Author: William A. Stein
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% Visibility of Shafarevich-Tate groups of modular abelian varieties %
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% William A. Stein and Amod Agashe. %
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\title{\sc Visibility of Shafarevich-Tate groups of modular abelian varieties
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and the Birch and Swinnerton-Dyer conjecture}
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\author{Amod Agashe\footnote{Warning: Agashe has not yet
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read this draft of our preprint.} \and William A.~Stein}
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\begin{document}
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\UseTips
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\maketitle
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\begin{abstract}
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We give examples of abelian subvarieties of $J_0(N)$ that have
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nontrivial visible Shafarevich-Tate groups. These examples provide
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evidence for the Birch and Swinnerton-Dyer conjecture, and the methods
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used to compute them may lead to new connections between this
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conjecture and the theory of congruences between modular forms.
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\end{abstract}
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\section{Introduction}\label{sec:intro}
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Let~$N$ be a positive integer and let $J_0(N)$ be the Jacobian of the
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modular curve $X_0(N)$ (see, e.g., \cite{diamond-im}). Let~$A$ be an
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abelian subvariety of $J_0(N)$. Then the inclusion $A\hookrightarrow 49 J_0(N)$ induces a map on Galois cohomology $H^1(\Q,A) \ra 50 H^1(\Q,J_0(N))$, which restricts to a map $\Sha(A) \ra \Sha(J_0(N))$.
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Following \cite{cremona-mazur}, we call an element $c\in\Sha(A)$
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\defn{visible} in $J_0(N)$ if it lies in the kernel of this map.
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When~$c$ is visible the torsor attached to~$c$ is realized as a
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subvariety of $J_0(N)$; it is a fiber of the quotient map $J_0(N)\ra 55 J_0(N)/A.$
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In~\cite{cremona-mazur}, Cremona and Mazur ask the following question:
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If $E\subset J_0(N)$ is a elliptic curve, how much of $\Sha(E)$ is
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visible in an abelian surface~$B$ that is itself a subvariety
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of~$J_0(N)$? Among the $52$ examples of odd nontrivial $\Sha$ that
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they consider, $\Sha$ is invisible in at least~$8$ cases. However,
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they warn that the elliptic curves they consider have small conductor,
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and the general situation is probably much different. Standard
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conjectures imply that when $\ell$ is sufficiently large, elements of
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order~$\ell$ in $\Sha(E)$ can not be visible in an abelian surface; it
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is thus essential to consider visibility in higher dimensional abelian
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varieties.
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We generalize the experiment of Cremona and
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Mazur by removing the constraint that only elliptic curves be
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considered. The numbers become large at much smaller conductor, and
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the proportion of~$\Sha$ that is visible declines accordingly.
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In Section~\ref{sec:vis} we give a general definition of visibility,
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which is motivated by a restriction-of-scalars construction of
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torsors. Then we prove a theorem that gives a criterion for
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the existence of nontrivial visible elements of Shafarevich-Tate
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groups. Section~\ref{sec:computing} describes the algorithms we use to
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enumerate modular abelian varieties, compute conjectural
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orders of their Shafarevich-Tate groups, and verify the visibility
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criterion of Section~\ref{sec:vis} when possible. Section~\ref{sec:tables}
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presents the results of extensive numerical investigations. It provides
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the impetus for the conjectures we make and questions we ask in
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Section~\ref{sec:conj}.
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{\bf Acknowledgment.} It is a pleasure to thank B.~Mazur for his
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lectures and conversations about visibility, K.~Ribet for explaining
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congruences to us, and R.~Coleman for helping us to understand
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component groups. The authors would also like to thank B.~Conrad,
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J.~Ellenberg, R.~Greenberg, D.~Gross, L.~Merel, B.~Poonen, and R.~Taylor
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for many helpful discussions.
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\section{Visible cohomology classes}\label{sec:vis}
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In Section~\ref{sec:torsors} we define the notion of visibility and
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observe that every cohomology class
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arises from a torsor that can be constructed geometrically inside of
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an appropriate restriction of scalars, then we define visible
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cohomology classes. Section~\ref{sec:visthm}, which should perhaps be
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omitted from a first reading, gives a criterion for the existence of
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visible elements of Shafarevich-Tate groups.
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\subsection{Geometric realization of torsors}\label{sec:torsors}
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Let~$A$ be an abelian variety over a field~$K$. The following
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definition is due to Mazur.
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\begin{definition}[Visible]
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Let $\iota:A\hookrightarrow J$
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be an embedding of~$A$ into an abelian variety~$J$. Then the \defn{visible part
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of $H^1(K,A)$ with respect to the embedding~$\iota$} is
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$$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\ra{}H^1(K,J)).$$
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\end{definition}
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The Galois cohomology group $H^1(K,A)$ has a geometric interpretation
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as the group of classes of torsors~$X$ for~$A$ (see~\cite{lang-tate}).
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To a cohomology class $c\in H^1(K,A)$, there is a corresponding
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variety~$X$ over~$K$ and a map $A\cross X \ra X$ that satisfies axioms
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similar to those for a simply transitive group action.
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Suppose $\iota: A\ra J$ is an embedding and $c\in \Vis_J(H^1(K,A))$.
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We have an exact sequence of abelian varieties
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$0\ra A\ra J\ra B\ra 0$. A piece of the
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associated long exact sequence of Galois cohomology is
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$$\cdots \ra J(K)\ra B(K) \ra H^1(K,A) \ra H^1(K,J) \ra \cdots,$$
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so, because $\iota_*(c)=0$, there is a point $x\in B(K)$ that maps
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to $c\in H^1(K,A)$. Then the fiber~$X$ over~$x$ is a subvariety of~$J$,
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which, when equipped with its natural action of~$A$, lies in the class
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of torsors corresponding to~$c$.
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\begin{proposition}
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Every element of $H^1(K,A)$ is visible in some abelian variety~$J$.
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\end{proposition}
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\begin{proof}
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Fix $c\in H^1(K,A)$. There is a finite extension~$L$ of~$K$ such that
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$\res_L(c) = 0\in H^1(L,A)$. Let $J=\Res_{L/K}(A_L)$ be the
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restriction of scalars down to~$K$ of the abelian variety~$A$, which
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we view as an abelian variety over~$L$ (see \cite[\S7.6]{neronmodels}).
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Thus~$J$ is an abelian variety over~$K$ of dimension $[L:K]\cdot \dim(A)$, and for any
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scheme~$S$ over~$K$, we have a natural bijection $J(S) \ncisom A_L(S_L)$.
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In particular, $J(A) = A_L(A_L)$, and there is an
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injection $\iota:A\hookrightarrow J$ attached to $\id_{A_L}\in A_L(A_L)$.
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Using the Shapiro lemma, one finds that there is a canonical isomorphism
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$H^1(L,A) \isom H^1(K,J)$ and that
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$\iota_*(c) = 0\in H^1(K,J)$.
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\end{proof}
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\begin{remark}
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In \cite{cremona-mazur}, J.~de Jong gives a sophisticated proof of
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the above proposition in the special case when~$A$ is an elliptic curve.
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His proof involves Azumaya algebras.
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\end{remark}
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\subsection{Visible elements of Shafarevich-Tate groups}\label{sec:visthm}
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The Shafarevich-Tate group of an abelian variety over a number
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field measures the failure of the local-to-global principle
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for its torsors.
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\begin{definition}[Shafarevich-Tate group]
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Let~$A$ be an abelian variety over~$\Q$.
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The \defn{Shafarevich-Tate group} of $A$ is
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$$\Sha(A) = \Ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right),$$
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where the product is over all places of~$\Q$.
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\end{definition}
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If $\iota:A\hookrightarrow J$ is an embedding, then the
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visible part of $\Sha(A)$ is
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$$\Vis_J(\Sha(A)) = \Sha(A) \intersect \Vis_J(H^1(\Q,A)) = \Ker(\Sha(A)\ra \Sha(J)).$$
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We use the following theorem to produce examples of visible
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elements of Shafarevich-Tate groups.
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\begin{theorem}\label{thm:shaexists}
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Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
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that $A\intersect B$ is finite and $A(\Q)$ is finite.
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Let~$N$ be an integer divisible by the primes of bad reduction for~$J$.
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Assume that~$B$ has purely toric reduction at each prime dividing~$N$.
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Suppose~$p$ is a prime such that
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$$p\nmid 2N\cdot \#(J/B)(\Q)_{\tor}\cdot\#B(\Q)_{\tor}\cdot 173 \prod_{\ell\mid N} \#\Phi_{A,\ell}(\Fbar_\ell)\cdot 174 \#\Phi_{B,\ell}(\F_\ell).$$
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Suppose furthermore that $B[p] \subset A\intersect B$.
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Then
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$$B(\Q)/pB(\Q)\hookrightarrow \Vis_J(\Sha(A)).$$
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\end{theorem}
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The proof proceeds in four steps. First the torsion and
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$p$-congruence hypothesis is used to produce an injection $B(\Q)/p 181 B(\Q)\hookrightarrow \Vis_J(H^1(\Q,A))$. Next we perform a local
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analysis at each place~$v$ of~$\Q$, which proceeds in three steps. At
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places~$v$ of bad reduction, we use the Mumford-Tate uniformization;
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at odd primes of good reduction we apply an exactness theorem about
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N\'eron models; when~$2$ is a place of bad reduction, we modify the
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situation by a $2$-isogeny and apply another exactness theorem.
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\begin{proof}
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The quotient $J/A$ is an abelian variety~$C$. The long exact sequence
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of Galois cohomology associated to the short exact sequence
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$$0 \ra A \ra J \ra C \ra 0$$
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begins
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$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,} 195 H^1(\Q,A) \ra \cdots.$$
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Let~$\phi$ be map $B\ra C$, which is obtained by composing
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the inclusion $B\hookrightarrow J$ with the quotient map $J\ra C$.
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Since $B[p]\subset A$, we see that~$\phi$ factors through multiplication by~$p$.
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We thus obtain the following commutative diagram:
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$$\xymatrix{ 201 & B\ar[d] \ar[r]^{p}& B\ar[d]\\ 202 A\ar[r]&J\ar[r]&C.}$$
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Using that $B(\Q)[p]=0$, we
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obtain the following diagram, all of whose rows and columns are exact:
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$$\xymatrix{ 206 & K_0\ar[d] & K_1\ar[d]& K_2\ar[d]\\ 207 0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d] 208 & B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\ 209 0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\ 210 & K_3, 211 }$$
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where $K_0$, $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the
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indicated cokernel.
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The snake lemma gives an exact sequence
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$$K_0\ra K_1 \ra K_2 \ra K_3.$$
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Because $B\ra C$ is an isogeny, $K_1\subset B(\Q)_{\tor}$.
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Since $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group, the map
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$K_1\ra K_2$ is the~$0$ map.
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The quotient
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$J(\Q)/B(\Q)$ has no $p$-torsion because
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it is a subgroup of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group,
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so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion, and the map
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$K_2\ra K_3$ must be the~$0$ map.
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We conclude that $K_2=0$.
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The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of
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$H^1(\Q,A)$; however, the latter group contains infinitely many elements of
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order~$p$, whereas $\Sha(A)[p]$ is a finite group, so we must work
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harder in order to deduce that $B(\Q)/p B(\Q)$ actually lies in
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$\Sha(A)[p]$. Let $x\in B(\Q)$; we must show
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that $\pi(x)\in \Sha(A)[p]$. It suffices to show that
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$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.
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At the archimedian place $v=\infty$, the restriction
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$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,
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hence $\res_v(\pi(x))=0$.
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Suppose that~$v$ is a place at which~$J$ has bad reduction.
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By hypothesis, $B$ has purely toric reduction,
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so over $\Q_v^{\ur}$
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there is an isomorphism $B\isom\Gm^d/\Gamma$
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of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,
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for some lattice'' $\Gamma$. For example, when
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$\dim B=1$, this is the Tate curve representation of~$B$.
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Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$
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equals the order of the cokernel of the valuation
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map $\Gamma\ra \Z^d$. Choose a representative
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$P=(x_1,\ldots,x_d)\in\Gm^d$ for the point~$x$.
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Let $n'=\#\Phi_{B,v}(\F_v)$.
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Then since~$P$ is rational over $\Q_v$,
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$n'P$ can be adjusted by elements of~$\Gamma$
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so that each of its components $x_i\in\Gm$ has valuation~$0$;
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since~$p$ does not equal the residue characteristic of~$v$,
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it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$
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such that $pQ = n'P$.
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Thus the cohomology class $\res_v(\pi(n'x))$ is unramified at~$v$.
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By \cite[Prop.~3.8]{milne:duality},
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$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur})) 259 =H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fpbar)),$$
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where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
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Since~$p$ does not divide $\#\Phi_{A,v}(\Fpbar)$,
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and $\pi(n'x)$ has order~$p$, it follows that
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$$0=\res_v(\pi(n'x))=n'\res_v(\pi(x)).$$
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Since the order of $\res_v(\pi(x))$ is coprime to~$n'$,
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we conclude that $\res_v(\pi(x))=0$.
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Next suppose that~$J$ has good reduction at~$\ell$
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and that~$\ell$ is {\em odd}.
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Let $\cA$, $\cJ$, $\cC$, be the N\'eron models
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of~$A$,~$J$,~$C$, respectively.
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Since~$\ell$ is odd, $1=e<\ell-1$, so we may apply
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\cite[Thm.~7.5.4]{neronmodels} to conclude that
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the sequence of group schemes
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$$0\ra \cA \ra \cJ\ra \cC \ra 0$$
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is exact, in the sense that it
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is exact as a sequence of sheaves on the
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\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).
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Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}
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the sequence
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$$0\ra \cA(\Z_v^{\ur})\ra \cB(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$
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is exact. By the N\'eron mapping property, the sequence
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$$0\ra A(\Q_v^{\ur})\ra B(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$
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is also exact, so
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$\res_v(\pi(x))$ is unramified.
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By \cite[Prop.~I.3.8]{milne:duality},
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$$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v))=0,$$
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since~$A$ has good reduction at~$v$.
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Hence $\res_v(\pi(x))=0$.
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If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.
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Consider the case when~$J$ has good reduction at~$2$. The
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absolute ramification index~$e$ of $\Z_2$ is~$1$, which is
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{\em not} less than $2-1=1$, so we can not apply \cite[Thm.~7.5.4]{neronmodels}.
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However, we can modify everything by an isogeny of degree a power
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of~$2$ and apply a different theorem, as follows.
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The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is rational
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over~$\Q$. The abelian varieties
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$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and
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$\tilde{B}=B/\Psi$ also satisfy the hypothesis of
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the theorem we wish to prove.
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By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of
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N\'eron models
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$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$
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is exact, so the sequence
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$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur}) 306 \ra\tilde{C}(\Q_v^{\ur})\ra 0$$
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is exact. Thus the image of
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$\res_v(\pi(x))$ in $H^1(\Q_v,\tilde{A})$ is unramified.
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It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},
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since the component group of $\tilde{A}$ at~$v$ has order a power
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of~$2$, whereas $\pi(x)$ has odd prime order~$p$.
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Thus $\res_v(\pi(x))=0$, since
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the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a
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finite group of $2$-power order.
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\end{proof}
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\subsubsection{Visibility when $A$ also has positive rank}
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In Theorem~\ref{thm:shaexists}, if the condition that $A(\Q)$ has rank~$0$ is removed,
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then the proof can be easily modified to show that the kernel of
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$B(\Q)/p B(\Q) \ra \Vis_J(\Sha(A))$
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has dimension at most the rank of $A(\Q)$.
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According to \cite{cremona:algs},
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the smallest conductor elliptic curve~$E$ of rank~$3$ is found in
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$J=J_0(5077)$. The number $5077$ is prime, and~$J$ decomposes
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up to isogeny as
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$A \cross B \cross E,$
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where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$
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associated to newforms, which have
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dimensions $205$, $216$, and~$1$, respectively.
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The modular degree of~$E$ is $1984=2^6\cdot 31$, and
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the sign of the Atkin-Lehner involution on~$E$ is the same
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as its sign on~$A$, so $E\subset A$.
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The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so
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$31$ is coprime to the orders of any relevant component groups or torsion.
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Thus $\Vis_J(\Sha(A))$ contains $(\Z/31\Z)^2$.
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\section{Guide to computing on $J_0(N)$}\label{sec:computing}
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The Jacobian $J_0(N)$ is equipped with an action of the Hecke algebra~$\T$.
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Let $f\in S_2(\Gamma_0(N))$ be a newform, and let $I_f\subset\T$
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be the annihilator of~$f$. The abelian variety~$A_f$ attached
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to~$f$ is the quotient $J_0(N)/I_f J_0(N)$. Thus $A_f$ is an
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abelian variety of dimension equal to the number of Galois
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conjugates of~$f$ and equipped with a faithful action of $\T/I_f$.
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For the remainder of this section, $A=A_f$ denotes the optimal quotient
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of $J_0(N)$ attached to the annihilator~$I=I_f$ of a newform~$f$.
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350
\subsection{The Birch and Swinnerton-Dyer conjecture}
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The Birch and Swinnerton-Dyer conjecture, as generalized by
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Tate in~\cite{tate:bsd}, furnishes a conjectural formula
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for the order of the Shafarevich-Tate group of any new
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optimal quotient~$A$.
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In general, it is difficult given~$A$ to compute
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the conjectural order of~$\Sha(A)$.
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However, the situation is more optimistic
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when~$A$ is a new modular abelian variety
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such that $L(A,1)\neq 0$.
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For these~$A$ we have devised an algorithm
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that we use to compute the odd part of the
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conjectural order of $\Sha(A)$ in many cases.
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The following is a special case of a much more general conjecture.
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\begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsd}
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Suppose $L(A,1)\neq 0$. Then
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$$\frac{L(A,1)}{\Omega_A} = 367 \frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p} 368 {\# A(\Q)\cdot\#\Adual(\Q)}.$$
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\end{conjecture}
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When $L(A,1)\neq 0$, work of Kolyvagin and Logachev
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\cite{kolyvagin-logachev:finiteness,kolyvagin-logachev:totallyreal}
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implies that $A(\Q)$, $\Adual(\Q)$, and $\Sha(A)$ are all finite,
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so the quantities appearing in the above formula make sense.
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Here $c_p=\#\Phi_{A,p}(\F_p)$, the positive real number~$\Omega_A$ is
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the measure of~$A(\R)$ with respect to a basis of differentials on
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the N\'eron model of~$A$, and~$\Adual$ is the abelian variety
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dual of~$A$.
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The algorithms described below enable us in my cases to compute the
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conjectural order of $\Sha(A)$. However, for question of visibility,
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we instead need to compute the order of $\Sha(\Adual)$. This is no
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different because the Cassels-Tate pairing implies that
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$\#\Sha(A) = \#\Sha(\Adual)$.
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\subsection{Modular symbols}\label{modsym}
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It is not possible to compute very much about $J_0(N)$ without
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modular symbols, which provide a finite presentation for the homology
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group $H_1(X_0(N),\Z)$ in terms of paths between elements of
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$\P^1(\Q) = \Q\union \{\infty\}$.
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The \defn{modular symbol} defined by a pair $\alpha,\beta\in\P^1(\Q)$
391
is denoted $\{\alpha,\beta\}$. This modular symbol should be viewed as
392
the homology class, relative to the cusps,
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of a geodesic path from~$\alpha$ to~$\beta$ in $\h^*$.
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The homology group relative to the cusps is a slight enlargement
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of the usual homology group, in that
396
we allow paths with endpoints in $\P^1(\Q)$ instead of restricting
397
to closed loops.
398
We declare that modular symbols satisfy
399
the following homology relations:
400
if $\alpha,\beta,\gamma \in \Q\union\{\infty\}$, then
401
$$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0.$$
402
Furthermore, the space of modular symbols is torsion free, so, e.g.,
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$\{\alpha,\alpha\} = 0$ and
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$\{\alpha,\beta\} = -\{\beta,\alpha\}$.
405
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Denote by~$\sM_2$ the free abelian group with basis the set of
407
symbols $\{\alpha,\beta\}$ modulo the three-term homology relations
408
above and modulo any torsion.
409
There is a left action of $\GL_2(\Q)$ on $\sM_2$, whereby
410
a matrix~$g$ acts by
411
$$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$
412
and~$g$ acts on~$\alpha$ and~$\beta$ by a linear fractional
413
transformation.
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The space $\sM_2(N)$ of \defn{modular symbols for $\Gamma_0(N)$}
415
is the quotient of $\sM_2$ by the submodule
416
generated by the infinitely many elements
417
of the form $x - g(x)$, for~$x$ in ~$\sM_2$
418
and~$g$ in $\Gamma_0(N)$, and modulo any torsion.
419
A \defn{modular symbol for $\Gamma_0(N)$} is an element of
420
this space. We frequently denote the equivalence
421
class that defines a modular symbol by giving a
422
representative element.
423
424
In \cite{manin:parabolic}, Manin proved that there is
425
a natural injection $H_1(X_0(N),\Z)\hookrightarrow \sM_2(N)$.
426
The image of $H_1(X_0(N),\Z)$ in $\sM_2(N)$ can be identified as follows.
427
Let $\sB_2(N)$ denote the free abelian group whose basis is the finite set
428
$\Gamma_0(N)\backslash \P^1(\Q)$.
429
The \defn{boundary map} $\delta: \sM_2(N)\ra \sB_2(N)$
430
sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$
431
denotes the basis element of $\sB_2(N)$ corresponding to $\beta\in\P^1(\Q)$.
432
The kernel $\sS_2(N)$ of~$\delta$ is the subspace of
433
\defn{cuspidal} modular symbols.
434
An element of $\sS_2(N)$ can be thought of as a linear
435
combination of paths
436
in $\h^*$ whose endpoints are cusps, and whose images in $X_0(N)$
437
are a linear combination of loops.
438
We thus obtain a canonical isomorphism $\varphi:\sS_2(N)\ra H_1(X_0(N),\Z)$.
439
440
Part of the utility of modular symbols comes from the classical Abel-Jacobi
441
theorem, which allows us to view $J_0(N)(\C)$ as the quotient
442
$\C^g/H_1(X_0(N),\Z)$,
443
where $H_1(X_0(N),\Z)$ is embedded in
444
$\C^d\ncisom\Hom(S_2(\Gamma_0(N)),\C)$ using the integration pairing.
445
Thus modular symbols give an explicit description of $J_0(N)(\C)$
446
and of its constituent parts as modules over the Hecke algebra.
447
We can also compute Hecke operators using modular symbols.
448
449
For further introductory remarks on modular symbols, see~\cite{stein:modsyms},
450
and for detailed instructions as to how to compute the space of modular symbols
451
and the action of Hecke operators on it, see~\cite{cremona:algs}.
452
453
\subsection{Computing with quotients and subvarieties of $J_0(N)$}
454
First, we describe how to enumerate the newforms of level~$N$. Then
455
we define the modular degree, whose square annihilates the visible part
456
of~$\Sha$. Finally, we describe how to intersect abelian subvarieties
457
of $J_0(N)$.
458
459
\subsubsection{Enumerating quotients}
460
Let $H_1(X_0(N),\Z)^+$ denote the $+1$-eigenspace for the action
461
of the involution induced by complex conjugation.
462
We list all newforms of a given level~$N$ by decomposing the new
463
subspace of $H_1(X_0(N),\Q)^+$ under the action of the the
464
Hecke operators. First we compute the characteristic polynomial of~$T_2$,
465
and use it to break up the full space. We apply this process
466
recursively with $T_3, T_5, \ldots$ until either we have exceeded the
467
bound coming from~\cite{sturm:cong}, or we have found a Hecke
468
operator~$T_n$ whose characteristic polynomial is irreducible. After
469
computing the decomposition, we order the newforms in a way that
470
extends the systematic ordering in~\cite{cremona:algs}: First sort by
471
dimension, with smallest dimension first; within each dimension, sort
472
in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$,
473
$++-$, $+-+$, $+--$, $-++$, etc. When two forms have the same sign
474
sequence, order by $|\Tr(a_p)|$ with ties broken by taking the
475
positive trace first.
476
477
We denote a Galois conjugacy class of newforms by a bold symbol such
478
as $\mathbf{389E}$, which consists of the level and the isogeny class,
479
where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second,
480
and so on.
481
482
As discussed in \cite[pg.~5]{cremona:algs}, for certain small levels
483
the above ordering when restricted to elliptic curves does not agree
484
with the ordering used in Cremona's tables. For example, in the
485
present paper our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$.
486
487
\subsubsection{The modular degree\label{modpolar}}
488
A \defn{polarization}~$\lambda$ of an abelian variety~$A$ over~$\Q$ is an isogeny
489
$\lambda:A\ra \Adual$ such that $\lambda_{\Qbar}$
490
arises from an ample invertible sheaf on $A_{\Qbar}$ (see, e.g.,
491
\cite[\S13]{milne:abvars}).
492
Since $J_0(N)$ is a Jacobian, it possesses a canonical
493
polarization arising from the $\theta$-divisor, and this
494
polarization induces the \defn{modular polarization}
495
$\theta: \Adual\ra A$ of $\Adual$.
496
$$=3pc{ 497 {\Adual}{^(->}[r]^{\pi^{\vee}\qquad}\ar[dr]^{\theta} & 498 J_0(N)^{\vee} \isom J_0(N)\,\,\,{->>}[d]^{\pi}\\ 499 &A.}$$
500
If we view $\Adual$ as an abelian subvariety of $J_0(N)$, then the
501
kernel of~$\theta$ is the intersection of $\Adual$ with $I J_0(N)$; thus
502
the kernel of $\theta$ measures intersections between $\Adual$ and other
503
factors of $J_0(N)$.
504
\begin{definition}[Modular degree]
505
The \defn{modular degree} $m_A$ of $A$ is
506
$\sqrt{\deg(\theta)}$.
507
\end{definition}
508
\noindent{}By \cite[Thm.~13.3]{milne:abvars}, $\deg(\theta)$ is a perfect
509
square, so $m_A$ is an integer.
510
For an algorithm to compute $m_A$, see~\cite{kohel-stein:ants4}.
511
512
The modular degree is of interest because its square
513
annihilates the visible cohomology classes.
514
\begin{proposition}
515
$\displaystyle \Vis_{J_0(N)}(H^1(\Q,\Adual))\subset H^1(\Q,\Adual)[m_A^2]$
516
\end{proposition}
517
\begin{proof}
518
Let $\delta$ be the composite map
519
$\Adual \ra J_0(N)\ra A$. There is a map $\hat{\delta}:A\ra \Adual$
520
such that $\hat{\delta}\circ\delta$ is multiplication by $\deg(\delta)=m_A^2$.
521
Thus
522
$\Ker(H^1(\Q,\Adual)\ra H^1(\Q,J_0(N)))$
523
is contained in $H^1(\Q,\Adual)[m_A^2]$.
524
\end{proof}
525
\begin{remark}
526
When~$A$ has dimension one, the visible part of $H^1(\Q,\Adual)$ is contained
527
in $H^1(\Q,A)[m_A]$. Is this true for~$A$ of all dimensions?
528
\end{remark}
529
530
\subsubsection{Intersecting complex tori}\label{sec:intersect}
531
Consider a complex torus $J=V/\Lambda$, and let
532
$A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ be subtori whose
533
intersection $A\intersect B$ is finite.
534
Here $V_A$ and $V_B$ are subspaces of~$V$ and $\Lambda_A$ and $\Lambda_B$
535
are submodules of~$\Lambda$.
536
\begin{proposition}\label{prop:intersection}
537
There is a natural isomorphism of groups
538
$$A\intersect B \isom 539 \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$
540
\end{proposition}
541
\begin{proof}
542
There is an exact sequence
543
$$0\ra A\intersect B \ra A \oplus B \ra J.$$
544
Consider the diagram
545
$$\xymatrix{ 546 & {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]& 547 {\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\ 548 & {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\ 549 {A\intersect B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B).}$$
550
The snake lemma\index{Snake lemma} gives an exact sequence
551
$$0 \ra 552 A\intersect B \ra 553 \Lambda/(\Lambda_A + \Lambda_B) \ra 554 V/(V_A+V_B).$$
555
Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion
556
part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$.
557
No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could
558
map to~$0$, because if it did then $A\intersect B$ would not
559
be finite. The lemma follows.
560
\end{proof}
561
562
The following formula for the intersection of~$n$
563
subtori is obtained in a similar way.
564
\begin{proposition}
565
For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of
566
$J=V/\Lambda$, and assume that each pairwise intersection
567
$A_i \intersect A_j$ is finite.
568
Then
569
$$A_1\intersect \cdots \intersect A_n 570 \isom 571 \left(\frac{\Lambda\oplus \cdots \oplus \Lambda} 572 {f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right),$$
573
where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.
574
\end{proposition}
575
576
577
\subsection{Computing the conjectural order of $\BigSha(A)$}
578
In this section, we describe how in many cases we can compute the
579
conjectural order of $\Sha(A)$ when $L(A,1)\neq 0$, at least up to a
580
power of~$2$.
581
582
In Section~\ref{sec:torsion}, we bound $\#A(\Q)$ and $\#\Adual(\Q)$.
583
We compute each $c_p$ in Section~\ref{sec:tamagawa},
584
for each~$p$ with $p\mid\mid N$. When $p^2\mid{}N$, it is
585
possible to bound $c_p$; see, e.g., \cite[Cor.~15.2.1]{silverman:aec}
586
where one finds that when $\dim A=1$ and $p^2\mid N$, we have $c_p\leq 4$.
587
In Section~\ref{sec:bsdratio}, we use modular symbols to compute
588
the rational number $L(A,1)/\Omega_A$, up to a bounded Manin constant.
589
590
\subsubsection{Torsion subgroup}\label{sec:torsion}
591
We obtain an upper bound on $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$
592
as follows.
593
The characteristic polynomial $\chi_p(X)$ of the Hecke operator~$T_p$
594
acting on~$A$ is a monic polynomial having integer coefficients
595
and degree equal to the dimension of~$A$.
596
\begin{proposition}
597
Both $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$ divide
598
$$\gcd \{ \chi_p(p+1) : (p,2N)=1,\, \text{\rm p prime} \}.$$
599
\end{proposition}
600
\begin{proof}
601
Use the Eichler-Shimura relation and that for primes~$p$ for which $p\nmid 2N$ the maps
602
$A(\Q)_{\tor} \ra \tilde{A}(\F_p)$ and
603
$\Adual(\Q)_{\tor} \ra \tilde{A}^{\vee}(\F_p)$
604
are both injective, and that
605
$\#\Adual(\F_p)_{\tor}=\#\tilde{A}^{\vee}(\F_p)$.
606
\end{proof}
607
608
The difference of two cusps $\alp,\beta \in{} X_0(N)$ defines
609
a point $(\alp)-(\beta) \in J_0(N)(\C)$. Manin observed
610
in \cite{manin:parabolic} that $(0)-(\infty)$ is rational.
611
The order of the image of $(0)-(\infty)$ in $A(\Q)$ can be computed as follows.
612
Let
613
$$V=\Hom(S_2(\Gamma_0(N)),\C),$$
614
and $V_I = \Hom(S_2(\Gamma_0(N))[I],\C)$.
615
The integration pairing $\langle f, \gamma \rangle = 2\pi i \int_\gamma f(z)dz$
616
between homology and cusp forms gives rise to a map $P: H_1(X_0(N),\Q)\ra V_I$.
617
By the Abel-Jacobi theory (see, e.g., \cite[Thm~IV.2.2]{lang:modular}),
618
$A(\C) \isom V_I/P(H_1(X_0(N),\Z))$.
619
\begin{proposition}
620
The order of the image of $(\alp)-(\beta)$ in $A(\C)$
621
equals the order of the image of the modular symbol $\{\alp,\beta\}$
622
in $P(H_1(X_0(N),\Q))/P(H_1(X_0(N),\Z)).$
623
\end{proposition}
624
The quotient appearing in the proposition can be computed algebraically
625
by replacing~$P$ by a map with the same kernel as~$P$. Such a map can
626
be computed using the Hecke operators (see \cite[\S3.7]{stein:phd}).
627
628
\subsubsection{Tamagawa numbers}\label{sec:tamagawa}
629
Suppose~$p$ is a prime that exactly divides~$N$ and let $\Phi_{A,p}$
630
denote the component group of~$A$ at~$p$.
631
We have an exact sequence,
632
$$0\ra \cA_{\Fp}^0\ra \cA_{\Fp} \ra \Phi_{A,p}\ra 0,$$
633
where $\cA_{\Fp}$ is the closed fiber of the N\'eron model of~$A$ over $\Z_p$ and $\cA_{\Fp}^0$
634
is the component of $\cA_{\Fp}$ that contains the identity.
635
A formula for $\#\Phi_{A,p}(\Fbar_p)$ and, up to a power of $2$,
636
for $\#\Phi_{A,p}(\F_p)$,
637
is given in \cite{kohel-stein:ants4} and \cite{stein:compgroup}.
638
639
\subsubsection{Rational part of the special value}\label{sec:bsdratio}
640
As in Section~\ref{sec:torsion},
641
let $P : H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N))[I],\C)$
642
be the map induced by integration.
643
Let $P(H_1(X_0(N),\Z))^+$ denote the $+1$-eigenspace for the action
644
of the involution induced by complex conjugation on the image of~$P$.
645
646
\begin{theorem}\label{thm:ratpart}
647
$$\frac{L(A,1)}{\Omega_{A}} 648 = [P(H_1(X_0(N),\Z))^+ : P(\T\{0,\infty\})]/(c_\infty\cdot c_A),$$
649
where $c_\infty$ is the number of components of $A(\R)$ and $c_A$
650
is the Manin constant of~$A$, as defined below.
651
\end{theorem}
652
In order to define the Manin constant of~$A$,
653
let $\cA$ denote the N\'eron model of~$A$ over~$\Z$.
654
\begin{definition}[Manin constant]
655
The \defn{Manin constant}~$c_A$ of~$A$ is the index
656
$$c_A := [S_2(\Gamma_0(N);\Z)[I]:H^0(\cA,\Omega_{\cA/\Z})].$$
657
\end{definition}
658
In the definition, we have implicitly mapped $H^0(\cA,\Omega_{\cA/\Z})$ into
659
$S_2(\Gamma_0(N);\Q)$ using the composition of the following maps:
660
$$H^0(\cA,\Omega_{\cA/\Z}) \ra 661 H^0(\cJ,\Omega_{\cJ/\Z})[I] \ra 662 H^0(J,\Omega_{J/\Q})[I] \ra 663 S_2(\Gamma_0(N);\Q)[I].$$
664
For a discussion of why $H^0(\cA,\Omega_{\cA/\Z})$ is in fact
665
contained in $S_2(\Gamma_0(N);\Z)[I]$, see \cite{agashe-stein:manin}.
666
\begin{theorem}
667
If $\ell \mid c_A$ then $\ell^2 \mid 4N$.
668
\end{theorem}
669
\begin{proof}
670
See~\cite[\S4]{mazur:rational} when~$A$ has dimension~$1$,
671
and \cite{agashe-stein:manin} in general.
672
\end{proof}
673
674
We now give the proof of Theorem~\ref{thm:ratpart}.
675
\begin{proof}[Proof of Theorem~\ref{thm:ratpart}]
676
Let $H=H_1(X_0(N),\Z)$ and $S=S_2(\Gamma_0(N))$.
677
There is a perfect pairing $\T \cross S \ra \Z$ given by
678
$\langle T_n, f\rangle = a_n(f)$, which
679
induces a canonical isomorphism of rings $\T\isom \Hom_\Z(S,\Z)$,
680
where $\Hom_\Z(S,\Z)$ is a ring under multiplication of functions.
681
The subring $W=\Hom_\Z(S[I],\Z)$ of $\Hom_\Z(S[I],\R)$ is
682
isomorphic to $\T/I$, since $S[I]$ is saturated in~$S$.
683
Thus
684
\begin{eqnarray*}
685
[W : P(\{0,\infty\})W] &=& [W:P(\T \{0,\infty\})]\\
686
&=& [W:P(H)^+] \cdot [P(H)^+ : P(\T \{0,\infty\})].
687
\end{eqnarray*}
688
To complete the proof, observe that
689
that $\Omega_A = [W:P(H)^+] \cdot c_\infty\cdot c_A$ and observe
690
that multiplication by $P(\{0,\infty\})$ has determinant
691
$\prod_{i=1}^d 2\pi i \int_{\{0,\infty\}} f^{(i)} = \pm L(A,1)$.
692
\end{proof}
693
694
\subsection{Emerton's work}
695
When~$N$ is prime, M.~Emerton has proved in \cite{emerton:myconj}
696
that $\#A_f(\Q)$ and $c_p(A_f)$ divide the numerator of $(N-1)/12$.
697
698
\section{Visibility tables}\label{sec:tables}
699
The tables in this section guide and motivate the conjectures
700
and questions of Section~\ref{sec:conj}.
701
702
In {\bf Table~\ref{table:invisible}}, we list each of the $8$ invisible odd
703
Shafarevich-Tate groups found in \cite{cremona-mazur}, and
704
prove\footnote{This computation is currently only partially complete.} that
705
they are visible in some $J_0(Nq)$.
706
707
{\bf Table~\ref{table:prime}} lists every quotient $A_f$ of $J_0(p)$
708
with $p\leq 2593$ and $L(A_f,1)\neq 0$ such that the BSD conjecture
709
predicts that $\#\Sha(\Adual_f)$ is divisible by an odd
710
prime. In addition, the table contains data that can frequently be
711
used in conjuction with Theorem~\ref{thm:shaexists} to deduce that
712
there are visible elements of $\Sha(\Adual_f)$. When the {\bf B}
713
column is labeled NONE then there is definitely nothing in
714
$\Sha(\Adual_f)$ of the predicted order. When the {\bf B} column
715
contains an elliptic curve, its rank has been computed and is~$2$, so
716
there are visible elements of $\Sha(\Adual_f)$. When the {\bf B}
717
column contains an abelian variety of dimension greater than~$1$, we
718
have verified that $L(B,1)=0$, so the BSD conjecture predicts that
719
$B(\Q)$ is infinite; however, we have not proved that $B(\Q)$ is
720
infinite. If we assume that $B(\Q)$ is infinite, it follows in these cases that
721
$\Sha(A)$ is visible in $J_0(p)$. Note that~$B$ has rank~$2$ over the
722
Hecke algebra here, so the results of \cite{gross-zagier} say
723
nothing about $B(\Q)$.
724
725
{\bf Table~\ref{table:prime2}} continues the computations of Table~\ref{table:prime}
726
up to level $5647$. For each prime~$p$ between $2609$ and $5674$, we computed each
727
factor $A$ such that $L(A,1)\neq 0$ and the odd part of
728
$\#\Shaan(A)$ is nontrivial. We then found all factors~$B$ such that
729
$L(B,1)=0$ and there is a mod~$\ell$ congruence between~$A$ and~$B$,
730
where $\ell\mid \#\Shaan(A)$. The column labeled~$N$ gives the level,
731
the column labeled $d(A)$ gives the dimension of~$A$, the column labeled
732
$d(B)$ gives the dimension of~$B$, and the column labeled cong'' gives
733
the odd part of $\gcd(\# A\intersect B, \#\Shaan(A))$.
734
735
{\bf Table~\ref{table:mordell-weil}}
736
lists every quotient $A_f$ of $J_0(N)$ with $N\leq 1642$ such
737
that $L(A_f,1)=0$ but the sign in the functional equation for~$f$ is $+1$.
738
For each such $A_f$, we looked for an abelian variety~$B$ such that~$B$
739
has rank~$0$ and $A_f^{\vee}$ probably gives rise to odd visible
740
elements of $\Sha(B)$. This table contains initial data towards the idea of constructing
741
points on high-rank abelian varieties by constructing visible elements
742
of Shafarevich-Tate groups using, e.g., Euler system methods.
743
For example, to prove that $A=\mathbf{1061B}$ really has positive rank,
744
we consider the variety $B=\mathbf{1061D}$.
745
To prove that $A(\Q)\neq 0$, it suffices to construct an appropriate element of $\Sha(B)$
746
and show that this element is visible in $A+B\subset J_0(1061)$.
747
748
{\bf Table~\ref{table:motive}} suggests a first tenuous step towards a
749
computational theory of motives attached to modular forms of weight
750
greater than two. This table is organized like
751
Table~\ref{table:prime}, except that the abelian varieties are
752
replaced by motives attached to weight~$4$ modular forms. For
753
example, at prime level~$127$ there is a $17$-dimensional motive $\cM$
754
such that $\Sha(\cM)(2)$ seems to contain elements of order~$43$. The
755
computations used to suggest this conclusion were carried out using
756
algorithms for higher weight modular symbols as described in
757
\cite{merel:1585}, \cite{stein:phd}, and \cite{stein-verrill:periods}.
758
759
760
\subsection{Odd invisible $\BigSha$ in \cite{cremona-mazur}}
761
\label{table:invisible}
762
$$763 \begin{array}{lccll} 764 \mbox{\rm\bf E}&\#\Sha(E)& \text{mod deg}(E) & \mbox{\rm\bf F} 765 & \text{Where \Sha(E) is visible}\\ 766 & & & & \vspace{-3ex} \\ 767 \mbox{\rm\bf 2849A}& 3^2 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 3\cdot 2849}\\ 768 \mbox{\rm\bf 3364C}& 7^2 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& \text{visible using a 3-dimensional F at level 3364}\\ 769 \mbox{\rm\bf 4229A}& 3^2 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& \text{not visible at level 4299,}\\ 770 &&&&\text{???}\\ 771 \mbox{\rm\bf 4343B}& 3^2 &2^4\cdot1583 &\mbox{\rm\bf NONE}& ???\\ 772 \mbox{\rm\bf 4914N}& 3^2 &2^4\cdot 3^5 &\mbox{\rm\bf none}& ??? \\ 773 \mbox{\rm\bf 5054C}& 3^2 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& ???\\ 774 \mbox{\rm\bf 5073D}& 3^2 &2^5\cdot 3\cdot 5\cdot7\cdot23 775 &\mbox{\rm\bf none}& ???\\ 776 \mbox{\rm\bf 5389A}& 3^2 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 7\cdot 5389}\\ 777 \end{array} 778$$
779
780
\comment{\subsubsection*{Remarks}
781
The elliptic curve~$E$ denoted {\bf 3364C} is labeled none'' because there is no
782
{\em elliptic curve} that satisfies an appropriate $7$-congruence with
783
{\bf 3364C}. However, the modular degree is divisible by~$7$, so there
784
must be some abelian subvariety that satisfies a $7$-congruence with~$E$.
785
Computing, we find a $3$-dimensional abelian variety~$A$ such that $T_2$, $T_3$,
786
and $T_5$ satisfies the polynomials $x^3$, $x^3 + 5x^2 + 6x + 1$, and
787
$x^3 + 5x^2 + 6x + 1$ on~$A$, respectively. Furthermore, $L(A,1)=0$, so
788
the BSD conjecture strongly suggests that there are elements of $\Sha(E)$
789
of order~$7$ that are visible in $E+A \subset J_0(3364)$.
790
}
791
792
\newpage
793
\subsection{Visibility of $\BigSha$ at prime level}\label{table:prime}
794
The entries in the columns mod deg'' and $\Shaan$'' are only really
795
the odd parts of mod deg'' and $\Shaan$''. Theorem~\ref{thm:shaexists} does not
796
apply to the two entries marked with a $*$.\vspace{-.25ex}
797
{\small
798
$$799 \begin{array}{lccclcc} 800 \mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan(A) & \mbox{\rm mod deg}(A) & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} (B)\\ 801 & & & & & & \vspace{-3ex} \\ 802 \mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ 803 \mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ 804 \mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ 805 \mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ 806 \mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ 807 \mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ 808 \mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ 809 \mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ 810 \mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ 811 \mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ 812 \mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ 813 \mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ 814 \mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ 815 \mbox{\rm\bf 1531D}*& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ 816 \mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ 817 \mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ 818 \mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ 819 \mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ 820 \mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ 821 \mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ 822 \mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ 823 \mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ 824 \mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ 825 \mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ 826 \mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ 827 \mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ 828 \mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A,C}& 1,2 &3\cdot5^{2}, 5^2\cdot 61\\ 829 \mbox{\rm\bf 1933C}*& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ 830 \mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ 831 \mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ 832 \mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ 833 \mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ 834 \mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ 835 \mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ 836 \mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ 837 \mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ 838 \mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ 839 \mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\ 840 \mbox{\rm\bf 2213C}& 101 & 3^4 & ? & \mbox{\rm NONE} & & \\ 841 \mbox{\rm\bf 2239B}& 110 & 11^4 & 1 & \mbox{\rm NONE} & & \\ 842 \mbox{\rm\bf 2251E}& 99 & 37^2 & 37 & \mbox{\rm\bf 2251A} & 1 & 37\\ 843 \mbox{\rm\bf 2273C}& 105 & 7^2 & ? & \mbox{\rm NONE}& & \\ 844 \mbox{\rm\bf 2287B}& 109 & 71^2 & 1 & \mbox{\rm NONE}& & \\ 845 \mbox{\rm\bf 2293C}& 96 & 479^2& 479 & \mbox{\rm\bf 2293A} & 2 & 479\\ 846 \mbox{\rm\bf 2311B}& 110 & 5^2 & 1 & \mbox{\rm NONE}& & \\ 847 \mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ 848 \mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ 849 \mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ 850 \mbox{\rm\bf 2593C}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213&\mbox {\bf 2593A}& 4 851 & 67 \cdot 2213\\ 852 \end{array} 853$$
854
}
855
856
\subsection{More $\Sha$ at prime level}\label{table:prime2}
857
Only odd parts of $\Shaan$ and congruences are given.
858
Observe that $\Shaan$ is only visible roughly 10 percent of the time!
859
As the level gets large, we find that there is almost always some
860
nontrivial $\Sha$ in a large-dimensional factor of $J_0(p)$, and that
861
this $\Sha$ is invisible.
862
(Warning: In making this table, $53$ primes below $5647$ were not analyzed.)
863
{\tiny
864
$$865 \hspace{-6em}\begin{array}{lcccc|} 866 N & d(A) & \Shaan & d(B) & cong\\ 867 \mathbf{2609} & 127 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 868 \mathbf{2617} & 114 & 11^{2}\cdot19^{2} & 2 & 11\cdot19 \\ 869 \mathbf{2647} & 117 & 13^{2} & \text{NONE} & \\ 870 \mathbf{2659} & 123 & 53^{2} & \text{NONE} & \\ 871 \mathbf{2663} & 132 & 43^{2} & \text{NONE} & \\ 872 \mathbf{2671} & 122 & 37^{2} & \text{NONE} & \\ 873 \mathbf{2677} & 115 & 3^{2} & 1 & 3 \\ 874 \mathbf{2693} & 122 & 3^{4} & \text{NONE} & \\ 875 \mathbf{2699} & 125 & 19^{2} & \text{NONE} & \\ 876 \mathbf{2707} & 119 & 5^{2} & \text{NONE} & \\ 877 \mathbf{2713} & 118 & 19^{2} & \text{NONE} & \\ 878 \mathbf{2731} & 124 & 53^{2} & \text{NONE} & \\ 879 \mathbf{2749} & 124 & 7^{2} & \text{NONE} & \\ 880 \mathbf{2767} & 125 & 5^{2} & \text{NONE} & \\ 881 \mathbf{2789} & 136 & 83^{2} & \text{NONE} & \\ 882 \mathbf{2791} & 135 & 29^{2} & \text{NONE} & \\ 883 \mathbf{2797} & 119 & 11^{2} & 1 & 11 \\ 884 \mathbf{2819} & 138 & 13^{2} & \text{NONE} & \\ 885 \mathbf{2837} & 128 & 23^{2} & 1 & 23 \\ 886 \mathbf{2843} & 129 & 3^{6}\cdot587^{2} & \text{NONE} & \\ 887 \mathbf{2851} & 129 & 7^{2} & \text{NONE} & \\ 888 \mathbf{2861} & 133 & 11^{4}\cdot61^{2} & 2 & 11\cdot61 \\ 889 \mathbf{2879} & 148 & 97^{2} & \text{NONE} & \\ 890 \mathbf{2903} & 150 & 643^{2} & \text{NONE} & \\ 891 \mathbf{2939} & 150 & 17^{2}\cdot19^{2} & \text{NONE} & \\ 892 \mathbf{2953} & 127 & 29^{2} & 1 & 29 \\ 893 \mathbf{2963} & 134 & 5^{2}\cdot31^{2}\cdot61^{2} & 2 & 31\cdot61 894 \\ 895 \mathbf{2969} & 136 & 103^{2} & \text{NONE} & \\ 896 \mathbf{2999} & 161 & 1459^{2} & \text{NONE} & \\ 897 \mathbf{3001} & 132 & 3^{4} & \text{NONE} & \\ 898 \mathbf{3011} & 146 & 5^{2}\cdot101^{2} & \text{NONE} & \\ 899 \mathbf{3019} & 130 & 3259^{2} & 2 & 3259 \\ 900 \mathbf{3041} & 147 & 103^{2} & \text{NONE} & \\ 901 \mathbf{3067} & 134 & 5^{4} & \text{NONE} & \\ 902 \mathbf{3079} & 148 & 131^{2} & \text{NONE} & \\ 903 \mathbf{3083} & 141 & 179^{2} & \text{NONE} & \\ 904 \mathbf{3089} & 135 & 5^{2}\cdot131^{2} & 2 & 5\cdot131 \\ 905 \mathbf{3109} & 136 & 5^{2} & \text{NONE} & \\ 906 \mathbf{3119} & 164 & 11^{2}\cdot59^{2} & \text{NONE} & \\ 907 \mathbf{3181} & 144 & 43^{2} & \text{NONE} & \\ 908 \mathbf{3187} & 139 & 3^{4} & \text{NONE} & \\ 909 \mathbf{3191} & 167 & 53^{2} & \text{NONE} & \\ 910 \mathbf{3203} & 143 & 13^{2} & \text{NONE} & \\ 911 \mathbf{3221} & 149 & 7^{2}\cdot41^{2} & \text{NONE} & \\ 912 \mathbf{3229} & 142 & 3^{2} & \text{NONE} & \\ 913 \mathbf{3251} & 166 & 3^{4} & \text{NONE} & \\ 914 \mathbf{3257} & 143 & 13^{2} & \text{NONE} & \\ 915 \mathbf{3271} & 146 & 7^{4}\cdot43^{2}\cdot71^{2} & 3 & 916 7\cdot43\cdot71 \\ 917 \mathbf{3299} & 164 & 6131^{2} & \text{NONE} & \\ 918 \mathbf{3301} & 145 & 5^{2} & \text{NONE} & \\ 919 \mathbf{3319} & 158 & 5^{4} & \text{NONE} & \\ 920 \mathbf{3323} & 155 & 179^{2} & \text{NONE} & \\ 921 \mathbf{3329} & 157 & 83^{2} & \text{NONE} & \\ 922 \mathbf{3331} & 152 & 937^{2} & \text{NONE} & \\ 923 \mathbf{3343} & 148 & 7^{2}\cdot53^{2} & \text{NONE} & \\ 924 \mathbf{3347} & 150 & 139^{2} & \text{NONE} & \\ 925 \mathbf{3359} & 174 & 67^{4} & \text{NONE} & \\ 926 \mathbf{3371} & 159 & 1259^{2} & \text{NONE} & \\ 927 \mathbf{3391} & 159 & 29^{2} & \text{NONE} & \\ 928 \mathbf{3407} & 170 & 499^{2} & \text{NONE} & \\ 929 \mathbf{3433} & 148 & 5^{4}\cdot7^{2} & \text{NONE} & \\ 930 \mathbf{3449} & 168 & 107^{2} & \text{NONE} & \\ 931 \mathbf{3461} & 167 & 83^{2} & \text{NONE} & \\ 932 \mathbf{3463} & 151 & 199^{2} & 2 & 199 \\ 933 \mathbf{3467} & 162 & 5^{4} & \text{NONE} & \\ 934 \mathbf{3469} & 151 & 47^{2} & \text{NONE} & \\ 935 \mathbf{3491} & 168 & 67^{2} & \text{NONE} & \\ 936 \mathbf{3511} & 166 & 37^{2} & \text{NONE} & \\ 937 \mathbf{3527} & 179 & 659^{2} & \text{NONE} & \\ 938 \mathbf{3529} & 153 & 79^{2} & \text{NONE} & \\ 939 \mathbf{3533} & 164 & 3^{4} & \text{NONE} & \\ 940 \mathbf{3539} & 170 & 1871^{2} & \text{NONE} & \\ 941 \mathbf{3541} & 156 & 5^{4} & \text{NONE} & \\ 942 \mathbf{3557} & 156 & 229^{2} & \text{NONE} & \\ 943 \mathbf{3559} & 170 & 1109^{2} & \text{NONE} & \\ 944 \end{array} 945 \begin{array}{|lcccc|} 946 N & d(A) & \Shaan & d(B) & cong\\ 947 \mathbf{3571} & 163 & 67^{2} & \text{NONE} & \\ 948 \mathbf{3583} & 161 & 3319^{2} & 2 & 3319 \\ 949 \mathbf{3607} & 159 & 7^{4}\cdot19^{2} & \text{NONE} & \\ 950 \mathbf{3613} & 156 & 7^{2} & \text{NONE} & \\ 951 \mathbf{3617} & 165 & 3^{2} & \text{NONE} & \\ 952 \mathbf{3623} & 172 & 3^{6} & \text{NONE} & \\ 953 \mathbf{3631} & 172 & 433^{2} & \text{NONE} & \\ 954 \mathbf{3643} & 160 & 5^{2} & \text{NONE} & \\ 955 \mathbf{3659} & 181 & 3^{2}\cdot11^{4} & \text{NONE} & \\ 956 \mathbf{3671} & 193 & 509^{2} & \text{NONE} & \\ 957 \mathbf{3691} & 166 & 353^{2} & \text{NONE} & \\ 958 \mathbf{3701} & 174 & 3^{4}\cdot281^{2} & 2 & 3^{2}\cdot281 \\ 959 \mathbf{3709} & 164 & 3^{12} & \text{NONE} & \\ 960 \mathbf{3719} & 188 & 13^{2}\cdot977^{2} & \text{NONE} & \\ 961 \mathbf{3739} & 166 & 83^{2} & \text{NONE} & \\ 962 \mathbf{3761} & 176 & 677^{2} & \text{NONE} & \\ 963 \mathbf{3769} & 168 & 13^{2} & \text{NONE} & \\ 964 \mathbf{3779} & 187 & 73^{2}\cdot149^{2} & 1 & 73 \\ 965 \mathbf{3797} & 172 & 19^{2} & \text{NONE} & \\ 966 \mathbf{3803} & 171 & 2531^{2} & \text{NONE} & \\ 967 \mathbf{3821} & 182 & 307^{2} & \text{NONE} & \\ 968 \mathbf{3823} & 173 & 7^{2} & \text{NONE} & \\ 969 \mathbf{3863} & 191 & 11^{2}\cdot23^{2}\cdot311^{2} & 970 \text{NONE} & \\ 971 \mathbf{3907} & 168 & 3^{4} & \text{NONE} & \\ 972 \mathbf{3919} & 182 & 71^{2} & \text{NONE} & \\ 973 \mathbf{3929} & 185 & 877^{2} & \text{NONE} & \\ 974 \mathbf{3931} & 174 & 31^{2} & \text{NONE} & \\ 975 \mathbf{3943} & 173 & 2479319^{2} & 4 & 2479319 \\ 976 \mathbf{3967} & 180 & 3^{6}\cdot13^{2} & 1 & 3\cdot13 \\ 977 \mathbf{4007} & 195 & 7321^{2} & \text{NONE} & \\ 978 \mathbf{4013} & 176 & 61^{2} & \text{NONE} & \\ 979 \mathbf{4019} & 186 & 3^{4}\cdot5^{2}\cdot7^{4} & \text{NONE} & \\ 980 \mathbf{4021} & 182 & 5^{4}\cdot71^{2} & \text{NONE} & \\ 981 \mathbf{4027} & 174 & 29^{2}\cdot79^{2} & 2 & 29\cdot79 \\ 982 \mathbf{4049} & 186 & 5^{2}\cdot3491^{2} & \text{NONE} & \\ 983 \mathbf{4057} & 173 & 103^{2} & \text{NONE} & \\ 984 \mathbf{4079} & 212 & 5^{2}\cdot157^{2}\cdot179^{2} & 985 \text{NONE} & \\ 986 \mathbf{4091} & 203 & 7^{4} & \text{NONE} & \\ 987 \mathbf{4093} & 174 & 3^{2}\cdot89^{4} & 2 & 89^{2} \\ 988 \mathbf{4099} & 185 & 3^{4}\cdot19^{2} & \text{NONE} & \\ 989 \mathbf{4111} & 190 & 229^{2} & \text{NONE} & \\ 990 \mathbf{4139} & 188 & 29^{2}\cdot67^{2} & 1 & 67 \\ 991 \mathbf{4153} & 177 & 7^{2} & \text{NONE} & \\ 992 \mathbf{4157} & 193 & 373^{2} & \text{NONE} & \\ 993 \mathbf{4159} & 188 & 997^{2} & \text{NONE} & \\ 994 \mathbf{4177} & 183 & 3^{2}\cdot17^{2} & \text{NONE} & \\ 995 \mathbf{4217} & 186 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 996 \mathbf{4219} & 190 & 71^{2} & \text{NONE} & \\ 997 \mathbf{4229} & 1 & 3^{2} & \text{NONE} & \\ 998 \mathbf{4229} & 194 & 3^{4} & \text{NONE} & \\ 999 \mathbf{4231} & 201 & 3^{6} & \text{NONE} & \\ 1000 \mathbf{4253} & 184 & 3^{6}\cdot2843^{2} & 3 & 3^{3}\cdot2843 \\ 1001 \mathbf{4261} & 185 & 5^{2} & \text{NONE} & \\ 1002 \mathbf{4271} & 210 & 163^{2}\cdot853^{2} & \text{NONE} & \\ 1003 \mathbf{4273} & 183 & 181^{2} & \text{NONE} & \\ 1004 \mathbf{4283} & 198 & 683^{2} & \text{NONE} & \\ 1005 \mathbf{4289} & 205 & 8807^{2} & \text{NONE} & \\ 1006 \mathbf{4339} & 196 & 17^{2} & \text{NONE} & \\ 1007 \mathbf{4349} & 191 & 127^{2} & \text{NONE} & \\ 1008 \mathbf{4357} & 187 & 7^{2}\cdot13^{2}\cdot17^{2} & 1 & 7\cdot13 1009 \\ 1010 \mathbf{4373} & 199 & 3^{12}\cdot29^{2} & \text{NONE} & \\ 1011 \mathbf{4391} & 222 & 5^{4}\cdot372037^{2} & \text{NONE} & \\ 1012 \mathbf{4409} & 200 & 157^{2} & \text{NONE} & \\ 1013 \mathbf{4421} & 206 & 1523^{2} & \text{NONE} & \\ 1014 \mathbf{4423} & 200 & 3^{6}\cdot587^{2} & \text{NONE} & \\ 1015 \mathbf{4441} & 198 & 59^{2}\cdot101^{2} & \text{NONE} & \\ 1016 \mathbf{4451} & 213 & 809^{2} & \text{NONE} & \\ 1017 \mathbf{4457} & 199 & 337^{2} & \text{NONE} & \\ 1018 \mathbf{4463} & 213 & 8951^{2} & \text{NONE} & \\ 1019 \mathbf{4483} & 193 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 1020 \mathbf{4517} & 201 & 181^{2} & \text{NONE} & \\ 1021 \mathbf{4519} & 202 & 2503^{2} & \text{NONE} & \\ 1022 \mathbf{4547} & 205 & 73^{2} & 1 & 73 \\ 1023 \mathbf{4549} & 203 & 19^{2}\cdot53^{2} & \text{NONE} & \\ 1024 \mathbf{4591} & 215 & 6317^{2} & \text{NONE} & \\ 1025 \end{array} 1026 \begin{array}{|lcccc} 1027 N & d(A) & \Shaan & d(B) & cong\\ 1028 \mathbf{4597} & 200 & 7^{2}\cdot17^{2} & \text{NONE} & \\ 1029 \mathbf{4603} & 198 & 829^{2} & \text{NONE} & \\ 1030 \mathbf{4621} & 196 & 13^{2} & \text{NONE} & \\ 1031 \mathbf{4639} & 218 & 89^{2} & \text{NONE} & \\ 1032 \mathbf{4649} & 215 & 751^{2} & \text{NONE} & \\ 1033 \mathbf{4651} & 210 & 13^{4} & \text{NONE} & \\ 1034 \mathbf{4673} & 207 & 11^{2}\cdot197^{2} & \text{NONE} & \\ 1035 \mathbf{4691} & 216 & 43^{2} & \text{NONE} & \\ 1036 \mathbf{4729} & 204 & 673^{2} & \text{NONE} & \\ 1037 \mathbf{4733} & 210 & 17^{2} & \text{NONE} & \\ 1038 \mathbf{4783} & 210 & 797^{2} & \text{NONE} & \\ 1039 \mathbf{4789} & 206 & 13^{4} & \text{NONE} & \\ 1040 \mathbf{4799} & 230 & 3^{2}\cdot7^{2}\cdot12203^{2} & 1 & 3\cdot7 1041 \\ 1042 \mathbf{4801} & 213 & 60271^{2} & \text{NONE} & \\ 1043 \mathbf{4813} & 207 & 3^{2}\cdot6883^{2} & \text{NONE} & \\ 1044 \mathbf{4817} & 214 & 283^{2} & \text{NONE} & \\ 1045 \mathbf{4831} & 217 & 1151^{2} & \text{NONE} & \\ 1046 \mathbf{4861} & 216 & 204749^{2} & \text{NONE} & \\ 1047 \mathbf{4877} & 219 & 3^{4}\cdot103^{2} & \text{NONE} & \\ 1048 \mathbf{4931} & 240 & 17^{2}\cdot37^{2}\cdot43^{2} & 1049 \text{NONE} & \\ 1050 \mathbf{4933} & 211 & 239^{2} & \text{NONE} & \\ 1051 \mathbf{4957} & 212 & 5^{2} & \text{NONE} & \\ 1052 \mathbf{4967} & 236 & 7^{2}\cdot53881^{2} & \text{NONE} & \\ 1053 \mathbf{4969} & 220 & 11^{4} & \text{NONE} & \\ 1054 \mathbf{4973} & 223 & 5^{2}\cdot11^{2} & \text{NONE} & \\ 1055 \mathbf{4993} & 215 & 4013^{2} & \text{NONE} & \\ 1056 \mathbf{4999} & 224 & 985121^{2} & \text{NONE} & \\ 1057 \mathbf{5003} & 220 & 97^{2}\cdot1861^{2} & 3 & 97\cdot1861 \\ 1058 \mathbf{5009} & 223 & 23^{2}\cdot977^{2} & \text{NONE} & \\ 1059 \mathbf{5011} & 229 & 11^{4} & \text{NONE} & \\ 1060 \mathbf{5021} & 225 & 1609^{2} & \text{NONE} & \\ 1061 \mathbf{5023} & 221 & 51431^{2} & \text{NONE} & \\ 1062 \mathbf{5039} & 251 & 166363^{2} & \text{NONE} & \\ 1063 \mathbf{5051} & 239 & 13^{2}\cdot2633^{2} & \text{NONE} & \\ 1064 \mathbf{5059} & 229 & 5^{2}\cdot13^{2}\cdot31^{2} & \text{NONE} & \\ 1065 \mathbf{5077} & 216 & 283^{2} & \text{NONE} & \\ 1066 \mathbf{5081} & 240 & 19^{2}\cdot149^{2} & \text{NONE} & \\ 1067 \mathbf{5099} & 251 & 7^{4}\cdot11^{2}\cdot461^{2} & 1068 \text{NONE} & \\ 1069 \mathbf{5113} & 223 & 19^{2}\cdot61^{2} & \text{NONE} & \\ 1070 \mathbf{5119} & 232 & 53^{2}\cdot103^{2} & \text{NONE} & \\ 1071 \mathbf{5153} & 223 & 3^{4}\cdot41^{2} & \text{NONE} & \\ 1072 \mathbf{5167} & 231 & 367^{2} & \text{NONE} & \\ 1073 \mathbf{5171} & 249 & 73^{2}\cdot773^{2} & 1 & 73 \\ 1074 \mathbf{5179} & 226 & 7^{2}\cdot13^{2} & \text{NONE} & \\ 1075 \mathbf{5189} & 240 & 83^{2} & \text{NONE} & \\ 1076 \mathbf{5197} & 223 & 37^{2} & \text{NONE} & \\ 1077 \mathbf{5209} & 227 & 181^{2}\cdot1471^{2} & \text{NONE} & \\ 1078 \mathbf{5227} & 232 & 3^{2}\cdot7717^{2} & \text{NONE} & \\ 1079 \mathbf{5231} & 255 & 4507^{2} & \text{NONE} & \\ 1080 \mathbf{5233} & 223 & 163^{2} & \text{NONE} & \\ 1081 \mathbf{5237} & 229 & 7^{2} & \text{NONE} & \\ 1082 \mathbf{5261} & 239 & 24103^{2} & \text{NONE} & \\ 1083 \mathbf{5273} & 227 & 17389^{2} & \text{NONE} & \\ 1084 \mathbf{5279} & 263 & 120431^{2} & \text{NONE} & \\ 1085 \mathbf{5281} & 232 & 67^{2} & \text{NONE} & \\ 1086 \mathbf{5297} & 238 & 397^{2} & \text{NONE} & \\ 1087 \mathbf{5303} & 247 & 13^{2}\cdot73^{2}\cdot15467^{2} & 1088 \text{NONE} & \\ 1089 \mathbf{5309} & 247 & 1822693^{2} & \text{NONE} & \\ 1090 \mathbf{5323} & 233 & 3^{4}\cdot120563^{2} & 3 & 120563 \\ 1091 \mathbf{5333} & 237 & 967^{2} & \text{NONE} & \\ 1092 \mathbf{5347} & 231 & 3643^{2} & \text{NONE} & \\ 1093 \mathbf{5501} & 250 & 163^{2} & \text{NONE} & \\ 1094 \mathbf{5503} & 241 & 7^{2}\cdot17^{2} & \text{NONE} & \\ 1095 \mathbf{5507} & 252 & 103^{2}\cdot233^{2} & \text{NONE} & \\ 1096 \mathbf{5519} & 278 & 61^{2}\cdot211469^{2} & \text{NONE} & \\ 1097 \mathbf{5521} & 244 & 5^{4} & \text{NONE} & \\ 1098 \mathbf{5531} & 253 & 977^{2} & \text{NONE} & \\ 1099 \mathbf{5563} & 246 & 3^{4}\cdot1213^{2} & \text{NONE} & \\ 1100 \mathbf{5569} & 239 & 3^{4}\cdot5^{2}\cdot13^{2} & \text{NONE} & \\ 1101 \mathbf{5573} & 247 & 9901^{2} & \text{NONE} & \\ 1102 \mathbf{5581} & 242 & 28927^{2} & \text{NONE} & \\ 1103 \mathbf{5591} & 282 & 3^{2}\cdot13^{4}\cdot1061^{2} & 1104 \text{NONE} & \\ 1105 \mathbf{5639} & 278 & 229717^{2} & \text{NONE} & \\ 1106 \mathbf{5641} & 244 & 41^{2}\cdot431^{2} & \text{NONE} & \\ 1107 \mathbf{5647} & 245 & 4463^{2} & \text{NONE} & \\ 1108 \end{array} 1109$$
1110
}
1111
1112
\subsection{Mordell-Weil groups of positive even rank and the $\BigSha$ they probably induce}
1113
\label{table:mordell-weil}
1114
1115
\noindent
1116
$$\begin{array}{lclccc} 1117 L(1)=0 & d & 1118 \hspace{-.5em}L(1)\neq 0 & 1119 \hspace{-.6em}d & 1120 \hspace{-.8em}\text{cong} & 1121 \hspace{-.7em} 1122 L(1)/\Omega \cdot 2^*\\ 1123 & & & & & \vspace{-2ex}\\ 1124 \mbox{\bf 389A} & 1 & \mbox{\bf 389E} & 20 & 5 & 25/97\\ 1125 \mbox{\bf 433A} & 1 & \mbox{\bf 433D} & 16 & 7 & 49/9\\ 1126 \mbox{\bf 446B} & 1 & \mbox{\bf 446F} & 8 & 11 & 121/3\\ 1127 \mbox{\bf 563A} & 1 & \mbox{\bf 563E} & 31 & 13 & 169/281\\ 1128 \mbox{\bf 571B} & 1 & \mbox{\bf 571D} & 2 & 3 & 9\\ 1129 \mbox{\bf 643A} & 1 & \text{NONE} & & & \\ 1130 \mbox{\bf 655A} & 1 & \mbox{\bf 655D} & 13 & 9 & 81\\ 1131 \mbox{\bf 664A} & 1 & \mbox{\bf 664F} & 8 & 5 & 25\\ 1132 \mbox{\bf 681C} & 1 & \mbox{\bf 681B} & 1 & 3 & 9\\ 1133 \mbox{\bf 707A} & 1 & \mbox{\bf 707G} & 15 & 13 & 169\\ 1134 \mbox{\bf 718B} & 1 & \mbox{\bf 718F} & 7 & 7 & 49\\ 1135 \mbox{\bf 794A} & 1 & \mbox{\bf 794G} & 12 & 11 & 121/3\\ 1136 \mbox{\bf 817A} & 1 & \mbox{\bf 817E} & 15 & 7 & 49/5\\ 1137 \mbox{\bf 916C} & 1 & \mbox{\bf 916G} & 9 & 11 & 121\\ 1138 \mbox{\bf 944E} & 1 & \mbox{\bf 944O} & 6 & 7 & 49\\ 1139 \mbox{\bf 997B} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ 1140 \mbox{\bf 997C} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ 1141 \mbox{\bf 1001C} & 1 & \mbox{\bf 1001F} & 3 & 3 & 27\\ 1142 \mbox{\bf 1001C} & 1 & \mbox{\bf 1001L} & 7 & 7 & 49\\ 1143 \mbox{\bf 1028A} & 1 & \mbox{\bf 1028E} & 14 & 11 & 3267\\ 1144 \mbox{\bf 1034A} & 1 & \text{NONE} & & & \\ 1145 \mbox{\bf 1041B} & 2 & \mbox{\bf 1041E} & 4 & 5 & 25\\ 1146 \mbox{\bf 1041B} & 2 & \mbox{\bf 1041J} & 13 & 25 & 625\\ 1147 \mbox{\bf 1058C} & 1 & \mbox{\bf 1058D} & 1 & 5 & 25\\ 1148 \mbox{\bf 1061B} & 2 & \mbox{\bf 1061D} & 46 & 151 & 22801/265\\ 1149 \mbox{\bf 1070A} & 1 & \mbox{\bf 1070M} & 7 & 15 & 75\\ 1150 \mbox{\bf 1073A} & 1 & \text{NONE} & & & \\ 1151 \mbox{\bf 1077A} & 1 & \mbox{\bf 1077J} & 15 & 9 & 81\\ 1152 \mbox{\bf 1088J} & 1 & \mbox{\bf 1088R} & 2 & 3 & 9\\ 1153 \mbox{\bf 1094A} & 1 & \mbox{\bf 1094F} & 13 & 11 & 121/3\\ 1154 \mbox{\bf 1102A} & 1 & \mbox{\bf 1102K} & 4 & 3 & 9\\ 1155 \mbox{\bf 1126A} & 1 & \mbox{\bf 1126F} & 11 & 11 & 121\\ 1156 \mbox{\bf 1132A} & 1 & \mbox{\bf 1132F} & 12 & 5 & 225\\ 1157 \mbox{\bf 1137A} & 1 & \mbox{\bf 1137C} & 14 & 9 & 81\\ 1158 \mbox{\bf 1141A} & 1 & \mbox{\bf 1141I} & 22 & 7 & 1524537/41\\ 1159 \end{array} 1160 \begin{array}{lclccc} 1161 & & & & & \\ 1162 & & & & & \vspace{-2ex}\\ 1163 \mbox{\bf 1143C} & 1 & \mbox{\bf 1143J} & 9 & 11 & 121\\ 1164 \mbox{\bf 1147A} & 1 & \mbox{\bf 1147H} & 23 & 5 & 225/19\\ 1165 \mbox{\bf 1171A} & 1 & \mbox{\bf 1171D} & 53 & 11 & 121/195\\ 1166 \mbox{\bf 1246C} & 1 & \mbox{\bf 1246B} & 1 & 5 & 25\\ 1167 \mbox{\bf 1309B} & 1 & \text{NONE} & & & \\ 1168 \mbox{\bf 1324A} & 1 & \mbox{\bf 1324E} & 14 & 9 & 6561\\ 1169 \mbox{\bf 1325E} & 1 & \mbox{\bf 1325T} & 11 & 9 & 2187\\ 1170 \mbox{\bf 1363B} & 2 & \mbox{\bf 1363F} & 25 & 31 & 961/5\\ 1171 \mbox{\bf 1431A} & 1 & \mbox{\bf 1431L} & 14 & 3 & 9\\ 1172 \mbox{\bf 1436A} & 1 & \text{NONE} & & & \\ 1173 \mbox{\bf 1443C} & 1 & \mbox{\bf 1443G} & 5 & 7 & 49\\ 1174 \mbox{\bf 1446A} & 1 & \mbox{\bf 1446N} & 7 & 3 & 9\\ 1175 \mbox{\bf 1466B} & 1 & \mbox{\bf 1466H} & 23 & 13 & \hspace{-2em}\mbox{{\tiny 4331806939187/367}}\\ 1176 \mbox{\bf 1477A} & 1 & \mbox{\bf 1477C} & 24 & 13 & 169\\ 1177 \mbox{\bf 1480A} & 1 & \mbox{\bf 1480G} & 5 & 7 & 49\\ 1178 \mbox{\bf 1483A} & 1 & \mbox{\bf 1483D} & 67 & 15 & 225/247\\ 1179 \mbox{\bf 1525C} & 1 & \mbox{\bf 1525O} & 16 & 7 & 49\\ 1180 \mbox{\bf 1531A} & 1 & \mbox{\bf 1531D} & 73 & 3 & 3/85\\ 1181 \mbox{\bf 1534B} & 1 & \mbox{\bf 1534J} & 6 & 3 & 3\\ 1182 \mbox{\bf 1567B} & 3 & \mbox{\bf 1567D} & 69 & 287 & 82369/261\\ 1183 \mbox{\bf 1570B} & 1 & \mbox{\bf 1570J} & 6 & 11 & 121\\ 1184 \mbox{\bf 1576A} & 1 & \mbox{\bf 1576E} & 14 & 11 & 121\\ 1185 \mbox{\bf 1591A} & 1 & \mbox{\bf 1591F} & 35 & 31 & 6727/19\\ 1186 \mbox{\bf 1594A} & 1 & \mbox{\bf 1594J} & 17 & 3 & 3370648239/19\\ 1187 \mbox{\bf 1608A} & 1 & \mbox{\bf 1608J} & 6 & 13 & 169\\ 1188 \mbox{\bf 1611D} & 1 & \mbox{\bf 1611O} & 11 & 9 & 81\\ 1189 \mbox{\bf 1613A} & 1 & \mbox{\bf 1613D} & 75 & 5 & 25/403\\ 1190 \mbox{\bf 1615A} & 1 & \mbox{\bf 1615J} & 13 & 9 & 102141\\ 1191 \mbox{\bf 1621A} & 1 & \mbox{\bf 1621C} & 70 & 17 & 289/135\\ 1192 \mbox{\bf 1627A} & 1 & \mbox{\bf 1627C} & 73 & 9 & 81/271\\ 1193 \mbox{\bf 1633A} & 3 & \mbox{\bf 1633D} & 27 & 189 & 35721\\ 1194 \mbox{\bf 1639B} & 1 & \mbox{\bf 1639G} & 34 & 17 & 680017/25\\ 1195 \mbox{\bf 1641B} & 1 & \mbox{\bf 1641J} & 24 & 23 & 529\\ 1196 \mbox{\bf 1642A} & 1 & \mbox{\bf 1642D} & 14 & 7 & 49\\ 1197 & & & & & \\ 1198 \end{array}$$
1199
1200
\mbox{}\par\noindent
1201
{{\bf 643A}, {\bf 1034A}, {\bf 1073A}, {\bf 1309B} all have modular degree
1202
a power of~$2$; {\bf 1436A} has modular degree divisible by~$3$.
1203
1204
\newpage
1205
\subsection{Conjecturally visible $\BigSha$ of modular motives of weight~$4$}
1206
\label{table:motive}
1207
Suppose $f$ and $g$ are elements of $S_4(\Gamma_0(N))$ such that $p^2 \mid L(\sM_f,2)/\Omega$
1208
and $L(\sM_g,2)=0$. If~$f$ and~$g$ satisfy a $p$-congruence'',
1209
does~$p$ then divide the visible part'' of $\Sha(\sM_f(2))$?
1210
1211
$$1212 \begin{array}{lcclcl} 1213 %\hspace{2em}\chi\hspace{2em}\mbox{} & 1214 \sM_f \hspace{1em}\mbox{}& 1215 \text{dim} 1216 & 1217 \hspace{2em}p^2\hspace{2em}\mbox{} 1218 & \sM_g & \text{dim}\\ 1219 & & & & & \vspace{-2ex} \\ 1220 1221 %(1,1) & \mbox{\bf 99C} & 8 & 19^2 & \mbox{\bf 99B} & 2 \\ 1222 \mbox{\bf 127k4C} & 17 & 43^2 & \mbox{\bf 127k4A} & 1\\ 1223 \mbox{\bf 159k4E} & 8 & 23^2 & \mbox{\bf 159k4B} & 1\\ 1224 %(0,0,1) & \mbox{\bf 200E} & 8 & 7^2 & \mbox{\bf 200G} & 2\\ 1225 \mbox{\bf 365k4E} & 18 & 29^2 & \mbox{\bf 365k4A} & 1\\ 1226 \mbox{\bf 369k4I} & 9 & 13^2 & \mbox{\bf 369k4A} & 1\\ 1227 \mbox{\bf 453k4E} & 23 & 17^2 & \mbox{\bf 453k4A} & 1\\ 1228 \mbox{\bf 465k4H} & 7 & 11^2 & \mbox{\bf 465k4A} & 1\\ 1229 \mbox{\bf 477k4L} & 12 & 73^2 & \mbox{\bf 477k4A} & 1\\ 1230 \mbox{\bf 567k4G} & 8 & 13^2, 23^2 & \mbox{\bf 567k4A} & 1\\ 1231 \mbox{\bf 581k4E} & 34 & 19^2 & \mbox{\bf 581k4A} & 1 1232 \end{array} 1233$$
1234
%Here~$\chi$ is the common nebentypus character of~$f$ and~$g$.
1235
1236
\section{Questions and conjectures}\label{sec:conj}
1237
The following questions and conjectures were motivated by the
1238
tables above and the computations that went into creating them.
1239
The first conjecture suggests a generalization of a result of
1240
Ribet on level raising. The second conjecture asserts that
1241
$\Sha$ is always visible in an appropriate modular Jacobian.
1242
The third question suggests a new approach to the long-standing
1243
open problem of constructing points on abelian varieties of
1244
analytic rank greater than~$1$ over the Hecke algebra.
1245
1246
\subsection{Level raising nonvanishing conjecture}
1247
Let $f\in S_2(\Gamma_0(N))$ be a newform such that the sign
1248
of the functional equation of $L(A_f,s)$ is equal to $+1$,
1249
and fix a prime~$\lambda$ such
1250
that the associated Galois representation $\rho_{f,\lambda}=A_f[\lambda]$
1251
is irreducible.
1252
For each prime~$q$ not dividing~$N$, let $\delta: J_0(N)\ra J_0(Nq)$ be
1253
the injection obtained from the sum of the two degeneracy maps.
1254
Ribet's construction in \cite{ribet:raising}
1255
produces infinitely many primes~$q$ and newforms
1256
$g\in S_2(\Gamma_0(qN))$ such that
1257
$$\delta(\Adual_f[\lambda]) \subset \delta(\Adual_f)\intersect \Adual_g$$
1258
and the Tamagawa number $c_q$ of $\Adual_g$ is a power of~$2$.
1259
\begin{conjecture}
1260
Fix~$f$ and~$\lambda$.
1261
\begin{enumerate}
1262
\item Then there is a~$g$ among those constructed by Ribet
1263
such that $L(A_g,1)\neq 0$.
1264
\item If~$\lambda$ is in the support of the $\T$-module
1265
$[P(H_1(X_0(N),\Z))^+: P(\T\{0,\infty\})]$
1266
(see Theorem~\ref{thm:ratpart}),
1267
then there is a~$g$ as above such that $L(A_g,1)=0$.
1268
\end{enumerate}
1269
\end{conjecture}
1270
1271
\subsection{Eventual visibility conjecture}
1272
Let~$S$ be the set of all square-free positive integers.
1273
If $M, N\in S$ with $M\mid N$ then there is
1274
a natural injection $J_0(M)\hookrightarrow J_0(N)$, and hence a map
1275
$\Sha(J_0(M))\ra \Sha(J_0(N))$.
1276
These maps are compatible, so the collection of groups $\Sha(J_0(N))$, with
1277
$N\in S$, forms a directed system. Let
1278
$\lim_{N\in S} \Sha(J_0(N))$
1279
be the direct limit of the $\Sha(J_0(N))$.
1280
\begin{conjecture}
1281
$\lim_{N\in S} \Sha(J_0(N))=0$
1282
\end{conjecture}
1283
If true, this would imply that if $A\subset J_0(N)$, then
1284
each element of $\Sha(A)$ is visible in some $J_0(N')$,
1285
for some multiple~$N'$ of~$N$.
1286
1287
\subsection{Euler systems}
1288
\label{sec:constructing}
1289
In \cite{kolyvagin:structureofsha} and \cite{mccallum:kolyvagin}
1290
one finds a construction using the Heegner point Euler system of
1291
Kolyvagin of the Shafarevich-Tate groups of certain abelian varieties.
1292
Under an unverified hypothesis on Heegner points, the construction
1293
gives much of $\Sha(A_f/K)$, where~$K$ is a suitable imaginary
1294
quadratic field. Is it possible to verify the unverified hypothesis,
1295
construct $\Vis_J(\Sha(A_f/K))$, and thus prove that $\Sha(\Adual_f/K)$
1296
contains visible elements, when the BSD conjecture suggests that it should? If so,
1297
it would follow that there is a congruent $B_f^{\vee}$ having positive
1298
algebraic rank, as predicted by the BSD conjecture. Thus a construction of
1299
visible elements of $\Sha(\Adual_f)$ also leads to a construction of points
1300
on abelian varieties of positive analytic rank.
1301
1302
\bibliography{biblio}
1303
1304
\end{document}
1305
1306