% $Header: /home/was/papers/thesis/RCS/bsd.tex,v 1.27 2001/07/08 20:11:41 was Exp $ \comment{ $Log: bsd.tex,v $ Revision 1.27 2001/07/08 20:11:41 was nothing. Revision 1.26 2000/05/21 03:05:38 was bsd Revision 1.25 2000/05/15 07:31:56 was added I. Revision 1.24 2000/05/15 07:28:22 was Changed F_p to F_v in proof of main theorem! Revision 1.23 2000/05/15 05:11:14 was quick fix Revision 1.22 2000/05/15 04:46:48 was away from level. Revision 1.21 2000/05/15 04:46:08 was got rid of a_p Revision 1.20 2000/05/15 04:45:32 was third fix; added Weierstrass equation but didn't change subsequent text! Revision 1.19 2000/05/15 04:44:23 was quick fix: typo 2; geometric component group of F! Revision 1.18 2000/05/15 04:43:26 was quick fix. a_p typo Revision 1.17 2000/05/12 01:52:07 was oops! Revision 1.16 2000/05/10 20:32:49 was done. Revision 1.15 2000/05/10 09:39:43 was weakend explantory claim. Revision 1.14 2000/05/10 09:35:36 was added\index{Mazur} Revision 1.13 2000/05/10 09:33:08 was eliminated \index{toric reduction} Revision 1.12 2000/05/10 07:12:25 was Added index's to tables. Revision 1.11 2000/05/09 20:21:57 was added a label to subsec{Notation} Revision 1.10 2000/05/08 15:42:17 was fixed up index some. Revision 1.9 2000/05/08 02:52:01 was Fixed mispelling of "isogeneous". Revision 1.8 2000/05/08 02:00:23 was Fixed up 1913. Made lots and lots of refinements and clarifcations of the tables. Revision 1.7 2000/05/07 21:57:57 was Added cremona's model for 5389A. Revision 1.6 2000/05/07 19:51:10 was Added a section on when Sha becomes visible. Revision 1.5 2000/05/06 20:55:41 was Worked on cleaning up the descriptions of the tables. } \chapter{The Birch and Swinnerton-Dyer conjecture} \label{chap:bsd}\index{Birch and Swinnerton-Dyer conjecture|see{BSD conjecture}}% \index{Conjecture!Birch and Swinnerton-Dyer|see{BSD conjecture}}% Now that the Shimura-Taniyama% \index{Shimura-Taniyama conjecture}\index{Conjecture!Shimura and Taniyama} conjecture has been proved, many experts consider the Birch and Swinnerton-Dyer conjecture (BSD conjecture) to be one of the main outstanding problems in the field (see~\cite[pg.~549]{darmon-bsd} and \cite[Intro.]{cime-1997}). This conjecture ties together many of the arithmetic and analytic invariants of an elliptic curve. At present, there is no general class of elliptic curves for which the full BSD conjecture\index{BSD conjecture!is still unknown} is known, though a slightly weakened form is known for a fairly broad class of complex multiplication elliptic curves of analytic rank~$0$ (see~\cite{rubin:main-conjectures}), and several deep partial results have been obtained during the last twenty years (see, e.g.,~\cite{gross-zagier} and \cite{kolyvagin:mordellweil}). Approaches to the BSD conjecture\index{BSD conjecture} that rely on congruences between\index{Congruences!and BSD conjecture} modular forms\index{Modular forms!and BSD} are likely to require a deeper understanding of the analogue of the BSD conjecture\index{BSD conjecture!in higher dimensions} for higher-dimensional abelian varieties. As a first step, this chapter presents theorems and explicit computations of some of the arithmetic invariants of modular abelian varieties. The reader is urged to also read A.~Agashe's 2000 Berkeley Ph.D.\ thesis which cover similar themes.\index{Agashe} The paper of Cremona and Mazur's~\cite{cremona-mazur}\index{Mazur} paints a detailed experimental picture of the way in which congruences link Mordell-Weil and Shafarevich-Tate groups of elliptic curves. \index{Congruences!and BSD conjecture} \section{The BSD conjecture}\index{BSD conjecture|textit} By~\cite{breuil-conrad-diamond-taylor} we now know that every elliptic curve over~$\Q$ is a quotient of the curve~$X_0(N)$, whose complex points are the isomorphism classes of pairs consisting of a (generalized) elliptic curve and a cyclic subgroup of order~$N$. Let~$J_0(N)$ denote the Jacobian\index{Jacobian} of $X_0(N)$; this is an abelian variety of dimension equal to the genus of~$X_0(N)$ whose points correspond to the degree~$0$ divisor classes on~$X_0(N)$. The survey article~\cite{diamond-im} is a good guide to the facts and literature about the family of abelian varieties $J_0(N)$. Following Mazur~\cite{mazur:rational}\index{Mazur}, we make the following definition. \begin{definition}[Optimal quotient]\index{Optimal quotient|textit} An {\em optimal quotient} of $J_0(N)$ is a quotient~$A$ of $J_0(N)$ by an abelian subvariety. \end{definition} Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$. By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A)$ are both finite. The BSD conjecture\index{BSD conjecture!statement of}% asserts that $$\frac{L(A,1)}{\Omega_A} = \frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p} {\# A(\Q)\cdot\#\Adual(\Q)}.$$ Here the Shafarevich-Tate group\index{Shafarevich-Tate group} $$\Sha(A) := \ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right)$$ is a measure of the failure of the local-to-global principle\index{Local-to-global principle}; the Tamagawa numbers~$c_p$\index{Tamagawa numbers} are the orders of the groups of rational points of the component groups of~$A$ (see Chapter~\ref{chap:compgroups}); the real number~$\Omega_A$ is the measure of~$A(\R)$ with respect to a basis of differentials having everywhere nonzero good reduction (see Section~\ref{sec:realmeasure}); and~$\Adual$ is the abelian variety dual to~$A$ (see \cite[\S9]{milne:abvars}). This chapter makes a small contribution to the long-term goal of verifying the above conjecture for many specific abelian varieties on a case-by-case basis. In a large list of examples, we compute the conjectured order of $\Sha(A)$, up to a power of $2$, and then show that $\Sha(A)$ is at least as big as conjectured. We also discuss methods to obtain upper bounds on $\#\Sha(A)$, but do not carry out any computations in this direction. This is the first step in a program to verify the above conjecture for an infinite family of quotients of~$J_0(N)$. \subsection{The ratio $L(A,1)/\Omega_A$} Extending classical work on elliptic curves, A.~Agashe\index{Agashe} and the author proved the following theorem. \begin{theorem}\label{thm:ratpart} Let~$m$ be the largest square dividing~$N$. The ratio $L(A,1)/\Omega_A$ is a rational number that can be explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$. \end{theorem} \begin{proof} The proof uses modular symbols\index{Modular symbols} combined with an extension of the argument used by Mazur\index{Mazur} in~\cite{mazur:rational} to bound the Manin constant\index{Manin constant}. The modular symbols part of the proof for $L$-functions attached to newforms of weight $k\geq 2$ is given in Section~\ref{sec:rationalvals}; it involves expressing the ratio $L(A,1)/\Omega_A$ as the lattice index\index{Lattice index} of two modules over the Hecke algebra\index{Hecke algebra}. The bound on the Manin constant\index{Manin constant} is given in Section~\ref{sec:maninconstant}. \end{proof} The author has computed $L(A,1)/\Omega_A$ for all simple optimal quotients of level $N\leq 1500$; this table can be obtained from the author's web page. \begin{remark} The method of proof should also give similar results for special values of twists of $L(A,s)$, just as it does in the case $\dim A=1$ (see~\cite[Prop.~2.11.2]{cremona:algs}). \end{remark} \subsection{Torsion subgroup\index{Torsion subgroup}} We can compute upper and lower bounds on $\#A(\Q)_{\tor}$, see Section~\ref{sec:torsionsubgroup}; these frequently determine $\#A(\Q)_{\tor}$. These methods, combined with the method used to obtain Theorem~\ref{thm:ratpart}, yield the following corollary, which supports the expected cancellation between torsion and~$c_p$ coming from the reduction map sending rational points to their image in the component group of~$A$. The corollary also generalizes to higher weight forms, thus suggesting a geometric way to think about reducibility of modular Galois representations. \begin{corollary} Let~$n$ be the order of the image of $(0)-(\infty)$ in $A(\Q)$, and let~$m$ be the largest square dividing~$N$. Then $n\cdot L(A,1)/\Omega_A \in \Z[1/(2m)].$ \end{corollary} For the proof, see Corollary~\ref{cor:denominator} in Chapter~\ref{chap:computing}. \subsection{Tamagawa numbers\index{Tamagawa numbers}} We prove the following theorem in Chapter~\ref{chap:compgroups}. \begin{theorem}\label{thm:tamagawa} When $p^2\nmid N$, the number~$c_p$ can be explicitly computed (up to a power of~$2$). \end{theorem} We can compute the order~$c_p$ of the group of rational points of the component group, but not its structure as a group. When $p^2 \mid N$ it may be possible to compute~$c_p$ using the Drinfeld-Katz-Mazur model of~$X_0(N)$, but we have not yet done this. There are also good bounds on the primes that can divide $c_p$ when $p^2\mid N$. Systematic computations (see Section~\ref{sec:compgroupconjectures}) using this formula suggest the following conjectural refinement of a result of Mazur~\cite{mazur:eisenstein}\index{Mazur}. \begin{conjecture} Suppose~$N$ is prime and~$A$ is an optimal quotient of $J_0(N)$ corresponding to a newform~$f$. Then $A(\Q)_{\tor}$ is generated by the image of $(0)-(\infty)$ and $c_p = \#A(\Q)_{\tor}$. Furthermore, the product of the~$c_p$ over all simple optimal quotients corresponding to newforms equals the numerator of $(N-1)/12$. \end{conjecture} I have checked this conjecture for all $N\leq 997$ and, up to a power of~$2$, for all $N\leq 2113$. The first part is known when~$A$ is an elliptic curve (see~\cite{mestre-oesterle:crelle}). Upon hearing of this conjecture, Mazur\index{Mazur} reportedly proved it when all ``$q$-Eisenstein quotients'' are simple. There are three promising approaches to finding a complete proof. One involves the explicit formula of Theorem~\ref{thm:tamagawa}; another is based on Ribet's\index{Ribet} level lowering theorem (see~\cite{ribet:modreps}), and a third makes use of a simplicity result of Merel\index{Merel} (see~\cite{merel:weil}). The formula that lies behind Theorem~\ref{thm:tamagawa} probably has a natural analogue in weight greater than~$2$. One could then guess that it produces Tamagawa numbers\index{Tamagawa numbers} of motifs\index{Motifs} attached to eigenforms of higher weight; however, we have no idea if this is really the case. These numbers appear in the conjectures of Bloch and Kato, \index{Conjecture!Bloch and Kato}% \index{Bloch and Kato conjecture}% which generalize the BSD conjecture\index{BSD conjecture!generalization of} to motifs (see~\cite{bloch-kato}). Anyone wishing to try to compute them should be aware of Neil Dummigan's paper~\cite{dummigan:cp}, which gives some information about the Tamagawa numbers\index{Tamagawa numbers} of motifs\index{Motifs} attached by Scholl in~\cite{scholl:motivesinvent} to modular eigenforms. \subsection{Upper bounds on $\#\Sha(A)$} V.~Kolyvagin (see \cite{kolyvagin:structureofsha}) and K.~Kato (see, e.g., \cite{scholl:kato}) constructed Euler systems\index{Euler system} that were used to prove that $\Sha(A)$ is {\em finite} when $L(A,1)\neq 0$. To verify the full BSD conjecture\index{BSD conjecture!verification of} for certain abelian varieties, we must make the Kolyvagin-Kato finiteness bound explicit. Kolyvagin's bounds involve computations with Heegner points\index{Heegner points}, and Kato's involve a study of the Galois representations associated to~$A$. \subsubsection{Kolyvagin's bounds}% \index{Bound of!Kolyvagin} In~\cite{kolyvagin:mordellweil}, Kolyvagin obtains explicit upper bounds for $\#\Sha(A)$ for a certain (finite) list of elliptic curves~$A$ by computing the index in $A(K)$ of the subgroup generated by the Heegner point, where~$K$ is a suitable imaginary quadratic extension. In~\cite{kolyvagin-logachev:totallyreal}, Kolyvagin and Logachev generalize Kolyvagin's earlier results; in Section~1.6, ``Unsolved problems'', they say that: ``If one were to compute the height of a Heegner point~$y$ [...] considered in the present paper, then one would have succeeded in obtaining an upper bound for $\#\Sha$ for this curve.'' (By ``curve'' they mean abelian variety.) This suggests that explicit computations should yield upper bounds on the order of $\Sha(A)$, but that they had not yet figured out how to carry out such computations. \subsubsection{Kato's bounds}% \index{Bound of!Kato} Kato has constructed Euler systems\index{Euler system} coming from $K_2$-groups of modular curves. These can be used to prove the following theorem (see, e.g., \cite[Cor.~3.5.19]{rubin:book}). \begin{theorem}[Kato] Suppose~$E$ is an elliptic curve over~$\Q$ without complex multiplication that~$E$ has conductor~$N$, that~$E$ has good reduction at~$p$, that~$p$ does not divide $2r_E\prod_{q\mid N} L_q(q^{-1})\#E(\Q_q)_{\tor}$, and the Galois representation $\rho_{E,p}:\GQ\ra\Aut(E[p])$ is surjective. Then $$\#\Sha(E)_{p^{\infty}}\text{ divides } \frac{L(E,1)}{\Omega_E}.$$ \end{theorem} Here $L_q(x)$ is the local Euler factor at~$q$ and the constant $r_E$ arises in the construction of Kato's Euler system. Rubin suggests that computing $r_E$ is not very difficult (private communication). Appropriate variants of Kato's arguments give similar results for quotients of $J_0(N)$ of arbitrary dimension, though these have not been written down. \comment{ >How mysterious is the constant r_E in, for example, Corollary 3.5.19 of >your Euler Systems book? I think it's not too bad. Certainly nothing like Heegner points are involved. When I wrote that part of my book, and the similar paper in the Durham proceedings, I did not really know what it was because Kato hadn't written anything. I don't have Scholl's paper here so I'm not certain, but I suspect that the only contribution to $r_E$ comes from passing from the modular curve to $E$, and perhaps some extra 2's and 3's. Karl } \subsection{Lower bounds on $\#\Sha(A)$} One approach to showing that~$\Sha(A)$ is as {\em at least} as large as predicted by the BSD conjecture\index{BSD conjecture!and $\Sha$} is suggested by Mazur's\index{Mazur} notion of the visible part $\Sha(A)^{\circ}$ of~$\Sha(A)$ (see~\cite{cremona-mazur, mazur:visthree}). Let~$\Adual\subset J_0(N)$ be the dual to~$A$. The \defn{visible part}\index{Visibility!of $\Sha$|textit}% \index{Shafarevich-Tate group!visible part of} of $\Sha(\Adual)$ is the kernel of the natural map $\Sha(\Adual)\ra \Sha(J_0(N))$. Mazur\index{Mazur} observed that if an element of order~$p$ in~$\Sha(\Adual)$ is visible, then it is explained by a ``jump in the rank of Mordell-Weil'' in the sense that there is another abelian subvariety $B\subset J_0(N)$ such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive. Mazur's\index{Mazur} observation can be turned around: if there is another abelian variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$, then, under mild hypotheses (see Theorem~\ref{thm:shaexists}), there is an element of~$\Sha(\Adual)$ of order~$p$. From a computational point of view it is easy to understand the intersections $\Adual\intersect B$; see Section~\ref{sec:intersection}. From a theoretical point of view, nontrivial intersections ``correspond'' to congruences between modular forms. Thus the well-developed theory of congruences between modular forms% \index{Modular forms!congruences between}% \index{Congruences!and lower bounds on $\Sha$} can be used to obtain a lower bound on~$\#\Sha(\Adual)$. \subsubsection{Invisible elements of $\#\Sha(\Adual)$} \index{Shafarevich-Tate group!invisible elements of} \index{Invisible elements of $\Sha$} Numerical experiments suggest that as $\Adual$ varies, $\Sha(\Adual)$ is often {\em not} visible inside of~$J_0(N)$. For example (see Table~\ref{table:primesha}), the BSD conjecture\index{BSD conjecture!predicts invisible elements} predicts the existence of invisible elements of odd order in~$\Sha(\Adual)$ for almost half of the~$37$ optimal quotients of prime level $\leq 2113$. \subsubsection{Visibility at higher level} \index{Shafarevich-Tate group!visibility at higher level} \index{Visibility!at higher level} For every integer~$M$ (Ribet~\cite{ribet:raising}\index{Ribet} tells us which~$M$ to choose), we can ask whether $\Sha(\Adual)$ maps to~$0$ under one of the natural maps $\Adual\ra J_0(NM)$; that is, we can ask whether $\Sha(\Adual)$ ``becomes visible at level $NM$.'' We have been unable to prove in any particular case that $\Sha(\Adual)$ is not visible at level~$N$, but becomes visible at some level $NM$. See Section~\ref{sec:higherlevel} for some computations which strongly indicate that such examples exist. \subsubsection{Visibility in some Jacobian}% \index{Visibility!in some Jacobian}% \index{Jacobian!visibility in}% Johan de Jong proved that if~$E$ is an elliptic curve over a number field~$K$ and $c\in H^1(K,E)$ then there is a Jacobian~$J$ and an imbedding $E\hookrightarrow J$ such that~$c$ maps to~$0$ under the natural map $H^1(K,E)\ra H^1(K,J)$ (see Remark~3 in~\cite{cremona-mazur}); de Jong's proof appears to generalize when~$E$ is replaced by an abelian variety, but time does not permit going into the details here. \subsection{Motivation for considering abelian varieties} If~$A$ is an elliptic curve, then explaining~$\Sha(A)$ using only congruences between elliptic curves will probably fail. \index{Congruences!between elliptic curves} This is because pairs of non-isogenous elliptic curves with isomorphic $p$-torsion for large~$p$ are, according to E.~Kani's\index{Kani} conjecture, extremely rare.% \index{Conjecture!Kani} It is crucial to understand what happens in all dimensions. Within the range accessible by computer, abelian varieties exhibit more richly textured structure than elliptic curves. For example, there is a visible element of prime order $83341$ in the Shafarevich-Tate group\index{Shafarevich-Tate group} of an abelian variety of prime conductor~$2333$; in contrast, over all optimal elliptic curves of conductor up to $5500$, it appears that the largest order of an element of a Shafarevich-Tate group is~$7$ (see the discussion in~\cite{cremona-mazur}). \section{Existence of nontrivial visible elements of $\Sha(A)$}% The reader who wants to see tables of Shafarevich-Tate groups can safely skip to the next section. Without relying on any unverified conjectures, we will use the following theorem to exhibit abelian varieties~$A$ such that the visible part of $\Sha(A)$ is nonzero. In the following theorem we do {\em not} assume that~$J$ is the Jacobian\index{Jacobian} of a curve. \begin{theorem}\label{thm:shaexists}\index{Visibility!existence theorem} Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such that $A\intersect B$ is finite and $A(\Q)$ is finite. Assume that~$B$ has purely toric reduction at each prime at which~$J$ has bad reduction. Let~$p$ be an odd prime at which~$J$ has good reduction, and assume that~$p$ does not divide the orders of any of the (geometric) component groups\index{Component group!geometric} of~$A$ and~$B$, or the orders of the torsion subgroups of $(J/B)(\Q)$ and $B(\Q)$. Suppose further that $B[p] \subset A\intersect B$. Then there exists an injection $$B(\Q)/pB(\Q)\hookrightarrow \Sha(A)^{\circ}$$ of $B(\Q)/p B(\Q)$ into the visible part of $\Sha(A)$. \end{theorem} \begin{proof} Let $C=J/A$. The long exact sequence of Galois cohomology associated to the short exact sequence $$0 \ra A \ra J \ra C \ra 0$$ begins $$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,} H^1(\Q,A) \ra \cdots.$$ Because $B[p]\subset A$, the map $B\ra C$, obtained by composing the inclusion $B\hookrightarrow J$ with $J\ra C$, factors through multiplication-by-$p$, giving the following commutative diagram: $$\xymatrix{ & B\ar[d] \ar[r]^{p}& B\ar[d]\\ A\ar[r]&J\ar[r]&C.}$$ Because $B(\Q)[p]=0$ and $B(\Q)\intersect A(\Q)=0$, we deduce the following commutative diagram with exact rows: $$\xymatrix{ & 0\ar[d] & K_1\ar[d]& K_2\ar[d]\\ 0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d] & B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\ 0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\ & K_3, }$$ where $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the cokernel. We have the snake lemma exact sequence $$0\ra K_1 \ra K_2 \ra K_3.$$ Because $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group, we have $K_1=0$. The quotient $J(\Q)/B(\Q)$ has no $p$-torsion because it is a subgroup of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group of order coprime to~$p$, so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion. Thus $K_2=0$. The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of $H^1(\Q,A)$. However, $H^1(\Q,A)$ contains infinitely many elements of order~$p$ (see~\cite{shafarevich:exp}), whereas $\Sha(A)[p]$ is a finite group, so we must work harder to deduce that $B(\Q)/p B(\Q)$ lies in $\Sha(A)[p]$. Fix $x\in B(\Q)$. We must show that $\pi(x)$ lies in $\Sha(A)[p]$; equivalently, that $\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$. At the archimedean place $v=\infty$, the restriction $\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$, hence $\res_v(\pi(x))=0$. Suppose that~$v$ is a place at which~$J$ has bad reduction. By hypothesis, $B$ has purely toric reduction\index{Purely toric reduction}, so over the maximal unramified extension $\Q_v^{\ur}$ of $\Q_v$ there is an isomorphism $B\isom\Gm^d/\Gamma$ of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules, for some ``lattice'' $\Gamma$. For example, when $\dim B=1$, this is the Tate curve representation of~$B$. Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$ equals the order of the cokernel of the valuation map $\Gamma\ra \Z^d$. Choose a representative $P=(x_1,\ldots,x_d)\in\Gm^d$ for the point~$x$. Then $nP$ can be adjusted by elements of~$\Gamma$ so that each of its components $x_i\in\Gm$ has valuation~$0$. By assumption,~$p$ is a prime at which~$J$ has good reduction, so $p\neq v$; it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$ such that $pQ = nP$. Thus the cohomology class $\res_v(\pi(nx))$ is unramified at~$v$. By \cite[Prop.~I.3.8]{milne:duality}, $$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur})) =H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v)),$$ where $\Phi_{A,v}$ is the component group of~$A$ at~$v$. Since\item{There is a mistake here, but it is easy to fix.} the component group $\Phi_{A,v}(\Fbar_v)$ has order~$n$, it follows that $$\res_v(\pi(nx))=n\res_v(\pi(x))=0.$$ Since the order~$p$ of $\res_v(\pi(x))$ is coprime to~$n$, we conclude that $\res_v(\pi(x))=0$. Next suppose that~$J$ has good reduction at~$v$ and that~$v$ is {\em odd}, in the sense that the residue characteristic of~$v$ is odd. To simplify notation in this paragraph, since~$v$ is a non-archimedean place of $\Q$, we will also let~$v$ denote the odd prime number which is the residue characteristic of~$v$. Let $\cA$, $\cJ$, $\cC$, be the N\'eron models of~$A$,~$J$, and~$C$, respectively (for more on N\'eron models, see Chapter~\ref{chap:compgroups}). Let $A$, $J$, $C$, also denote the sheaves on the \'etale-site over $\Spec(\Z_v)$ determined by the group schemes $\cA$, $\cJ$, and $\cC$, respectively. Since~$v$ is odd, $1=e2161$. The sporadic examples appear at the bottom of the table below a horizontal line. \subsubsection{Notation} The columns of the table contain the following information. The abelian varieties~$A$ for which $\Shaan(A)$ is greater than~$1$ are laid out in the first column of Table~\ref{table:primesha}. The second column contains the dimensions of the abelian varieties in the first column. The third column contains the {\em odd part} (i.e., largest odd divisor) of the order of the Shafarevich-Tate group, as predicted by the BSD conjecture.\index{BSD conjecture!predicted order of $\Sha$} Column four contains the odd parts of the modular degrees of the abelian varieties in the first column. The fifth column contains an optimal quotient~$B$ of $J_0(p)$ of positive analytic rank, such that the $\ell$-torsion of $\Bdual$ is contained in~$\Adual$, when one exists, where $\ell$ is a divisor of $\Shaan(A)$. We computed this intersection using the algorithm described in Section~\ref{sec:intersection}. Such a~$B$ is called an \defn{explanatory factor}.\index{Explanatory factor} When no such abelian varieties exists, we write ``NONE'' in the fifth column. The sixth column contains the dimensions of the abelian varieties in the fifth column, and the seventh column contains the odd parts of the modular degrees of the abelian varieties in the fifth column. \subsubsection{Ranks of the explanatory factors} That the explanatory factors have positive analytic rank follows from our modular symbols computation of $L(B,1)/\Omega_B$. This is supported by the table in~\cite{brumer:rank}, except in the case {\bf 2333A}, where there is a mistake in \cite{brumer:rank} (see below). The explanatory factor {\bf 389A} is the first elliptic curve of rank~$2$. The table in \cite{brumer:rank} suggests that the explanatory factor {\bf 1061B} is the first $2$-dimensional abelian variety (attached to a newform) whose Mordell-Weil group when tensored with the field of fractions~$F$ of the corresponding ring of Fourier coefficients, is of dimension~$2$ over~$F$. Similarly {\bf 1567B} appears to be the first $3$-dimensional one of rank~$2$, and {\bf 2333A} is the first $4$-dimensional one of rank~$2$. It thus seems very likely that the ranks of each explanatory factor is exactly~$2$, though we have not proved this. \subsubsection{Discussion of the data} There are~$23$ examples in which~$\Sha(A)$ is visible and~$18$ in which~$\Sha(A)$ is invisible. The largest visible $\Sha(A)$ found occurs at level $2333$ and has order at least $83341^2$ ($83341$ is prime). The largest invisible\index{Invisible elements of $\Sha$} $\Sha(A)$ occurs in a $112$-dimensional abelian variety at level $2111$ and has order at least $211^2$. The example {\bf 1283C} demonstrates that $\Shaan(A)$ can divide the modular degree, yet be {\em invisible}. In this case~$5$ divides $\Shaan(A)$. Since~$5$ divides the modular degree, it follows that there must be other non-isogenous subvarieties of $J_0(1283)$ that intersect {\bf 1283C} in a subgroup of order divisible by~$5$. In this case, the only such subvariety is {\bf 1283A}, which has dimension~$2$ and whose $5$-torsion is contained in {\bf 1283C}. However {\bf 1283A} has analytic (hence algebraic) rank~$0$, so $\Shaan(A)$ cannot be visible. The cases {\bf 1483D}, {\bf 1567D}, {\bf 2029C}, and {\bf 2593B} are interesting because {\em all} of~$\Sha$, even though it has two nontrivial $p$-primary components in each of these cases, is made visible in a single~$B$. In the case {\bf 1913E} only the $5$-primary component of $\Sha$ is visible in {\bf 1913A}, but still {\em both} the $5$-primary and $61$-primary components of $\Sha$ are visible in {\bf 1913C}. Examples {\bf 1091C} and {\bf 1429B} were first found in \cite{agashe} and {\bf 1913B} in \cite{cremona-mazur}. \subsubsection{Errata to Brumer's paper} Contrary to our computations, \cite{brumer:rank} suggests that {\bf 2333A} has rank~$0$. However, the author pointed the discrepancy out to Brumer who replied: \begin{quote} I looked in vain for information about $\theta$-relations on~$2333$ and have concluded that I never ran the interval from~$2300$ to~$2500$ or else had lost all info by the time I wrote up the paper. The punchline:~$4$ relations for~$2333$ and~$2$ relations for~$2381$ (also missing from the table). \end{quote} \comment{ Date: Wed, 08 Sep 1999 18:24:10 -0400 From: armand brumer To: William Arthur Stein CC: ab Subject: Re: The rank of J_0(2333) Dear William, I just found your 3 emails (including one from the end of June) sitting on a mail server I did not know existed until a few days ago (the university did not tell us that the two addresses were on different servers!!) I then looked in vain for information about theta relations on 2333 and have concluded that I never ran the interval from 2300 to 2500 or else had lost all info by the time I wrote up the paper. The punchline:4 relations for 2333 and 2 relations for 2381 (also missing from the table). I may try to check as much as possible in the background and would be grateful if you mention this errata when you write up your stuff (I don't know any other way of publicizing the correction). Best regards and hope you did not think I was ``blowing you off" as my son would say! Armand } \subsection{Tables~\ref{table:newvis}--\ref{table:shacompgps}: New visible Shafarevich-Tate groups} Let~$n$ denote the largest odd square dividing the numerator of $L(A,1)/\Omega_A$. Table~\ref{table:newvis} lists those~$A$ such that for some $p\mid n$ there exists a quotient~$B$ of $J_0(N)$, corresponding to a {\em newform} and having positive analytic rank, such that $(\Adual\intersect B^{\vee})[p]\neq 0$. Our search was systematic up to level $1001$, but there might be omitted examples between levels $1001$ and $1028$. Table~\ref{table:explain} contains further arithmetic information about each explanatory factor. Table~\ref{table:shacompgps} gives the quantities involved in the formula of Chapter~\ref{chap:compgroups} for Tamagawa numbers, for each of the abelian varieties~$A$ in Table~\ref{table:newvis}. \subsubsection{Notation} Most of the notation is the same as in Table~\ref{table:primesha}. However the additional columns $w_q$ and $c_p$ contain the signs of the Atkin-Lehner involutions and the Tamagawa numbers, respectively. These are given in order, from smallest to largest prime divisor of~$N$. In each case~$B$ has dimension~$1$. When $4\mid N$, we write ``$a$'' for $c_2$ to remind us that we did not compute $c_2$ because the reduction at~$2$ is additive. Again only {\em odd parts} of the invariants are given. Section~\ref{sec:compgrptables} contains a discussion of the notation used in the headings of Table~\ref{table:shacompgps}. \subsubsection{Remarks on the data} The explanatory factors~$B$ of level $\leq 1028$ are {\em exactly} the set of rank~$2$ elliptic curves of level $\leq 1028$. \section{Further visibility computations} \subsection{Does $\Sha$ become visible at higher level?} \label{sec:higherlevel} This section is concerned with whether or not the examples of invisible elements of Shafarevich-Tate groups of elliptic curves of level~$N$ that are given in \cite{cremona-mazur} become visible in abelian surfaces inside appropriate $J_0(Np)$. We analyze each of the cases in Table~1 of \cite{cremona-mazur}. For the reader's convenience, the part of this table which concerns us is reproduced as Table~\ref{table:cm}. The most interesting examples we give are {\bf 2849A} and {\bf 5389A}. As in \cite{cremona-mazur}, the assertions below assume the truth of the BSD conjecture.\index{BSD conjecture} \begin{table}\index{Table of!odd invisible $\Sha_E$} \ssp \caption{Odd invisible $|\Sha_E|>1$, all $N\leq 5500$ (from Table~1 of~\cite{cremona-mazur})} \label{table:cm} $$ \begin{array}{lcclcl} \mbox{\rm\bf E}&\sqrt{|\Sha_E|}& m_E & \mbox{\rm\bf F} & m_F & \text{Remarks}\\ & & & & & \vspace{-3ex} \\ \mbox{\rm\bf 2849A}& 3 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& - &\\ \mbox{\rm\bf 3364C}& 7 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& - &\\ \mbox{\rm\bf 4229A}& 3 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& - &\\ \mbox{\rm\bf 4343B}& 3 &2^4\cdot1583 &\mbox{\rm\bf NONE}& -&\\ \mbox{\rm\bf 4914N}& 3 &2^4\cdot 3^5 &\mbox{\rm\bf none}& - &\text{{\bf E} has rational $3$-torsion}\\ \mbox{\rm\bf 5054C}& 3 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& - &\\ \mbox{\rm\bf 5073D}& 3 &2^5\cdot 3\cdot 5\cdot7\cdot23 &\mbox{\rm\bf none}& - & \\ \mbox{\rm\bf 5389A}& 3 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& - &\\ \end{array} $$ \end{table} \subsubsection{How we found the explanatory curves} We use a naive heuristic observation to find possible explanatory curves of higher level, even though their conductors are out of the range of Cremona's tables. Note that we have not proved that these factors are actually explanatory in any cases, and expect that in some cases they are not. First we recall some of the notation from Table~1 of~\cite{cremona-mazur}, which is partially reproduced below. The ``NONE'' label in the row corresponding to an elliptic curve~$E$ indicates that there are elements in $\Sha(E)$ whose order does not divide the modular degree of~$E$, and hence they must be invisible. The label ``none'' indicates only that no suitable explanatory elliptic curves could be found, so $\Sha(E)$ is not visible in an {\em abelian surface} inside $J_0(N)$; it could, however, be visible in the full abelian variety $J_0(N)$. Studying the Weierstrass equations corresponding to the curves in \cite{cremona-mazur} reveals that the elliptic curves labeled ``NONE'' have unusually large height, as compared to their conductors. However, the explanatory factors often have unusually small height. Motivated by this purely heuristic observation, we make a table of all elliptic curves of the form $$y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x+ a_6,$$ with $a_1, a_2, a_3 \in \{-1,0,1\}$, $|a_4|, |a_6| < 1000$, and conductor bounded by $50000$. The bound on the conductor is required only so that the table will fit within computer storage. This table took a few days to generate. \subsubsection{2849A} Mazur\index{Mazur} and Adam Logan\index{Logan} found the first known example of an {\em invisible} Shafarevich-Tate group\index{Shafarevich-Tate group!first invisible example}. This was $\Sha(E)$, where~$E$ is the elliptic curve {\bf 2849A}, which has minimal Weierstrass equation $$E:\quad y^2 + xy + y = x^3 + x^2 - 53484x - 4843180.$$ Consulting our table of curves of small height, we find an elliptic curve~$F$ of conductor $8547=2839\cdot 3$ such that $f_E \con f_F \pmod{3}$, where $f_E$ and $f_F$ are the newforms attached to~$E$ and~$F$, respectively. This is a congruence for {\em all} eigenvalues $a_p$ attached to~$E$ and~$F$. The elliptic curve~$F$ is defined by the equation $$F:\quad y^2 + xy + y = x^3 + x^2 - 154x - 478.$$ Cremona's program {\tt mwrank} reveals that the Mordell-Weil group of~$F$ has rank~$2$. Thus maybe $\Sha(E)$ becomes visible at level~$8547$. Unfortunately, visibility of $\Sha(E)$ is not implied by Theorem~\ref{thm:shaexists} because the geometric component group of~$F$ at~$3$ has order divisible by~$3$. \subsubsection{4343B} Consider the elliptic curve $E$ labeled {\bf 4343B}. According to Table~1 of \cite{cremona-mazur}, $\Sha(E)$ has order~$9$, but the modular degree prevents~$\Sha(E)$ from being visible in $J_0(4343)$. At level $21715 = 5\cdot 4343$ there is an elliptic curve~$F$ of rank~$1$ that is congruent to~$E$. Its equation is $$F:\quad y^2 - xy - y = x^3 - x^2 + 78x - 256.$$ \subsubsection{5389A} The last curve labeled ``NONE'' in the table is curve {\bf 5389A}, which has minimal Weierstrass equation $$y^2+xy+y =x^3 - 35590x-2587197.$$ The main theorem of~\cite{ribet:raising} implies that there exists a newform that is congruent modulo~$3$ to the newform corresponding to {\bf 5389A} and of level $3\cdot 5389$. This is because $(-2)^2 = (3+1)^2 \pmod{3}$. However, our table of curves of small height does not contain any curve of conductor $3\cdot 5389$. Next we observe that $(-2)^2 \con (7+1)^2 \pmod{3}$, so using Ribet's\index{Ribet} theorem we can instead augment the level by~$7$. Our table of small-height curves contains just one (up to isogeny) elliptic curve of conductor~$37723$, and {\em luckily} the corresponding newform is congruent modulo~$3$ to the newform corresponding to {\bf 5389A} (away from primes dividing the level)! The Weierstrass equation of this curve is $$F:\quad y^2 - y = x^3 + x^2 + 34x - 248.$$ According to Cremona's program {\tt mwrank}, the rank of~$F$ is~$2$. \subsubsection{3364C, 4229A, 5073D} Perhaps $\Sha(E)$ is already visible in some of the cases in which the curve is labeled ``none'', because the method fails in most of these cases. Each of the curves {\bf 3364C}, {\bf 4229A}, and {\bf 5073D} is labeled ``none''. In none of these 3 cases are we able to find an explanatory factor at higher level, within the range of our table of elliptic curves of small height. \subsubsection{4194N, 5054C} The curve {\bf 4914N} is labeled ``none'' and we find the remark ``$E$ has rational $3$-torsion''. There is a congruent curve~$F$ of conductor $24570$ given by the equation $$F: \quad y^2 - xy = x^3 - x^2 - 15x - 75,$$ and $F(\Q) = \{0\}$. The curve {\bf 5054C} is labeled ``none'' and its Shafarevich-Tate group contains invisible elements of order~$3$. We find a congruent curve of level~$25270$ and rank~$1$. The equation of the congruent curve is $$F: y^2 - xy = x^3 + x^2 - 178x + 882.$$ \subsection{Positive rank example} The abelian varieties with nontrivial $\Sha(A)$ that one finds in both ours and Cremona's tables all have rank~$0$. In this section we outline a computation which sugggests, but does not prove, that there is a positive-rank abelian subvariety $A$ of $J_0(5077)$ such that $\Sha(A)$ possesses a nontrivial visible element of order~$31$. According to \cite{cremona:algs}, the smallest conductor elliptic curve~$E$ of rank~$3$ is found in $J=J_0(5077)$. The number $5077$ is prime, and the isogeny decomposition of~$J$ is\footnote{ This decomposition was found in about one minute using the Mestre-Oesterl\'e\index{Mestre} method of graphs (see~\cite{mestre:graphs}).} $$J \sim A \cross B \cross E,$$ where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$ associated to newforms, which have dimensions $205$, $216$, and~$1$, respectively. Using Remark~\ref{rem:moddegmestre} or \cite{zagier:parametrizations}, we find that the modular degree of~$E$ is $1984=2^6\cdot 31$. The sign of the Atkin-Lehner involution on~$E$ is the same as its sign on~$A$, so $E[31]\subset A$. We have $E(\Q)\isom \Z\cross\Z\cross\Z$, and the component group of~$E$ is trivial. The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so \cite{mazur:eisenstein} implies that none of the abelian varieties above have $31$-torsion. It might be possible to find an analogue of Theorem~\ref{thm:shaexists} that applies when~$A$ has positive rank, and deduce in this case that $\Sha(A)$ contains visible elements of order~$31$. %\section{Tables} \begin{table}\index{Table of!$\Sha$ at prime level} \ssp \caption{Shafarevich-Tate groups at prime level. (The entries in the columns ``mod deg'' and ``$\Shaan$'' are only really the odd parts of ``mod deg'' and ``$\Shaan$''.)\label{table:primesha}} \vspace{-.25in}$$ \begin{array}{lccclcc} \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)\\ & & & & & & \vspace{-3ex} \\ \mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ \mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ \mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ \mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ \mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ \mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ \mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ \mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ \mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ \mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ \mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ \mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ \mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ \mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ \mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ \mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ \mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ \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\\ \mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ \mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ \mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ \mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ \mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\\hline \mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ \mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2593B}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213 &\mbox {\bf 2593A}& 4 & 67 \cdot 2213\\ \end{array} $$ \end{table} \begin{table}\index{Table of!new visible $\Sha$} \ssp \caption{New visible Shafarevich-Tate groups\label{table:newvis}}\vspace{-2ex} $$\begin{array}{lcccccccl} \mathbf{A} &\text{dim} &\Shaan &w_q& c_p &\#A(\Q) &\frac{\#A(\Q)\cdot L(A,1)}{\Omega_A} &\mbox{\rm mod deg(A)} & \quad\mathbf{B}\\ \vspace{-2ex} & & &&& & & \\ \mathbf{389E} & 20&5^2&-&97&97&5^2&5&\mathbf{389A}\\ \mathbf{433D} & 16&7^2&-&3^2&3^2&7^2&3\cdot 7\cdot 37&\mathbf{433A}\\ \mathbf{446F}&8&11^2&+- &1,3&3&11^2&11\cdot359353&\mathbf{446B}\\ \mathbf{563E}&31 & 13^2 & - & 281 &281 &13^2 &13 &\mathbf{563A}\\ \mathbf{571D}&2 & 3^2 & - & 1 &1 & 3^2 & 3^2\cdot 127&\mathbf{571B}\\ \mathbf{655D}&13 & 3^4 &+- & 1,1 & 1 & 3^4 & 3^2\cdot 19\cdot 515741&\mathbf{655A}\\ \mathbf{664F} & 8&5^2 &-+ &a,1&1& 5^2 &5 & \mathbf{664A}\\ % Sha dim Wq c_p T TL/O delta B \mathbf{681B}&1 & 3^2 &+- & 1,1 & 1 & 3^2 & 3\cdot 5^3 & \mathbf{681C}\\ \mathbf{707G}& 15& 13^2 &+- & 1,1 & 1 & 13^2 & 13\cdot 800077& \mathbf{707A}\\ \mathbf{709C}&30& 11^2 &- & 59 &59 & 11^2 & 11 & \mathbf{709A}\\ \mathbf{718F}&7& 7^2 &+- & 1,1 & 1 & 7^2 &7\cdot 151\cdot 35573 & \mathbf{718B}\\ \mathbf{794G}&14& 11^2 &+- & 3,1 & 3 & 11^2 &3\cdot7\cdot11\cdot47\cdot35447& \mathbf{794A}\\ \mathbf{817E}& 15& 7^2 &+- & 1,5 & 5 & 7^2 & 7\cdot 79 & \mathbf{817A}\\ \mathbf{916G}&9& 11^2 &-+ & a,1 & 1 & 11^2 & 3^9\cdot 11\cdot 17\cdot 239 & \mathbf{916C}\\ \mathbf{944O} &6& 7^2 &+- & a,1 & 1 & 7^2 & 7 & \mathbf{944E}\\ \mathbf{997H}&42& 3^4 &- & 83 & 83 & 3^4 & 3^2 & \mathbf{997BC}\\ \mathbf{1001L}&7& 7^2 &+-+& 1,1,1 & 1 & 7^2 & 7\cdot19\cdot47\cdot2273&\mathbf{1001C}\\ \mathbf{1028E}&14& 11^2&-+ & a,1 & 3 & 3^4\cdot 11^2 & 3^{13}\cdot 11 & \mathbf{1028A}\\ \end{array}$$ \end{table} \begin{table}\index{Table of!explanatory factors} \ssp \caption{Explanatory factors\label{table:explain}}\vspace{-2ex} $$\begin{array}{lcccccc} \mathbf{B}&\text{rank}&w_q&c_p&\#A(\Q)&\mbox{\rm mod deg(A)}&\text{Comments}\\ \vspace{-2ex} & & && & & \\ \mathbf{389A}& 2 &-&1 &1&5&\text{first curve of rank $2$}\\ \mathbf{433A}&2 &-&1&1&7&\\ \mathbf{446B}&2 &+-&1,1& 1 &11&\text{called $\mathbf{446D}$ in \cite{cremona:algs}}\\ \mathbf{563A}&2 &- & 1 & 1 & 13 & \\ \mathbf{571B}&2 &- & 1 & 1 & 3 & \\ \mathbf{655A}&2 &+-& 1,1 & 1 & 3^2 & \\ \mathbf{664A}&2 &-+& 1,1 & 1 & 5 & \\ % RANK wq g_p Tor delta comments \mathbf{681C} & 2 & +- & 1,1 & 1 & 3 & \\ \mathbf{707A} & 2 & +- & 1,1 & 1 & 13 & \\ \mathbf{709A} & 2 & - & 1 & 1 & 11 & \\ \mathbf{718B} & 2 & +- & 1,1 & 1 & 7 & \\ \mathbf{794A} & 2 & +- & 1,1 & 1 & 11 & \\ \mathbf{817A} & 2 & +- & 1,1 & 1 & 7 & \\ \mathbf{916C} & 2 & -+ & 3,1 & 1 & 3\cdot 11 & \\ \mathbf{944E} & 2 & +- & 1,1 & 1 & 7 & \\ \mathbf{997B} & 2 & - & 1 & 1 & 3 & \\ \mathbf{997C} & 2 & - & 1 & 1 & 3 & \\ \mathbf{1001C} & 2 & +-+& 1,3,1& 1 & 3^2\cdot 7 & \\ \mathbf{1028A} & 2 & -+ & 3,1 & 1 & 3\cdot 11& \text{intersects $\mathbf{1028E}$ mod $11$}\\ \end{array}$$ \end{table} \begin{table}\index{Table of!factorizations} \ssp \caption{Factorizations\label{table:factor}}\vspace{-1ex} $$\begin{array}{llll} \mathbf{446}=2\cdot 223& \mathbf{655}=5\cdot 131& \mathbf{664}=2^3\cdot 83& \mathbf{681}=3\cdot 227\\ \mathbf{707}=7\cdot 101& \mathbf{718}=2\cdot 359& \mathbf{794}=2\cdot 397& \mathbf{817}=19\cdot 43\\ \mathbf{916}=2^2\cdot 229& \mathbf{944}=2^4\cdot 59& \mathbf{1001}=7\cdot 11\cdot 13& \mathbf{1028}=2^2\cdot 257\\ \end{array}$$ \end{table} \begin{table}\index{Table of!component groups of explanatory factors} \ssp \caption{Component groups\label{table:shacompgps}}\vspace{-2ex} $$\begin{array}{lcccccc} \vspace{-2ex}&&&&&&\\ \mathbf{A} &\text{dim} & p & w_q &\#\Phi_{X,p} &m_{X,p} &\#\Phi_{A,p}(\Fpbar) \\ \vspace{-2ex}& & & & & \\ \mathbf{389E}&20& 389&-& 97 & 5\cdot 97 & 97 \\ \mathbf{433D}&16& 433&-& 3^2& 3^3\cdot 7\cdot 37 & 3^2 \\ \mathbf{446F}&8 & 223&-& 3 & 3\cdot 11\cdot 359353 & 3 \\ && 2 &+ & 3 & 3\cdot 11& 3\cdot 359353\\ \mathbf{563E}&31& 563&-& 281& 13\cdot 281& 281 \\ \mathbf{571D}&2 & 571&-& 1 & 3^2\cdot 127 &1 \\ \mathbf{655D}&13& 131&-& 1 & 3^{2}\cdot19\cdot515741 & 1\\ && 5&+& 1 & 3^2 & 19\cdot 515741 \\ \mathbf{664F}&8 & 83&+& 1 & 5 & 1 \\ \mathbf{681B}&1 & 227&-& 1 & 3\cdot 5^3 & 1 \\ & & 3&+& 1 & 3\cdot 5^2& 5 \\ \mathbf{707G}&15& 101&-& 1 & 13\cdot800077 & 1 \\ && 7&+& 1 & 13& 800077 \\ \mathbf{709C}&30& 709&-& 59& 11\cdot 59 & 59 \\ \mathbf{718F}&7 & 359&-& 1 & 7\cdot 151\cdot 35573 &1 \\ && 2 &+& 1 & 7 & 151\cdot 35573 \\ \mathbf{794G}&14& 397&-& 3 & 3^2\cdot7\cdot11\cdot47\cdot35447 & 3 \\ && 2&+& 3 & 3\cdot11& 3^2\cdot 7\cdot 47\cdot 35447 \\ \mathbf{817E}&15& 43&- & 5 & 5\cdot7\cdot 79 & 5 \\ && 19&+ &1 & 7 & 79 \\ \mathbf{916G}&9 & 229&+ &1 & 3^9\cdot 11\cdot 17\cdot 239 & 1 \\ \mathbf{944O}&6 & 59&-& 1 & 7 & 1\\ \mathbf{997H}&42& 997&-& 83& 3^2\cdot 83 & 83 \\ \mathbf{1001L}&7& 13&+& 1& 7\cdot 19\cdot 47\cdot 2273& 1\\ && 11&-& 1& 7\cdot19\cdot47\cdot2273 & 1 \\ && 7&+& 1& 7\cdot 19\cdot 47 & 2273 \\ \mathbf{1028E}&14&257&+& 1 & 3^{13}\cdot 11 & 1 \\ \end{array}$$ \end{table} \comment{ \begin{table}\index{Table of!odd square numerators} \caption{Square roots of odd square divisors of $L(A,1)/\Omega_A$\label{table:oddnumer}}\vspace{2ex} $$\begin{array}{lc} \mathbf{305D7}&3\\ \mathbf{309D8}&5\\ \mathbf{335E11}&3^2\\ \mathbf{389E20}&5\\ \mathbf{394A2}&5\\ \mathbf{399G5}&3^4\\ \mathbf{433D16}&7\\ \mathbf{435G2}&3\\ \mathbf{436C4}&3\\ \mathbf{446E7}&3\\ \mathbf{446F8}&11\\ \mathbf{455D4}&3\\ \mathbf{473F9}&3\\ \mathbf{500C4}&3\\ \mathbf{502E6}&11\\ \mathbf{506I4}&5\\ \mathbf{524D4}&3\\ \mathbf{530G4}&7\\ \mathbf{538E7}&3\\ \mathbf{551H18}&3\\ \mathbf{553D13}&3\\ \mathbf{555E2}&3\\ \mathbf{556C7}&3\\ \mathbf{563E31}&13\\ \mathbf{564C3}&3\\ \mathbf{571D2}&3\\ \end{array}\qquad \begin{array}{lc} \mathbf{579G13}&15\\ \mathbf{597E14}&19\\ \mathbf{602G3}&3\\ \mathbf{604C6}&3 \\ \mathbf{615F6}&5 \\ \mathbf{615G8}&7 \\ \mathbf{620D3}&3\\ \mathbf{620E4}&3\\ \mathbf{626F12}&5\\ \mathbf{629G15}&3\\ \mathbf{642D2}&3\\ \mathbf{644C5}&3\\ \mathbf{644D5}&3\\ \mathbf{655D13}&3^2\\ \mathbf{660F2}&3\\ \mathbf{662E10}\!&\!\!43\\ \mathbf{664F8}&5\\ \mathbf{668B5}&3\\ \mathbf{678I2}&3\\ \mathbf{681B1}&3\\ \mathbf{681I10}&3\\ \mathbf{682I6}&11\\ \mathbf{707G15}&13\\ \mathbf{709C30}&11\\ \mathbf{718F7}&7\\ \mathbf{721F14}&3^2\\ \end{array}\qquad \begin{array}{lc} \mathbf{724C8}&3\\ \mathbf{756G2}&3\\ \mathbf{764A8}&3\\ \mathbf{765M4}&3\\ \mathbf{766B4}&3\\ \mathbf{772C9}&3\\ \mathbf{790H6}&3\\ \mathbf{794G12}\!&\!\!11\\ \mathbf{794H14}&5^2\\ \mathbf{796C8}&3\\ \mathbf{817E15}&7\\ \mathbf{820C4}&3\\ \mathbf{825E2}&3\\ \mathbf{844C10}\!&\!\!3^2\\ \mathbf{855M4}&3\\ \mathbf{860D4}&3\\ \mathbf{868E5}&3\\ \mathbf{876E5}&3\\ \mathbf{878C2}&3\\ \mathbf{884D6}&3\\ \mathbf{885L9}&3^2\\ \mathbf{894H2}&3\\ \mathbf{902I5}&3\\ \mathbf{913G17}&3\\ \mathbf{916G9}&11\\ \mathbf{918O2}&5\\ \end{array}\qquad \begin{array}{lc} \mathbf{918P2}&3\\ \mathbf{925K7}&3\\ \mathbf{932B13}&3^2\\ \mathbf{933E14}&19\\ \mathbf{934I12}&7\\ %-+ \mathbf{944O6}&7\\ \mathbf{946K7}&3\\ \mathbf{949B2}&3\\ \mathbf{951D19}&3\\ \mathbf{959D24}&3\\ \mathbf{964C12}&3^2\\ % -+, same as EC 964A but that has rank=0. \mathbf{966J1}&3\\ \mathbf{970I5}&3\\ \mathbf{980F1}&3\\ \mathbf{980J2}&3\\ \mathbf{986J7}&5\\ \mathbf{989E22}&5\\ \mathbf{993B3}&3^2\\ \mathbf{996E4}&3\\ \mathbf{997H42}&3^2\\ \mathbf{998A2}&3\\ % ++ \mathbf{998H9}&3\\ \mathbf{999J10}&3\\ &\\ &\\ &\\ \end{array}$$ \end{table} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \comment{ \begin{lemma} Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such that $\Phi=A\intersect B$ is finite and and $A(\Q)$ is finite. Suppose that~$p$ is a prime such that neither $(J/B)(\Q)$ nor $B(\Q)$ have any $p$-torsion and such that $B[p]\subset \Phi$. Then $B(\Q)/p B(\Q)$ is a subgroup of $(J/B)(\Q)/J(\Q)$. \end{lemma} \begin{proof} \end{proof} \section{Explanatory factors at higher level} Consider one of the items in Table~\ref{table:primesha} for which $\Sha$ is invisible. It is natural to ask whether these elements of~$\Sha$ ``become visible somewhere.'' For example, Mazur~\cite{mazur:visthree}\index{Mazur} proved that if $E$ is an elliptic curve and $c\in\Sha(E)$ has order $3$ then there is some abelian surface $A$ and an injection $\iota: E\hookrightarrow A$ such that $\iota_*(c)=0\in H^1(\Q,A)$. T. Klenke has proved a partial statement in this direction for elements of order $2$ as part of his Harvard Ph.D. thesis. J.~de Jong (see \cite[Remark 3]{cremona-mazur}) showed that every element of the Shafarevich-Tate group of an elliptic curve is visible in some Jacobian.\index{Jacobian} Consider an abelian variety $A$, taken from Table~\ref{table:primesha}, for which $\Sha$ has an invisible element $c$. Thus $A$ sits inside $J_0(p)$ for some prime $p$, and we ask ``is there a prime $q$ such that $\delta(c)=0$ for one of the degeneracy maps $\delta : J_0(p)\ra J_0(pq)$?'' The author has no idea\footnote{Lo\"\i{}c Merel\index{Merel} suspects the answer might be yes whereas Richard Taylor is more skeptical.}. To get a feeling for what might happen we consider in detail abelian variety $A=A_f$ at level $p=1091$ in which $\Shaan$ is divisible by $7$. There is a prime $\lambda$ of the ring $\Z[f] = \Z[\ldots a_n\ldots]$ attached to $A$. The Fourier coefficients of $f$ modulo $\lambda$ are $$\begin{array}{rcccccccccccccccc} p= &2 &3 &5 &7 &11 &13 &17 &19 &23 &29 &31 &37 &41 &43 &47 &53\\ a_p= &3 &0 &1 &5 &0 &2 &0 &5 &4 &6 &3 &3 &5 &5 &6 &5 \end{array}$$ These were computed by finding an eigenvector in $H_1(X_0(N);\F_7)$. [[SAY MORE ABOUT THE TRICK FOR FINDING ALL RIBET $q$'s.]] According to Ribet's\index{Ribet} level raising theorem \cite{ribet:raising} there is a newform $g$ of level $1091\ell$ such that $f\con g$ modulo [[something]] if $a_\ell = \pm (\ell+1)\pmod{\lambda}$. Fortunately this criterion is already satisfied for $\ell=2$. Looking closely at level $2\cdot 1091$ (for example, in Cremona's online tables \cite{cremona:onlinetables}) we find an elliptic curve $E$ whose corresponding newform $g=\sum b_n q^n$ has Fourier coefficients $$\begin{array}{rcccccccccccccccc} p = &2 & 3 & 5 & 7& 11& 13& 17& 19& 23& 29& 31& 37&41&43&47&53\\ b_p=&1 & 0 & 1 &-2& 0 & -5& 0 & -2& 4& 6& -4& -4&-9&-9&-8&-2\\ \end{array}$$ This is convincing evidence that one of the two images of~$A$ in $J_0(2\cdot 1091)$ shares some $7$-torsion with the elliptic curve \abvar{2182B}. This can be [[WILL BE!!, EASILY]] established by a direct computation with the period lattices. This is at first disconcerting because the rank of this elliptic curve is {\em not} $2$. However, the rank is still positive; it is $1$ with Mordell-Weil group $\E(\Q)=Z$. I would not be at all surprised if your $7$-torsion in Sha does become visible in $J_0(2\cdot1091)$. The curve \abvar{2182B}, which shares 7-torsion with $A$ is \begin{verbatim} e=ellinit([1,-1,1,-67,67]); The Tamagawa number c_2 is 14 (!!) ? elllocalred(e,2) %2 = [1, 18, [1, 0, 0, 0], 14] The Tamagawa number c_1091 is 1. ? elllocalred(e,1091) %3 = [1, 5, [1, 0, 0, 0], 1] I have this feeling that the right statement about congruence and mordell-Weil is really something like congruence ==> "Selmer + Comp group"'s are identified. Anyway, the extra component group of order 7 may perhaps account for the other nontrivial element of Sha. This might just be wild speculation. Good luck. william Barry, Amod asked me to investigate whether his element of order 7 in the winding quotient J_e at level 1091 becomes visible at higher level. Luckily, Ribet's\index{Ribet} level raising theorem predicts the existence of a form at level 2*1091 congruent mod a prime over 7 to the form corresponding to J_e. Even more luckily, one of the two rational newforms does the trick. Thus an image of J_e in J_0(2*1091) shares 7-torsion with an elliptic curve E (2182B in Cremona's tables). This elliptic curve has: E(Q) = Z Sha(E/Q) = 0 c_2 = 14, c_1091 = 1 L^(1)(f,1)/1! = 4.27332686791516 So there is reasonable hope that the elements of order 7 in Sha(J_e) are visible at this higher level, even though they are invisible even in J_1(1091). Best, William Dear William, This is terrific. I assume that you will be showing that for J_e the winding (not quite quotient, but more conveniently sub-thing) in J_0(1091), the image of Sha(J_e) ---> Sha(J_0(2*1091)) just dies? Since our working hope, I think, is that for any N there is an M so that Sha(J_0(N)) ---. Sha(J_0(N.M)) dies, this suggests returning to the (mod 3) N=2849 example, where I "know" that there must exist such an M (because all three-torsion in Sha on elliptic curves is visible in some appropriate abelian surface which is isogenous to a product of two elliptic curves, and therefore, is abelian surface is probably "modular"). But I do not know a specific M. Barry \end{verbatim} \comment{ \begin{remark} One reason we must assume~$p$ is odd, is because when~$B$ has good reduction at~$2$, in the proof we change~$J$ by an isogeny of $2$-power degree in order to apply~\cite[\S7.5, Prop.~3]{neronmodels} at $p=2$. When~$B$ has purely toric reduction,\index{Toric reduction} at~$2$ we use Tate uniformization to directly verify that points of $B(\Q)$ map into $\Sha(A)$, thereby avoiding exactness properties of N\'eron models\index{N\'eron model}. \end{remark}} \subsubsection{Table~\ref{table:oddnumer}: Odd square divisors of $L(A,1)/\Omega_A$} In order to find candidate~$A$ with nontrivial visible $\Sha(A)$, we first enumerated those~$A$ for which the numerator of $L(A,1)/\Omega_A$ is divisible by an odd square~$n>1$. For $N<1000$, these are given in Table~\ref{table:oddnumer}. Any odd visible $\Sha(A)$ coprime to primes dividing torsion and $c_p$ must show up as a divisor of the numerator; it should show up as a square divisor because the Mordell-Weil rank of the explanatory factor should be even. It would be interesting to compute the conjectural order of $\Sha(A)$ for each abelian variety in this table, but not in table 1, and show (when possible) that the visible $\Sha(A)$ is old. }