\documentclass{article}1\title{Visibility of Shafarevich-Tate groups of modular2abelian varieties}3\author{William Stein}4\date{July 2000}5\include{macros}6\DeclareMathOperator{\Vis}{Vis}7\DeclareMathOperator{\Res}{Res}8\usepackage[all]{xy}9\begin{document}10\maketitle11\begin{abstract}12These are the notes for my July 2000 Durham talk on visibility13of Shafarevich-Tate groups. I typed them in quickly the night14before my talk, in order to make precise what I would say.15\end{abstract}1617\section{Modular abelian varieties}18A central class of abelian varieties of interest in number theory19are the modular abelian varieties of $\GL_2$-type. It is now known20that every elliptic curve over~$\Q$ is a member of this class.21Such abelian varieties are constructed as follows.22Let $f=\sum a_n q^n\in S_2(\Gamma_0(N),\C)$ be a newform. The23Hecke algebra $\T=\Z[\ldots T_n \ldots]$ acts on~$f$, and24there is a surjective homomorphism25$\T\ra \O_f = \Z[\ldots a_n \ldots]$ sending $T_n$ to $a_n$.26The kernel $I_f$ of this homomorphism is the annihilator of~$f$ in $\T$.27Attached to $I_f$ there is a quotient $A_f = J_0(N)/I_f(J_0(N))$,28where $J_0(N)$ is the Jacobian of the modular curve $X_0(N)$.29The abelian variety $A_f$ has dimension $d=[\Q(\ldots a_n\ldots):\Q]$,30and the dual $A_f^{\vee}$ is an abelian subvariety of $J_0(N)$.3132There are two groups attached to an abelian variety~$A$ over $\Q$.33One is the Mordell-Weil group $A(\Q)$, which is a finitely generated34abelian group, and the other is the Shafarevich-Tate group35$\Sha(A) = \ker(H^1(\Q,A) \ra \prod_v H^1(\Q_v,A))$,36which is ({\em conjecturally}!) a finite abelian group. This talk37is about a link between these two groups, which is frequently signaled38by the term ``visibility''. Before proceeding further, we state39a fundamental conjecture that both guides and stimulates our work.4041\begin{conjecture}[Birch and Swinnerton-Dyer]42Let $A$ be an abelian variety over~$\Q$. Then $\Sha(A)$ is finite,43and the following formula holds:44$$L(A,1) = \frac{\prod c_p \cdot \Omega_A}{\#A(\Q)\cdot \#A^{\vee}(\Q)} \cdot \#\Sha(A),$$45where the right hand side should be interpreted as~$0$ if $A(\Q)$46is infinite.47(There is a refinement when $L(A,1)=0$.)48\end{conjecture}4950\section{Mazur: Visualize Sha!}5152Let~$A$ be an abelian variety over~$\Q$. To give an elements of $\Sha(A)$53is the same as giving an algebraic variety $X$ over~$\Q$ equipped with54an action $A\cross X \ra X$ satisfying analogues of the usual axioms55for a simply transitive group action. Where can we find these varieties~$X$?56As translates of $A$ inside a bigger abelian variety~$J$!5758Fix an embedding of $A$ into an abelian variety $J$, and let $C$ be the59quotient:60$$A \hookrightarrow J \onto C.$$61Taking cohomology, reveals the following diagram:62$$\xymatrix{63& & & {\Vis_J(\Sha(A))}\[email protected]{^(->}[r] & {\Sha(A)}\[email protected]{^(->}[d] \ar[r] & {\Sha(J)}\[email protected]{^(->}[d]\\640\ar[r] & A(\Q) \ar[r] & J(\Q) \ar[r] & C(\Q) \ar[r] & H^1(\Q,A) \ar[r] & H^1(\Q,J) }$$6566\begin{definition}[Visible part]67The \defn{visible part of $\Sha(A)$} with respect to the embedding $A\hookrightarrow J$68is the subgroup69$$\Vis_J(A) = \ker(\Sha(A) \ra \Sha(J)).$$70Likewise, the visible part of $H^1(\Q,A)$ is71$$\Vis_J(H^1(\Q,A)) = \ker(H^1(\Q,A) \ra H^1(\Q,J)).$$72\end{definition}7374\begin{question}[Mazur]75Is every element $c\in H^1(\Q,A)$ visible somewhere?76\end{question}77Johan de Jong answered this question in the affirmative, when $\dim A = 1$; however,78his proof, which exploits Azamuya algebras over $\Spec(\Z)$, is not elementary.79Fortunately, Mazur's question has an elementary answer.80\begin{proposition}[Folklore?]81Given $c\in H^1(\Q,A)$ there is an abelian variety $J$ and an embedding $\iota:A\hookrightarrow J$82such that $\iota_*(c) = 0 \in H^1(\Q,J)$.83\end{proposition}84\begin{proof}85Because cochains are continuous, there is a finite extension $K$ of $\Q$86such that $\res_K(c) = 0 \in H^1(K,A)$. The Shapiro lemma implies that87$H^1(K,A)$ is canonically isomorphic to $H^1(\Q,\Res_{K/\Q} A_K)$, where88$\Res_{K/\Q} A_K$ is the Weil restriction of scalars of $A_K$. Furthermore,89there is a canonical embedding $\iota: A\hookrightarrow J=\Res_{K/\Q} A_K$,90and this embedding sends~$c$ to $\res_K(c) = 0$.91\end{proof}9293\begin{corollary}94Every $c\in \Sha(E)[2]$ is visible in an abelian surface, and95every $c\in \Sha(E)[3]$ is visible in a three-dimensional abelian variety.96\end{corollary}97Using more sophisticated techniques, Mazur proved that every $c\in \Sha(E)[3]$ is visible98in an abelian surface.99100For the rest of this talk, we return to the case where $A$ is a attached to a101modular form.102\begin{question}[Mazur]103Let~$f$ be a newform of level~$N$.104Can one visualize the elements $c\in \Sha(A_f^{\vee})$ inside $J_0(N)$? Inside $J_0(M)$105for some multiple $M$ of $N$?106\end{question}107108\section{Some data}109Mazur, Cremona, and Logan collected data on visibility of Sha using only elliptic curves110and congruences between elliptic curves. The data is impressive, but probably only because111$\Sha$ is very small in the range considered. As Frey emphasized in his talk, there112are very few intersections between elliptic curves.113114The author made a table of rational numbers $L(A_f,1)/\Omega_{A_f}$ for115all $f \in S_2(\Gamma_0(p))$, for $p\leq 2161$.116To analyze the resulting data, let us momentarily assume the truth of the Birch and Swinnerton-Dyer117conjecture. We find~$38$ of the $A_f$ have the property that an odd prime~$\ell$ divides118$\#\Sha(A_f)$. Of these, the full odd part of $\Sha(A_f)$ is visible in $22$ cases, whereas119it is invisible in all of the remaining $16$ cases. A small {\em selection} of data is120recorded in the table below.121122\begin{center}123\begin{tabular}{lcll}124$p$ & $\dim A_f$ & $\sqrt{\#\Sha(A_f)^{\text{odd}}}$ & $\dim B$\\125389 & 20 & 5 (VIS)& 1\\126433 & 16 & 7 (VIS)& 1\\1271061 & 46 & 151 (VIS)& 2 \\1281091 & 62 & 7 (INV, so far) & ---\\1291429 & 64 & 5 (VIS at level $2\cdot 1429$) & 2\\1302111 & 112 & 211 (INV, so far) & ---\\1312333 & 101 & 83341 (VIS) & 4\\1322849 & 1 & 3 (VIS at level $3\cdot 2849$) & 1\\1335389 & 1 & 3 (VIS at level $7\cdot 3849$) & 1\\134\end{tabular}135\end{center}136137\section{Criterion for visibility}138\begin{theorem}[Stein]139Let $A,B \subset J$, and let $N$ be divisible exactly by the primes of bad reduction for~$J$.140Suppose that141\begin{itemize}142\item $(A \intersect B)(\Qbar)$ is finite,143\item $A(\Q)$ is finite,144\item $B$ has purely toric reduction at each $\ell\mid N$.145\end{itemize}146Let $p$ be a prime such that147\begin{enumerate}148\item149$$p\nmid 2N \cdot \# (\frac{J(\Q)}{A(\Q)})_{\tor} \cdot B(\Q)_{\tor} \cdot150\prod_{\ell \mid N} \#\Phi_{A,\ell}(\overline{\F})\cdot \#\Phi_{B,\ell}(\F_\ell),$$151\item $B[p]\subset A$ (this condition can be relaxed).152\end{enumerate}153Then154$$B(\Q)/p B(\Q) \hookrightarrow \Vis_J(\Sha(A)).$$155\end{theorem}156\begin{proof}[Sketch]157158Consider the diagram159$$\xymatrix{160& {A \intersect B} \ar[r]\ar[d] & B\ar[d] \ar[r]^p& B\ar[d]\\1610 \ar[r] & A \ar[r] & J \ar[r] & C \ar[r]& 0.162}$$163164Taking cohomology, we arrive at the following key diagram:165$$\xymatrix{1660\ar[r] & B(\Q)\ar[d] \ar[r]^{p} & B(\Q)\ar[d] \ar[r] & B(\Q)/p B(\Q)\ar[d] \ar[r] & 0\\1670\ar[r] & J(\Q)/A(\Q) \ar[r] & C(\Q) \ar[r] & \Vis_J(H^1(\Q,A)) \ar[r] & 0.168}$$169Upon applying the snake lemma and using the above hypothesis, we170deduce that171$B(\Q)/p B(\Q) \hookrightarrow \Vis_J(H^1(\Q,A))$. The172deduction that $B(\Q)/p B(\Q)\hookrightarrow \Vis_J(\Sha(A))$ involves173a local analysis at each prime $\ell$ of $\Q$.174\end{proof}175176\section{Visibility at higher level}177There is a $53$-dimensional abelian subvariety~$A$ of $J_0(1429)$178such that~$5$ divides the conjectural order of $\Sha(A)$. However, this part of179$\Sha$ can not be visible in $J_0(1429)$.180Ribet's level raising theorem supplies us with an infinite collection181of primes $p$ such that an image of $A$ in $J_0(pN)$ meets an abelian182variety $B$ in some $\m$-torsion, where $\m$ is an appropriate maximal ideal183of residue characteristic $5$ in $\T$.184For each of these $B$, {\em probably} (I haven't proved this)185$$L(B,1)\con L(A,1) = 0 \pmod{\m},$$186so either $5\mid \#\Sha(B)$ or $L(B,1)=0$. In the latter case, the Birch187and Swinnerton-Dyer conjecture predicts that $B(\Q)$ is infinite;188the theorem above then implies that189$$5\mid \Vis_{J_0(pN)}(\Sha(A)).$$190There are many similar examples, some of which have $\dim B=1$, and hence don't191rely on any conjectures. Based on this evidence, we make the following conjecture.192\begin{conjecture}[Stein]193Let $N$ be a prime and $A=A_f^{\vee}\subset J_0(N)$.194If $c\in \Sha(A)$, then there is a prime $p$ such that195$\delta_1^*(c)+ \delta_p^*(c) = 0 \in \Sha(J_0(pN))$.196\end{conjecture}197198\section{Speculation: constructing points?}199Constructing elements of $\Vis_{J_0(N)}(\Sha(A))$ is equivalent to200constructing points on~$B$. This observation might lead to an201approach to proving the implication $L(B,1)=0$ implies $B(\Q)$ is infinite.202At present, mathematicians seem to have no good construction of points203when $L(B,1)=L'(B,1)=0$. However, we do have Kolyvagin's Euler systems when $L(A,1)\neq 0$!204205\end{document}206