[Prop.~3.1(d), pg.~343]{grothendieck:monodromie}. The \'etale quasi-finite flat group scheme $E[p]_{/Y}$ can be characterized by the following features: \begin{itemize} \item[(i)] Its generic fiber is the group scheme $E[p]_{/\eta}\subset J_0(N)_{/\eta},$ \end{itemize} and (one has a choice here) either: \begin{itemize} \item[(ii)] $E[p]_{/Y} \subset J_0(N)_{/Y}$ is a closed \'etale quasi-finite flat) subgroup scheme, \end{itemize} or: \begin{itemize} \item[(ii')] $E[p]_{/Y}$ enjoys the N\'eronian property over the base $Y$. \end{itemize} Similar statements hold for $F[p]_{/X}$. Let~$\Phi$ be the (punctual) sheaf of abelian groups for the flat topology over~$X$ which fits into the exact sequence (of abelian sheaves over~$X$) \begin{equation}\label{eqn:m1} 0 \to E^o \to E \to \Phi \to 0. \end{equation} We will use the same notation to indicate the corresponding exact sequence of sheaves for the \'etale topology over~$X$. Since $E^0$ and~$E$ are smooth group schemes, the long exact sequences of cohomology derived from the short exact sequence (\ref{eqn:m1}), viewed either as sheaves of abelian groups for the flat or \'etale topology, coincide; cf Section~11 {\em Appendice: Un th{\'e}or{\`e}me de comparaison de la cohomologie {\'e}tale et de la cohomologie fppf}\hspace{1ex}in \cite{grothendieck:brauer3}. Thinking now of $\Phi$ as a sheaf for the \'etale topology, denote by $\Phi_\ell$ its stalk at the prime~$\ell$. So $\Phi_\ell$ is representable as a finite \'etale group scheme over the field $\F_\ell$. We have that $$ \Phi = \bigoplus_{\ell \ | \ N} (i_{\ell})_*\Phi_{\ell}, $$ where $i_\ell: {\rm Spec }{\bf F}_{\ell} \hookrightarrow X$ is the natural closed immersion. We have an exact sequence \begin{equation}\label{eqn:m2} 0 \to E^o(X) \to E({\bf Q}) \to H^0(X,\Phi) \to H^1(X,E^o) \to H^1(X,E) \to H^1(X,\Phi), \end{equation} where cohomology is computed for the \'etale topology. We have, for either topology, $$ H^i(X,\Phi) = \bigoplus_{\ell \ | \ N} H^i({\rm Spec}({\bf F}_\ell),\Phi_{\ell}). $$ Viewing (\ref{eqn:m1}) as an exact sequence of sheaves for the flat topology, and passing to the associated cohomology sequence we see that (\ref{eqn:m2}) may be thought of, ambiguously as computed for either the \'etale or the flat topology. If~$p$ is an odd prime number, the $p$-primary component of the Shafarevich-Tate group of~$E$ is the $p$-primary component of the image of $H^1(X,E^o) \to H^1(X,E)$ (see the appendix to \cite{mazur:towers}), or equivalently the intersection of the kernels of $$ H^1(X,E) \to H^1({\rm Spec}({\bf F}_\ell),\Phi_{\ell}). $$ Let~$p$ be an odd prime number. Let $E' \subset E$ be the open subgroup scheme of $E$ which is the inverse image of $p\Phi \subset \Phi$, so that we have an exact sequence of sheaves for the flat (or {\' e}tale) topology: \begin{equation}\label{eqn:m3} 0 \to E' \to E \to \Phi/p\Phi \to 0, \end{equation} and if~$p$ is a prime of semistable reduction for~$E$ (equivalently: $p^2$ doesn't divide~$N$) we have an exact sequence of flat group schemes \begin{equation}\label{eqn:m4} 0 \to E[p] \to E \to E' \to 0. \end{equation} Put $$ E[p]^o_{/X}:= E[p]_{/X}\bigcap E^o_{/X}. $$ Then $E[p]^o_{/X}$ is an open (quasi-finite) subgroup scheme of $E[p]_{/X}$. Let ${\tilde E[p]}_{/X}$ be any ``intermediate'' open (quasi-finite) subgroup scheme $$ E[p]^o_{/X} \subset {\tilde E[p]}_{/X} \subset E[p]_{/X} $$ so that we have the exact sequence of sheaves for the finite flat topology \begin{equation}\label{eqn:m5} 0 \to {\tilde E[p]}_{/X} \to E[p]_{/X} \to \Psi \to 0, \end{equation} with~$\Psi$ a subquotient of~$\Phi$. Consider the following hypothesis: \begin{quote} \begin{itemize} \item[${\bf A}(E,p,\ell)$:] The Galois module $\Phi_{\ell}/p\Phi_{\ell}$ is either trivial, or else is a nonconstant cyclic Galois module over ${\bf F}_\ell$. \end{itemize} \end{quote} Let ${\bf A}(E,p)$ denote the conjunction of Hypotheses ${\bf A}(E,p,\ell)$ for all prime numbers~$\ell$, or equivalently, for all~$\ell$ dividing~$N$. \begin{lemma} These are equivalent formulations of Hypothesis ${\bf A}(E,p)$. \begin{itemize} \item[(a)] $\Phi/p\Phi$ is cohomologically trivial; that is, $ H^0(X,\Phi/p\Phi ) = H^1(X,\Phi/p\Phi ) = 0.$ \item[(b)] If $\Psi$ is any subquotient of $\Phi$, $\Psi$ is ``$p$-cohomologically trivial'' in the sense that the $p$-primary components of $H^i(X,\Psi)$ vanish for all~$i$. \end{itemize} Moreover, if $p \ge 5$, or if $p=3$ and~$E$ has no N{\'e}ron fibers of type IV or IV*, the above conditions are equivalent to: \begin{itemize} \item[(c)] For every~$\ell$ at which~$E$ has split multiplicative reduction,~$p$ does not divide the order of the group of connected components of the N{\'e}ron fiber of~$E$ at~$\ell$. \end{itemize} \end{lemma} \begin{proof} The equivalence of Hypothesis ${\bf A}(E,p)$ with (a) and with (b) is straightforward using standard exact sequences plus the fact that the $p$-primary components of the (underlying abelian group of) $\Phi_{\ell}$ is cyclic since $p > 2$; and noting that a (finite) $G$-module of prime order with nontrivial $G$-action has trivial cohomology. For (c) we are using that if $p >2$ the $p$-primary component of $\Phi_{\ell}$ vanishes for all primes~$\ell$ of additive reduction for~$E$ except when $p=3$ and the N{\'e}ron fiber type of~$E$ at $\ell$ is IV or IV*. \end{proof} A morphism $G_1 \to G_2$ of flat (commutative, finite type) groups schemes over~$X$ will be said to {\em induce an isomorphism on $p$-cohomology} if the induced mappings $$ H^i(X, G_1)\otimes {\bf Z}_p \to H^i(X, G_2)\otimes {\bf Z}_p $$ are isomorphisms for all $i \ge 0$, where cohomology is computed for the flat topology. \begin{lemma} Let~$p$ be an odd prime number for which ${\bf A}(E,p)$ holds. We have that the natural morphisms $$ E^o_{/X} \to E'_{/X} \qquad \text{ and }\qquad E'_{/X} \to E_{/X} $$ induce isomorphisms on $p$-cohomology. If~$p$ is of semistable reduction for~$E$, we also have that $$ {\tilde E[p]}_{/X} \to E[p]_{/X} $$ induces isomorphisms on $p$-cohomology, for any of the open subgroup schemes ${\tilde E[p]}_{/X}$ in $E[p]_{/X}$ described above. \end{lemma} \begin{proof} These all sit in short exact sequences of sheaves of abelian groups for the flat topology over $X$ where the third sheaf is $p$-cohomologically trivial. \end{proof} \begin{corollary} If $p > 2$ and ${\bf A}(E,p)$ holds we have natural isomorphisms $$ H^0(X,E^o)\otimes{\bf Z}_p\cong H^0(X,E')\otimes{\bf Z}_p\cong E({\bf Q})\otimes{\bf Z}_p $$ and $$ \Sha(E)\otimes{\bf Z}_p \cong H^1(X,E^o)\otimes{\bf Z}_p\cong H^1(X,E') \otimes{\bf Z}_p\cong H^1(X,E)\otimes{\bf Z}_p. $$ \end{corollary} \begin{corollary} Let $p$ be an odd prime number, semistable for $E$, and suppose that ${\bf A}(E,p)$ holds. \begin{itemize} \item[(i)] The image of the natural (injective) coboundary mapping $$ 0 \to E(\Q)/ p E(\Q) \hookrightarrow H^1(G_\Q, E[p]) $$ attached to the Kummer sequence is contained in the image of the natural injection $$ H^1(X, E[p]^o) \hookrightarrow H^1(G_\Q, E[p]). $$ \item[(ii)] We have an exact sequence $$ 0 \to E(\Q)/ p E(\Q) \to H^1(X, {\tilde E[p]}) \to \Sha(E)[p] \to 0 $$ for any of the open subgroup schemes ${\tilde E[p]}_{/X} \subset E[p]_{/X}$ defined above. \item[(iii)] The image of $H^1(X, {\tilde E[p]}) \hookrightarrow H^1(G_\Q, E[p])$ is equal to the $p$-Selmer subgroup, $$ S_p(E) \subset H^1(G_\Q, E[p]). $$ \end{itemize} \end{corollary} \begin{proof} All this follows from straightforward calculations using the cohomological exact sequences associated to the exact sequences (1)--(5) in the light of the previous discussion. \end{proof} To set things up for our application, let us record the following: \begin{corollary}\label{cor:3} Let $E_{/\Q}$ and $F_{/\Q}$ be elliptic curves over~$\Q$. Let~$p$ be an odd prime number of semistable reduction for~$E$ and~$F$, and for which ${\bf A}(E,p)$ and ${\bf A}(F,p)$ both hold. Define ${\tilde E}[p]_{/X} \subset E[p]_{/X}$ to be the open quasi-finite subgroup scheme whose restriction to~$Y$ is equal to $E[p]_{/Y}$ and whose fiber at $\F_p$ is equal to $E[p]^o_{/\F_p} = (E[p]\bigcap E^o)_{/{\bf F}_p}$. Define ${\tilde F}[p]_{/X}$ similarly. Suppose, finally, that we have an isomorphism of $G_\Q$-modules $\iota: F[p]_{/\bf Q}\cong E[p]_{/\bf Q}$ which extends to an injection of quasi-finite flat group schemes $$ {\tilde F}[p]_{/X}\hookrightarrow {\tilde E}[p]_{/X}. $$ Letting $$ H:= H^1(G_\Q, E[p])= H^1(G_\Q, F[p]) $$ (making the identification via $\iota$) we have that the $p$-Selmer groups $S_p(E) \subset H$ and $S_p(F) \subset H$ are the same. \end{corollary} \begin{proposition} Let $(E,F,p)$ be a triple which is an entry of Table~1 of \cite{cremona-mazur}. Suppose further that~$p$ is of semistable reduction for~$E$ and for~$F$. Then, with the notation of the previous corollary, the $p$-Selmer groups $S_p(E) \subset H$ and $S_p(F) \subset H$ are the same. In the terminology of \cite{cremona-mazur} the Shafarevich-Tate group of~$E$ is explained by the Mordell-Weil group of~$F$. \end{proposition} \begin{proof} As mentioned above, we have checked that $E[p]_{/\bf Q}= F[p]_{/\bf Q} \subset J_0(N)$ whenever the pair $E$ and $F$ (appearing as entry of Table~1 of \cite{cremona-mazur}) have the same conductor. We have checked that $E[p]_{/\bf Q}\cong F[p]_{/\bf Q}$ for the three entries where $E$ and $F$ have different conductor ($E=$ \nf{2932A}, \nf{3306B}, and \nf{5136B}). We have checked that Hypothesis $A(E,p, \ell)$ and $A(F,p, \ell)$ hold for all quadruples $(E,F,p, \ell)$ such that $(E,F,p)$ occurs as an entry in Table~1 of \cite{cremona-mazur} (even when $p$ is not semistable for $E$ and $F$) with the exception of the entry $(E,F, p, \ell) = (\nf{2366D}, \nf{2366E}, 3, 13)$. \begin{sublemma} Under the hypotheses of our proposition, the isomorphism of $G_\Q$-modules $\iota: E[p]_{/\bf Q}\cong F[p]_{/\bf Q}$ extends to an injection of quasi-finite flat group schemes $$ {\tilde E}[p]_{/X}\hookrightarrow {\tilde F}[p]_{/X} $$ which is an isomorphism except in two instances ($E=$ \nf{3306B}, and \nf{5136B}). \end{sublemma} \begin{proof} First, since ${\tilde E}[p]_{/Y} = E[p]_{/Y}$, ${\tilde F}[p]_{/Y} = F[p]_{/Y}$, and, as we mentioned at the beginning, both of these quasi-finite, flat ({\' e}tale) group schemes $F[p]_{/Y}$ and $E[p]_{/Y}$ enjoy the N{\' e}ronian property, the isomorphism $\iota$ extends to an isomorphism ${\tilde E}[p]_{/Y}\cong {\tilde F}[p]_{/Y}$. The remaining question is then local about~$p$. If~$p$ is of good reduction for~$E$, then ${\tilde E}[p]_{/X_p}$ and ${\tilde F}[p]_{/X_p}$ are both finite flat group schemes of odd order, so by Fontaine's Theorem \cite{fontaine:groupes}, the isomorphism between their generic fibers extends to an isomorphism over $X_p$. (Compare: Theorem I.1.4 in \cite{mazur:eisenstein}.) A standard result allows us to {\it patch} the isomorphism extending~$\iota$ over~$Y$ with the isomorphism (``extending~$\iota$") over $X_p$ to get the extension of~$\iota$ to an isomorphism of group schemes over~$X$, ${\tilde E}[p]_{/X}\cong {\tilde F}[p]_{/X}$. Now consider the case where $p$ is of bad reduction. By the assumptions of our proposition, $p$ is then of multiplicative reduction for~$E$, and hence the fiber of~$E$ over $\F_p$ is a finite multiplicative type group scheme of order $p$. We therefore have that ${\tilde E}[p]_{/X_p}$ sits in an exact sequence \begin{equation}\label{eqn:m6} 0 \to {\calC}_{/X_p} \to {\tilde E}[p]_{/X_p} \to {\calE}_{/X_p} \to 0 \end{equation} where ${\calC}_{/X_p}$ is a finite flat group scheme of order~$p$ (and with fiber of multiplicative type in characteristic~$p$) and where ${\calE}_{/X_p}$ is an {\' e}tale quasi-finite group scheme, with trivial fiber in characteristic~$p$. Let us take a moment to recall (see \cite[Lem.~1.1]{mazur:rational}) the construction of such an exact sequence (\ref{eqn:m6}): working in the category of formal schemes, let ${\hat X}_p:= \Spf(\Z_p)$, and let ${\hat{\calC}}_{/\hat{X}_p}$ be the formal completion of the zero-section in ${\tilde E}[p]_{/X_p}$. One checks that ${\hat{\calC}}_{/\hat{X}_p}$ may be identified with a finite flat formal group scheme over ${\hat{X}_p}$ which admits a closed immersion into the formal group scheme over ${\hat{X}_p}$ associated to ${\tilde E}[p]_{/X_p}$. A standard algebrization argument establishes that there is a (unique) finite flat subgroup scheme ${\calC}_{/X_p}\subset {\tilde E}[p]_{/X_p}$ whose associated formal group scheme over ${\hat{X}_p}$ is ${\hat{\calC}}_{/\hat{X}_p}$. The exact sequence (\ref{eqn:m6}) is then obtained by letting ${\calE}_{/X_p}$ be the evident quotient (quasi-finite flat) group scheme, and noting that, by construction, its special fiber is trivial. Now let us return to the proof of the sublemma. Since the restriction of $\iota$ to ${\calC}_{/X_p}$ (a finite flat multiplicative type group scheme of order~$p$) is injective over the generic point, it follows (by elementary considerations, or by Fontaine's Theorem cited above) that $\iota$ restricted to ${\calC}_{/X_p}$ is an injection over $X_p$. Since ${\calE}_{/X_p}$ has trivial fiber in characteristic~$p$, $\iota$ is an injection as was to be proved. In all cases under consideration, then, $$ \iota: {\tilde E}[p]_{/X}\hookrightarrow {\tilde F}[p]_{/X} $$ is an injection. If $E$ is of good reduction at $p$, or if $F$ is of bad reduction at $p$, $\iota$ is therefore an isomorphism. The cases remaining are when $E$ is of bad reduction at $p$ and $F$ is of good reduction (i.e., $E=$ \nf{3306B}, and \nf{5136B}) in which case we can only assert that $\iota$ is an injection. \end{proof} Returning to our proposition, suppose that ${\tilde F}[p]_{/X_p}$ is finite flat (which happens in the two cases signalled above: $E=$ \nf{3306B}, and \nf{5136B}). Then the isomorphism induced by $\iota$ on generic fibers $$ {\tilde E}[p]_{/\Q_p} \cong {\tilde F}[p]_{/{\Q}_p} $$ restricted to the $G_{\Q_p}$-stable subgroup ${\calC}_{/\Q_p} \subset {\tilde E}[p]_{/\Q_p}$ extends to a morphism of the finite flat group scheme ${\calC}_{/X_p}$ into ${\tilde F}[p]_{/X_p}$. This extended morphism $j: {\calC}_{/X_p} \to {\tilde F}[p]_{X_p}$ is necessarily a closed immersion since ${\calC}_{/X_p}$ is a multiplicative type finite flat group scheme. Since ${\calE}_{/X_p}$ has trivial fiber in characteristic $p$ an application of the standard patching argument (as used in the previous case) allows us to put together the isomorphism of group schemes over $Y$ extending $\iota$ with the closed immersion $j$ over $X_p$ to get a closed immersion $$ {\tilde E}[p]_{/X} \hookrightarrow {\tilde F}[p]_{/X}. $$ Finally suppose that both $E$ and $F$ have multiplicative reduction at $p$. We then have exact sequences (\ref{eqn:m6}) for each of our quasi-finite flat group schemes ${\tilde E}[p]_{/X_p}$ and ${\tilde F}[p]_{/X_p}$. Let~$V$ denote their common generic fiber (identified via~$\iota$) considered as two-dimensional ${\bf F}_p$-vector space with $G_{\bf Q_p}$-action. Let ${\calC}(E) \subset V$ and ${\calC}(F) \subset V$ denote the one-dimensional subspaces given by the generic fibers of the finite flat subgroup schemes ${\calC}_{/X_p}$ corresponding to the exact sequence (\ref{eqn:m6}) for for~$E$ and for~$F$ respectively. Suppose, first, that these one-dimensional $\F_p$-subspaces ${\calC}(E)$ and ${\calC}(F)$ are different. It then follows that the $G_{\Qp}$-representation~$V$ splits as the direct sum of ${\calC}(E)$ and ${\calC}(F)$, both ${\bf F}_p$-subspaces being isomorphic, as $I_{\bf Q_p}$-modules to $\mu_p$, where $I_{\bf Q_p} \subset G_{\bf Q_p}$ is the inertia subgroup of $G_{\Qp}$. But this contradicts the fact that $V$ is self-Cartier dual (under the Weil pairing). Consequently, ${\calC}(E)={\calC}(F) \subset V$. From the above discussion it follows that we can extend $\iota$ to an isomorphism ${\tilde E}[p]_{/X_p}\cong {\tilde F}[p]_{/X_p}$. Our proposition then follows (from Corollary~\ref{cor:3}) for all entries in Table~1 of \cite{cremona-mazur} where~$p$ is of semistable reduction for $E$ once we produce special arguments to cover the three special cases $E=$ \nf{3306B}, \nf{5136B} and \nf{2366D}. The first two of these cases are ``special'' because we only have an injection ${\tilde E}[p]_{/X_p}\hookrightarrow {\tilde F}[p]_{/X_p}$ and not an isomorphism. However, the cokernel of this morphism restricted to the fiber in characteristic~$3$ is, in both of these cases, a cyclic group with nontrivial $G_{\bf Q_3}$-action and hence is $3$-cohomologically trivial. In particular, the injection ${\tilde E}[p]_{/X_p}\hookrightarrow {\tilde F}[p]_{/X_p}$ induces an isomorphism on flat cohomology over~$X$, and the argument for these two cases proceeds as before. This leaves $(E,F, p) = (\nf{2366D}, \nf{2366E}, 3)$ which is the only example of an entry $(E,F,p)$ in our table, where $E$ has a $\Q$-rational point of order~$p$, and (this is no accident) where Hypothesis ${\bf A}(E,p)$ and Hypothesis ${\bf A}(F,p)$ fail. (Indeed there are no other failures of Hypothesis ${\bf A}(F,p)$ for any of the $(E,F,p)$'s occurring in Table~1 of \cite{cremona-mazur} and only one other failure of Hypothesis ${\bf A}(E,p)$, which is for $(E,p, \ell)=(\nf{2932A},3,2)$.) Let us now deal with the case $(E,F, p) = (\nf{2366D},\nf{2366E}, 3)$. The subgroup~$C$ of $\Q$-rational points of order~$3$ on~$E$ specialize in characteristic $13$ to yield an isomorphism $$ C \cong \Phi_{13} $$ and the same for the subgroup of $\Q$-rational points of order~$3$ on~$F$. We make use of this information to cut down the group schemes ${\tilde E}[3]_{/X}$ and ${\tilde F}[3]_{/X}$ and define open subgroup schemes: ${\tilde{\tilde E}}[3]_{/X} \subset {\tilde E}[3]_{/X}$ and ${\tilde{\tilde E}}[3]_{/X} \subset {\tilde E}[3]_{/X}$ by requiring that these closed immersions of subgroup schemes be isomorphisms outside characteristic $13$, and that the ``double-tilded'' group schemes each have trivial fiber in characteristic $13$. We get via the above argument an isomorphism of group schemes ${\tilde{\tilde E}}[3]_{/X} \cong {\tilde{\tilde F}}[3]_{/X}$ extending~$\iota$, and an identification of the $3$-Selmer groups of~$E$ and~$F$ with $H^1(X, {\tilde{\tilde E}}[3])$ and $H^1(X, {\tilde{\tilde F}}[3])$ respectively. Our proposition is proved. \end{proof} It remains to say a few words about why, in the~$7$ cases of entries $(E,F,p)$ in our Table~1 of \cite{cremona-mazur} for which~$p$ is a prime of additive reduction for~$E$ {\em some} nontrivial elements of the Shafarevich-Tate group of~$E$ are explained by the Mordell-Weil group of~$F$. Briefly, the reason is as follows. By the {\em inflated} $p$-Selmer group of~$E$ (and of~$F$) let us mean the subgroup of~$H$ obtained by insisting upon all the local Selmer conditions at primes different from~$p$, but putting no condition at~$p$. The $p$-Selmer group of~$E$ (and of~$F$) are, in all~$7$ instances, ${\bf F}_p$-vector spaces of dimension~$2$ and therefore, the inflated $p$-Selmer groups are of dimensions either~$2$ or~$3$. Working over~$Y$ rather than over~$X$, the above argument applied to these~$7$ remaining cases gives us an identification of the {\it inflated} $p$-Selmer groups of~$E$ and of~$F$ in~$H$. 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