Sharedwww / Tables / visshatalk.texOpen in CoCalc
Author: William A. Stein
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% tucsontalk.tex
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\documentclass[12pt]{article}
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\pagestyle{empty}
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\title{Visibility of Shafarevich-Tate Groups\\
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of Modular Abelian Varieties}
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\include{macros}
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\author{William Stein\\
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{\tt was@math.berkeley.edu}\\
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{\tt http://shimura.math.berkeley.edu/\~{}was}}
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\date{March 1999}
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\begin{document}
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\large
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\maketitle
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\tableofcontents
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\Large
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\newpage
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\section{BSD Conjecture}
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\begin{itemize}
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\item $A/\Q$ abelian variety, $\dim A=d$
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\item $L(A,s)$ associated $L$-function
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\end{itemize}
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\vspace{.3in}
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\begin{conjecture} [Birch, Swinnerton-Dyer, Tate]
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\par\noindent
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\begin{enumerate}
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\item $L(A,s)$ is holomorphic on all $\C$
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\item $$\frac{L(A,1)}{\Omega} = \frac{|\Sha(A)|\cdot \prod c_p}
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{|A(\Q)|\cdot|A^{\vee}(\Q)|}$$
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where
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\begin{itemize}
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\item $\Sha(A)=\Ker(H^1(\Q,A)\ra \prod_v H^1(\Q_v,A))$
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\item $c_p $ = rational components of special
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fiber of Neron model $\A/\Fp$
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\item $\Omega = \int_{A(\R)}|\omega|$, where $H^0(\cA,\Omega^d)=\Z\omega$
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\end{itemize}
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\end{enumerate}
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\end{conjecture}
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\newpage
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\section{Visibility}
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Fix
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$$i:A\hookrightarrow J$$
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{\bf Visible Sha:}
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$$\Sha^0(A) := \Ker(\Sha(A)\xrightarrow{i_*} \Sha(J))$$
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\par \noindent We can {\em see} $\Sha^0(A)$:\\
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The long exact sequence associated to
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$$ 0\ra A \ra J \ra B \ra 0$$
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gives
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$$\begin{matrix}
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0 & \ra & B(\Q)/J(\Q) & \ra & H^1(\Q,A) & \ra & H^1(\Q,J) & \ra \cdots\\
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& & \cup & & \cup & & \cup\\
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0 & \ra & \Sha^0(A) & \ra & \Sha(A) & \ra & \Sha(J) \\
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\end{matrix}$$
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{\bf Remark:} (Ogg) Mordell-Weil implies $\Sha^0(A)$ is finite.
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\newpage
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\section{Congruences}
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Fix $A$.\\
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Find $B$ such that:
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$$A[p]\isom B[p] \qquad \text{ as $\Gal(\Qbar/\Q)$-modules}$$
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Then
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$$H^1(\Q,A[p])\isom H^1(\Q,B[p]).$$
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\\
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Kummer and Selmer:
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$$\begin{matrix}
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0 &\ra& A(\Q)/pA(\Q) &\ra & H^1(\Q,A[p]) & \ra & H^1(\Q,A)[p] & \ra & 0 \\
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& & || & & \cup & & || & & \\
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0 &\ra& A(\Q)/pA(\Q) &\ra & \Selmer_p(A/\Q) & \ra & \Sha(A)[p] & \ra & 0 \\
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& & & & || \text{ if lucky!} & & & \\
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0 &\ra& B(\Q)/pB(\Q) &\ra & \Selmer_p(B/\Q) & \ra & \Sha(B)[p] & \ra & 0 \\
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& & || & & \cap & & || & & \\
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0 &\ra& B(\Q)/pB(\Q) &\ra & H^1(\Q,B[p]) & \ra & H^1(\Q,B)[p] & \ra & 0 \\
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\end{matrix}$$
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If $A(\Q)=0$ and $\Sha(B)[p]=0$ then
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$$\Sha(A)[p] = B(\Q)/p B(\Q).$$
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``Luck'' understandable in terms of Mazur's flat cohomology.
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\newpage
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\section{Modular Rank $0$ BSD}
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$A_f$ optimal quotient of $J_0(N)$, corresponding to
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$$f = \sum a_n q^n \in S_2(\Gamma_0(N),\C).$$
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Have
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$$0 \ra C \ra J_0(N) \ra A_f \ra 0.$$
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\begin{theorem}[Hecke]
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$$L(f,s) =\sum a_n n^{-s}$$
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holomorphic on $\C$.
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\end{theorem}
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\begin{theorem}[Shimura]
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$$L(A_f,s) = \prod_i L(f_i,s),$$
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product over conjugates of $f$.
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\end{theorem}
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\begin{theorem}[Kolyvagin-Logachev]
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\mbox{ }\par\noindent
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$L(A_f,1)\neq 0$ $\implies$ $A_f(\Q)$ and $\Sha(A_f)$ both finite.
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(Heegner points.)
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\end{theorem}
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\newpage
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\section{Formula for $L(A_f,1)/\Omega$}
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Abel-Jacobi map:
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$$H_1(X_0(N),\Z) \xrightarrow{\Phi} \C^d \ra A_f \ra 0$$
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$$\Phi(\gamma)= (\int_\gamma f_1, \ldots ,\int_\gamma f_d)$$
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\vspace{.3in}
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\begin{theorem}[Agashe]
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\mbox{}\\
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\mbox{}\\
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$\e=-\{0,\infty\}\in H_1(X_0(N),\Q)$\\
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$\T=\text{ Hecke algebra }$
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\begin{eqnarray*}
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\frac{L(A_f,1)}{\Omega}
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&=& \frac{[\Phi(H_1^+(X_0(N),\Z)) : \Phi(\T\e)]}
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{c_{\infty} \cdot c_M} \\
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c_\infty &=& \text{ number of components of } A_f(\R)\\
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c_M &=& \text{ a Manin constant (conj = 1)}
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\end{eqnarray*}
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\end{theorem}
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\vspace{.3in}
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\begin{corollary}
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Evidence for the BSD conjecture!
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\begin{itemize}
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\item $L(A_f,1)/\Omega \in\Q$
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\item Bounds on denomonitor of $L(A_f,1)/\Omega$.
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\end{itemize}
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\end{corollary}
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\newpage
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\section{Modular Degree}
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Autoduality of $J_0(N)$ gives:\\
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\mbox{}\\
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\begin{center}
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\begin{picture}(150,80)
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\put(0,45){$\begin{matrix}
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\Ker(\delta_f) & = & A_f^{\vee}\intersect C & \hookrightarrow & C \\
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& & \cap & & \cap \\
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& & A_f^{\vee} & \xrightarrow{\pi^{\vee}} & J\\
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& & & \delta_f & \downarrow \pi\\
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& & & & A_f
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\end{matrix}$}
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\put(125,30){\vector(2,-1){55}}
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\end{picture}
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\end{center}
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\
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\begin{eqnarray*}
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\delta_f &=& \text{ modular map}\\
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\Ker(\delta_f) &=& \text{ congruences between $A_f^{\vee}$ and $C$ }\\
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\deg(\delta_f) &=& \text{ (generalized) {\bf modular degree} }
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\end{eqnarray*}
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{\bf WARNING:} square of the usual one for elliptic curves!
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\vspace{.2in}
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\begin{theorem}[Formula for $\Ker(\delta)$]
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Let $\p_f = \Ann_{\T}(f)$. Then
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$$\Ker(\delta)
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\isom \frac{\Phi(H_1(X_0(N),\Z)) }
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{\Phi(H_1(X_0(N),\Z)[\p_f])}$$
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and
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$$0\ra \Hom(H_1,\Z)[\p] \ra \Hom(H_1[\p],\Z) \ra \Ker(\delta) \ra 0$$
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\end{theorem}
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\vspace{.2in}
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\begin{proposition}
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$$\Sha^0(A_f^{\vee}) \subset \Sha(A_f^{\vee})[\deg(\delta)]$$
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\end{proposition}
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\comment{\begin{question}[Mazur]
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$$t_f= \pi^{\vee}\circ \hat{\delta}\circ \pi \in \End(J_0(N))$$
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Is $t_f\in\T$? What is it?
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\end{question}}
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\newpage
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\section{Experiment: Dimension 1 (Cremona-Mazur)}
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Analyze all nontrivial (analytic) $\Sha(E)$ for optimal $E$ of
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level $N\leq 5500$.\\
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Given:
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\begin{enumerate}
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\item $E\subset J_0(N)$
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\item $p|\#\Sha(E)$
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\end{enumerate}
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Search for an elliptic curve $F\subset J_0(N)$ such that
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\begin{enumerate}
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\item $E[p]=F[p]\subset J_0(N)[p]$
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\item rank $F$ = 2
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\end{enumerate}
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For odd $p$: {\em do} find $F$ except
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\begin{enumerate}
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\item $N=2849, 4343, 5389$: where $p$ doesn't divide the modular degree.
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\item $N=2932, 3364, 4229, 4914, 5054, 5073$, there exists some congruence,
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but not with a 1 dimensional factor. (No further analysis.)
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\end{enumerate}
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\newpage
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\section{Experiment: Dimension $>1$ (Agashe-Stein)}
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Even assuming BSD, don't really know how to compute analytic $|\Sha|$!
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Assume $L(A_f,1)\neq 0$. Can compute
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$$\L(f) = [\Phi(H_1^+):\Phi(\T\e)] =\text{(conjecture)}=
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\frac{|\Sha|\cdot \prod c_p \cdot c_{\infty} \cdot c_M}
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{|A_f(\Q)|\cdot |A_f^{\vee}(\Q)|}$$
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Odd part of $|\Sha|$ is a square, so let
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$$s(f) = \text{largest odd square dividing numer}(\L(f)).$$
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{\bf Warning:} $p\mid s(f)$ need not imply $p\mid \Sha$. Example {\bf 980E1}.
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\vspace{.3in}
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\par\noindent
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{\bf Experiment.}
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\begin{enumerate}
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\item For each newform $f$ of level $N\leq 1500$ compute $s(f)$.
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\item When $s(f)\neq 1$ compute $\deg(\delta_f)$, then ...
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\item ... list all $B=A_g\subset J_0(N)$ such that
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$$A_f[p]\intersect A_g[p]\neq\{0\}$$
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for some $p\mid s(f)$.
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\item Analyze the results of 1: conjecture something!
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\end{enumerate}
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\newpage
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\section{Examples}
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{\bf Notation:} Level - Isogeny Class - Dimension
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\begin{center}
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\begin{tabular}{|lccl|}\hline
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&&&\\
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{\bf A} & $s(f)$\qquad & $\text{odd part}(\deg(\delta_f))$
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& {\bf B}(analytic rk / $\T$) \\
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& & & \\ \hline
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{\bf 305D7} & $3^2$ & $3^4$& {\bf 61A1}(1) \\
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{\bf 309D8} & $5^2$ & $5^4$& {\bf 103A2}($>0$) \\
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{\bf 335E11} & $3^4$ &$3^8$ & {\bf 67B2}($>0$) \\
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{\bf 389E20} & $5^2$ & $5^2$& {\bf 389A1}(2) \\
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$\cdots$ & $\cdots $ &$\cdots$ & $\cdots $ \\
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{\bf 446F8} & $11^2$ & $11^2\cdot 359353^2$ & {\bf 446A1}(2) \\
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$\cdots$ & $\cdots$ &$\cdots$ & $\cdots$ \\
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{\bf 1061D46} & $151^2$ & $61^2\cdot 151^2\cdot 179^2$ & {\bf 1061A2}(2)\\
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$\cdots$ & $\cdots$ &$\cdots$ & $\cdots$ \\
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{\bf 1091C62} & $7^2$ & $1$ & invisible! (Agashe)\\
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$\cdots$ & $\cdots$ &$\cdots$ & $\cdots$ \\
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{\bf 2849A1} & $3^4$ & $5^2\cdot 61^2$ & invisible! (Mazur)\\\hline
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\end{tabular}
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\end{center}
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\vspace{.05in}
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\begin{eqnarray*}
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305&=&5\cdot 61\\
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309&=&3\cdot 103\\
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335&=&5\cdot 67\\
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446 &=& 2\cdot 223\\
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2849 &=& 7\cdot 11\cdot 37
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\end{eqnarray*}
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\end{document}
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