Sharedwww / talks / 2004-05-22-UIUC / visibility.texOpen in CoCalc
Author: William A. Stein
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%opening
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\title{\Huge\bf\dblue Modularity of Shafarevich-Tate Groups}
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\author{\rd{\LARGE William Stein}\\
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{\tt http://modular.fas.harvard.edu}}
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\date{May 22, 2004\vspace{1ex}\\
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\includegraphics[width=0.7\textwidth]{sha_fractal.eps}}
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\begin{document}
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\maketitle
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\begin{slide}
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\h{\rd{Goal}}
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The goal of this 40-minute talk is to explain the meaning of
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the following conjecture and give evidence for it.
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{\Large{\bf Conjecture (--).} If $A$ is a modular abelian
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variety, then the Shafarevich-Tate
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group \rd{$\Sha(A)$} is \blue{modular}.}
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\end{slide}
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\begin{slide}
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\h{Table of Contents}
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\begin{enumerate}
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\item Elliptic Curves and \rd{Modular Abelian Varieties}
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\item The \rd{Birch and Swinnerton-Dyer} Conjecture
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\item \mbox{}\rd{Visibility} of Shafarevich-Tate groups
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\item \mbox{}\rd{Modularity} of Shafarevich-Tate groups
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\item Some \rd{Data}
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\end{enumerate}
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\end{slide}
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\begin{slide}
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\h{1. Elliptic Curves and Abelian Varieties}
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\begin{tabular}{ll}
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\hspace{-.8ex}\begin{minipage}{6in}
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\vspace{-2in}
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\rd{Elliptic curve over $\Q$:} $y^2=x^3+ax+b$
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with $a,b\in\Q$
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and $\Delta=-16(4a^3+27b^2)\neq 0$
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\end{minipage}&
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\hspace{0.5in}
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\includegraphics[width=3in]{elliptic6_floating_2.eps}
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\end{tabular}\vspace{-.7in}
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\rd{Abelian variety:} Any complete group variety.
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\rd{Examples:} Jacobians of curves. Elliptic curves
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are the abelian varieties of dimension one.
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\rd{Modular abelian varieties}.
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\end{slide}
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\begin{slide}
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\h{Modular Curves, Modular Forms}
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\rd{Congruence Subgroup:}
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$$
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\Gamma_0(N) = \left\lbrace
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\mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c
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\right\rbrace.
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$$
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\vspace{-1ex}
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\rd{Modular Curve:}
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$X_0(N) = \Gamma_0(N) \setminus \text{(upper half plane)} \cup \text{(cusps)}$
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\vspace{-1ex}
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\rd{Example:}{\dblue $X_0(39)$}
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\begin{center}\vspace{-.4in}
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\includegraphics[width=0.6\textwidth]{torus39.eps}
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\end{center}
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\vspace{-4ex}
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\rd{Algebraic structure over $\Q$:}
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$$
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y^2 = (x^4-7x^3+11x^2-7x+1)(x^4+x^3-x^2+x+1).
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$$
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\end{slide}
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\begin{slide}
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\h{Modular Abelian Varieties}
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\rd{Modular Jacobian: } $J_0(N) = \Jac(X_0(N))$
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\rd{Modular Abelian Variety:} Any abelian variety quotient of $J_0(N)$ (or
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of $J_1(N)$, where $J_1(N)$ is defined using $\Gamma_1(N)$).
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\rd{Theorem (Wiles et al.):} All elliptic curves over $\Q$ are modular.
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\rd{Cusp Forms: }
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$S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1)$
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\rd{Hecke Algebra:} $\T = \Z[T_1,T_2,T_3, \ldots] \subset \End(S_2(\Gamma_0(N)))$
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\rd{Shimura:}
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$\T$-eigenform $f \in S_2(\Gamma_0(N))$
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gives $A_f = J_0(N)/I_f J_0(N)$, where $I_f=\Ann_\T(f)$.
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We have $\dim A_f = [\Q(a_2(f),\ldots):\Q]$.
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\end{slide}
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\begin{slide}
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\h{2. The Birch and Swinnerton-Dyer Conjecture}
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Let $A$ be an abelian variety over $\Q$ (e.g., $A=A_f$ modular).
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\vspace{-1in}
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\begin{center}
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\begin{minipage}{7in}
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\rd{\LARGE Conjecture of \includegraphics[height=1.1in]{bsd.eps}:}\\
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\begin{enumerate}
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\item $r = \ds \rank A(\Q) \ce \ord_{s=1}L(A,s)$, and\\
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\item
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$\ds
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\frac{L^{(r)}(A,1)}{r!} =
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\frac{\prod c_p \cdot \Omega_A \cdot \Reg(A)
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\cdot \#\Sha(A)}
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{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}}.
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$
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\end{enumerate}
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\end{minipage}
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\end{center}
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Results: Kolyvagin, Kato, Rubin, etc.
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\end{slide}
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\begin{slide}
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\h{Shafarevich-Tate Group \includegraphics[width=2.2in]{sha_fractal2.eps}}
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\rd{Definition:}
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$$
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\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_{\text{all }v} \H^1(K_v,A)\right).
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$$
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$\H^1(K,A)$ is \rd{Galois cohomology}.
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Interpret geometrically as the \rd{Weil-Chatalet group}:
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$$
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\text{WC}(A/K) = \{\,\text{principal homogenous spaces }X\text{ for }A\,\}/\sim.
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$$
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$\Sha(A)$ is the subgroup of {\dblue locally trivial classes}.
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\rd{Example:}
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$$3x^3+4y^3+5z^3=0
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\in \Sha(x^3+y^3+60z^3=0)[3].$$
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\end{slide}
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\begin{slide}
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\h{3. Visibility of Sha \includegraphics[width=1.4in,angle=-10]{mazur.eps}}
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\vspace{-1in}
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{\LARGE$$0 \to A\xrightarrow{i} B \to C \to 0$$}
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1998 -- Barry Mazur introduced \rd{Visibility}:
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\begin{align*}
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\Vis_B(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\
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&\isom \Coker(B(K) \to C(K))\\
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& \\
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\Vis_B(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).
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\end{align*}
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\end{slide}
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\begin{slide}
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%\h{\begin{tabular}{lcr}
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%\begin{minipage}{2in}\mbox{}\\Visibility\end{minipage}
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%&&\includegraphics[width=3in]{eye.eps}
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%\end{tabular}}
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% The eye graphic is from
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% http://www.littlebeast.com/ images/eye.jpg
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% and was found using the google image search.
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\begin{center}
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\pspicture(-1,-1)(5,1)
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\rput(-4,0){\h{Visibility \hspace{2in}\mbox{}}}
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\rput(20,3){\includegraphics[width=3in]{eye.eps}}
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\endpspicture
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\end{center}
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\vspace{-.7in}
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$$
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\xymatrix{
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&&X=\pi^{-1}(P)\[email protected]{-}[d]\ar[r]&P\[email protected]{-}[d]\[email protected]{|->}[r]&c\[email protected]{-}[d]\\
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0\ar[r]&A(K)\ar[r]&B(K)\ar[r]^{\pi}&C(K) \ar[r]&{\Vis_B(\H^1(K,A))}\ar[r]&0
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}
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$$
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\rd{Why?} To write down $X$ using equations is terrifying;
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to give $P$ is just to give a rational point. Visibility concisely
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encodes connections between Mordell-Weil and Shafarevich-Tate groups.
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Give nonzero $c\in \Sha(A)[5]$ with $\dim(A)=20$ by giving
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$$(0,0)\in [y^2 + y = x^3 + x^2 - 2x]$$
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\end{slide}
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\begin{slide}
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\h{Everything is Visible Somewhere}
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\rd{Theorem (--):}
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If $c\in \H^1(K,A)$ then there exists
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$B$
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such that $i:A\hra B$ and $c\in\Vis_B(\H^1(K,A))$.
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\rd{Proof.} Let $L$ be such that $\res_{L/K}(c)=0$. Then
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$$
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\xymatrix{
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{c} \ar\[email protected]{|->}[rr] && {\res_{L/K}(c)=0}\\
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{\H^1(K,A)}\ar[rr]\ar[ddrr] && {\H^1(K,\rd{\Res_{L/K}(A_L)})}\ar[dd]\\
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& \\
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&& {\H^1(L,A)}
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}$$
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Note: If $A/\Q$ is modular and $L$ is abelian,
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then $B$ is modular.
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\end{slide}
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\begin{slide}
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\h{4. Modularity of Sha}
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\rd{\bf Definition (Modular):} An element $c\in \Sha(A)$ is
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{\em modular} if it is visible in a modular abelian variety.
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I.e., if there is a factor $B$ of some $J_0(N)$ (or $J_1(N)$)
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and an inclusion $i:A\hra B$ such that $i_*(c)=0$.
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Torsor corresponding to $c$ is modular in ``usual''
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sense.
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\rd{Modularity Conjecture (--):}\\
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\mbox{}\hspace{.6in}{\dblue If $A$ is modular, then every element of
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$\Sha(A)$ is modular.}
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%{\red\bf Should you believe my conjecture?}
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\rd{Theorem (Klenke, Mazur, Stein):} If $c\in \Sha(E)[p]$ with $p=2,3$ and $E$
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an elliptic curve, then $c$ is modular.
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\rd{Related questions:}\\
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1. Levels $N$ such that such that $c$
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is modular of level $N$?\\
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2. Which genus one curves are modular?
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%2. If an element
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%$c\in \Sha(A)$ is modular, does that imply that
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%$A$ is modular?
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\end{slide}
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\begin{slide}
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\h{\dred Why Care about Modularity?}
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$\bullet$ It could yield results about {\dblue
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structure of $\Sha(A)$}, e.g., finiteness.
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$\bullet$ It would give a nice ``explanation'' of
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{\dblue where all $\Sha(A)$
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``comes from''} --- it all comes from Mordell-Weil groups.
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$\bullet$ It {\dblue motivates proving new results} about the arithmetic
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of modular abelian varieties.
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$\bullet$ It provides {\dblue powerful computational tools}
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for explicitly working with Shafarevich-Tate groups.
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\end{slide}
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\begin{slide}
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\h{Visibility Construction}
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\rd{Theorem (Agashe, --):} {\em Suppose $A, B\subset J_0(N)$,
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that $B[p]\subset A$, and other technical hypotheses.
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Then
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$$ B(\Q)/p B(\Q) \hra \Vis_{J_0(N)}(\Sha(A)). $$}
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\rd{Proof.} Use the following diagram, chase some exact sequences, and
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apply subtle properties of N\'eron models. (Also more general
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Hecke-equivariant version, proved recently with Jetchev.)
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$$
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{\dgreen\[email protected]=0.9in{
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{B[p]}\ar[r]\ar[d]& B \ar[r]^p\ar[d] & B \ar[d]\\
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{A}\ar[r] & {J_0(N)} \ar[r] & C.}
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}$$
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\end{slide}
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\begin{slide}
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%\h{5. Some Data}
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\begin{center}
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\pspicture(-1,-1)(5,1)
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\rput(-1,0){\h{\hspace{-2in}5. Some Data}}
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\rput(20,3){\includegraphics[width=4in]{tables.eps}}
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\endpspicture
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\end{center}\vspace{-1.5in}
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{\LARGE $$\text{Suppose } A \subset J_0(N) $$}
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\begin{itemize}
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\item[(a)] Visibility of $\Sha(A)$ in $J_0(N)$, when $A$ is an elliptic
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curve.
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\item[(b)] Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
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\item[(c)] Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$. (Modularity)
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\end{itemize}
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\end{slide}
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\begin{slide}
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%\voffset=-1in
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\h{(a) Elliptic Curve tables from Cremona-Mazur}
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\begin{center}\Large
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Visibility of $\Sha(A)$ in $J_0(N)$, when $A=E$ is an \rd{elliptic
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curve}.
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\end{center}
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\begin{center}
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\includegraphics[width=2in]{cremona_table_1.eps}
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$\qquad$\includegraphics[width=2in]{cremona_table_1b.eps}
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$\qquad$\includegraphics[width=2in]{cremona_table_1c.eps}
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\includegraphics[width=1in,angle=-10]{cremona5.eps}
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\end{center}
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\end{slide}
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\begin{slide}
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\voffset=-1in
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\begin{center}
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\includegraphics[height=1.3\textheight]{cm1.ps}
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\end{center}
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\begin{center}
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\includegraphics[height=1.3\textheight]{cm2.ps}
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\end{center}
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\end{slide}
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\begin{slide}
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\voffset=-1in
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\h{(b) Abelian Variety Tables from Agashe-Stein}
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\begin{center}\Large
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Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
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\end{center}
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\begin{center}
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\includegraphics[height=1.5\textheight]{as1.ps}
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\end{center}
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\begin{center}
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\includegraphics[height=1.5\textheight]{as2.ps}
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\end{center}
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\begin{center}
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\includegraphics[height=1.5\textheight]{as3.ps}
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\end{center}
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\begin{center}
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\includegraphics[height=1.5\textheight]{as4.ps}
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\end{center}
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\end{slide}
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\begin{slide}
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\h{(c) Visibility at Higher Level -- Evidence for Modularity Conjecture}
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\begin{center}\Large
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Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$.
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\end{center}
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Recall Ribet's theorem...
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\end{slide}
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\begin{slide}
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\vspace{-0.5in}
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\h{Ribet Level Raising \includegraphics[width=1.5in,angle=-5]{ribet.eps}}
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\vspace{-0.1in}Suppose\vspace{-0.4in}
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\begin{itemize}\setlength{\itemsep}{0in}
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\item $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ a newform
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\item $\lambda\subset \Z[a_1,a_2,\ldots]$ a nonzero prime ideal
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s.t. $A_f[\lambda]$ irreducible.
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\end{itemize}\vspace{-0.3in}
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\rd{\bf Theorem:} {\dblue\large $
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a_p + p + 1 \con 0\pmod{\lambda} \implies
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$}
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there exists a $p$-newform $g\in S_2(\Gamma_0(Np))$ such that
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\vspace{-0.4in}
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\begin{itemize}\setlength{\itemsep}{0in}
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\item $i(A_f[\lambda]) = A_g[\lambda]$ some
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$i:J_0(N)\to J_0(N p)$, and
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\item sign of functional equations for $L(f,s)$ and $L(g,s)$
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same.
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\end{itemize}
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%{\small Note: If instead $a_p - (p+1) \con 0\pmod{\lambda}$
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%then there is such a $g$, but the sign of the functional
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%equation changes, and the new Tamagawa number of
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%$A_g$ at $p$ are divisible by $\lambda$.}
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\end{slide}
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\begin{slide}
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\rd{\bf Big Computation:} For every level $N$ up to $5000$ (and
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more), use my modular forms package in MAGMA to provably compute:
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\begin{enumerate}
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\item Each newform $f=\sum a_n q^n \in S_2(\Gamma_0(N))$.
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\item Whether or not $L(f,1)=0$.
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\item Whether or not $\ord_{s=1} L(f,s)$ is even.
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\item Characteristic polynomials of $a_2, a_3, a_5, \ldots, a_{19}$.
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\end{enumerate}
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(I hope to redo this computation
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using only open-source software that I'm currently
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writing.)
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\end{slide}
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\begin{slide}
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\h{Probable Modularity}
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$\bullet$ Two forms $f=\sum a_n q^n$ and $g=\sum b_n q^n$
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are {\em\dred probably congruent mod $\ell$} (away from level) if
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for $p<20$ with $p\nmid N_f N_g$ we have
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$$
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\ell \mid \text{resultant}(\charpoly(a_p),\charpoly(b_p)).
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$$
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%(Congruence would imply this, but not conversely in general.)
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$\bullet$ If $A=A_g\subset J_0(N)$, then there is
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{\em\dred probably a nonzero element in
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$\Sha(A)[\ell]$ modular of level $N p$} if
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there is $f$ of level $N p$ such that:\vspace{-.4in}
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\begin{enumerate}\setlength{\itemsep}{0in}
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\item $f$ and $g$ are probably congruent modulo $\ell$, and
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\item $\ord_{s=1} L(f,s)$ is positive and even.
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\end{enumerate}
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\end{slide}
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\begin{slide}
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\voffset=-1.3in
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\begin{center}
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\includegraphics[height=1.7\textheight]{higher.ps}
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\end{center}
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\end{slide}
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\begin{slide}
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\h{Questions}
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\begin{center}
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\includegraphics[width=3in]{questions.eps}
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%\Huge {\dred ?}
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\end{center}
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\end{document}
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\begin{slide}
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Let $N$ be a positive integer and consider the congruence
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subgroup
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$$
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\Gamma_0(N) = \left\lbrace
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\mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c
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\right\rbrace.
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$$
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(Almost everything in this talk also makes sense with $\Gamma_0(N)$
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replaced by $\Gamma_1(N)$.)
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The \textit{modular curve}
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$$
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X_0(N) = \Gamma_0(N) \setminus \left(\left\lbrace
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z\in \C : \Im(z) > 0\right\rbrace
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\cup \Q \cup \left\lbrace \infty\right\rbrace \right)
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$$
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is a Riemann surface that is the set of complex
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points of an algebraic curve over $\Q$.
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We will not use that
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$$
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X_0(N)(\C) = \left\lbrace
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\text{ isomorphism classes of }(E,C)\,\,
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\right\rbrace
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\cup
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\left\lbrace
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\text{ cusps }
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\right\rbrace.
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$$
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Our primary interest is the Jacobian
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$$
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J_0(N) = \Jac(X_0(N))
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$$
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which is an abelian variety over $\Q$ of dimension equal
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to the genus of $X_0(N)$. The points on the Jacobian
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parametrize, in a natural way, the divisor classes
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of degree $0$ on $X_0(N)$.
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Let
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$
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S_2(\Gamma_0(N))
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$
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be the cusp forms of weight $2$ for $\Gamma_0(N)$.
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This is the finite-dimensional complex vector space
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of holomorphic functions on the upper half plane
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such that
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$$f(z)dz = f(\gamma(z))d(\gamma(z))$$
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for all $\gamma\in\Gamma_0(N)$, and which ``vanish
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at the cusps''.
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The map $f(z) \mapsto f(z)dz$ induces
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$$
548
S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1)
549
$$
550
so $S_2(\Gamma_0(N))$ has dimension the genus
551
of $X_0(N)$.
552
553
The \textit{Hecke algebra} is a commutative ring
554
$$
555
\T = \Z[T_1,T_2,T_3, \ldots]
556
$$
557
which acts on $S_2(\Gamma_0(N))$ and $J_0(N)$.
558
A \textit{newform}
559
$$
560
f = \sum_{n=1}^{\infty} a_n q^n \in S_2(\Gamma_0(N))
561
$$
562
is an eigenvector for every element of $\T$ normalized
563
so $a_1 = 1$, which does not ``come from'' any lower level.
564
Attached to $f$ there is an ideal
565
$$
566
I_f = \Ann_{\T}(f) = \Ker(\T \to \Z[a_1,a_2,\ldots]),
567
$$
568
and (following Shimura) to this ideal we attach an abelian variety $A_f$ and an $L$-function $L(A_f,s)$.
569
570
Let
571
$$
572
A_f = J_0(N)[I_f]^0 = \left( \bigcap_{\vphi \in I_f} \Ker(\vphi) \right)^0
573
$$
574
be the connected component of the intersections of the kernels
575
of elements of $I_f$.
576
Then $A_f$ has dimension $[K_f:\Q] = [\Q(a_1,a_2,\ldots):\Q)]$, and
577
is define over $\Q$.
578
579
Let
580
$$
581
L(A_f,s) = \prod_{i=1}^d L(f_i,s)
582
$$
583
where $d=[K_f:\Q]$ and the $f_i$ are the Galois conjugates
584
of $f$. Also,
585
$$
586
L(f,s) = \sum_{n=1}^\infty \dfrac{a_n}{n^s}.
587
$$
588
Hecke proved that $L(f,s)$ is entire and satisfies
589
a functional equation.
590
591
The abelian varieties $A_f$ are a rich class of abelian
592
varieties. The elliptic curves over~$\Q$ are
593
all isogenous to some $A_f$ (the Wiles-Breuil-Conrad-Diamond-Taylor
594
modularity theorem).
595
596
\section{The Birch and Swinnerton-Dyer Conjecture}
597
\subsection{Conjecture}
598
\begin{conjecture}[Birch and Swinnerton-Dyer]\mbox{}\vspace{-4ex}\\
599
\begin{enumerate}
600
\item $\rank A_f(\Q) = \ord_{s=1}L(A_f,s)$
601
\item
602
$\ds
603
\frac{L^{(r)}(A_f,1)}{r!} =
604
\frac{\prod c_p \cdot \Omega_{A_f} \cdot \Reg_{A_f}
605
\cdot \#\Sha(A_f)}
606
{\#A_f(\Q)_{\tor}\cdot \#A_f^{\vee}(\Q)_{\tor}}.
607
$
608
\end{enumerate}
609
\end{conjecture}
610
Remarks: Part of the conjecture is that $\Sha(A_f)$ is finite.
611
There is also a conjecture for arbitrary abelian varieties
612
over global fields.
613
Clay Math Problem: \$1000000 prize for proof of (1) in case $\dim(A_f)=1$
614
615
Here:
616
\begin{itemize}
617
\item $c_p$ is the \textit{Tamagawa number} at the prime $p$, and the
618
product is over the prime divisors of $N$.
619
\item $\Omega_{A_f}$ is the canonical N\'eron measure
620
of $A_f(\R)$.
621
\item $\Reg_{A_f}$ is the regulator (absolute value
622
of N\'eron-Tate canonical height pairing matrix).
623
\item $A_f(\Q)_{\tor}$ is the torsion subgroup of $A_f(\Q)$.
624
\item $\Sha(A_f)$ is the Shafarevich-Tate group.
625
\end{itemize}
626
\subsection{Evidence}
627
\begin{itemize}
628
\item Rubin: results in CM Case
629
\item Kolyvagin, Logachev,
630
Gross-Zagier, et al.: If $\ord_{s=1}L(f,s)=0$ or $1$,
631
then (1) true and $\Sha(A_f)$ finite.
632
\item Cremona: Compute $\Sha(A_f)_{?}$ (=conjectural order) for tens
633
of thousands of $A_f$ of dimension $1$ and get approximate square order.
634
(Theorem of Cassels: if $E$ an elliptic curve and
635
$\Sha(E)$ finite then order a perfect square. Note that the analogue for
636
abelian varieties is false; for exampe, I've constructed examples for
637
each odd prime $p<25000$ of abelian varieties $A$ of dimension $p-1$
638
such that $\Sha(A) = p\cdot n^2$.)
639
\end{itemize}
640
641
In this talk I will focus on $A_f$ of possibly large dimension with $L(A_f,1)\neq 0$, since computation
642
of $\Reg_{A_f}$ is difficult (impossible?) when one can't even reasonably
643
hope to write down $A_f$ explicitly with equations.
644
645
\section{Visibility of Shafarevich-Tate Groups}
646
\subsection{Definitions}
647
It is easy to write down a point on an elliptic curve $E$. You simply write down a pair of rational numbers, which are a solution to a Weierstrass equation. In contrast, imagine describing explicitly an element of $\Sha(E)$ of order $2003$. The most direct way would be to give a genus one curve (with principal homogeneous space structure), embedded in $\P^3$ of degree at least $2003$ (!), hence very complicated.
648
649
The idea of visibility of Shafarevich-Tate groups was introduced
650
by Barry Mazur around 1998 to unify various constructions of
651
elements of Shafarevich-Tate groups.
652
\begin{definition}[Shafarevich-Tate Group]\label{defn:sha}
653
$$
654
\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_v \H^1(K_v,A)\right).
655
$$
656
\end{definition}
657
Here $\H^1(K,A)$ is the first Galois cohomology, which can
658
be interpreted geometrically as the Weil-Chatalet group
659
$$
660
\text{WC}(A/K) = \{\text{ principal homogenous spaces }X\text{ for }A\,\}/\sim.
661
$$
662
663
Then $\Sha(A)$ is the subgroup of locally trivial classes of
664
homogenous spaces. For example
665
$$3x^3+4y^3+5z^3=0
666
\in \Sha(x^3+y^3+60z^3=0)[3].$$
667
668
Fix an inclusion $i:A\hra B$ of abelian varieties and let $\pi: B\to C$
669
be the quotient of~$B$ by the image of $A$, so we have an exact sequence
670
$$
671
0 \to A \to B \to C \to 0
672
$$
673
of abelian varieties.
674
\begin{definition}[Visible Subgroup]
675
\begin{align*}
676
\Vis_i(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\
677
&=\Coker(B(K) \to C(K))
678
\end{align*}
679
and
680
$$\Vis_i(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).$$
681
\end{definition}
682
\begin{enumerate}
683
\item The visible subgroup is finite because
684
$B(K)$ is finitely generated and $\Vis_i(\H^1(K,A))$
685
is torsion.
686
\item If $c\in\Vis_i(\H^1(K,A))$, then $c$ is
687
also ``visible'' in the sense that if $c$ is the image
688
of a point $x\in C(K)$, and if $X=\pi^{-1}(x)\subset B$, then
689
$[X]\in\text{WC}(A)$ corresponds to $c$.
690
\item The visibile subgroups depends on the choice of embedding
691
$i:A\hra B$. I've also considered defining
692
$\Vis_B(\H^1(K,A))$ to be the subgroup generated by all visible
693
subgroups with respect to all embeddings $A\to B$, but I'm not
694
sure what properties this definition has.
695
\end{enumerate}
696
697
698
\subsection{Theorems}
699
700
``Everything is visible somewhere.''
701
\begin{theorem}[Stein]
702
If $c\in \H^1(K,A)$ then there exists
703
$B=\Res_{L/K}(A_L)$
704
such that $i:A\hra B$ and $c\in\Vis_i(\H^1(K,A))$.
705
(Here $L$ is such that $\res_{L/K}(c)=0$.)
706
\end{theorem}
707
708
\noindent``Visibility construction.''
709
\begin{theorem}[Agashe-Stein]\label{vis:const}
710
Suppose $A,B\subset C$ over $\Q$, that $A+B=C$, that
711
$A\cap B$ is finite. Suppose $N$ is divisible by all
712
bad primes for $C$, and $p$ is a prime such that
713
\begin{itemize}
714
\item $B[p]\subset A$
715
\item $\ds p\nmid 2\cdot N\cdot \#B(\Q)_{\tor}\cdot \#(C/B)(\Q)_{\tor}\cdot
716
\prod_{p\mid N} c_{A,p} \cdot c_{B,p}.$
717
\end{itemize}
718
If $A$ has rank $0$, then there is a natural inclusion
719
$$
720
B(\Q)/p B(\Q) \hra \Vis_C(\Sha(A)).
721
$$
722
(And certain generalizations...)
723
\end{theorem}
724
725
726
727
\subsection{Example}
728
729
\begin{example}
730
For $N=389$, take $B$ the (first ever) rank $2$ elliptic curve, and
731
$A$ the $20$-dimensional rank $0$ factor.
732
$$
733
\xymatrix{ & B\ar[d] \\
734
A \ar[r] & {J_0(389)}
735
}
736
$$
737
Gives
738
$$
739
(\Z/5\Z)^2 \isom B(\Q)/5 B(\Q) \hra \Sha(A).
740
$$
741
Part 2 of the Birch and Swinnerton-Dyer conjecture predicts that
742
$$
743
\Sha(A) = 5^2 \cdot 2^{?},
744
$$
745
so this gives evidence.
746
\end{example}
747
748
\section{Visibility in Modular Jacobians}
749
Suppose now $A=A_f\subset J_0(N)$ is attached to a newform.
750
\begin{definition}[Modular of level $M$]
751
An element $c\in\Sha(A)[p]$ is \textit{modular of level}~$M$ if
752
$c \in \Vis_{M}^p(\Sha(A))$,
753
where $\Vis_{M}^p(\Sha(A))$ is
754
the subgroup generated by all
755
kernels of maps $\Sha(A)[p^\infty]\to \Sha(J_0(M))[p^\infty]$
756
induced by homomorphisms $A\to J_0(M)$
757
of degree coprime to $p$.
758
\end{definition}
759
Note that $M$ must be a multiple of $N$.
760
761
\begin{question}[Mazur]
762
Suppose $E\subset J_0(N)$ is an elliptic curve of conductor $N$. How much of $\Sha(E)$ is modular of level $N$?
763
\end{question}
764
Answer: In examples, surprisingly much. Expect not all visible, since
765
$$
766
\Vis_{N}(\Sha(E)) \subset \Sha(E)[\text{modular degree}],
767
$$
768
and modular degree annihilates symmetric square Selmer
769
group (work of Flach).
770
771
\subsection{Data and Experiments}
772
\begin{itemize}
773
\item {\bf Cremona-Mazur:}
774
There are $52$ elliptic curves $E\subset J_0(N)$ with $N<5500$ such that $p\mid \#\Sha(E)_?$.
775
Cremona-Mazur show that for $43$ of these that $\Sha(E)$
776
``probably'' is modular of level $N$, and for $3$ that it is definitely not:
777
$N=2849, 4343, 5389$. (``Probably'' was made ``provably'' in many cases
778
in subsequent work.)
779
780
\item {\bf Agashe-Stein:}
781
Same question as Cremona-Mazur for $A_f\subset J_0(N)$ of
782
any dimension. Using results of my Ph.D. thesis, MAGMA packages,
783
etc. I computed a divisor and multiple of $\#\Sha(A_f)_?$
784
for the following:
785
\begin{itemize}
786
\item $10360$ abelian varieties $A_f\subset J_0(N)$ with $L(A_f,1)\neq 0$.
787
\item Found $168$ with $\#\Sha(A_f)_?$ definitely divisible by an odd prime.
788
\item For $39$ of these, prove that all $\#\Sha(A_f)_?^{\text{odd}}$
789
elements are modular of level $N$, and $106$ probably are. This gives
790
strong evidence for the BSD conjecture, and a sense that maybe
791
something further is going on.
792
\item Of these $168$, at least $62$ have odd conjectural $\Sha$ that
793
is definitely {\em not} modular of level $N$. Big mystery?
794
Where is this $\Sha$ modular? Is it modular at all? Is it even there??
795
(Perhaps a good place to look for counterexample to BSD.)
796
\end{itemize}
797
798
\end{itemize}
799
800
\section{Visibility at Higher Level}
801
\begin{definition}
802
Let $c\in\Sha(A_f)$. The {\em modularity levels} of $c$ are the
803
set of integers
804
$$\mathcal{N}(c) = \{M: c\in\Vis_{M}(\Sha(A_f))\}.$$
805
\end{definition}
806
807
\begin{conjecture}[Stein]
808
For any $c\in\Sha(A_f)$ we have
809
$$
810
\mathcal{N}(c) \neq \emptyset,
811
$$
812
i.e., every element of $\Sha(A_f)$ is modular.
813
\end{conjecture}
814
Motivation: This is a working hypothesis that makes \textit{computing}
815
with modular abelian varieties easier.
816
Also, if there were a common level at which all of $\Sha(A_f)$ were modular,
817
then $\Sha(A_f)$ would be finite, and conversely (assuming the conjecture).
818
819
\subsection{Ribet Level Raising}
820
Suppose that $f=\sum a_n q^n\in S_2(\Gamma_0(N))$
821
is a newform and $\p$ is a nonzero prime ideal
822
of $\Z[a_1,a_2,\ldots]$ such that $A_f[\p]$
823
is irreducible. If
824
$$
825
a_\ell + \ell + 1 \con 0\pmod{\p}
826
$$
827
then there exists an $\ell$-newform
828
$g\in S_2(\Gamma_0(N\ell))$
829
such that $i(A_f[\p]) = A_g[\p]$ for an appropriate
830
$i:J_0(N)\to J_0(N\ell)$ of degree coprime to $\text{char}(\p)$
831
and the sign
832
of the functional equations for $L(f,s)$ and $L(g,s)$
833
are the same.
834
835
If we instead require that $a_\ell - (\ell+1)\con 0\pmod{\p}$
836
then there is such a $g$, but the sign of the functional
837
equation changes, and the new Tamagawa numbers of
838
$A_g$ at $\ell$ will (or tends to be?) divisible by $\p$.
839
840
\subsection{Evidence for Conjecture}
841
I defined a precise notion of ``probably modular'' motivated
842
by Theorem~\ref{vis:const} and what I can compute. In many cases
843
I could do extra work and actually prove modularity; however, at this stage it is more interesting to gather data to see what is going on, in order to have a sense for what
844
to conjecture.
845
846
Mazur proved that everything in $\Sha(E)[3]$, for $E$ an elliptic curve, is visible
847
in an abelian surface, which, together with the modularity theorem, {\em might}
848
imply modularity of $\Sha(E)[3]$ at higher level. Same for $2$, proved by me and by a
849
different method by Thomas Klenke.
850
851
852
\section{Some Tables}
853
The first two pages of tables below give some of the data that
854
I computed about visibility of Shafarevich-Tate groups
855
at level $N$. The third table gives the new data about
856
visibility at higher level.
857
858
\newpage
859
\begin{center}
860
{\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups}
861
$\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline
862
\quad A& \dim& S_l & S_u & \moddeg(A)^{\op}
863
& \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline
864
865
\mathbf{389E}*&20&5^{2}&=&5&\mathbf{389A}&1&[20^{2}]&5^{2} \\
866
\mathbf{433D}*&16&7^{2}&=&7\!\cdot\!\mbox{\tiny $111$}&\mathbf{433A}&1&[14^{2}]&7^{2} \\
867
\mathbf{446F}*&8&11^{2}&=&11\!\cdot\!\mbox{\tiny $359353$}&\mathbf{446B}&1&[11^{2}]&11^{2} \\
868
\mathbf{551H}&18&3^{2}&=&\mbox{\tiny $169$}&\text{NONE} & & & \\
869
\hline
870
\mathbf{563E}*&31&13^{2}&=&13&\mathbf{563A}&1&[26^{2}]&13^{2} \\
871
\mathbf{571D}*&2&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $127$}&\mathbf{571B}&1&[3^{2}]&3^{2} \\
872
\mathbf{655D}*&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $9799079$}&\mathbf{655A}&1&[36^{2}]&3^{4} \\
873
\mathbf{681B}&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $125$}&\mathbf{681C}&1&[3^{2}]&- \\
874
\hline
875
\mathbf{707G}*&15&13^{2}&=&13\!\cdot\!\mbox{\tiny $800077$}&\mathbf{707A}&1&[13^{2}]&13^{2} \\
876
\mathbf{709C}*&30&11^{2}&=&11&\mathbf{709A}&1&[22^{2}]&11^{2} \\
877
\mathbf{718F}*&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $5371523$}&\mathbf{718B}&1&[7^{2}]&7^{2} \\
878
\mathbf{767F}&23&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
879
\hline
880
\mathbf{794G}*&12&11^{2}&=&11\!\cdot\!\mbox{\tiny $34986189$}&\mathbf{794A}&1&[11^{2}]&- \\
881
\mathbf{817E}*&15&7^{2}&=&7\!\cdot\!\mbox{\tiny $79$}&\mathbf{817A}&1&[7^{2}]&- \\
882
\mathbf{959D}&24&3^{2}&=&\mbox{\tiny $583673$}&\text{NONE} & & & \\
883
\mathbf{997H}*&42&3^{4}&=&3^{2}&\mathbf{997B}&1&[12^{2}]&3^{2} \\
884
\hline
885
&&& && \mathbf{997C}&1&[24^{2}]&3^{2} \\
886
\mathbf{1001F}&3&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $1269$}&\mathbf{1001C}&1&[3^{2}]&- \\
887
&&& && \mathbf{91A}&1&[3^{2}]&- \\
888
\mathbf{1001L}&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $2029789$}&\mathbf{1001C}&1&[7^{2}]&- \\
889
\hline
890
\mathbf{1041E}&4&5^{2}&=&5^{2}\!\cdot\!\mbox{\tiny $13589$}&\mathbf{1041B}&2&[5^{2}]&- \\
891
\mathbf{1041J}&13&5^{4}&=&5^{3}\!\cdot\!\mbox{\tiny $21120929983$}&\mathbf{1041B}&2&[5^{4}]&- \\
892
\mathbf{1058D}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $483$}&\mathbf{1058C}&1&[5^{2}]&- \\
893
\mathbf{1061D}&46&151^{2}&=&151\!\cdot\!\mbox{\tiny $10919$}&\mathbf{1061B}&2&[2^{2}302^{2}]&- \\
894
\hline
895
\mathbf{1070M}&7&3 \!\cdot\! 5^{2}&3^{2} \!\cdot\! 5^{2}&3 \!\cdot\! 5\!\cdot\!\mbox{\tiny $1720261$}&\mathbf{1070A}&1&[15^{2}]&- \\
896
\mathbf{1077J}&15&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $1227767047943$}&\mathbf{1077A}&1&[9^{2}]&- \\
897
\mathbf{1091C}&62&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
898
\mathbf{1094F}*&13&11^{2}&=&11^{2}\!\cdot\!\mbox{\tiny $172446773$}&\mathbf{1094A}&1&[11^{2}]&11^{2} \\
899
\hline
900
\mathbf{1102K}&4&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $31009$}&\mathbf{1102A}&1&[3^{2}]&- \\
901
\mathbf{1126F}*&11&11^{2}&=&11\!\cdot\!\mbox{\tiny $13990352759$}&\mathbf{1126A}&1&[11^{2}]&11^{2} \\
902
\mathbf{1137C}&14&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $64082807$}&\mathbf{1137A}&1&[9^{2}]&- \\
903
\mathbf{1141I}&22&7^{2}&=&7\!\cdot\!\mbox{\tiny $528921$}&\mathbf{1141A}&1&[14^{2}]&- \\
904
\hline
905
\mathbf{1147H}&23&5^{2}&=&5\!\cdot\!\mbox{\tiny $729$}&\mathbf{1147A}&1&[10^{2}]&- \\
906
\mathbf{1171D}*&53&11^{2}&=&11\!\cdot\!\mbox{\tiny $81$}&\mathbf{1171A}&1&[44^{2}]&11^{2} \\
907
\mathbf{1246B}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $81$}&\mathbf{1246C}&1&[5^{2}]&- \\
908
\mathbf{1247D}&32&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $2399$}&\mathbf{43A}&1&[36^{2}]&- \\
909
\hline
910
\mathbf{1283C}&62&5^{2}&=&5\!\cdot\!\mbox{\tiny $2419$}&\text{NONE} & & & \\
911
\mathbf{1337E}&33&3^{2}&=&\mbox{\tiny $71$}&\text{NONE} & & & \\
912
\mathbf{1339G}&30&3^{2}&=&\mbox{\tiny $5776049$}&\text{NONE} & & & \\
913
\mathbf{1355E}&28&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $2224523985405$}&\text{NONE} & & & \\
914
\hline
915
\mathbf{1363F}&25&31^{2}&=&31\!\cdot\!\mbox{\tiny $34889$}&\mathbf{1363B}&2&[2^{2}62^{2}]&- \\
916
\mathbf{1429B}&64&5^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
917
\mathbf{1443G}&5&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $18525$}&\mathbf{1443C}&1&[7^{1}14^{1}]&- \\
918
\mathbf{1446N}&7&3^{2}&=&3\!\cdot\!\mbox{\tiny $17459029$}&\mathbf{1446A}&1&[12^{2}]&- \\
919
\hline
920
921
\end{array}$
922
\end{center}
923
924
\newpage
925
\begin{center}
926
{\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups}
927
$\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline
928
\quad A& \dim& S_l & S_u & \moddeg(A)^{\op}
929
& \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline
930
931
\mathbf{1466H}*&23&13^{2}&=&13\!\cdot\!\mbox{\tiny $25631993723$}&\mathbf{1466B}&1&[26^{2}]&13^{2} \\
932
\mathbf{1477C}*&24&13^{2}&=&13\!\cdot\!\mbox{\tiny $57037637$}&\mathbf{1477A}&1&[13^{2}]&13^{2} \\
933
\mathbf{1481C}&71&13^{2}&=&\mbox{\tiny $70825$}&\text{NONE} & & & \\
934
\mathbf{1483D}*&67&3^{2} \!\cdot\! 5^{2}&=&3 \!\cdot\! 5&\mathbf{1483A}&1&[60^{2}]&3^{2} \!\cdot\! 5^{2} \\
935
\hline
936
\mathbf{1513F}&31&3&3^{4}&3\!\cdot\!\mbox{\tiny $759709$}&\text{NONE} & & & \\
937
\mathbf{1529D}&36&5^{2}&=&\mbox{\tiny $535641763$}&\text{NONE} & & & \\
938
\mathbf{1531D}&73&3&3^{2}&3&\mathbf{1531A}&1&[48^{2}]&- \\
939
\mathbf{1534J}&6&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $635931$}&\mathbf{1534B}&1&[6^{2}]&- \\
940
\hline
941
\mathbf{1551G}&13&3^{2}&=&3\!\cdot\!\mbox{\tiny $110659885$}&\mathbf{141A}&1&[15^{2}]&- \\
942
\mathbf{1559B}&90&11^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
943
\mathbf{1567D}&69&7^{2} \!\cdot\! 41^{2}&=&7 \!\cdot\! 41&\mathbf{1567B}&3&[4^{4}1148^{2}]&- \\
944
\mathbf{1570J}*&6&11^{2}&=&11\!\cdot\!\mbox{\tiny $228651397$}&\mathbf{1570B}&1&[11^{2}]&11^{2} \\
945
\hline
946
\mathbf{1577E}&36&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $15$}&\mathbf{83A}&1&[6^{2}]&- \\
947
\mathbf{1589D}&35&3^{2}&=&\mbox{\tiny $6005292627343$}&\text{NONE} & & & \\
948
\mathbf{1591F}*&35&31^{2}&=&31\!\cdot\!\mbox{\tiny $2401$}&\mathbf{1591A}&1&[31^{2}]&31^{2} \\
949
\mathbf{1594J}&17&3^{2}&=&3\!\cdot\!\mbox{\tiny $259338050025131$}&\mathbf{1594A}&1&[12^{2}]&- \\
950
\hline
951
\mathbf{1613D}*&75&5^{2}&=&5\!\cdot\!\mbox{\tiny $19$}&\mathbf{1613A}&1&[20^{2}]&5^{2} \\
952
\mathbf{1615J}&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $13317421$}&\mathbf{1615A}&1&[9^{1}18^{1}]&- \\
953
\mathbf{1621C}*&70&17^{2}&=&17&\mathbf{1621A}&1&[34^{2}]&17^{2} \\
954
\mathbf{1627C}*&73&3^{4}&=&3^{2}&\mathbf{1627A}&1&[36^{2}]&3^{4} \\
955
\hline
956
\mathbf{1631C}&37&5^{2}&=&\mbox{\tiny $6354841131$}&\text{NONE} & & & \\
957
\mathbf{1633D}&27&3^{6} \!\cdot\! 7^{2}&=&3^{5} \!\cdot\! 7\!\cdot\!\mbox{\tiny $31375$}&\mathbf{1633A}&3&[6^{4}42^{2}]&- \\
958
\mathbf{1634K}&12&3^{2}&=&3\!\cdot\!\mbox{\tiny $3311565989$}&\mathbf{817A}&1&[3^{2}]&- \\
959
\mathbf{1639G}*&34&17^{2}&=&17\!\cdot\!\mbox{\tiny $82355$}&\mathbf{1639B}&1&[34^{2}]&17^{2} \\
960
\hline
961
\mathbf{1641J}*&24&23^{2}&=&23\!\cdot\!\mbox{\tiny $1491344147471$}&\mathbf{1641B}&1&[23^{2}]&23^{2} \\
962
\mathbf{1642D}*&14&7^{2}&=&7\!\cdot\!\mbox{\tiny $123398360851$}&\mathbf{1642A}&1&[7^{2}]&7^{2} \\
963
\mathbf{1662K}&7&11^{2}&=&11\!\cdot\!\mbox{\tiny $16610917393$}&\mathbf{1662A}&1&[11^{2}]&- \\
964
\mathbf{1664K}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $7$}&\mathbf{1664N}&1&[5^{2}]&- \\
965
\hline
966
\mathbf{1679C}&45&11^{2}&=&\mbox{\tiny $6489$}&\text{NONE} & & & \\
967
\mathbf{1689E}&28&3^{2}&=&3\!\cdot\!\mbox{\tiny $172707180029157365$}&\mathbf{563A}&1&[3^{2}]&- \\
968
\mathbf{1693C}&72&1301^{2}&=&1301&\mathbf{1693A}&3&[2^{4}2602^{2}]&- \\
969
\mathbf{1717H}*&34&13^{2}&=&13\!\cdot\!\mbox{\tiny $345$}&\mathbf{1717B}&1&[26^{2}]&13^{2} \\
970
\hline
971
\mathbf{1727E}&39&3^{2}&=&\mbox{\tiny $118242943$}&\text{NONE} & & & \\
972
\mathbf{1739F}&43&659^{2}&=&659\!\cdot\!\mbox{\tiny $151291281$}&\mathbf{1739C}&2&[2^{2}1318^{2}]&- \\
973
\mathbf{1745K}&33&5^{2}&=&5\!\cdot\!\mbox{\tiny $1971380677489$}&\mathbf{1745D}&1&[20^{2}]&- \\
974
\mathbf{1751C}&45&5^{2}&=&5\!\cdot\!\mbox{\tiny $707$}&\mathbf{103A}&2&[505^{2}]&- \\
975
\hline
976
\mathbf{1781D}&44&3^{2}&=&\mbox{\tiny $61541$}&\text{NONE} & & & \\
977
\mathbf{1793G}*&36&23^{2}&=&23\!\cdot\!\mbox{\tiny $8846589$}&\mathbf{1793B}&1&[23^{2}]&23^{2} \\
978
\mathbf{1799D}&44&5^{2}&=&\mbox{\tiny $201449$}&\text{NONE} & & & \\
979
\mathbf{1811D}&98&31^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
980
\hline
981
\mathbf{1829E}&44&13^{2}&=&\mbox{\tiny $3595$}&\text{NONE} & & & \\
982
\mathbf{1843F}&40&3^{2}&=&\mbox{\tiny $8389$}&\text{NONE} & & & \\
983
\mathbf{1847B}&98&3^{6}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
984
\mathbf{1871C}&98&19^{2}&=&\mbox{\tiny $14699$}&\text{NONE} & & & \\
985
\hline
986
987
\end{array}$
988
\end{center}
989
990
991
\newpage
992
\large
993
{\bf\Large Visibility at Higher Level\vspace{3ex}}
994
\begin{center}
995
\begin{tabular}{|l|l|}\hline
996
&\vspace{-2ex}\\
997
$A_f$ with odd invisible $\Sha_{\an}[\ell]$& All $\ell$-congruent\\
998
& $A_g\subset J_0(Np)_{\new}$\\
999
&with $Np\leq 5000$ and \\
1000
& $\ord_{s=1}L(g,s)\geq 0$\\
1001
& (and higher $Np$ if known)\\
1002
&\vspace{-2ex}\\
1003
% data is autogenerated by table.py
1004
\first{\sha{551}{18}{3}}
1005
\add{\higher{2}{1}{2}}
1006
\add{\higher{3}{1}{2}}
1007
\add{\higher{5}{25}{0}}
1008
\first{\sha{767}{23}{3}}
1009
\add{\higher{2}{1}{2}}
1010
\add{\higher{7}{1}{2}}
1011
\add{\higher{7}{52}{0}}
1012
\first{\sha{959}{24}{3}}
1013
\add{\higher{2}{1}{2}}
1014
\first{\sha{1091}{62}{7}}
1015
\add{\higher{7}{2}{2}}
1016
\first{\sha{1283}{62}{5}}
1017
\add{\higher{3}{2}{2}}
1018
\first{\sha{1337}{33}{3}}
1019
\add{\higher{2}{1}{2}}
1020
\first{\sha{1339}{30}{3}}
1021
\add{\higher{2}{1}{2}}
1022
\first{\sha{1355}{28}{3}}
1023
\add{\higher{2}{1}{2}}
1024
\first{\sha{1429}{64}{5}}
1025
\add{\higher{2}{2}{2}}
1026
\add{\higher{3}{66}{0}}
1027
\first{\sha{1481}{71}{13}}
1028
\add{Nothing in range}
1029
\first{\sha{1513}{31}{3}}
1030
\add{\higher{2}{1}{2}}
1031
\first{\sha{1529}{36}{5}}
1032
\add{\higher{7}{1}{2}}
1033
\first{\sha{1559}{90}{11}}
1034
\add{Nothing in range}
1035
\first{\sha{1589}{35}{3}}
1036
\add{Nothing in range}
1037
\first{\sha{1631}{37}{5}}
1038
\add{\higher{2}{1}{2}}
1039
\first{\sha{1679}{45}{11}}
1040
\add{\higher{2}{2}{2}}
1041
\first{\sha{1727}{39}{3}}
1042
\add{\higher{2}{1}{2}}
1043
\first{\sha{2849}{1}{3}}
1044
\add{\higher{3}{1}{2}}
1045
\first{\sha{4343}{1}{3}}
1046
\add{Nothing in range}
1047
\first{\sha{5389}{1}{3}}
1048
\add{\higher{7}{1}{2}}
1049
\hline\end{tabular}
1050
\end{center}
1051
\vspace{3ex}
1052
1053
\noindent When the second column contains an $A_g$ of rank~$2$,
1054
then $\Sha(A_f)[\ell]$ is ``very likely'' to be visible of level $M=Np$.
1055
This is the case for most examples. The ``Nothing in range'' note
1056
means that the smallest~$p$ for which there exists~$g$ of even
1057
analytic rank congruent to~$f$ is beyond the range of my current
1058
tables. The examples of level 2849, 4343, and 5389 are the odd and
1059
definitely invisible examples in Cremona and Mazur's original paper on
1060
visibility.
1061
1062
1063
1064
\end{document}
1065