Sharedwww / talks / durham / durham.texOpen in CoCalc
Author: William A. Stein
1\documentclass{article}
2\title{Visibility of Shafarevich-Tate groups of modular
3abelian varieties}
4\author{William Stein}
5\date{July 2000}
6\include{macros}
7\DeclareMathOperator{\Vis}{Vis}
8\DeclareMathOperator{\Res}{Res}
9\usepackage[all]{xy}
10\begin{document}
11\maketitle
12\begin{abstract}
13These are the notes for my July 2000 Durham talk on visibility
14of Shafarevich-Tate groups.  I typed them in quickly the night
15before my talk, in order to make precise what I would say.
16\end{abstract}
17
18\section{Modular abelian varieties}
19A central class of abelian varieties of interest in number theory
20are the modular abelian varieties of $\GL_2$-type.  It is now known
21that every elliptic curve over~$\Q$ is a member of this class.
22Such abelian varieties are constructed as follows.
23Let $f=\sum a_n q^n\in S_2(\Gamma_0(N),\C)$ be a newform.  The
24Hecke algebra $\T=\Z[\ldots T_n \ldots]$ acts on~$f$, and
25there is a surjective homomorphism
26$\T\ra \O_f = \Z[\ldots a_n \ldots]$ sending $T_n$ to $a_n$.
27The kernel $I_f$ of this homomorphism is the annihilator of~$f$ in $\T$.
28Attached to $I_f$ there is a quotient $A_f = J_0(N)/I_f(J_0(N))$,
29where $J_0(N)$ is the Jacobian of the modular curve $X_0(N)$.
30The abelian variety $A_f$ has dimension $d=[\Q(\ldots a_n\ldots):\Q]$,
31and the dual $A_f^{\vee}$ is an abelian subvariety of $J_0(N)$.
32
33There are two groups attached to an abelian variety~$A$ over $\Q$.
34One is the Mordell-Weil group $A(\Q)$, which is a finitely generated
35abelian group, and the other is the Shafarevich-Tate group
36$\Sha(A) = \ker(H^1(\Q,A) \ra \prod_v H^1(\Q_v,A))$,
37which is ({\em conjecturally}!) a finite abelian group.  This talk
38is about a link between these two groups, which is frequently signaled
39by the term visibility''.  Before proceeding further, we state
40a fundamental conjecture that both guides and stimulates our work.
41
42\begin{conjecture}[Birch and Swinnerton-Dyer]
43Let $A$ be an abelian variety over~$\Q$.  Then $\Sha(A)$ is finite,
44and the following formula holds:
45 $$L(A,1) = \frac{\prod c_p \cdot \Omega_A}{\#A(\Q)\cdot \#A^{\vee}(\Q)} \cdot \#\Sha(A),$$
46where the right hand side should be interpreted as~$0$ if $A(\Q)$
47is infinite.
48(There is a refinement when $L(A,1)=0$.)
49\end{conjecture}
50
51\section{Mazur: Visualize Sha!}
52
53Let~$A$ be an abelian variety over~$\Q$.   To give an elements of $\Sha(A)$
54is the same as giving an algebraic variety $X$ over~$\Q$ equipped with
55an action $A\cross X \ra X$ satisfying analogues of the usual axioms
56for a simply transitive group action.  Where can we find these varieties~$X$?
57As translates of $A$ inside a bigger abelian variety~$J$!
58
59Fix an embedding of $A$ into an abelian variety $J$, and let $C$ be the
60quotient:
61    $$A \hookrightarrow J \onto C.$$
62Taking cohomology, reveals the following diagram:
63$$\xymatrix{ 64 & & & {\Vis_J(\Sha(A))}\[email protected]{^(->}[r] & {\Sha(A)}\[email protected]{^(->}[d] \ar[r] & {\Sha(J)}\[email protected]{^(->}[d]\\ 650\ar[r] & A(\Q) \ar[r] & J(\Q) \ar[r] & C(\Q) \ar[r] & H^1(\Q,A) \ar[r] & H^1(\Q,J) }$$
66
67\begin{definition}[Visible part]
68The \defn{visible part of $\Sha(A)$} with respect to the embedding $A\hookrightarrow J$
69is the subgroup
70$$\Vis_J(A) = \ker(\Sha(A) \ra \Sha(J)).$$
71Likewise, the visible part of $H^1(\Q,A)$ is
72$$\Vis_J(H^1(\Q,A)) = \ker(H^1(\Q,A) \ra H^1(\Q,J)).$$
73\end{definition}
74
75\begin{question}[Mazur]
76Is every element $c\in H^1(\Q,A)$ visible somewhere?
77\end{question}
78Johan de Jong answered this question in the affirmative, when $\dim A = 1$; however,
79his proof, which exploits Azamuya algebras over $\Spec(\Z)$, is not elementary.
80Fortunately, Mazur's question has an elementary answer.
81\begin{proposition}[Folklore?]
82Given $c\in H^1(\Q,A)$ there is an abelian variety $J$ and an embedding $\iota:A\hookrightarrow J$
83such that $\iota_*(c) = 0 \in H^1(\Q,J)$.
84\end{proposition}
85\begin{proof}
86Because cochains are continuous, there is a finite extension $K$ of $\Q$
87such that $\res_K(c) = 0 \in H^1(K,A)$.  The Shapiro lemma implies that
88$H^1(K,A)$ is canonically isomorphic to $H^1(\Q,\Res_{K/\Q} A_K)$, where
89$\Res_{K/\Q} A_K$ is the Weil restriction of scalars of $A_K$.   Furthermore,
90there is a canonical embedding $\iota: A\hookrightarrow J=\Res_{K/\Q} A_K$,
91and this embedding sends~$c$ to $\res_K(c) = 0$.
92\end{proof}
93
94\begin{corollary}
95Every $c\in \Sha(E)[2]$ is visible in an abelian surface, and
96every $c\in \Sha(E)[3]$ is visible in a three-dimensional abelian variety.
97\end{corollary}
98Using more sophisticated techniques, Mazur proved that every $c\in \Sha(E)[3]$ is visible
99in an abelian surface.
100
101For the rest of this talk, we return to the case where $A$ is a attached to a
102modular form.
103\begin{question}[Mazur]
104Let~$f$ be a newform of level~$N$.
105Can one visualize the elements $c\in \Sha(A_f^{\vee})$ inside $J_0(N)$?  Inside $J_0(M)$
106for some multiple $M$ of $N$?
107\end{question}
108
109\section{Some data}
110Mazur, Cremona, and Logan collected data on visibility of Sha using only elliptic curves
111and congruences between elliptic curves.  The data is impressive, but probably only because
112$\Sha$ is very small in the range considered.  As Frey emphasized in his talk, there
113are very few intersections between elliptic curves.
114
115The author made a table of rational numbers $L(A_f,1)/\Omega_{A_f}$ for
116all $f \in S_2(\Gamma_0(p))$, for $p\leq 2161$.
117To analyze the resulting data, let us momentarily assume the truth of the Birch and Swinnerton-Dyer
118conjecture.  We find~$38$ of the $A_f$ have the property that an odd prime~$\ell$ divides
119$\#\Sha(A_f)$.   Of these, the full odd part of $\Sha(A_f)$ is visible in $22$ cases, whereas
120it is invisible in all of the remaining $16$ cases.  A small {\em selection} of data is
121recorded in the table below.
122
123\begin{center}
124\begin{tabular}{lcll}
125$p$ & $\dim A_f$ & $\sqrt{\#\Sha(A_f)^{\text{odd}}}$ & $\dim B$\\
126389 & 20 & 5 (VIS)& 1\\
127433 & 16 & 7 (VIS)& 1\\
1281061 & 46 & 151 (VIS)& 2 \\
1291091 & 62 & 7 (INV, so far) & ---\\
1301429 & 64 & 5 (VIS at level $2\cdot 1429$) & 2\\
1312111 & 112 & 211 (INV, so far) & ---\\
1322333 & 101 & 83341 (VIS) & 4\\
1332849 & 1 & 3 (VIS at level $3\cdot 2849$) & 1\\
1345389 & 1 & 3 (VIS at level $7\cdot 3849$) & 1\\
135\end{tabular}
136\end{center}
137
138\section{Criterion for visibility}
139\begin{theorem}[Stein]
140Let $A,B \subset J$, and let $N$ be divisible exactly by the primes of bad reduction for~$J$.
141Suppose that
142\begin{itemize}
143\item $(A \intersect B)(\Qbar)$ is finite,
144\item $A(\Q)$ is finite,
145\item $B$ has purely toric reduction at each $\ell\mid N$.
146\end{itemize}
147Let $p$ be a prime such that
148\begin{enumerate}
149\item
150  $$p\nmid 2N \cdot \# (\frac{J(\Q)}{A(\Q)})_{\tor} \cdot B(\Q)_{\tor} \cdot 151 \prod_{\ell \mid N} \#\Phi_{A,\ell}(\overline{\F})\cdot \#\Phi_{B,\ell}(\F_\ell),$$
152\item $B[p]\subset A$  (this condition can be relaxed).
153\end{enumerate}
154Then
155$$B(\Q)/p B(\Q) \hookrightarrow \Vis_J(\Sha(A)).$$
156\end{theorem}
157\begin{proof}[Sketch]
158
159Consider the diagram
160$$\xymatrix{ 161 & {A \intersect B} \ar[r]\ar[d] & B\ar[d] \ar[r]^p& B\ar[d]\\ 1620 \ar[r] & A \ar[r] & J \ar[r] & C \ar[r]& 0. 163}$$
164
165Taking cohomology, we arrive at the following key diagram:
166$$\xymatrix{ 1670\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\\ 1680\ar[r] & J(\Q)/A(\Q) \ar[r] & C(\Q) \ar[r] & \Vis_J(H^1(\Q,A)) \ar[r] & 0. 169}$$
170Upon applying the snake lemma and using the above hypothesis, we
171deduce that
172$B(\Q)/p B(\Q) \hookrightarrow \Vis_J(H^1(\Q,A))$.   The
173deduction that $B(\Q)/p B(\Q)\hookrightarrow \Vis_J(\Sha(A))$ involves
174a local analysis at each prime $\ell$ of $\Q$.
175\end{proof}
176
177\section{Visibility at higher level}
178There is a $53$-dimensional abelian subvariety~$A$ of $J_0(1429)$
179such that~$5$ divides the conjectural order of $\Sha(A)$.  However, this part of
180$\Sha$ can not be visible in $J_0(1429)$.
181Ribet's level raising theorem supplies us with an infinite collection
182of primes $p$ such that an image of $A$ in $J_0(pN)$ meets an abelian
183variety $B$ in some $\m$-torsion, where $\m$ is an appropriate maximal ideal
184of residue characteristic $5$ in $\T$.
185For each of these $B$, {\em probably} (I haven't proved this)
186$$L(B,1)\con L(A,1) = 0 \pmod{\m},$$
187so either $5\mid \#\Sha(B)$ or $L(B,1)=0$.  In the latter case, the Birch
188and Swinnerton-Dyer conjecture predicts that $B(\Q)$ is infinite;
189the theorem above then implies that
190$$5\mid \Vis_{J_0(pN)}(\Sha(A)).$$
191There are many similar examples, some of which have $\dim B=1$, and hence don't
192rely on any conjectures.  Based on this evidence, we make the following conjecture.
193\begin{conjecture}[Stein]
194Let $N$ be a prime and $A=A_f^{\vee}\subset J_0(N)$.
195If $c\in \Sha(A)$, then there is a prime $p$ such that
196$\delta_1^*(c)+ \delta_p^*(c) = 0 \in \Sha(J_0(pN))$.
197\end{conjecture}
198
199\section{Speculation: constructing points?}
200Constructing elements of $\Vis_{J_0(N)}(\Sha(A))$ is equivalent to
201constructing points on~$B$.   This observation might lead to an
202approach to proving the implication $L(B,1)=0$ implies $B(\Q)$ is infinite.
203At present, mathematicians seem to have no good construction of points
204when $L(B,1)=L'(B,1)=0$.  However, we do have Kolyvagin's Euler systems when $L(A,1)\neq 0$!
205
206\end{document}