Sharedwww / tables / visshatalk.texOpen in CoCalc
Author: William A. Stein
1% tucsontalk.tex
2\documentclass[12pt]{article}
3\pagestyle{empty}
4\title{Visibility of Shafarevich-Tate Groups\\
5       of Modular Abelian Varieties}
6\include{macros}
7\author{William Stein\\
8{\tt was@math.berkeley.edu}\\
9{\tt http://shimura.math.berkeley.edu/\~{}was}}
10\date{March 1999}
11\begin{document}
12\large
13\maketitle
14\tableofcontents
15\Large
16\newpage
17\section{BSD Conjecture}
18\begin{itemize}
19\item $A/\Q$ abelian variety, $\dim A=d$
20\item $L(A,s)$ associated $L$-function
21\end{itemize}
22
23\vspace{.3in}
24\begin{conjecture} [Birch, Swinnerton-Dyer, Tate]
25\par\noindent
26\begin{enumerate}
27\item $L(A,s)$ is holomorphic on all $\C$
28\item $$\frac{L(A,1)}{\Omega} = \frac{|\Sha(A)|\cdot \prod c_p} 29 {|A(\Q)|\cdot|A^{\vee}(\Q)|}$$
30where
31\begin{itemize}
32\item $\Sha(A)=\Ker(H^1(\Q,A)\ra \prod_v H^1(\Q_v,A))$
33\item $c_p$ = rational components of special
34     fiber of Neron model $\A/\Fp$
35\item $\Omega = \int_{A(\R)}|\omega|$, where $H^0(\cA,\Omega^d)=\Z\omega$
36\end{itemize}
37\end{enumerate}
38\end{conjecture}
39
40\newpage
41\section{Visibility}
42Fix
43$$i:A\hookrightarrow J$$
44
45{\bf Visible Sha:}
46$$\Sha^0(A) := \Ker(\Sha(A)\xrightarrow{i_*} \Sha(J))$$
47
48\par \noindent We can {\em see} $\Sha^0(A)$:\\
49The long exact sequence associated to
50$$0\ra A \ra J \ra B \ra 0$$
51gives
52$$\begin{matrix} 530 & \ra & B(\Q)/J(\Q) & \ra & H^1(\Q,A) & \ra & H^1(\Q,J) & \ra \cdots\\ 54 & & \cup & & \cup & & \cup\\ 550 & \ra & \Sha^0(A) & \ra & \Sha(A) & \ra & \Sha(J) \\ 56\end{matrix}$$
57
58{\bf Remark:} (Ogg) Mordell-Weil implies $\Sha^0(A)$ is finite.
59
60\newpage
61\section{Congruences}
62Fix $A$.\\
63Find $B$ such that:
64                  $$A[p]\isom B[p] \qquad \text{ as \Gal(\Qbar/\Q)-modules}$$
65Then
66$$H^1(\Q,A[p])\isom H^1(\Q,B[p]).$$
67\\
68Kummer and Selmer:
69$$\begin{matrix} 700 &\ra& A(\Q)/pA(\Q) &\ra & H^1(\Q,A[p]) & \ra & H^1(\Q,A)[p] & \ra & 0 \\ 71 & & || & & \cup & & || & & \\ 720 &\ra& A(\Q)/pA(\Q) &\ra & \Selmer_p(A/\Q) & \ra & \Sha(A)[p] & \ra & 0 \\ 73 & & & & || \text{ if lucky!} & & & \\ 740 &\ra& B(\Q)/pB(\Q) &\ra & \Selmer_p(B/\Q) & \ra & \Sha(B)[p] & \ra & 0 \\ 75 & & || & & \cap & & || & & \\ 760 &\ra& B(\Q)/pB(\Q) &\ra & H^1(\Q,B[p]) & \ra & H^1(\Q,B)[p] & \ra & 0 \\ 77\end{matrix}$$
78
79If $A(\Q)=0$ and $\Sha(B)[p]=0$ then
80$$\Sha(A)[p] = B(\Q)/p B(\Q).$$
81
82Luck'' understandable in terms of Mazur's flat cohomology.
83
84\newpage
85\section{Modular Rank $0$ BSD}
86$A_f$ optimal quotient of $J_0(N)$, corresponding to
87    $$f = \sum a_n q^n \in S_2(\Gamma_0(N),\C).$$
88Have
89$$0 \ra C \ra J_0(N) \ra A_f \ra 0.$$
90
91\begin{theorem}[Hecke]
92$$L(f,s) =\sum a_n n^{-s}$$
93holomorphic on $\C$.
94\end{theorem}
95
96\begin{theorem}[Shimura]
97$$L(A_f,s) = \prod_i L(f_i,s),$$
98product over conjugates of $f$.
99\end{theorem}
100
101\begin{theorem}[Kolyvagin-Logachev]
102\mbox{ }\par\noindent
103$L(A_f,1)\neq 0$ $\implies$ $A_f(\Q)$ and $\Sha(A_f)$ both finite.
104(Heegner points.)
105\end{theorem}
106
107
108
109\newpage
110\section{Formula for $L(A_f,1)/\Omega$}
111Abel-Jacobi map:
112$$H_1(X_0(N),\Z) \xrightarrow{\Phi} \C^d \ra A_f \ra 0$$
113$$\Phi(\gamma)= (\int_\gamma f_1, \ldots ,\int_\gamma f_d)$$
114
115\vspace{.3in}
116
117\begin{theorem}[Agashe]
118\mbox{}\\
119\mbox{}\\
120$\e=-\{0,\infty\}\in H_1(X_0(N),\Q)$\\
121$\T=\text{ Hecke algebra }$
122\begin{eqnarray*}
123\frac{L(A_f,1)}{\Omega}
124    &=& \frac{[\Phi(H_1^+(X_0(N),\Z)) : \Phi(\T\e)]}
125           {c_{\infty} \cdot c_M} \\
126c_\infty &=& \text{ number of components of } A_f(\R)\\
127c_M &=& \text{ a Manin constant (conj = 1)}
128\end{eqnarray*}
129\end{theorem}
130
131\vspace{.3in}
132
133\begin{corollary}
134Evidence for the BSD conjecture!
135\begin{itemize}
136\item $L(A_f,1)/\Omega \in\Q$
137\item Bounds on denomonitor of $L(A_f,1)/\Omega$.
138\end{itemize}
139\end{corollary}
140
141
142\newpage
143\section{Modular Degree}
144Autoduality of $J_0(N)$ gives:\\
145\mbox{}\\
146
147\begin{center}
148\begin{picture}(150,80)
149\put(0,45){$\begin{matrix} 150 \Ker(\delta_f) & = & A_f^{\vee}\intersect C & \hookrightarrow & C \\ 151 & & \cap & & \cap \\ 152 & & A_f^{\vee} & \xrightarrow{\pi^{\vee}} & J\\ 153 & & & \delta_f & \downarrow \pi\\ 154 & & & & A_f 155\end{matrix}$}
156\put(125,30){\vector(2,-1){55}}
157\end{picture}
158\end{center}
159\
160\begin{eqnarray*}
161\delta_f &=& \text{ modular map}\\
162\Ker(\delta_f) &=& \text{ congruences between $A_f^{\vee}$ and $C$ }\\
163\deg(\delta_f) &=& \text{ (generalized) {\bf modular degree} }
164\end{eqnarray*}
165{\bf WARNING:} square of the usual one for elliptic curves!
166
167\vspace{.2in}
168\begin{theorem}[Formula for $\Ker(\delta)$]
169Let $\p_f = \Ann_{\T}(f)$. Then
170$$\Ker(\delta) 171 \isom \frac{\Phi(H_1(X_0(N),\Z)) } 172 {\Phi(H_1(X_0(N),\Z)[\p_f])}$$
173and
174$$0\ra \Hom(H_1,\Z)[\p] \ra \Hom(H_1[\p],\Z) \ra \Ker(\delta) \ra 0$$
175\end{theorem}
176
177\vspace{.2in}
178\begin{proposition}
179$$\Sha^0(A_f^{\vee}) \subset \Sha(A_f^{\vee})[\deg(\delta)]$$
180\end{proposition}
181
182\comment{\begin{question}[Mazur]
183$$t_f= \pi^{\vee}\circ \hat{\delta}\circ \pi \in \End(J_0(N))$$
184Is $t_f\in\T$? What is it?
185\end{question}}
186
187
188
189\newpage
190\section{Experiment: Dimension 1 (Cremona-Mazur)}
191Analyze all nontrivial (analytic) $\Sha(E)$ for optimal $E$ of
192level $N\leq 5500$.\\
193Given:
194\begin{enumerate}
195\item  $E\subset J_0(N)$
196\item $p|\#\Sha(E)$
197\end{enumerate}
198Search for an elliptic curve $F\subset J_0(N)$ such that
199\begin{enumerate}
200\item $E[p]=F[p]\subset J_0(N)[p]$
201\item rank $F$ = 2
202\end{enumerate}
203For odd $p$: {\em do} find $F$ except
204\begin{enumerate}
205\item $N=2849, 4343, 5389$: where $p$ doesn't divide the modular degree.
206\item $N=2932, 3364, 4229, 4914, 5054, 5073$, there exists some congruence,
207but not with a 1 dimensional factor. (No further analysis.)
208\end{enumerate}
209
210\newpage
211\section{Experiment: Dimension $>1$ (Agashe-Stein)}
212Even assuming BSD, don't really know how to compute analytic $|\Sha|$!
213
214Assume $L(A_f,1)\neq 0$.  Can compute
215$$\L(f) = [\Phi(H_1^+):\Phi(\T\e)] =\text{(conjecture)}= 216 \frac{|\Sha|\cdot \prod c_p \cdot c_{\infty} \cdot c_M} 217 {|A_f(\Q)|\cdot |A_f^{\vee}(\Q)|}$$
218
219Odd part of $|\Sha|$ is a square, so let
220$$s(f) = \text{largest odd square dividing numer}(\L(f)).$$
221{\bf Warning:} $p\mid s(f)$ need not imply $p\mid \Sha$. Example {\bf 980E1}.
222\vspace{.3in}
223\par\noindent
224{\bf Experiment.}
225\begin{enumerate}
226\item For each newform $f$ of level $N\leq 1500$ compute $s(f)$.
227\item When $s(f)\neq 1$ compute $\deg(\delta_f)$, then ...
228\item ... list all $B=A_g\subset J_0(N)$ such that
229         $$A_f[p]\intersect A_g[p]\neq\{0\}$$
230     for some $p\mid s(f)$.
231\item Analyze the results of 1: conjecture something!
232\end{enumerate}
233
234\newpage
235\section{Examples}
236
237{\bf Notation:} Level -  Isogeny Class - Dimension
238
239\begin{center}
240\begin{tabular}{|lccl|}\hline
241        &&&\\
242{\bf A} & $s(f)$\qquad & $\text{odd part}(\deg(\delta_f))$
243                       & {\bf B}(analytic rk / $\T$) \\
244        &  &                           &               \\ \hline
245{\bf 305D7} & $3^2$ & $3^4$& {\bf 61A1}(1) \\
246{\bf 309D8} & $5^2$ & $5^4$& {\bf 103A2}($>0$) \\
247{\bf 335E11} & $3^4$ &$3^8$ & {\bf 67B2}($>0$)  \\
248{\bf 389E20} & $5^2$ & $5^2$&  {\bf 389A1}(2)    \\
249$\cdots$       & $\cdots$ &$\cdots$ &   $\cdots$                 \\
250{\bf 446F8} & $11^2$ & $11^2\cdot 359353^2$ & {\bf 446A1}(2)    \\
251$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
252{\bf 1061D46} & $151^2$ & $61^2\cdot 151^2\cdot 179^2$  &  {\bf 1061A2}(2)\\
253$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
254{\bf 1091C62} & $7^2$ &  $1$   & invisible! (Agashe)\\
255$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
256{\bf 2849A1}  & $3^4$ & $5^2\cdot 61^2$ & invisible! (Mazur)\\\hline
257\end{tabular}
258\end{center}
259
260\vspace{.05in}
261\begin{eqnarray*}
262305&=&5\cdot 61\\
263309&=&3\cdot 103\\
264335&=&5\cdot 67\\
265446 &=& 2\cdot 223\\
2662849 &=& 7\cdot 11\cdot 37
267\end{eqnarray*}
268
269\end{document}
270
271
272