% $Header: /home/was/papers/thesis/RCS/bsd.tex,v 1.27 2001/07/08 20:11:41 was Exp $12\comment{3$Log: bsd.tex,v $4Revision 1.27 2001/07/08 20:11:41 was5nothing.67Revision 1.26 2000/05/21 03:05:38 was8bsd910Revision 1.25 2000/05/15 07:31:56 was11added I.1213Revision 1.24 2000/05/15 07:28:22 was14Changed F_p to F_v in proof of main theorem!1516Revision 1.23 2000/05/15 05:11:14 was17quick fix1819Revision 1.22 2000/05/15 04:46:48 was20away from level.2122Revision 1.21 2000/05/15 04:46:08 was23got rid of a_p2425Revision 1.20 2000/05/15 04:45:32 was26third fix; added Weierstrass equation but didn't change subsequent text!2728Revision 1.19 2000/05/15 04:44:23 was29quick fix: typo 2; geometric component group of F!3031Revision 1.18 2000/05/15 04:43:26 was32quick fix. a_p typo3334Revision 1.17 2000/05/12 01:52:07 was35oops!3637Revision 1.16 2000/05/10 20:32:49 was38done.3940Revision 1.15 2000/05/10 09:39:43 was41weakend explantory claim.4243Revision 1.14 2000/05/10 09:35:36 was44added\index{Mazur}4546Revision 1.13 2000/05/10 09:33:08 was47eliminated \index{toric reduction}4849Revision 1.12 2000/05/10 07:12:25 was50Added index's to tables.5152Revision 1.11 2000/05/09 20:21:57 was53added a label to subsec{Notation}5455Revision 1.10 2000/05/08 15:42:17 was56fixed up index some.5758Revision 1.9 2000/05/08 02:52:01 was59Fixed mispelling of "isogeneous".6061Revision 1.8 2000/05/08 02:00:23 was62Fixed up 1913. Made lots and lots of refinements63and clarifcations of the tables.6465Revision 1.7 2000/05/07 21:57:57 was66Added cremona's model for 5389A.6768Revision 1.6 2000/05/07 19:51:10 was69Added a section on when Sha becomes visible.7071Revision 1.5 2000/05/06 20:55:41 was72Worked on cleaning up the descriptions of the tables.7374}7576\chapter{The Birch and Swinnerton-Dyer conjecture}77\label{chap:bsd}\index{Birch and Swinnerton-Dyer conjecture|see{BSD conjecture}}%78\index{Conjecture!Birch and Swinnerton-Dyer|see{BSD conjecture}}%79Now that the Shimura-Taniyama%80\index{Shimura-Taniyama conjecture}\index{Conjecture!Shimura and Taniyama}81conjecture has been proved, many experts consider the Birch and82Swinnerton-Dyer conjecture (BSD conjecture)83to be one of the main outstanding problems in the field84(see~\cite[pg.~549]{darmon-bsd} and \cite[Intro.]{cime-1997}).85This conjecture ties together many of the arithmetic86and analytic invariants of an elliptic curve. At present, there is no87general class of elliptic curves for which the full BSD88conjecture\index{BSD conjecture!is still unknown} is89known, though a slightly weakened form is known for a fairly broad90class of complex multiplication elliptic91curves of analytic rank~$0$ (see~\cite{rubin:main-conjectures}), and92several deep partial results have been obtained during93the last twenty years (see, e.g.,~\cite{gross-zagier} and94\cite{kolyvagin:mordellweil}).9596Approaches to the BSD conjecture\index{BSD conjecture}97that rely on congruences between\index{Congruences!and BSD conjecture}98modular forms\index{Modular forms!and BSD}99are likely to require a deeper100understanding of the analogue of the BSD conjecture\index{BSD conjecture!in higher dimensions}101for higher-dimensional abelian varieties. As a first step, this chapter102presents theorems and explicit computations of some of the arithmetic103invariants of modular abelian varieties.104105The reader is urged to also read A.~Agashe's 2000106Berkeley Ph.D.\ thesis which cover similar themes.\index{Agashe} The paper of107Cremona and Mazur's~\cite{cremona-mazur}\index{Mazur} paints a detailed108experimental picture of the way in which congruences link109Mordell-Weil and Shafarevich-Tate groups of elliptic curves.110\index{Congruences!and BSD conjecture}111112\section{The BSD conjecture}\index{BSD conjecture|textit}113By~\cite{breuil-conrad-diamond-taylor} we now know114that every elliptic curve over~$\Q$ is115a quotient of the curve~$X_0(N)$, whose complex points116are the isomorphism classes of pairs consisting of a117(generalized) elliptic curve and a cyclic subgroup of order~$N$.118Let~$J_0(N)$ denote the Jacobian\index{Jacobian} of $X_0(N)$; this is an abelian119variety of dimension equal to the genus of~$X_0(N)$ whose points120correspond to the degree~$0$ divisor classes on~$X_0(N)$.121The survey article~\cite{diamond-im} is a good122guide to the facts and literature123about the family of abelian varieties $J_0(N)$.124125Following Mazur~\cite{mazur:rational}\index{Mazur}, we make the following definition.126\begin{definition}[Optimal quotient]\index{Optimal quotient|textit}127An {\em optimal quotient} of $J_0(N)$ is a quotient~$A$ of128$J_0(N)$ by an abelian subvariety.129\end{definition}130Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.131By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A)$132are both finite.133The BSD conjecture\index{BSD conjecture!statement of}%134asserts that135$$\frac{L(A,1)}{\Omega_A} =136\frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p}137{\# A(\Q)\cdot\#\Adual(\Q)}.$$138Here the Shafarevich-Tate group\index{Shafarevich-Tate group}139$$\Sha(A) := \ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right)$$140is a measure of the failure141of the local-to-global principle\index{Local-to-global principle};142the Tamagawa numbers~$c_p$\index{Tamagawa numbers} are the143orders of the groups of rational points of the144component groups of~$A$ (see Chapter~\ref{chap:compgroups});145the real number~$\Omega_A$ is146the measure of~$A(\R)$ with respect to a basis of differentials having147everywhere nonzero good reduction (see Section~\ref{sec:realmeasure});148and~$\Adual$ is the abelian variety dual to~$A$ (see \cite[\S9]{milne:abvars}).149This chapter makes a small contribution to the long-term goal150of verifying the above conjecture for many specific abelian varieties151on a case-by-case basis. In a large list of examples, we compute152the conjectured order of $\Sha(A)$, up to a power of $2$, and then153show that $\Sha(A)$ is at least as big as conjectured.154We also discuss methods to obtain upper bounds on $\#\Sha(A)$, but do155not carry out any computations in this direction.156This is the first step in a program to verify the above157conjecture for an infinite family of quotients of~$J_0(N)$.158159\subsection{The ratio $L(A,1)/\Omega_A$}160Extending classical work on elliptic curves,161A.~Agashe\index{Agashe} and the author proved the following162theorem.163\begin{theorem}\label{thm:ratpart}164Let~$m$ be the largest square dividing~$N$.165The ratio $L(A,1)/\Omega_A$ is a rational number that can be166explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.167\end{theorem}168\begin{proof}169The proof uses modular symbols\index{Modular symbols}170combined with an extension of the argument171used by Mazur\index{Mazur} in~\cite{mazur:rational} to bound172the Manin constant\index{Manin constant}.173The modular symbols part of the proof for $L$-functions attached174to newforms of weight $k\geq 2$ is given in Section~\ref{sec:rationalvals};175it involves expressing the176ratio $L(A,1)/\Omega_A$ as the lattice177index\index{Lattice index} of178two modules over the Hecke algebra\index{Hecke algebra}.179The bound on the Manin constant\index{Manin constant} is given in180Section~\ref{sec:maninconstant}.181\end{proof}182183The author has computed $L(A,1)/\Omega_A$ for all simple optimal184quotients of level $N\leq 1500$; this table can be185obtained from the author's web page.186187\begin{remark}188The method of proof should also give similar results for special189values of twists of $L(A,s)$, just as it does in the case $\dim A=1$190(see~\cite[Prop.~2.11.2]{cremona:algs}).191\end{remark}192193194\subsection{Torsion subgroup\index{Torsion subgroup}}195We can compute upper and lower bounds on $\#A(\Q)_{\tor}$,196see Section~\ref{sec:torsionsubgroup};197these frequently determine $\#A(\Q)_{\tor}$.198199These methods, combined with the method used200to obtain Theorem~\ref{thm:ratpart},201yield the following corollary, which supports the expected202cancellation between torsion and~$c_p$ coming from the reduction203map sending rational points to their image in the component204group of~$A$. The corollary also generalizes to higher weight forms,205thus suggesting a geometric way to think about reducibility206of modular Galois representations.207\begin{corollary}208Let~$n$ be the order of the image of209$(0)-(\infty)$ in $A(\Q)$, and210let~$m$ be the largest square dividing~$N$.211Then212$n\cdot L(A,1)/\Omega_A \in \Z[1/(2m)].$213\end{corollary}214For the proof, see Corollary~\ref{cor:denominator}215in Chapter~\ref{chap:computing}.216217\subsection{Tamagawa numbers\index{Tamagawa numbers}}218We prove the following theorem in Chapter~\ref{chap:compgroups}.219\begin{theorem}\label{thm:tamagawa}220When $p^2\nmid N$, the number~$c_p$ can be explicitly computed221(up to a power of~$2$).222\end{theorem}223224We can compute the order~$c_p$ of the group of rational225points of the component group, but not226its structure as a group.227When $p^2 \mid N$ it may be possible228to compute~$c_p$ using the229Drinfeld-Katz-Mazur model230of~$X_0(N)$, but we have not yet done this.231There are also good bounds on the primes that can divide $c_p$ when232$p^2\mid N$.233234Systematic computations (see Section~\ref{sec:compgroupconjectures})235using this formula suggest the236following conjectural refinement of a result237of Mazur~\cite{mazur:eisenstein}\index{Mazur}.238\begin{conjecture}239Suppose~$N$ is prime and~$A$240is an optimal quotient of $J_0(N)$ corresponding241to a newform~$f$. Then $A(\Q)_{\tor}$ is242generated by the image of $(0)-(\infty)$243and $c_p = \#A(\Q)_{\tor}$. Furthermore,244the product of the~$c_p$ over all simple optimal quotients245corresponding to newforms equals the numerator of $(N-1)/12$.246\end{conjecture}247I have checked this conjecture for all $N\leq 997$ and,248up to a power of~$2$, for all $N\leq 2113$.249The first part is known when~$A$ is an elliptic250curve (see~\cite{mestre-oesterle:crelle}).251Upon hearing of this conjecture, Mazur\index{Mazur} reportedly252proved it when all ``$q$-Eisenstein quotients'' are simple.253There are three promising approaches to finding254a complete proof. One involves the explicit255formula of Theorem~\ref{thm:tamagawa};256another is based on Ribet's\index{Ribet} level lowering theorem257(see~\cite{ribet:modreps}),258and a third makes use of a simplicity result of Merel\index{Merel}259(see~\cite{merel:weil}).260261262The formula that lies behind Theorem~\ref{thm:tamagawa} probably263has a natural analogue in weight greater than~$2$.264One could then guess that it produces Tamagawa numbers\index{Tamagawa numbers}265of motifs\index{Motifs} attached to eigenforms of higher weight; however,266we have no idea if this is really the case. These numbers appear267in the conjectures of Bloch and Kato,268\index{Conjecture!Bloch and Kato}%269\index{Bloch and Kato conjecture}%270which generalize the BSD conjecture\index{BSD conjecture!generalization of} to271motifs (see~\cite{bloch-kato}).272Anyone wishing to273try to compute them should be aware of Neil Dummigan's274paper~\cite{dummigan:cp}, which gives some information275about the Tamagawa numbers\index{Tamagawa numbers}276of motifs\index{Motifs} attached by277Scholl in~\cite{scholl:motivesinvent}278to modular eigenforms.279280\subsection{Upper bounds on $\#\Sha(A)$}281V.~Kolyvagin (see \cite{kolyvagin:structureofsha})282and K.~Kato (see, e.g., \cite{scholl:kato})283constructed Euler systems\index{Euler system} that284were used to prove that $\Sha(A)$ is {\em finite}285when $L(A,1)\neq 0$.286To verify the full BSD conjecture\index{BSD conjecture!verification of}287for certain abelian varieties, we must make the Kolyvagin-Kato288finiteness bound explicit.289Kolyvagin's bounds involve computations with Heegner290points\index{Heegner points},291and Kato's involve a study of the Galois representations292associated to~$A$.293294\subsubsection{Kolyvagin's bounds}%295\index{Bound of!Kolyvagin}296In~\cite{kolyvagin:mordellweil}, Kolyvagin obtains explicit upper297bounds for $\#\Sha(A)$ for a certain (finite) list of elliptic curves~$A$298by computing the index in $A(K)$ of the subgroup299generated by the Heegner point, where~$K$300is a suitable imaginary quadratic extension.301In~\cite{kolyvagin-logachev:totallyreal}, Kolyvagin and Logachev302generalize Kolyvagin's earlier results; in Section~1.6, ``Unsolved303problems'', they say that: ``If one were to compute the304height of a Heegner point~$y$ [...]305considered in the present paper, then one would have succeeded in306obtaining an upper bound for $\#\Sha$ for this curve.''307(By ``curve'' they mean abelian variety.)308This suggests that explicit computations should yield upper309bounds on the order of $\Sha(A)$, but that they had not yet310figured out how to carry out such computations.311312\subsubsection{Kato's bounds}%313\index{Bound of!Kato}314Kato has constructed Euler systems\index{Euler system} coming315from $K_2$-groups of modular316curves. These can be used to prove the following theorem (see, e.g.,317\cite[Cor.~3.5.19]{rubin:book}).318\begin{theorem}[Kato]319Suppose~$E$ is an elliptic curve over~$\Q$ without complex320multiplication that~$E$ has conductor~$N$,321that~$E$ has good reduction at~$p$, that~$p$ does not divide322$2r_E\prod_{q\mid N} L_q(q^{-1})\#E(\Q_q)_{\tor}$, and323the Galois representation $\rho_{E,p}:\GQ\ra\Aut(E[p])$324is surjective. Then325$$\#\Sha(E)_{p^{\infty}}\text{ divides }326\frac{L(E,1)}{\Omega_E}.$$327\end{theorem}328Here $L_q(x)$ is the local Euler factor at~$q$ and the constant329$r_E$ arises in the construction of Kato's Euler system.330Rubin suggests that computing $r_E$ is not very331difficult (private communication).332Appropriate variants of Kato's arguments333give similar results for quotients of $J_0(N)$ of arbitrary334dimension, though these have not been written down.335336\comment{337>How mysterious is the constant r_E in, for example, Corollary 3.5.19 of338>your Euler Systems book?339340I think it's not too bad. Certainly nothing like Heegner points are341involved. When I wrote that part of my book, and the similar paper in342the Durham proceedings, I did not really know what it was because Kato343hadn't written anything. I don't have Scholl's paper here so I'm not344certain, but I suspect that the only contribution to $r_E$ comes from345passing from the modular curve to $E$, and perhaps some extra 2's and3463's.347348Karl349}350351352\subsection{Lower bounds on $\#\Sha(A)$}353One approach to showing that~$\Sha(A)$ is as {\em at least} as354large as predicted355by the BSD conjecture\index{BSD conjecture!and $\Sha$}356is suggested by Mazur's\index{Mazur} notion of357the visible part $\Sha(A)^{\circ}$358of~$\Sha(A)$ (see~\cite{cremona-mazur, mazur:visthree}).359Let~$\Adual\subset J_0(N)$ be the dual to~$A$.360The \defn{visible part}\index{Visibility!of $\Sha$|textit}%361\index{Shafarevich-Tate group!visible part of}362of $\Sha(\Adual)$ is the363kernel of the natural map364$\Sha(\Adual)\ra \Sha(J_0(N))$.365Mazur\index{Mazur} observed that if an element of order~$p$366in~$\Sha(\Adual)$ is visible,367then it is explained by a ``jump in the rank of Mordell-Weil''368in the sense that there is another abelian subvariety $B\subset J_0(N)$369such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.370371Mazur's\index{Mazur} observation can be turned around: if there is another abelian372variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,373then, under mild hypotheses (see Theorem~\ref{thm:shaexists}), there374is an element of~$\Sha(\Adual)$ of order~$p$. From a computational375point of view it is easy to understand the intersections376$\Adual\intersect B$; see Section~\ref{sec:intersection}.377From a theoretical point of view, nontrivial378intersections ``correspond'' to congruences between modular forms.379Thus the well-developed380theory of congruences between modular forms%381\index{Modular forms!congruences between}%382\index{Congruences!and lower bounds on $\Sha$}383can be used to obtain a lower bound on~$\#\Sha(\Adual)$.384385\subsubsection{Invisible elements of $\#\Sha(\Adual)$}386\index{Shafarevich-Tate group!invisible elements of}387\index{Invisible elements of $\Sha$}388Numerical experiments suggest389that as $\Adual$ varies, $\Sha(\Adual)$ is390often {\em not} visible inside of~$J_0(N)$.391For example (see Table~\ref{table:primesha}), the392BSD conjecture\index{BSD conjecture!predicts invisible elements}393predicts the existence of invisible elements of odd394order in~$\Sha(\Adual)$395for almost half of the~$37$ optimal quotients396of prime level $\leq 2113$.397398\subsubsection{Visibility at higher level}399\index{Shafarevich-Tate group!visibility at higher level}400\index{Visibility!at higher level}401For every integer~$M$ (Ribet~\cite{ribet:raising}\index{Ribet}402tells us which~$M$403to choose), we can ask whether $\Sha(\Adual)$ maps to~$0$404under one of the natural maps $\Adual\ra J_0(NM)$; that is, we405can ask whether $\Sha(\Adual)$ ``becomes visible at406level $NM$.''407We have been unable to prove in any particular case that $\Sha(\Adual)$ is408not visible at level~$N$, but becomes visible at some level $NM$.409See Section~\ref{sec:higherlevel} for some computations which strongly410indicate that such examples exist.411412\subsubsection{Visibility in some Jacobian}%413\index{Visibility!in some Jacobian}%414\index{Jacobian!visibility in}%415Johan de Jong proved that if~$E$ is an elliptic curve416over a number field~$K$ and $c\in H^1(K,E)$ then there is a417Jacobian~$J$ and an imbedding $E\hookrightarrow J$ such that~$c$ maps418to~$0$ under the natural map $H^1(K,E)\ra H^1(K,J)$ (see Remark~3419in~\cite{cremona-mazur}); de Jong's proof appears to generalize420when~$E$ is replaced by an abelian variety, but time does not permit421going into the details here.422423\subsection{Motivation for considering abelian varieties}424If~$A$ is an elliptic curve, then explaining~$\Sha(A)$ using425only congruences between elliptic curves will probably fail.426\index{Congruences!between elliptic curves}427This is because pairs of non-isogenous elliptic curves with isomorphic428$p$-torsion for large~$p$ are, according to E.~Kani's\index{Kani}429conjecture, extremely rare.%430\index{Conjecture!Kani}431It is crucial to understand what happens in all dimensions.432433Within the range accessible by computer, abelian varieties exhibit434more richly textured structure than elliptic curves. For example, there435is a visible element of prime order $83341$ in the436Shafarevich-Tate group\index{Shafarevich-Tate group} of an abelian437variety of prime conductor~$2333$; in contrast, over all optimal438elliptic curves of conductor up to $5500$, it appears that the largest439order of an element of a Shafarevich-Tate group is~$7$ (see the440discussion in~\cite{cremona-mazur}).441442\section{Existence of nontrivial visible elements of $\Sha(A)$}%443The reader who wants to see tables of Shafarevich-Tate groups can444safely skip to the next section.445446Without relying on any unverified conjectures,447we will use the following theorem to exhibit abelian varieties~$A$448such that the visible part of $\Sha(A)$ is nonzero.449In the following theorem we do {\em not} assume that~$J$ is the450Jacobian\index{Jacobian} of a curve.451\begin{theorem}\label{thm:shaexists}\index{Visibility!existence theorem}452Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such453that $A\intersect B$ is finite and $A(\Q)$ is finite.454Assume that~$B$ has purely toric reduction455at each prime at which~$J$ has456bad reduction.457Let~$p$ be an odd prime at which~$J$ has good reduction, and458assume that~$p$ does not divide the orders of any of459the (geometric) component groups\index{Component group!geometric}460of~$A$ and~$B$,461or the orders of the torsion subgroups of $(J/B)(\Q)$ and $B(\Q)$.462Suppose further that $B[p] \subset A\intersect B$.463Then there exists an injection464$$B(\Q)/pB(\Q)\hookrightarrow \Sha(A)^{\circ}$$465of $B(\Q)/p B(\Q)$ into the visible part of $\Sha(A)$.466\end{theorem}467468\begin{proof}469Let $C=J/A$.470The long exact sequence of Galois cohomology471associated to the short exact sequence472$$0 \ra A \ra J \ra C \ra 0$$473begins474$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,}475H^1(\Q,A) \ra \cdots.$$476Because $B[p]\subset A$, the map $B\ra C$, obtained by composing477the inclusion478$B\hookrightarrow J$ with $J\ra C$, factors through multiplication-by-$p$,479giving the following commutative diagram:480$$\xymatrix{481& B\ar[d] \ar[r]^{p}& B\ar[d]\\482A\ar[r]&J\ar[r]&C.}$$483Because $B(\Q)[p]=0$ and $B(\Q)\intersect A(\Q)=0$, we484deduce the following commutative diagram with exact485rows:486$$\xymatrix{487& 0\ar[d] & K_1\ar[d]& K_2\ar[d]\\4880 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d]489& B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\4900 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\491& K_3,492}$$493where $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the cokernel.494We have the snake lemma exact sequence495$$0\ra K_1 \ra K_2 \ra K_3.$$496Because $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group,497we have $K_1=0$.498The quotient $J(\Q)/B(\Q)$ has no $p$-torsion because499it is a subgroup of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group500of order coprime to~$p$,501so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion. Thus $K_2=0$.502503The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of504$H^1(\Q,A)$. However, $H^1(\Q,A)$ contains infinitely many elements of505order~$p$ (see~\cite{shafarevich:exp}),506whereas $\Sha(A)[p]$ is a finite group, so we must work507harder to deduce that $B(\Q)/p B(\Q)$ lies in508$\Sha(A)[p]$. Fix $x\in B(\Q)$. We must show509that $\pi(x)$ lies in $\Sha(A)[p]$; equivalently, that510$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.511512At the archimedean place $v=\infty$, the restriction513$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,514hence $\res_v(\pi(x))=0$.515516Suppose that~$v$ is a place at which~$J$ has bad reduction.517By hypothesis, $B$ has purely toric reduction\index{Purely toric reduction},518so over the maximal unramified extension $\Q_v^{\ur}$519of $\Q_v$ there is an isomorphism $B\isom\Gm^d/\Gamma$520of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,521for some ``lattice'' $\Gamma$.522For example, when523$\dim B=1$, this is the Tate curve representation of~$B$.524Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$525equals the order of the cokernel of the valuation526map $\Gamma\ra \Z^d$. Choose a representative $P=(x_1,\ldots,x_d)\in\Gm^d$527for the point~$x$. Then $nP$ can be adjusted by elements of~$\Gamma$528so that each of its components $x_i\in\Gm$ has valuation~$0$.529By assumption,~$p$ is a prime at which~$J$ has good reduction, so530$p\neq v$;531it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$ such that532$pQ = nP$.533Thus the cohomology class $\res_v(\pi(nx))$ is unramified534at~$v$. By \cite[Prop.~I.3.8]{milne:duality},535$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur}))536=H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v)),$$537where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.538Since\item{There is a mistake here, but it is easy to fix.}539the component group $\Phi_{A,v}(\Fbar_v)$ has540order~$n$, it follows that $$\res_v(\pi(nx))=n\res_v(\pi(x))=0.$$541Since the order~$p$ of $\res_v(\pi(x))$ is coprime to~$n$,542we conclude that $\res_v(\pi(x))=0$.543544Next suppose that~$J$ has good reduction at~$v$545and that~$v$ is {\em odd}, in the sense that the546residue characteristic of~$v$ is odd. To simplify notation in547this paragraph, since~$v$ is a non-archimedean place548of $\Q$, we will also let~$v$ denote the odd prime number549which is the residue characteristic of~$v$.550Let $\cA$, $\cJ$, $\cC$, be the N\'eron models551of~$A$,~$J$, and~$C$, respectively (for more on N\'eron552models, see Chapter~\ref{chap:compgroups}).553Let $A$, $J$, $C$, also denote the sheaves on554the \'etale-site over $\Spec(\Z_v)$ determined555by the group schemes $\cA$, $\cJ$, and $\cC$, respectively.556Since~$v$ is odd, $1=e<v-1$, so we may apply557\cite[Thm.~7.5.4]{neronmodels} to conclude that558the sequence of group schemes559$$0\ra \cA \ra \cJ\ra \cC \ra 0$$560is exact; in particular, it561is exact as a sequence of sheaves on the562\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).563Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}564the sequence565$$0\ra \cA(\Z_v^{\ur})\ra \cJ(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$566is exact; by the N\'eron mapping property the sequence567$$0\ra A(\Q_v^{\ur})\ra J(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$568is also exact.569Thus $\res_v(\pi(x))$ in unramified,570so it arises by inflation from571an element of $H^1(\Q_v^{\ur}/\Q_v,A)$.572By \cite[Prop.~I.3.8]{milne:duality},573$$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}),$$574where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.575Since~$A$ has good reduction, $\Phi_{A,v}=0$, hence576$\res_v(\pi(x))=0$.577578If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.579Consider the case when~$J$ has good reduction at~$2$. Because the580absolute ramification index~$e$ of $\Z_2$ is~$1$, which is581{\em not} less than $v-1=1$, we can not apply \cite[Thm.~7.5.4]{neronmodels}.582However, we can modify our situation by an isogeny of degree a power583of~$2$, then apply a different theorem as follows.584The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is585rational as a subgroup over~$\Q$, in the sense that the586conjugates of any point in $\Psi$ are587also contained in $\Psi$.588The abelian varieties589$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and590$\tilde{B}=B/\Psi$ also satisfy the hypothesis of591the theorem we are proving.592By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of593N\'eron models594$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$595is exact, so the sequence596$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur})597\ra\tilde{C}(\Q_v^{\ur})\ra 0$$598is exact. Thus the image of599$\res_v(\pi(x))$ in $H^1(\Q_v,\tilde{A})$ is unramified.600It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},601since the component group of $\tilde{A}$ at~$v$ has order a power602of~$2$ (in fact it is trivial, since $\tilde{A}$ has603good reduction at~$2$), whereas $\pi(x)$ has odd prime order~$p$.604Thus $\res_v(\pi(x))=0$, since605the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a606finite group of $2$-power order.607\end{proof}608609610611612\section{Description of tables}613In this section we describe our tables of optimal quotients of614$J_0(N)$, which have nontrivial Shafarevich-Tate group.615The tables, which can be found on616pages~\pageref{table:primesha}--\pageref{table:shacompgps},617were computed using a combination of618\hecke{}~\cite{stein:hecke}, {\sc LiDIA}, {\sc NTL}, {\sc Pari}, and619most successfully \magma{}~\cite{magma}. The component group620computations at non-prime level rely on Kohel's quaternion621algebra package, which was also written in \magma{}.622623We compute the conjectural order of the Shafarevich-Tate group of an624abelian variety~$A$, and then make assertions about the625Shafarevich-Tate group of~$\Adual$. This is justified because the626order of $\Sha(\Adual)$ equal the order of~$\Sha(A)$, since both are627finite and the Cassells-Tate pairing sets up a nondegenerate duality628between them.629630\subsection{Notation}\label{sec:optquo-notation}631Each optimal quotient~$A$ of $J_0(N)$ is denoted by a label, such as632{\bf 389E}, which consists of a level~$N$ and a letter indicating the633isogeny class. In the labeling,~$N$ is a positive integer and the634isogeny class is given by a letter: the first isogeny class is labeled635{\bf A}, the second is labeled {\bf B}, the third labeled {\bf C}, and636so on. Thus {\bf 389E} is the fifth isogeny class of optimal quotient637of $J_0(389)$, corresponding to a Galois-conjugacy class of newforms.638The isogeny classes that we consider are in bijection with the639Galois-conjugacy classes of newforms in $S_2(\Gamma_0(N))$. The640classes of newforms are ordered as described in Section~\ref{sec:sorting}.641642{\bf WARNING:} The {\em odd part} of a rational number $x$ is $x/2^v$,643where $v=\ord_2(x)$. In the tables, only the {\bf odd parts} of the644arithmetic invariants of~$A$ are given.645646\subsection{Table~\ref{table:primesha}: Shafarevich-Tate647groups at prime level}648649Table~\ref{table:primesha} was constructed as follows. Using the650results of Section~\ref{sec:ratpartformula}, we computed the odd part651of the conjectural order $\Shaan(A)$ of the Shafarevich-Tate group of652every optimal quotient of $J_0(p)$ that corresponds to a single Galois653conjugacy-class of eigenforms and has analytic rank~$0$, for~$p$ a654prime with $p\leq 2161$. We also computed a few sporadic examples of655prime level~$p$ with $p>2161$. The sporadic examples appear at the656bottom of the table below a horizontal line.657658\subsubsection{Notation}659The columns of the table contain the following information. The660abelian varieties~$A$ for which $\Shaan(A)$ is greater than~$1$ are661laid out in the first column of Table~\ref{table:primesha}. The662second column contains the dimensions of the abelian varieties in the663first column. The third column contains the {\em odd part} (i.e.,664largest odd divisor) of the order of the Shafarevich-Tate group, as665predicted by the BSD conjecture.\index{BSD conjecture!predicted order of $\Sha$}666Column four contains the odd parts667of the modular degrees of the abelian varieties in the first column.668669The fifth column contains an optimal quotient~$B$ of $J_0(p)$ of670positive analytic rank, such that the $\ell$-torsion of $\Bdual$ is671contained in~$\Adual$, when one exists, where $\ell$ is a divisor672of $\Shaan(A)$. We computed this673intersection using the algorithm described in674Section~\ref{sec:intersection}. Such a~$B$ is called an675\defn{explanatory factor}.\index{Explanatory factor}676When no such abelian varieties exists, we write ``NONE'' in677the fifth column. The sixth column contains the dimensions of the678abelian varieties in the fifth column, and the seventh column contains679the odd parts of the modular degrees of the abelian varieties in the680fifth column.681682\subsubsection{Ranks of the explanatory factors}683That the explanatory factors have positive analytic rank follows from684our modular symbols computation of $L(B,1)/\Omega_B$.685This is supported by the table in~\cite{brumer:rank}, except686in the case {\bf 2333A}, where there is a mistake in \cite{brumer:rank}687(see below).688689The explanatory factor {\bf 389A} is the first elliptic curve of690rank~$2$. The table in \cite{brumer:rank} suggests that the691explanatory factor {\bf 1061B} is the first $2$-dimensional abelian692variety (attached to a newform) whose Mordell-Weil group when tensored693with the field of fractions~$F$ of the corresponding ring of Fourier694coefficients, is of dimension~$2$ over~$F$. Similarly695{\bf 1567B} appears to be the first $3$-dimensional one of rank~$2$, and696{\bf 2333A} is the first $4$-dimensional one of rank~$2$.697It thus seems very likely that the ranks of each explanatory factor698is exactly~$2$, though we have not proved this.699700\subsubsection{Discussion of the data}701There are~$23$ examples in which~$\Sha(A)$ is702visible and~$18$ in which~$\Sha(A)$703is invisible. The largest visible704$\Sha(A)$ found occurs at level $2333$ and has order at least $83341^2$705($83341$ is prime).706The largest invisible\index{Invisible elements of $\Sha$} $\Sha(A)$707occurs in a $112$-dimensional abelian variety at level708$2111$ and has order at least $211^2$.709710The example {\bf 1283C} demonstrates that $\Shaan(A)$ can divide the711modular degree, yet be {\em invisible}. In this case~$5$ divides712$\Shaan(A)$. Since~$5$ divides the713modular degree, it follows that there must be714other non-isogenous subvarieties of $J_0(1283)$ that715intersect {\bf 1283C} in a subgroup of order divisible716by~$5$. In this case, the only such subvariety is717{\bf 1283A}, which has dimension~$2$ and whose $5$-torsion is contained718in {\bf 1283C}. However {\bf 1283A} has analytic (hence algebraic) rank~$0$,719so $\Shaan(A)$ cannot be visible.720721The cases {\bf 1483D}, {\bf 1567D}, {\bf 2029C}, and {\bf 2593B} are722interesting because {\em all} of~$\Sha$, even though it has two723nontrivial $p$-primary components in each of these cases, is made724visible in a single~$B$. In the case {\bf 1913E} only725the $5$-primary component of $\Sha$ is visible in {\bf 1913A}, but726still {\em both} the $5$-primary and727$61$-primary components of $\Sha$ are visible in {\bf 1913C}.728729Examples {\bf 1091C} and {\bf 1429B} were first found in730\cite{agashe} and {\bf 1913B} in \cite{cremona-mazur}.731732\subsubsection{Errata to Brumer's paper}733Contrary to our computations, \cite{brumer:rank} suggests that734{\bf 2333A} has rank~$0$. However, the author pointed the discrepancy out735to Brumer who replied:736\begin{quote}737I looked in vain for information about $\theta$-relations on~$2333$738and have concluded that I never ran the interval from~$2300$ to~$2500$739or else had lost all info by the time I wrote up the paper. The740punchline:~$4$ relations for~$2333$ and~$2$ relations for~$2381$ (also741missing from the table).742\end{quote}743744\comment{745Date: Wed, 08 Sep 1999 18:24:10 -0400746From: armand brumer <brumer@murray.fordham.edu>747To: William Arthur Stein <was@math.berkeley.edu>748CC: ab <brumer@murray.fordham.edu>749Subject: Re: The rank of J_0(2333)750751Dear William,752I just found your 3 emails (including one from the end of June)753sitting on a mail server I did not know existed until a few days ago (the754university did not tell us that the two addresses were on different755servers!!)756757I then looked in vain for information about theta relations on 2333 and have758concluded that I never ran the interval from 2300 to 2500 or else had lost759all info by the time I wrote up the paper. The punchline:4 relations for 2333760and 2 relations for 2381 (also missing from the table). I may try to check as761much as possible in the background and would be grateful if you mention this762errata when you write up your stuff (I don't know any other way of763publicizing the correction).764765Best regards and hope you did not think I was ``blowing you off" as my son766would say!767768Armand769}770771772\subsection{Tables~\ref{table:newvis}--\ref{table:shacompgps}:773New visible Shafarevich-Tate groups}774775Let~$n$ denote the largest odd square dividing the numerator of776$L(A,1)/\Omega_A$. Table~\ref{table:newvis} lists those~$A$ such that777for some $p\mid n$ there exists a quotient~$B$ of $J_0(N)$,778corresponding to a {\em newform} and having positive analytic rank,779such that $(\Adual\intersect B^{\vee})[p]\neq 0$. Our search was780systematic up to level $1001$, but there might be omitted examples781between levels $1001$ and $1028$.782Table~\ref{table:explain} contains783further arithmetic information about each explanatory factor.784Table~\ref{table:shacompgps} gives the quantities involved in the785formula of Chapter~\ref{chap:compgroups} for Tamagawa numbers, for786each of the abelian varieties~$A$ in Table~\ref{table:newvis}.787788789\subsubsection{Notation}790Most of the notation is the same as in Table~\ref{table:primesha}.791However the additional columns $w_q$ and $c_p$ contain the signs792of the Atkin-Lehner involutions and the Tamagawa numbers, respectively.793These are given in order, from smallest to largest prime divisor794of~$N$.795796In each case~$B$ has dimension~$1$. When $4\mid N$, we write ``$a$''797for $c_2$ to remind us that we did not compute $c_2$ because the798reduction at~$2$ is additive. Again only799{\em odd parts} of the invariants are given.800Section~\ref{sec:compgrptables} contains a discussion of801the notation used in the802headings of Table~\ref{table:shacompgps}.803804\subsubsection{Remarks on the data}805The explanatory factors~$B$ of level $\leq 1028$ are {\em exactly} the806set of rank~$2$ elliptic curves of level $\leq 1028$.807808809\section{Further visibility computations}810811812813\subsection{Does $\Sha$ become visible at higher level?}814\label{sec:higherlevel}815816This section is concerned with whether or not the examples of invisible817elements of Shafarevich-Tate groups of elliptic curves of level~$N$818that are given in \cite{cremona-mazur} become visible in abelian819surfaces inside appropriate $J_0(Np)$. We analyze each of the cases820in Table~1 of \cite{cremona-mazur}. For the reader's convenience, the821part of this table which concerns us is reproduced as822Table~\ref{table:cm}.823The most interesting examples we give824are {\bf 2849A} and {\bf 5389A}. As in825\cite{cremona-mazur}, the assertions below assume the826truth of the BSD conjecture.\index{BSD conjecture}827828829\begin{table}\index{Table of!odd invisible $\Sha_E$}830\ssp831\caption{Odd invisible $|\Sha_E|>1$, all $N\leq 5500$ (from Table~1 of~\cite{cremona-mazur})}832\label{table:cm}833$$834\begin{array}{lcclcl}835\mbox{\rm\bf E}&\sqrt{|\Sha_E|}& m_E & \mbox{\rm\bf F}836& m_F & \text{Remarks}\\837& & & & & \vspace{-3ex} \\838\mbox{\rm\bf 2849A}& 3 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& - &\\839\mbox{\rm\bf 3364C}& 7 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& - &\\840\mbox{\rm\bf 4229A}& 3 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& - &\\841\mbox{\rm\bf 4343B}& 3 &2^4\cdot1583 &\mbox{\rm\bf NONE}& -&\\842\mbox{\rm\bf 4914N}& 3 &2^4\cdot 3^5 &\mbox{\rm\bf none}& - &\text{{\bf E} has rational $3$-torsion}\\843\mbox{\rm\bf 5054C}& 3 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& - &\\844\mbox{\rm\bf 5073D}& 3 &2^5\cdot 3\cdot 5\cdot7\cdot23845&\mbox{\rm\bf none}& - & \\846\mbox{\rm\bf 5389A}& 3 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& - &\\847\end{array}848$$849\end{table}850851852\subsubsection{How we found the explanatory curves}853We use a naive heuristic observation to find possible explanatory854curves of higher level, even though their conductors are out of the855range of Cremona's tables. Note that we have not proved that856these factors are actually explanatory in any cases, and expect857that in some cases they are not.858859First we recall some of the notation from Table~1860of~\cite{cremona-mazur}, which is partially reproduced below.861The ``NONE'' label in the row corresponding862to an elliptic curve~$E$ indicates that there are elements in863$\Sha(E)$ whose order does not divide the modular degree of~$E$, and864hence they must be invisible. The label ``none'' indicates only that865no suitable explanatory elliptic curves could be found, so $\Sha(E)$ is866not visible in an {\em abelian surface} inside $J_0(N)$; it could,867however, be visible in the full abelian variety $J_0(N)$.868869870Studying the Weierstrass equations corresponding to the curves in871\cite{cremona-mazur} reveals that the elliptic curves labeled872``NONE'' have unusually large height, as compared to their conductors.873However, the explanatory factors often have unusually small height.874Motivated by this purely heuristic observation, we make a table of875all elliptic curves of the form876$$y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x+ a_6,$$877with $a_1, a_2, a_3 \in \{-1,0,1\}$, $|a_4|, |a_6| < 1000$,878and conductor bounded by $50000$. The bound on the conductor879is required only so that the table will fit within880computer storage. This table took a few days to generate.881882\subsubsection{2849A} Mazur\index{Mazur} and Adam Logan\index{Logan}883found the first known example of an884{\em invisible} Shafarevich-Tate group\index{Shafarevich-Tate group!first invisible example}. This885was $\Sha(E)$, where~$E$ is the elliptic curve {\bf 2849A},886which has minimal Weierstrass equation887$$E:\quad y^2 + xy + y = x^3 + x^2 - 53484x - 4843180.$$888Consulting our table of curves of small height,889we find an elliptic curve~$F$ of conductor890$8547=2839\cdot 3$ such that $f_E \con f_F \pmod{3}$, where $f_E$891and $f_F$ are the newforms attached to~$E$ and~$F$, respectively.892This is a congruence for {\em all} eigenvalues $a_p$ attached to~$E$ and~$F$.893The elliptic curve~$F$ is defined by the equation894$$F:\quad y^2 + xy + y = x^3 + x^2 - 154x - 478.$$895Cremona's program {\tt mwrank} reveals that896the Mordell-Weil group of~$F$ has rank~$2$.897Thus maybe $\Sha(E)$ becomes visible at level~$8547$.898Unfortunately, visibility of $\Sha(E)$ is not899implied by Theorem~\ref{thm:shaexists} because900the geometric component group of~$F$ at~$3$ has order901divisible by~$3$.902903\subsubsection{4343B} Consider the elliptic curve $E$ labeled904{\bf 4343B}. According to Table~1 of \cite{cremona-mazur},905$\Sha(E)$ has order~$9$, but the modular degree prevents~$\Sha(E)$906from being visible in $J_0(4343)$.907At level $21715 = 5\cdot 4343$908there is an elliptic curve~$F$ of rank~$1$ that is909congruent to~$E$. Its equation is910$$F:\quad y^2 - xy - y = x^3 - x^2 + 78x - 256.$$911912\subsubsection{5389A} The last curve labeled ``NONE'' in the table is curve913{\bf 5389A}, which has minimal Weierstrass equation914$$y^2+xy+y =x^3 - 35590x-2587197.$$915916The main theorem of~\cite{ribet:raising} implies that there exists a917newform that is congruent modulo~$3$ to the newform corresponding to918{\bf 5389A} and of level $3\cdot 5389$. This is because $(-2)^2 =919(3+1)^2 \pmod{3}$. However, our table of curves of small height does920not contain any curve of conductor $3\cdot 5389$. Next we observe that921$(-2)^2 \con (7+1)^2 \pmod{3}$, so using Ribet's\index{Ribet} theorem we can922instead augment the level by~$7$. Our table of small-height curves923contains just one (up to isogeny) elliptic curve of924conductor~$37723$, and {\em luckily} the925corresponding newform is congruent modulo~$3$ to the newform926corresponding to {\bf 5389A} (away from primes dividing the level)!927The Weierstrass equation of this curve is928$$F:\quad y^2 - y = x^3 + x^2 + 34x - 248.$$929According to Cremona's program {\tt mwrank}, the rank of~$F$ is~$2$.930931\subsubsection{3364C, 4229A, 5073D}932Perhaps $\Sha(E)$ is already visible in some of the cases in which the933curve is labeled ``none'', because the method fails in most934of these cases. Each of the curves {\bf 3364C}, {\bf9354229A}, and {\bf 5073D} is labeled ``none''.936In none of these 3 cases are we able to find937an explanatory factor at higher level, within the range of our table938of elliptic curves of small height.939940\subsubsection{4194N, 5054C} The curve {\bf 4914N} is labeled ``none''941and we find the remark ``$E$ has rational $3$-torsion''.942There is a congruent curve~$F$ of conductor $24570$ given943by the equation944$$F: \quad y^2 - xy = x^3 - x^2 - 15x - 75,$$945and $F(\Q) = \{0\}$. The curve {\bf 5054C} is labeled ``none''946and its Shafarevich-Tate group contains invisible elements of947order~$3$. We find a congruent curve of level~$25270$ and rank~$1$.948The equation of the congruent curve is949$$F: y^2 - xy = x^3 + x^2 - 178x + 882.$$950951952\subsection{Positive rank example}953The abelian varieties with nontrivial $\Sha(A)$ that one954finds in both ours and Cremona's955tables all have rank~$0$. In this section we outline a computation956which sugggests, but does not prove, that there is a positive-rank abelian957subvariety $A$ of $J_0(5077)$ such that $\Sha(A)$958possesses a nontrivial visible element of order~$31$.959960According to \cite{cremona:algs},961the smallest conductor elliptic curve~$E$ of rank~$3$ is found in962$J=J_0(5077)$. The number $5077$ is prime, and the isogeny963decomposition of~$J$ is\footnote{964This decomposition was found in about one minute965using the Mestre-Oesterl\'e\index{Mestre}966method of graphs (see~\cite{mestre:graphs}).}967$$J \sim A \cross B \cross E,$$968where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$969associated to newforms, which have970dimensions $205$, $216$, and~$1$, respectively.971Using Remark~\ref{rem:moddegmestre} or972\cite{zagier:parametrizations},973we find that the modular degree of~$E$ is $1984=2^6\cdot 31$.974The sign of the Atkin-Lehner involution on~$E$ is the same975as its sign on~$A$, so $E[31]\subset A$.976We have $E(\Q)\isom \Z\cross\Z\cross\Z$, and the977component group of~$E$ is trivial.978The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so \cite{mazur:eisenstein}979implies that none of the abelian varieties above have $31$-torsion.980It might be possible to find an analogue of Theorem~\ref{thm:shaexists}981that applies when~$A$ has positive rank, and deduce in this case982that $\Sha(A)$ contains visible elements of order~$31$.983984%\section{Tables}985\begin{table}\index{Table of!$\Sha$ at prime level}986\ssp987\caption{Shafarevich-Tate groups at prime level.988(The entries in the columns989``mod deg'' and ``$\Shaan$'' are only really990the odd parts of ``mod deg'' and ``$\Shaan$''.)\label{table:primesha}}991\vspace{-.25in}$$992\begin{array}{lccclcc}993\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan(A) & \mbox{\rm mod deg}(A) & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} (B)\\994& & & & & & \vspace{-3ex} \\995\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\996\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\997\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\998\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\999\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\1000\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\1001\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\1002\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\1003\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\1004\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\1005\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\1006\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\1007\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\1008\mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\1009\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\1010\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\1011\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\1012\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\1013\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\1014\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\1015\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\1016\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\1017\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\1018\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\1019\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\1020\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\1021\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A,C}& 1,2 &3\cdot5^{2}, 5^2\cdot 61\\1022\mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\1023\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\1024\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\1025\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\1026\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\1027\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\1028\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\1029\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\1030\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\1031\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\1032\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\\hline1033\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\1034\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\1035\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\1036\mbox{\rm\bf 2593B}& 109 &67^2\cdot 2213^2 & 67 \cdot 22131037&\mbox {\bf 2593A}& 41038& 67 \cdot 2213\\1039\end{array}1040$$1041\end{table}10421043\begin{table}\index{Table of!new visible $\Sha$}1044\ssp1045\caption{New visible Shafarevich-Tate groups\label{table:newvis}}\vspace{-2ex}1046$$\begin{array}{lcccccccl}1047\mathbf{A} &\text{dim} &\Shaan &w_q& c_p &\#A(\Q)1048&\frac{\#A(\Q)\cdot L(A,1)}{\Omega_A}1049&\mbox{\rm mod deg(A)} & \quad\mathbf{B}\\1050\vspace{-2ex} & & &&& & & \\1051\mathbf{389E} & 20&5^2&-&97&97&5^2&5&\mathbf{389A}\\1052\mathbf{433D} & 16&7^2&-&3^2&3^2&7^2&3\cdot 7\cdot 37&\mathbf{433A}\\1053\mathbf{446F}&8&11^2&+- &1,3&3&11^2&11\cdot359353&\mathbf{446B}\\1054\mathbf{563E}&31 & 13^2 & - & 281 &281 &13^2 &13 &\mathbf{563A}\\1055\mathbf{571D}&2 & 3^2 & - & 1 &1 & 3^2 & 3^2\cdot 127&\mathbf{571B}\\1056\mathbf{655D}&13 & 3^4 &+- & 1,1 & 1 & 3^4 & 3^2\cdot 19\cdot 515741&\mathbf{655A}\\10571058\mathbf{664F} & 8&5^2 &-+ &a,1&1& 5^2 &5 & \mathbf{664A}\\1059% Sha dim Wq c_p T TL/O delta B1060\mathbf{681B}&1 & 3^2 &+- & 1,1 & 1 & 3^2 & 3\cdot 5^3 & \mathbf{681C}\\1061\mathbf{707G}& 15& 13^2 &+- & 1,1 & 1 & 13^2 & 13\cdot 800077& \mathbf{707A}\\1062\mathbf{709C}&30& 11^2 &- & 59 &59 & 11^2 & 11 & \mathbf{709A}\\1063\mathbf{718F}&7& 7^2 &+- & 1,1 & 1 & 7^2 &7\cdot 151\cdot 35573 & \mathbf{718B}\\1064\mathbf{794G}&14& 11^2 &+- & 3,1 & 3 & 11^2 &3\cdot7\cdot11\cdot47\cdot35447& \mathbf{794A}\\1065\mathbf{817E}& 15& 7^2 &+- & 1,5 & 5 & 7^2 & 7\cdot 79 & \mathbf{817A}\\1066\mathbf{916G}&9& 11^2 &-+ & a,1 & 1 & 11^2 & 3^9\cdot 11\cdot 17\cdot 239 & \mathbf{916C}\\1067\mathbf{944O} &6& 7^2 &+- & a,1 & 1 & 7^2 & 7 & \mathbf{944E}\\1068\mathbf{997H}&42& 3^4 &- & 83 & 83 & 3^4 & 3^2 & \mathbf{997BC}\\1069\mathbf{1001L}&7& 7^2 &+-+& 1,1,1 & 1 & 7^2 & 7\cdot19\cdot47\cdot2273&\mathbf{1001C}\\1070\mathbf{1028E}&14& 11^2&-+ & a,1 & 3 & 3^4\cdot 11^2 & 3^{13}\cdot 11 & \mathbf{1028A}\\1071\end{array}$$1072\end{table}10731074\begin{table}\index{Table of!explanatory factors}1075\ssp1076\caption{Explanatory factors\label{table:explain}}\vspace{-2ex}1077$$\begin{array}{lcccccc}1078\mathbf{B}&\text{rank}&w_q&c_p&\#A(\Q)&\mbox{\rm mod deg(A)}&\text{Comments}\\1079\vspace{-2ex} & & && & & \\1080\mathbf{389A}& 2 &-&1 &1&5&\text{first curve of rank $2$}\\1081\mathbf{433A}&2 &-&1&1&7&\\1082\mathbf{446B}&2 &+-&1,1& 1 &11&\text{called $\mathbf{446D}$ in \cite{cremona:algs}}\\1083\mathbf{563A}&2 &- & 1 & 1 & 13 & \\1084\mathbf{571B}&2 &- & 1 & 1 & 3 & \\1085\mathbf{655A}&2 &+-& 1,1 & 1 & 3^2 & \\1086\mathbf{664A}&2 &-+& 1,1 & 1 & 5 & \\1087% RANK wq g_p Tor delta comments1088\mathbf{681C} & 2 & +- & 1,1 & 1 & 3 & \\1089\mathbf{707A} & 2 & +- & 1,1 & 1 & 13 & \\1090\mathbf{709A} & 2 & - & 1 & 1 & 11 & \\1091\mathbf{718B} & 2 & +- & 1,1 & 1 & 7 & \\1092\mathbf{794A} & 2 & +- & 1,1 & 1 & 11 & \\1093\mathbf{817A} & 2 & +- & 1,1 & 1 & 7 & \\1094\mathbf{916C} & 2 & -+ & 3,1 & 1 & 3\cdot 11 & \\1095\mathbf{944E} & 2 & +- & 1,1 & 1 & 7 & \\1096\mathbf{997B} & 2 & - & 1 & 1 & 3 & \\1097\mathbf{997C} & 2 & - & 1 & 1 & 3 & \\1098\mathbf{1001C} & 2 & +-+& 1,3,1& 1 & 3^2\cdot 7 & \\1099\mathbf{1028A} & 2 & -+ & 3,1 & 1 & 3\cdot 11& \text{intersects $\mathbf{1028E}$ mod $11$}\\1100\end{array}$$1101\end{table}110211031104\begin{table}\index{Table of!factorizations}1105\ssp1106\caption{Factorizations\label{table:factor}}\vspace{-1ex}1107$$\begin{array}{llll}1108\mathbf{446}=2\cdot 223&1109\mathbf{655}=5\cdot 131&1110\mathbf{664}=2^3\cdot 83&1111\mathbf{681}=3\cdot 227\\1112\mathbf{707}=7\cdot 101&1113\mathbf{718}=2\cdot 359&1114\mathbf{794}=2\cdot 397&1115\mathbf{817}=19\cdot 43\\1116\mathbf{916}=2^2\cdot 229&1117\mathbf{944}=2^4\cdot 59&1118\mathbf{1001}=7\cdot 11\cdot 13&1119\mathbf{1028}=2^2\cdot 257\\1120\end{array}$$1121\end{table}11221123\begin{table}\index{Table of!component groups of explanatory factors}1124\ssp1125\caption{Component groups\label{table:shacompgps}}\vspace{-2ex}1126$$\begin{array}{lcccccc}1127\vspace{-2ex}&&&&&&\\1128\mathbf{A} &\text{dim} & p & w_q &\#\Phi_{X,p} &m_{X,p}1129&\#\Phi_{A,p}(\Fpbar) \\1130\vspace{-2ex}& & & & & \\1131\mathbf{389E}&20& 389&-& 97 & 5\cdot 97 & 97 \\1132\mathbf{433D}&16& 433&-& 3^2& 3^3\cdot 7\cdot 37 & 3^2 \\1133\mathbf{446F}&8 & 223&-& 3 & 3\cdot 11\cdot 359353 & 3 \\1134&& 2 &+ & 3 & 3\cdot 11& 3\cdot 359353\\1135\mathbf{563E}&31& 563&-& 281& 13\cdot 281& 281 \\1136\mathbf{571D}&2 & 571&-& 1 & 3^2\cdot 127 &1 \\1137\mathbf{655D}&13& 131&-& 1 & 3^{2}\cdot19\cdot515741 & 1\\1138&& 5&+& 1 & 3^2 & 19\cdot 515741 \\1139\mathbf{664F}&8 & 83&+& 1 & 5 & 1 \\1140\mathbf{681B}&1 & 227&-& 1 & 3\cdot 5^3 & 1 \\1141& & 3&+& 1 & 3\cdot 5^2& 5 \\1142\mathbf{707G}&15& 101&-& 1 & 13\cdot800077 & 1 \\1143&& 7&+& 1 & 13& 800077 \\1144\mathbf{709C}&30& 709&-& 59& 11\cdot 59 & 59 \\1145\mathbf{718F}&7 & 359&-& 1 & 7\cdot 151\cdot 35573 &1 \\1146&& 2 &+& 1 & 7 & 151\cdot 35573 \\1147\mathbf{794G}&14& 397&-& 3 & 3^2\cdot7\cdot11\cdot47\cdot35447 & 3 \\1148&& 2&+& 3 & 3\cdot11& 3^2\cdot 7\cdot 47\cdot 35447 \\1149\mathbf{817E}&15& 43&- & 5 & 5\cdot7\cdot 79 & 5 \\1150&& 19&+ &1 & 7 & 79 \\1151\mathbf{916G}&9 & 229&+ &1 & 3^9\cdot 11\cdot 17\cdot 239 & 1 \\11521153\mathbf{944O}&6 & 59&-& 1 & 7 & 1\\1154\mathbf{997H}&42& 997&-& 83& 3^2\cdot 83 & 83 \\1155\mathbf{1001L}&7& 13&+& 1& 7\cdot 19\cdot 47\cdot 2273& 1\\1156&& 11&-& 1& 7\cdot19\cdot47\cdot2273 & 1 \\1157&& 7&+& 1& 7\cdot 19\cdot 47 & 2273 \\1158\mathbf{1028E}&14&257&+& 1 & 3^{13}\cdot 11 & 1 \\1159\end{array}$$1160\end{table}1161116211631164\comment{11651166\begin{table}\index{Table of!odd square numerators}1167\caption{Square roots of odd square divisors of $L(A,1)/\Omega_A$\label{table:oddnumer}}\vspace{2ex}1168$$\begin{array}{lc}1169\mathbf{305D7}&3\\1170\mathbf{309D8}&5\\1171\mathbf{335E11}&3^2\\1172\mathbf{389E20}&5\\1173\mathbf{394A2}&5\\1174\mathbf{399G5}&3^4\\1175\mathbf{433D16}&7\\1176\mathbf{435G2}&3\\1177\mathbf{436C4}&3\\1178\mathbf{446E7}&3\\1179\mathbf{446F8}&11\\1180\mathbf{455D4}&3\\1181\mathbf{473F9}&3\\1182\mathbf{500C4}&3\\1183\mathbf{502E6}&11\\1184\mathbf{506I4}&5\\1185\mathbf{524D4}&3\\1186\mathbf{530G4}&7\\1187\mathbf{538E7}&3\\1188\mathbf{551H18}&3\\1189\mathbf{553D13}&3\\1190\mathbf{555E2}&3\\1191\mathbf{556C7}&3\\1192\mathbf{563E31}&13\\1193\mathbf{564C3}&3\\1194\mathbf{571D2}&3\\1195\end{array}\qquad1196\begin{array}{lc}1197\mathbf{579G13}&15\\1198\mathbf{597E14}&19\\1199\mathbf{602G3}&3\\1200\mathbf{604C6}&3 \\1201\mathbf{615F6}&5 \\1202\mathbf{615G8}&7 \\1203\mathbf{620D3}&3\\1204\mathbf{620E4}&3\\1205\mathbf{626F12}&5\\1206\mathbf{629G15}&3\\1207\mathbf{642D2}&3\\1208\mathbf{644C5}&3\\1209\mathbf{644D5}&3\\1210\mathbf{655D13}&3^2\\1211\mathbf{660F2}&3\\1212\mathbf{662E10}\!&\!\!43\\1213\mathbf{664F8}&5\\1214\mathbf{668B5}&3\\1215\mathbf{678I2}&3\\1216\mathbf{681B1}&3\\1217\mathbf{681I10}&3\\1218\mathbf{682I6}&11\\1219\mathbf{707G15}&13\\1220\mathbf{709C30}&11\\1221\mathbf{718F7}&7\\1222\mathbf{721F14}&3^2\\1223\end{array}\qquad1224\begin{array}{lc}1225\mathbf{724C8}&3\\1226\mathbf{756G2}&3\\1227\mathbf{764A8}&3\\1228\mathbf{765M4}&3\\1229\mathbf{766B4}&3\\1230\mathbf{772C9}&3\\1231\mathbf{790H6}&3\\1232\mathbf{794G12}\!&\!\!11\\1233\mathbf{794H14}&5^2\\1234\mathbf{796C8}&3\\1235\mathbf{817E15}&7\\1236\mathbf{820C4}&3\\1237\mathbf{825E2}&3\\1238\mathbf{844C10}\!&\!\!3^2\\1239\mathbf{855M4}&3\\1240\mathbf{860D4}&3\\1241\mathbf{868E5}&3\\1242\mathbf{876E5}&3\\1243\mathbf{878C2}&3\\1244\mathbf{884D6}&3\\1245\mathbf{885L9}&3^2\\1246\mathbf{894H2}&3\\1247\mathbf{902I5}&3\\1248\mathbf{913G17}&3\\1249\mathbf{916G9}&11\\1250\mathbf{918O2}&5\\1251\end{array}\qquad1252\begin{array}{lc}1253\mathbf{918P2}&3\\1254\mathbf{925K7}&3\\1255\mathbf{932B13}&3^2\\1256\mathbf{933E14}&19\\1257\mathbf{934I12}&7\\ %-+1258\mathbf{944O6}&7\\1259\mathbf{946K7}&3\\1260\mathbf{949B2}&3\\1261\mathbf{951D19}&3\\1262\mathbf{959D24}&3\\1263\mathbf{964C12}&3^2\\ % -+, same as EC 964A but that has rank=0.1264\mathbf{966J1}&3\\1265\mathbf{970I5}&3\\1266\mathbf{980F1}&3\\1267\mathbf{980J2}&3\\1268\mathbf{986J7}&5\\1269\mathbf{989E22}&5\\1270\mathbf{993B3}&3^2\\1271\mathbf{996E4}&3\\1272\mathbf{997H42}&3^2\\1273\mathbf{998A2}&3\\ % ++1274\mathbf{998H9}&3\\1275\mathbf{999J10}&3\\1276&\\1277&\\1278&\\1279\end{array}$$1280\end{table}128112821283}1284128512861287%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%12881289129012911292\comment{12931294\begin{lemma}1295Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such1296that $\Phi=A\intersect B$ is finite and and $A(\Q)$ is finite.1297Suppose that~$p$1298is a prime such that neither $(J/B)(\Q)$ nor $B(\Q)$ have any1299$p$-torsion and such that $B[p]\subset \Phi$.1300Then $B(\Q)/p B(\Q)$ is a subgroup of1301$(J/B)(\Q)/J(\Q)$.1302\end{lemma}1303\begin{proof}13041305\end{proof}1306130713081309\section{Explanatory factors at higher level}1310Consider one of the items in Table~\ref{table:primesha} for which1311$\Sha$ is invisible. It is natural to ask whether these1312elements of~$\Sha$ ``become visible somewhere.''1313For example, Mazur~\cite{mazur:visthree}\index{Mazur} proved that if1314$E$ is an elliptic curve and $c\in\Sha(E)$ has order $3$ then1315there is some abelian surface $A$ and an1316injection $\iota: E\hookrightarrow A$ such that1317$\iota_*(c)=0\in H^1(\Q,A)$. T. Klenke has proved1318a partial statement in this direction for elements of1319order $2$ as part of his Harvard Ph.D. thesis.1320J.~de Jong (see \cite[Remark 3]{cremona-mazur})1321showed that every element of the Shafarevich-Tate1322group of an elliptic curve is visible in some Jacobian.\index{Jacobian}13231324Consider an abelian variety $A$, taken1325from Table~\ref{table:primesha}, for which1326$\Sha$ has an invisible element $c$. Thus1327$A$ sits inside $J_0(p)$ for some prime $p$,1328and we ask ``is there a prime $q$ such that $\delta(c)=0$1329for one of the degeneracy maps1330$\delta : J_0(p)\ra J_0(pq)$?''13311332The author has no idea\footnote{Lo\"\i{}c Merel\index{Merel} suspects1333the answer might be yes whereas Richard Taylor is more skeptical.}.1334To get a feeling for what might happen we consider in detail abelian1335variety $A=A_f$ at level $p=1091$ in which $\Shaan$ is divisible by1336$7$.13371338There is a prime $\lambda$ of the ring1339$\Z[f] = \Z[\ldots a_n\ldots]$ attached to $A$.1340The Fourier coefficients of $f$ modulo $\lambda$ are1341$$\begin{array}{rcccccccccccccccc}1342p= &2 &3 &5 &7 &11 &13 &17 &19 &23 &29 &31 &37 &41 &43 &47 &53\\1343a_p= &3 &0 &1 &5 &0 &2 &0 &5 &4 &6 &3 &3 &5 &5 &6 &51344\end{array}$$1345These were computed by finding an eigenvector in $H_1(X_0(N);\F_7)$.1346[[SAY MORE ABOUT THE TRICK FOR FINDING ALL RIBET $q$'s.]]13471348According to Ribet's\index{Ribet} level raising theorem \cite{ribet:raising}1349there is a newform $g$ of level $1091\ell$ such that1350$f\con g$ modulo [[something]] if $a_\ell = \pm (\ell+1)\pmod{\lambda}$.1351Fortunately this criterion is already satisfied for $\ell=2$.1352Looking closely at level $2\cdot 1091$ (for example,1353in Cremona's online tables \cite{cremona:onlinetables})1354we find an elliptic curve $E$ whose corresponding newform1355$g=\sum b_n q^n$ has Fourier coefficients1356$$\begin{array}{rcccccccccccccccc}1357p = &2 & 3 & 5 & 7& 11& 13& 17& 19& 23& 29& 31& 37&41&43&47&53\\1358b_p=&1 & 0 & 1 &-2& 0 & -5& 0 & -2& 4& 6& -4& -4&-9&-9&-8&-2\\1359\end{array}$$13601361This is convincing evidence that one of the1362two images of~$A$ in $J_0(2\cdot 1091)$ shares some1363$7$-torsion with the elliptic curve \abvar{2182B}.1364This can be [[WILL BE!!, EASILY]] established by1365a direct computation with the period lattices.1366This is at first disconcerting because the rank of1367this elliptic curve is {\em not} $2$. However, the1368rank is still positive; it is $1$ with1369Mordell-Weil group $\E(\Q)=Z$.13701371I would not be at all surprised if your1372$7$-torsion in Sha does become visible in $J_0(2\cdot1091)$.1373The curve \abvar{2182B}, which shares 7-torsion with $A$ is13741375\begin{verbatim}1376e=ellinit([1,-1,1,-67,67]);1377The Tamagawa number c_2 is 14 (!!)1378? elllocalred(e,2)1379%2 = [1, 18, [1, 0, 0, 0], 14]1380The Tamagawa number c_1091 is 1.1381? elllocalred(e,1091)1382%3 = [1, 5, [1, 0, 0, 0], 1]13831384I have this feeling that the right statement about congruence1385and mordell-Weil is really something like1386congruence ==> "Selmer + Comp group"'s are identified.1387Anyway, the extra component group of order 7 may perhaps1388account for the other nontrivial element of Sha. This might1389just be wild speculation.13901391Good luck.13921393william1394Barry,13951396Amod asked me to investigate whether his element of order 71397in the winding quotient J_e at level 1091 becomes visible at1398higher level. Luckily, Ribet's\index{Ribet} level raising theorem predicts the1399existence of a form at level 2*1091 congruent mod a prime over14007 to the form corresponding to J_e. Even more luckily, one of the1401two rational newforms does the trick. Thus an image of J_e in1402J_0(2*1091) shares 7-torsion with an elliptic curve E (2182B1403in Cremona's tables). This elliptic curve has:14041405E(Q) = Z1406Sha(E/Q) = 01407c_2 = 14, c_1091 = 11408L^(1)(f,1)/1! = 4.27332686791516140914101411So there is reasonable hope that the elements of order 7 in1412Sha(J_e) are visible at this higher level, even though they1413are invisible even in J_1(1091).14141415Best,1416William141714181419Dear William,1420This is terrific. I assume that you will be showing that for J_e the1421winding1422(not quite quotient, but more conveniently sub-thing) in J_0(1091),1423the image of14241425Sha(J_e) ---> Sha(J_0(2*1091))14261427just dies? Since our working hope, I think, is that for any N there1428is an1429M so that1430Sha(J_0(N)) ---. Sha(J_0(N.M))143114321433dies, this suggests returning to the (mod 3) N=2849 example, where I1434"know" that there must exist such an M (because all three-torsion in Sha1435on elliptic curves is visible in some appropriate abelian surface which is1436isogenous to a product of two elliptic curves, and therefore, is abelian1437surface is probably "modular"). But I do not know a specific M.14381439Barry14401441\end{verbatim}144214431444\comment{1445\begin{remark}1446One reason we must assume~$p$ is odd, is because1447when~$B$ has good reduction at~$2$,1448in the proof we change~$J$ by an isogeny of $2$-power1449degree in order to apply~\cite[\S7.5, Prop.~3]{neronmodels} at $p=2$.1450When~$B$ has purely toric reduction,\index{Toric reduction}1451at~$2$ we use Tate1452uniformization to directly verify that points of $B(\Q)$ map into1453$\Sha(A)$, thereby avoiding exactness properties of1454N\'eron models\index{N\'eron model}.1455\end{remark}}14561457\subsubsection{Table~\ref{table:oddnumer}:1458Odd square divisors of $L(A,1)/\Omega_A$}1459In order to find candidate~$A$ with nontrivial visible1460$\Sha(A)$, we first enumerated those~$A$ for which the numerator1461of $L(A,1)/\Omega_A$ is divisible by an odd square~$n>1$.1462For $N<1000$, these are given in Table~\ref{table:oddnumer}.1463Any odd visible $\Sha(A)$ coprime to1464primes dividing torsion and $c_p$ must show up as a divisor1465of the numerator; it should show up as a square1466divisor because the Mordell-Weil rank of the explanatory factor1467should be even. It would be interesting to compute the conjectural1468order of $\Sha(A)$ for each abelian variety in this table, but1469not in table 1, and show (when possible) that the visible1470$\Sha(A)$ is old.1471}14721473147414751476147714781479