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\documentclass[12pt]{article}
\pagestyle{empty}
\title{Visibility of Shafarevich-Tate Groups\\
       of Modular Abelian Varieties}
\include{macros}
\author{William Stein\\
{\tt was@math.berkeley.edu}\\
{\tt http://shimura.math.berkeley.edu/\~{}was}}
\date{March 1999}
\begin{document}
\large
\maketitle
\tableofcontents
\Large
\newpage
\section{BSD Conjecture}
\begin{itemize}
\item $A/\Q$ abelian variety, $\dim A=d$
\item $L(A,s)$ associated $L$-function
\end{itemize}

\vspace{.3in}
\begin{conjecture} [Birch, Swinnerton-Dyer, Tate]
\par\noindent
\begin{enumerate}
\item $L(A,s)$ is holomorphic on all $\C$
\item $$\frac{L(A,1)}{\Omega} = \frac{|\Sha(A)|\cdot \prod c_p}
                                     {|A(\Q)|\cdot|A^{\vee}(\Q)|}$$
where
\begin{itemize}
\item $\Sha(A)=\Ker(H^1(\Q,A)\ra \prod_v H^1(\Q_v,A))$
\item $c_p $ = rational components of special
     fiber of Neron model $\A/\Fp$
\item $\Omega = \int_{A(\R)}|\omega|$, where $H^0(\cA,\Omega^d)=\Z\omega$
\end{itemize}
\end{enumerate}
\end{conjecture}

\newpage
\section{Visibility}
Fix
$$i:A\hookrightarrow J$$

{\bf Visible Sha:}
$$\Sha^0(A) := \Ker(\Sha(A)\xrightarrow{i_*} \Sha(J))$$

\par \noindent We can {\em see} $\Sha^0(A)$:\\
The long exact sequence associated to
$$ 0\ra A \ra J \ra B \ra 0$$
gives
$$\begin{matrix}
0 & \ra & B(\Q)/J(\Q) & \ra & H^1(\Q,A) & \ra & H^1(\Q,J) & \ra \cdots\\
  &     & \cup      &      & \cup      &   &   \cup\\
0 & \ra & \Sha^0(A) & \ra & \Sha(A) & \ra & \Sha(J) \\
\end{matrix}$$

{\bf Remark:} (Ogg) Mordell-Weil implies $\Sha^0(A)$ is finite.

\newpage
\section{Congruences}
Fix $A$.\\  
Find $B$ such that:
                  $$A[p]\isom B[p] \qquad \text{ as $\Gal(\Qbar/\Q)$-modules}$$
Then
$$H^1(\Q,A[p])\isom H^1(\Q,B[p]).$$
\\
Kummer and Selmer:
$$\begin{matrix}
0 &\ra& A(\Q)/pA(\Q) &\ra & H^1(\Q,A[p])    & \ra & H^1(\Q,A)[p] & \ra & 0 \\
  &   &    ||        &    &   \cup          &     &    ||      &     &    \\
0 &\ra& A(\Q)/pA(\Q) &\ra & \Selmer_p(A/\Q) & \ra & \Sha(A)[p] & \ra & 0 \\
  &   &              &    &  || \text{ if lucky!} &           &       &  \\
0 &\ra& B(\Q)/pB(\Q) &\ra & \Selmer_p(B/\Q) & \ra & \Sha(B)[p] & \ra & 0 \\
  &   &    ||        &    &   \cap          &     &    ||      &     &    \\
0 &\ra& B(\Q)/pB(\Q) &\ra & H^1(\Q,B[p])    & \ra & H^1(\Q,B)[p] & \ra & 0 \\
\end{matrix}$$

If $A(\Q)=0$ and $\Sha(B)[p]=0$ then 
$$\Sha(A)[p] = B(\Q)/p B(\Q).$$

``Luck'' understandable in terms of Mazur's flat cohomology.

\newpage
\section{Modular Rank $0$ BSD}
$A_f$ optimal quotient of $J_0(N)$, corresponding to 
    $$f = \sum a_n q^n \in S_2(\Gamma_0(N),\C).$$
Have
$$0 \ra C \ra J_0(N) \ra A_f \ra 0.$$

\begin{theorem}[Hecke] 
$$L(f,s) =\sum a_n n^{-s}$$
holomorphic on $\C$.
\end{theorem}

\begin{theorem}[Shimura]
$$L(A_f,s) = \prod_i L(f_i,s),$$ 
product over conjugates of $f$.
\end{theorem}

\begin{theorem}[Kolyvagin-Logachev]
\mbox{ }\par\noindent
$L(A_f,1)\neq 0$ $\implies$ $A_f(\Q)$ and $\Sha(A_f)$ both finite.
(Heegner points.)
\end{theorem}



\newpage
\section{Formula for $L(A_f,1)/\Omega$}
Abel-Jacobi map:
$$H_1(X_0(N),\Z) \xrightarrow{\Phi} \C^d \ra A_f \ra 0$$
$$\Phi(\gamma)= (\int_\gamma f_1, \ldots ,\int_\gamma f_d)$$

\vspace{.3in}

\begin{theorem}[Agashe]
\mbox{}\\
\mbox{}\\
$\e=-\{0,\infty\}\in H_1(X_0(N),\Q)$\\
$\T=\text{ Hecke algebra }$
\begin{eqnarray*}
\frac{L(A_f,1)}{\Omega}
    &=& \frac{[\Phi(H_1^+(X_0(N),\Z)) : \Phi(\T\e)]}  
           {c_{\infty} \cdot c_M} \\
c_\infty &=& \text{ number of components of } A_f(\R)\\
c_M &=& \text{ a Manin constant (conj = 1)}
\end{eqnarray*}
\end{theorem}

\vspace{.3in}

\begin{corollary}
Evidence for the BSD conjecture!
\begin{itemize}
\item $L(A_f,1)/\Omega \in\Q$
\item Bounds on denomonitor of $L(A_f,1)/\Omega$.  
\end{itemize}
\end{corollary}


\newpage
\section{Modular Degree}
Autoduality of $J_0(N)$ gives:\\
\mbox{}\\

\begin{center}
\begin{picture}(150,80)
\put(0,45){$\begin{matrix}
 \Ker(\delta_f) & = & A_f^{\vee}\intersect C & \hookrightarrow & C \\
                &   & \cap             &                 & \cap \\
                &   & A_f^{\vee}             & \xrightarrow{\pi^{\vee}} & J\\
                &   &                  &  \delta_f & \downarrow \pi\\
                &   &                  &                   & A_f 
\end{matrix}$}
\put(125,30){\vector(2,-1){55}}
\end{picture}
\end{center}
\
\begin{eqnarray*}
\delta_f &=& \text{ modular map}\\
\Ker(\delta_f) &=& \text{ congruences between $A_f^{\vee}$ and $C$ }\\
\deg(\delta_f) &=& \text{ (generalized) {\bf modular degree} }
\end{eqnarray*}
{\bf WARNING:} square of the usual one for elliptic curves!

\vspace{.2in}
\begin{theorem}[Formula for $\Ker(\delta)$]
Let $\p_f = \Ann_{\T}(f)$. Then
$$\Ker(\delta)
   \isom \frac{\Phi(H_1(X_0(N),\Z)) }
              {\Phi(H_1(X_0(N),\Z)[\p_f])}$$
and
$$0\ra \Hom(H_1,\Z)[\p] \ra \Hom(H_1[\p],\Z) \ra \Ker(\delta) \ra 0$$
\end{theorem}

\vspace{.2in}
\begin{proposition}
$$\Sha^0(A_f^{\vee}) \subset \Sha(A_f^{\vee})[\deg(\delta)]$$
\end{proposition}

\comment{\begin{question}[Mazur]
$$t_f= \pi^{\vee}\circ \hat{\delta}\circ \pi \in \End(J_0(N))$$
Is $t_f\in\T$? What is it?
\end{question}}



\newpage
\section{Experiment: Dimension 1 (Cremona-Mazur)}
Analyze all nontrivial (analytic) $\Sha(E)$ for optimal $E$ of 
level $N\leq 5500$.\\
Given:
\begin{enumerate}
\item  $E\subset J_0(N)$
\item $p|\#\Sha(E)$
\end{enumerate}
Search for an elliptic curve $F\subset J_0(N)$ such that
\begin{enumerate}
\item $E[p]=F[p]\subset J_0(N)[p]$
\item rank $F$ = 2
\end{enumerate}
For odd $p$: {\em do} find $F$ except 
\begin{enumerate}
\item $N=2849, 4343, 5389$: where $p$ doesn't divide the modular degree.
\item $N=2932, 3364, 4229, 4914, 5054, 5073$, there exists some congruence,
but not with a 1 dimensional factor. (No further analysis.)
\end{enumerate}

\newpage
\section{Experiment: Dimension $>1$ (Agashe-Stein)}
Even assuming BSD, don't really know how to compute analytic $|\Sha|$!

Assume $L(A_f,1)\neq 0$.  Can compute 
$$\L(f) = [\Phi(H_1^+):\Phi(\T\e)] =\text{(conjecture)}=
      \frac{|\Sha|\cdot \prod c_p \cdot c_{\infty} \cdot c_M}
           {|A_f(\Q)|\cdot |A_f^{\vee}(\Q)|}$$

Odd part of $|\Sha|$ is a square, so let
$$s(f) = \text{largest odd square dividing numer}(\L(f)).$$
{\bf Warning:} $p\mid s(f)$ need not imply $p\mid \Sha$. Example {\bf 980E1}.
\vspace{.3in}
\par\noindent
{\bf Experiment.}
\begin{enumerate}
\item For each newform $f$ of level $N\leq 1500$ compute $s(f)$.
\item When $s(f)\neq 1$ compute $\deg(\delta_f)$, then ...
\item ... list all $B=A_g\subset J_0(N)$ such that 
         $$A_f[p]\intersect A_g[p]\neq\{0\}$$
     for some $p\mid s(f)$.
\item Analyze the results of 1: conjecture something!
\end{enumerate}

\newpage
\section{Examples}

{\bf Notation:} Level -  Isogeny Class - Dimension

\begin{center}
\begin{tabular}{|lccl|}\hline
        &&&\\
{\bf A} & $s(f)$\qquad & $\text{odd part}(\deg(\delta_f))$ 
                       & {\bf B}(analytic rk / $\T$) \\ 
        &  &                           &               \\ \hline
{\bf 305D7} & $3^2$ & $3^4$& {\bf 61A1}(1) \\ 
{\bf 309D8} & $5^2$ & $5^4$& {\bf 103A2}($>0$) \\
{\bf 335E11} & $3^4$ &$3^8$ & {\bf 67B2}($>0$)  \\
{\bf 389E20} & $5^2$ & $5^2$&  {\bf 389A1}(2)    \\ 
$\cdots$       & $\cdots $ &$\cdots$ &   $\cdots $                 \\
{\bf 446F8} & $11^2$ & $11^2\cdot 359353^2$ & {\bf 446A1}(2)    \\ 
$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
{\bf 1061D46} & $151^2$ & $61^2\cdot 151^2\cdot 179^2$  &  {\bf 1061A2}(2)\\ 
$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
{\bf 1091C62} & $7^2$ &  $1$   & invisible! (Agashe)\\ 
$\cdots$       & $\cdots$  &$\cdots$ &   $\cdots$                  \\
{\bf 2849A1}  & $3^4$ & $5^2\cdot 61^2$ & invisible! (Mazur)\\\hline
\end{tabular}
\end{center}

\vspace{.05in}
\begin{eqnarray*}
305&=&5\cdot 61\\
309&=&3\cdot 103\\
335&=&5\cdot 67\\
446 &=& 2\cdot 223\\
2849 &=& 7\cdot 11\cdot 37
\end{eqnarray*}

\end{document}