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\title{\sc Visibility of Shafarevich-Tate groups of modular abelian varieties
and the Birch and Swinnerton-Dyer conjecture}
\author{Amod Agashe\footnote{Warning: Agashe has not yet
read this draft of our preprint.} \and William A.~Stein}

\begin{document}
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\begin{abstract}
We give examples of abelian subvarieties of $J_0(N)$ that have
nontrivial visible Shafarevich-Tate groups.  These examples provide
evidence for the Birch and Swinnerton-Dyer conjecture, and the methods
used to compute them may lead to new connections between this
conjecture and the theory of congruences between modular forms.
\end{abstract}

\section{Introduction}\label{sec:intro}
Let~$N$ be a positive integer and let $J_0(N)$ be the Jacobian of the
modular curve $X_0(N)$ (see, e.g., \cite{diamond-im}).  Let~$A$ be an
abelian subvariety of $J_0(N)$.  Then the inclusion $A\hookrightarrow J_0(N)$ induces a map on Galois cohomology $H^1(\Q,A) \ra H^1(\Q,J_0(N))$, which restricts to a map $\Sha(A) \ra \Sha(J_0(N))$.
Following \cite{cremona-mazur}, we call an element $c\in\Sha(A)$
\defn{visible} in $J_0(N)$ if it lies in the kernel of this map.
When~$c$ is visible the torsor attached to~$c$ is realized as a
subvariety of $J_0(N)$; it is a fiber of the quotient map $J_0(N)\ra J_0(N)/A.$

In~\cite{cremona-mazur}, Cremona and Mazur ask the following question:
If $E\subset J_0(N)$ is a elliptic curve, how much of $\Sha(E)$ is
visible in an abelian surface~$B$ that is itself a subvariety
of~$J_0(N)$?  Among the $52$ examples of odd nontrivial $\Sha$ that
they consider, $\Sha$ is invisible in at least~$8$ cases.  However,
they warn that the elliptic curves they consider have small conductor,
and the general situation is probably much different.  Standard
conjectures imply that when $\ell$ is sufficiently large, elements of
order~$\ell$ in $\Sha(E)$ can not be visible in an abelian surface; it
is thus essential to consider visibility in higher dimensional abelian
varieties.

We generalize the experiment of Cremona and
Mazur by removing the constraint that only elliptic curves be
considered.  The numbers become large at much smaller conductor, and
the proportion of~$\Sha$ that is visible declines accordingly.

In Section~\ref{sec:vis} we give a general definition of visibility,
which is motivated by a restriction-of-scalars construction of
torsors.  Then we prove a theorem that gives a criterion for
the existence of nontrivial visible elements of Shafarevich-Tate
groups.  Section~\ref{sec:computing} describes the algorithms we use to
enumerate modular abelian varieties, compute conjectural
orders of their Shafarevich-Tate groups, and verify the visibility
criterion of Section~\ref{sec:vis} when possible.  Section~\ref{sec:tables}
presents the results of extensive numerical investigations.  It provides
the impetus for the conjectures we make and questions we ask in
Section~\ref{sec:conj}.

{\bf Acknowledgment.} It is a pleasure to thank B.~Mazur for his
lectures and conversations about visibility, K.~Ribet for explaining
congruences to us, and R.~Coleman for helping us to understand
component groups.  The authors would also like to thank B.~Conrad,
J.~Ellenberg, R.~Greenberg, D.~Gross, L.~Merel, B.~Poonen, and R.~Taylor

\section{Visible cohomology classes}\label{sec:vis}
In Section~\ref{sec:torsors} we define the notion of visibility and
observe that every cohomology class
arises from a torsor that can be constructed geometrically inside of
an appropriate restriction of scalars, then we define visible
cohomology classes.  Section~\ref{sec:visthm}, which should perhaps be
omitted from a first reading, gives a criterion for the existence of
visible elements of Shafarevich-Tate groups.

\subsection{Geometric realization of torsors}\label{sec:torsors}
Let~$A$ be an abelian variety over a field~$K$.  The following
definition is due to Mazur.

\begin{definition}[Visible]
Let $\iota:A\hookrightarrow J$
be an embedding of~$A$ into an abelian variety~$J$.  Then the \defn{visible part
of $H^1(K,A)$ with respect to the embedding~$\iota$} is
$$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\ra{}H^1(K,J)).$$
\end{definition}

The Galois cohomology group $H^1(K,A)$ has a geometric interpretation
as the group of classes of torsors~$X$ for~$A$ (see~\cite{lang-tate}).
To a cohomology class $c\in H^1(K,A)$, there is a corresponding
variety~$X$ over~$K$ and a map $A\cross X \ra X$ that satisfies axioms
similar to those for a simply transitive group action.
Suppose $\iota: A\ra J$ is an embedding and $c\in \Vis_J(H^1(K,A))$.
We have an exact sequence of abelian varieties
$0\ra A\ra J\ra B\ra 0$.  A piece of the
associated long exact sequence of Galois cohomology is
$$\cdots \ra J(K)\ra B(K) \ra H^1(K,A) \ra H^1(K,J) \ra \cdots,$$
so, because $\iota_*(c)=0$, there is a point $x\in B(K)$ that maps
to $c\in H^1(K,A)$.  Then the fiber~$X$ over~$x$ is a subvariety of~$J$,
which, when equipped with its natural action of~$A$, lies in the class
of torsors corresponding to~$c$.

\begin{proposition}
Every element of $H^1(K,A)$ is visible in some abelian variety~$J$.
\end{proposition}
\begin{proof}
Fix $c\in H^1(K,A)$.  There is a finite extension~$L$ of~$K$ such that
$\res_L(c) = 0\in H^1(L,A)$.  Let $J=\Res_{L/K}(A_L)$ be the
restriction of scalars down to~$K$ of the abelian variety~$A$, which
we view as an abelian variety over~$L$ (see \cite[\S7.6]{neronmodels}).
Thus~$J$ is an abelian variety over~$K$ of dimension $[L:K]\cdot \dim(A)$, and for any
scheme~$S$ over~$K$, we have a natural bijection $J(S) \ncisom A_L(S_L)$.
In particular, $J(A) = A_L(A_L)$, and there is an
injection $\iota:A\hookrightarrow J$ attached to $\id_{A_L}\in A_L(A_L)$.
Using the Shapiro lemma, one finds that there is a canonical isomorphism
$H^1(L,A) \isom H^1(K,J)$ and that
$\iota_*(c) = 0\in H^1(K,J)$.
\end{proof}
\begin{remark}
In \cite{cremona-mazur}, J.~de Jong gives a sophisticated proof of
the above proposition in the special case when~$A$ is an elliptic curve.
His proof involves Azumaya algebras.
\end{remark}

\subsection{Visible elements of Shafarevich-Tate groups}\label{sec:visthm}
The Shafarevich-Tate group of an abelian variety over a number
field measures the failure of the local-to-global principle
for its torsors.
\begin{definition}[Shafarevich-Tate group]
Let~$A$ be an abelian variety over~$\Q$.
The \defn{Shafarevich-Tate group} of $A$ is
$$\Sha(A) = \Ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right),$$
where the product is over all places of~$\Q$.
\end{definition}

If $\iota:A\hookrightarrow J$ is an embedding, then the
visible part of $\Sha(A)$ is
$$\Vis_J(\Sha(A)) = \Sha(A) \intersect \Vis_J(H^1(\Q,A)) = \Ker(\Sha(A)\ra \Sha(J)).$$
We use the following theorem to produce examples of visible
elements of Shafarevich-Tate groups.
\begin{theorem}\label{thm:shaexists}
Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
that $A\intersect B$ is finite and $A(\Q)$ is finite.
Let~$N$ be an integer divisible by the primes of bad reduction for~$J$.
Assume that~$B$ has purely toric reduction at each prime dividing~$N$.
Suppose~$p$ is a prime such that
$$p\nmid 2N\cdot \#(J/B)(\Q)_{\tor}\cdot\#B(\Q)_{\tor}\cdot \prod_{\ell\mid N} \#\Phi_{A,\ell}(\Fbar_\ell)\cdot \#\Phi_{B,\ell}(\F_\ell).$$
Suppose furthermore that $B[p] \subset A\intersect B$.
Then
$$B(\Q)/pB(\Q)\hookrightarrow \Vis_J(\Sha(A)).$$
\end{theorem}
The proof proceeds in four steps.  First the torsion and
$p$-congruence hypothesis is used to produce an injection $B(\Q)/p B(\Q)\hookrightarrow \Vis_J(H^1(\Q,A))$.  Next we perform a local
analysis at each place~$v$ of~$\Q$, which proceeds in three steps.  At
places~$v$ of bad reduction, we use the Mumford-Tate uniformization;
at odd primes of good reduction we apply an exactness theorem about
N\'eron models; when~$2$ is a place of bad reduction, we modify the
situation by a $2$-isogeny and apply another exactness theorem.

\begin{proof}
The quotient $J/A$ is an abelian variety~$C$.  The long exact sequence
of Galois cohomology associated to the short exact sequence
$$0 \ra A \ra J \ra C \ra 0$$
begins
$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,} H^1(\Q,A) \ra \cdots.$$
Let~$\phi$ be map $B\ra C$, which is obtained by composing
the inclusion $B\hookrightarrow J$ with the quotient map $J\ra C$.
Since $B[p]\subset A$, we see that~$\phi$ factors through multiplication by~$p$.
We thus obtain the following commutative diagram:
$$\xymatrix{ & B\ar[d] \ar[r]^{p}& B\ar[d]\\ A\ar[r]&J\ar[r]&C.}$$
Using that $B(\Q)[p]=0$, we
obtain the following diagram, all of whose rows and columns are exact:
$$\xymatrix{ & K_0\ar[d] & K_1\ar[d]& K_2\ar[d]\\ 0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d] & B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\ 0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\ & K_3, }$$
where $K_0$, $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the
indicated cokernel.
The snake lemma gives an exact sequence
$$K_0\ra K_1 \ra K_2 \ra K_3.$$
Because $B\ra C$ is an isogeny, $K_1\subset B(\Q)_{\tor}$.
Since $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group, the map
$K_1\ra K_2$ is the~$0$ map.
The quotient
$J(\Q)/B(\Q)$ has no $p$-torsion because
it is a subgroup  of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group,
so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion, and the map
$K_2\ra K_3$ must be the~$0$ map.
We conclude that $K_2=0$.

The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of
$H^1(\Q,A)$; however, the latter group contains infinitely many elements of
order~$p$, whereas $\Sha(A)[p]$ is a finite group, so we must work
harder in order to deduce that $B(\Q)/p B(\Q)$ actually lies in
$\Sha(A)[p]$.   Let $x\in B(\Q)$; we must show
that $\pi(x)\in \Sha(A)[p]$.  It suffices to show that
$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.

At the archimedian place $v=\infty$, the restriction
$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,
hence $\res_v(\pi(x))=0$.

Suppose that~$v$ is a place at which~$J$ has bad reduction.
By hypothesis, $B$ has purely toric reduction,
so over $\Q_v^{\ur}$
there is an isomorphism $B\isom\Gm^d/\Gamma$
of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,
for some lattice'' $\Gamma$.  For example, when
$\dim B=1$, this is the Tate curve representation of~$B$.
Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$
equals the order of the cokernel of the valuation
map $\Gamma\ra \Z^d$.  Choose a representative
$P=(x_1,\ldots,x_d)\in\Gm^d$ for the point~$x$.
Let $n'=\#\Phi_{B,v}(\F_v)$.
Then since~$P$ is rational over $\Q_v$,
$n'P$ can be adjusted by elements of~$\Gamma$
so that each of its components $x_i\in\Gm$ has valuation~$0$;
since~$p$ does not equal the residue characteristic of~$v$,
it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$
such that $pQ = n'P$.
Thus the cohomology class $\res_v(\pi(n'x))$ is unramified at~$v$.
By \cite[Prop.~3.8]{milne:duality},
$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur})) =H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fpbar)),$$
where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
Since~$p$ does not divide $\#\Phi_{A,v}(\Fpbar)$,
and $\pi(n'x)$ has order~$p$, it follows that
$$0=\res_v(\pi(n'x))=n'\res_v(\pi(x)).$$
Since the order of $\res_v(\pi(x))$ is coprime to~$n'$,
we conclude that $\res_v(\pi(x))=0$.

Next suppose that~$J$ has good reduction at~$\ell$
and that~$\ell$ is {\em odd}.
Let $\cA$, $\cJ$, $\cC$, be the N\'eron models
of~$A$,~$J$,~$C$, respectively.
Since~$\ell$ is odd, $1=e<\ell-1$, so we may apply
\cite[Thm.~7.5.4]{neronmodels} to conclude that
the sequence of group schemes
$$0\ra \cA \ra \cJ\ra \cC \ra 0$$
is exact, in the sense that it
is exact as a sequence of sheaves on the
\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).
Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}
the sequence
$$0\ra \cA(\Z_v^{\ur})\ra \cB(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$
is exact. By the N\'eron mapping property, the sequence
$$0\ra A(\Q_v^{\ur})\ra B(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$
is also exact, so
$\res_v(\pi(x))$ is unramified.
By \cite[Prop.~I.3.8]{milne:duality},
$$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v))=0,$$
since~$A$ has good reduction at~$v$.
Hence $\res_v(\pi(x))=0$.

If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.
Consider the case when~$J$ has good reduction at~$2$.   The
absolute ramification index~$e$ of $\Z_2$ is~$1$, which is
{\em not} less than $2-1=1$, so we can not apply \cite[Thm.~7.5.4]{neronmodels}.
However, we can modify everything by an isogeny of degree a power
of~$2$ and apply a different theorem, as follows.
The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is rational
over~$\Q$.   The abelian varieties
$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and
$\tilde{B}=B/\Psi$ also satisfy the hypothesis of
the theorem we wish to prove.
By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of
N\'eron models
$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$
is exact, so the sequence
$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur}) \ra\tilde{C}(\Q_v^{\ur})\ra 0$$
is exact. Thus the image of
$\res_v(\pi(x))$  in $H^1(\Q_v,\tilde{A})$ is unramified.
It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},
since the component group of $\tilde{A}$ at~$v$ has order a power
of~$2$, whereas $\pi(x)$ has odd prime order~$p$.
Thus $\res_v(\pi(x))=0$, since
the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a
finite group of $2$-power order.
\end{proof}

\subsubsection{Visibility when $A$ also has positive rank}
In Theorem~\ref{thm:shaexists}, if the condition that $A(\Q)$ has rank~$0$ is removed,
then the proof can be easily modified to show that the kernel of
$B(\Q)/p B(\Q) \ra \Vis_J(\Sha(A))$
has dimension at most the rank of $A(\Q)$.

According to \cite{cremona:algs},
the smallest conductor elliptic curve~$E$ of rank~$3$ is found in
$J=J_0(5077)$.  The number $5077$ is prime, and~$J$ decomposes
up to isogeny as
$A \cross B \cross E,$
where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$
associated to newforms, which have
dimensions $205$, $216$, and~$1$, respectively.
The modular degree of~$E$ is $1984=2^6\cdot 31$, and
the sign of the Atkin-Lehner involution on~$E$ is the same
as its sign on~$A$, so $E[31]\subset A$.
The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so
$31$ is coprime to the orders of any relevant component groups or torsion.
Thus $\Vis_J(\Sha(A))$ contains $(\Z/31\Z)^2$.

\section{Guide to computing on $J_0(N)$}\label{sec:computing}
The Jacobian $J_0(N)$ is equipped with an action of the Hecke algebra~$\T$.
Let $f\in S_2(\Gamma_0(N))$ be a newform, and let $I_f\subset\T$
be the annihilator of~$f$.  The abelian variety~$A_f$ attached
to~$f$ is the quotient $J_0(N)/I_f J_0(N)$.  Thus $A_f$ is an
abelian variety of dimension equal to the number of Galois
conjugates of~$f$ and equipped with a faithful action of $\T/I_f$.
For the remainder of this section, $A=A_f$ denotes the optimal quotient
of $J_0(N)$ attached to the annihilator~$I=I_f$ of a newform~$f$.

\subsection{The Birch and Swinnerton-Dyer conjecture}
The Birch and Swinnerton-Dyer conjecture, as generalized by
Tate in~\cite{tate:bsd}, furnishes a conjectural formula
for the order of the Shafarevich-Tate group of any new
optimal quotient~$A$.
In general, it is difficult given~$A$ to compute
the conjectural order of~$\Sha(A)$.
However, the situation is more optimistic
when~$A$ is a new modular abelian variety
such that $L(A,1)\neq 0$.
For these~$A$ we have devised an algorithm
that we use to compute the odd part of the
conjectural order of $\Sha(A)$ in many cases.
The following is a special case of a much more general conjecture.
\begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsd}
Suppose $L(A,1)\neq 0$.   Then
$$\frac{L(A,1)}{\Omega_A} = \frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p} {\# A(\Q)\cdot\#\Adual(\Q)}.$$
\end{conjecture}
When $L(A,1)\neq 0$, work of Kolyvagin and Logachev
\cite{kolyvagin-logachev:finiteness,kolyvagin-logachev:totallyreal}
implies that $A(\Q)$, $\Adual(\Q)$, and $\Sha(A)$ are all finite,
so the quantities appearing in the above formula make sense.
Here $c_p=\#\Phi_{A,p}(\F_p)$, the positive real number~$\Omega_A$ is
the measure of~$A(\R)$ with respect to a basis of differentials on
the N\'eron model of~$A$, and~$\Adual$ is the abelian variety
dual of~$A$.
The algorithms described below enable us in my cases to compute the
conjectural order of $\Sha(A)$.  However, for question of visibility,
we instead need to compute the order of $\Sha(\Adual)$.  This is no
different because the Cassels-Tate pairing implies that
$\#\Sha(A) = \#\Sha(\Adual)$.

\subsection{Modular symbols}\label{modsym}
It is not possible to compute very much about $J_0(N)$ without
modular symbols, which provide a finite presentation for the homology
group $H_1(X_0(N),\Z)$ in terms of paths between elements of
$\P^1(\Q) = \Q\union \{\infty\}$.

The \defn{modular symbol} defined by a pair $\alpha,\beta\in\P^1(\Q)$
is denoted $\{\alpha,\beta\}$.  This modular symbol should be viewed as
the homology class, relative to the cusps,
of a geodesic path from~$\alpha$ to~$\beta$ in $\h^*$.
The homology group relative to the cusps is a slight enlargement
of the usual homology group, in that
we allow paths with endpoints in $\P^1(\Q)$ instead of restricting
to closed loops.
We declare that modular symbols satisfy
the following homology relations:
if $\alpha,\beta,\gamma \in \Q\union\{\infty\}$, then
$$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0.$$
Furthermore, the space of modular symbols is torsion free, so, e.g.,
$\{\alpha,\alpha\} = 0$ and
$\{\alpha,\beta\} = -\{\beta,\alpha\}$.

Denote by~$\sM_2$ the free abelian group with basis the set of
symbols $\{\alpha,\beta\}$ modulo the three-term homology relations
above and modulo any torsion.
There is a left action of $\GL_2(\Q)$ on $\sM_2$, whereby
a matrix~$g$ acts by
$$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$
and~$g$ acts on~$\alpha$ and~$\beta$ by a linear fractional
transformation.
The space $\sM_2(N)$ of \defn{modular symbols for $\Gamma_0(N)$}
is the quotient of $\sM_2$ by the submodule
generated by the infinitely many elements
of the form $x - g(x)$, for~$x$ in ~$\sM_2$
and~$g$ in $\Gamma_0(N)$, and modulo any torsion.
A \defn{modular symbol for $\Gamma_0(N)$} is an element of
this space.   We frequently denote the equivalence
class that defines a modular symbol by giving a
representative element.

In \cite{manin:parabolic}, Manin proved that there is
a natural injection $H_1(X_0(N),\Z)\hookrightarrow \sM_2(N)$.
The image of $H_1(X_0(N),\Z)$ in $\sM_2(N)$ can be identified as follows.
Let $\sB_2(N)$ denote the free abelian group whose basis is the finite set
$\Gamma_0(N)\backslash \P^1(\Q)$.
The \defn{boundary map} $\delta: \sM_2(N)\ra \sB_2(N)$
sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$
denotes the basis element of $\sB_2(N)$ corresponding to $\beta\in\P^1(\Q)$.
The kernel $\sS_2(N)$ of~$\delta$ is the subspace of
\defn{cuspidal} modular symbols.
An element of $\sS_2(N)$ can be thought of as a linear
combination of paths
in $\h^*$ whose endpoints are cusps, and whose images in $X_0(N)$
are a linear combination of loops.
We thus obtain a canonical isomorphism $\varphi:\sS_2(N)\ra H_1(X_0(N),\Z)$.

Part of the utility of modular symbols comes from the classical Abel-Jacobi
theorem, which allows us to view $J_0(N)(\C)$ as the quotient
$\C^g/H_1(X_0(N),\Z)$,
where $H_1(X_0(N),\Z)$ is embedded in
$\C^d\ncisom\Hom(S_2(\Gamma_0(N)),\C)$ using the integration pairing.
Thus modular symbols give an explicit description of $J_0(N)(\C)$
and of its constituent parts as modules over the Hecke algebra.
We can also compute Hecke operators using modular symbols.

For further introductory remarks on modular symbols, see~\cite{stein:modsyms},
and for detailed instructions as to how to compute the space of modular symbols
and the action of Hecke operators on it, see~\cite{cremona:algs}.

\subsection{Computing with quotients and subvarieties of $J_0(N)$}
First, we describe how to enumerate the newforms of level~$N$.  Then
we define the modular degree, whose square annihilates the visible part
of~$\Sha$.  Finally, we describe how to intersect abelian subvarieties
of $J_0(N)$.

\subsubsection{Enumerating quotients}
Let $H_1(X_0(N),\Z)^+$ denote the $+1$-eigenspace for the action
of the involution induced by complex conjugation.
We list all newforms of a given level~$N$ by decomposing the new
subspace of $H_1(X_0(N),\Q)^+$ under the action of the the
Hecke operators.  First we compute the characteristic polynomial of~$T_2$,
and use it to break up the full space.  We apply this process
recursively with $T_3, T_5, \ldots$ until either we have exceeded the
bound coming from~\cite{sturm:cong}, or we have found a Hecke
operator~$T_n$ whose characteristic polynomial is irreducible.  After
computing the decomposition, we order the newforms in a way that
extends the systematic ordering in~\cite{cremona:algs}: First sort by
dimension, with smallest dimension first; within each dimension, sort
in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$,
$++-$, $+-+$, $+--$, $-++$, etc.  When two forms have the same sign
sequence, order by $|\Tr(a_p)|$ with ties broken by taking the
positive trace first.

We denote a Galois conjugacy class of newforms by a bold symbol such
as $\mathbf{389E}$, which consists of the level and the isogeny class,
where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second,
and so on.

As discussed in \cite[pg.~5]{cremona:algs}, for certain small levels
the above ordering when restricted to elliptic curves does not agree
with the ordering used in Cremona's tables.  For example, in the
present paper our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$.

\subsubsection{The modular degree\label{modpolar}}
A \defn{polarization}~$\lambda$ of an abelian variety~$A$ over~$\Q$ is an isogeny
$\lambda:A\ra \Adual$ such that $\lambda_{\Qbar}$
arises from an ample invertible sheaf on $A_{\Qbar}$ (see, e.g.,
\cite[\S13]{milne:abvars}).
Since $J_0(N)$ is a Jacobian, it possesses a canonical
polarization arising from the $\theta$-divisor, and this
polarization induces the \defn{modular polarization}
$\theta: \Adual\ra A$ of $\Adual$.
$$\[email protected]=3pc{ {\Adual}\[email protected]{^(->}[r]^{\pi^{\vee}\qquad}\ar[dr]^{\theta} & J_0(N)^{\vee} \isom J_0(N)\,\,\,\[email protected]{->>}[d]^{\pi}\\ &A.}$$
If we view $\Adual$ as an abelian subvariety of $J_0(N)$, then the
kernel of~$\theta$ is the intersection of $\Adual$ with $I J_0(N)$; thus
the kernel of $\theta$ measures intersections between $\Adual$ and other
factors of $J_0(N)$.
\begin{definition}[Modular degree]
The \defn{modular degree} $m_A$ of $A$ is
$\sqrt{\deg(\theta)}$.
\end{definition}
\noindent{}By \cite[Thm.~13.3]{milne:abvars}, $\deg(\theta)$ is a perfect
square, so $m_A$ is an integer.
For an algorithm to compute $m_A$, see~\cite{kohel-stein:ants4}.

The modular degree is of interest because its square
annihilates the visible cohomology classes.
\begin{proposition}
$\displaystyle \Vis_{J_0(N)}(H^1(\Q,\Adual))\subset H^1(\Q,\Adual)[m_A^2]$
\end{proposition}
\begin{proof}
Let $\delta$ be the composite map
$\Adual \ra J_0(N)\ra A$.  There is a map $\hat{\delta}:A\ra \Adual$
such that $\hat{\delta}\circ\delta$ is multiplication by $\deg(\delta)=m_A^2$.
Thus
$\Ker(H^1(\Q,\Adual)\ra H^1(\Q,J_0(N)))$
is contained in $H^1(\Q,\Adual)[m_A^2]$.
\end{proof}
\begin{remark}
When~$A$ has dimension one, the visible part of $H^1(\Q,\Adual)$ is contained
in $H^1(\Q,A)[m_A]$.  Is this true for~$A$ of all dimensions?
\end{remark}

\subsubsection{Intersecting complex tori}\label{sec:intersect}
Consider a complex torus $J=V/\Lambda$, and let
$A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ be subtori whose
intersection $A\intersect B$ is finite.
Here $V_A$ and $V_B$ are subspaces of~$V$ and $\Lambda_A$ and $\Lambda_B$
are submodules of~$\Lambda$.
\begin{proposition}\label{prop:intersection}
There is a natural isomorphism of groups
$$A\intersect B \isom \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$
\end{proposition}
\begin{proof}
There is an exact sequence
$$0\ra A\intersect B \ra A \oplus B \ra J.$$
Consider the diagram
$$\xymatrix{ & {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]& {\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\ & {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\ {A\intersect B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B).}$$
The snake lemma\index{Snake lemma} gives an exact sequence
$$0 \ra A\intersect B \ra \Lambda/(\Lambda_A + \Lambda_B) \ra V/(V_A+V_B).$$
Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion
part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$.
No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could
map to~$0$, because if it did then $A\intersect B$ would not
be finite.  The lemma follows.
\end{proof}

The following formula for the intersection of~$n$
subtori is obtained  in a similar way.
\begin{proposition}
For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of
$J=V/\Lambda$, and assume that each pairwise intersection
$A_i \intersect A_j$ is finite.
Then
$$A_1\intersect \cdots \intersect A_n \isom \left(\frac{\Lambda\oplus \cdots \oplus \Lambda} {f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right),$$
where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.
\end{proposition}

\subsection{Computing the conjectural order of $\BigSha(A)$}
In this section, we describe how in many cases we can compute the
conjectural order of $\Sha(A)$ when $L(A,1)\neq 0$, at least up to a
power of~$2$.

In Section~\ref{sec:torsion}, we bound $\#A(\Q)$ and $\#\Adual(\Q)$.
We compute each $c_p$ in Section~\ref{sec:tamagawa},
for each~$p$ with $p\mid\mid N$. When $p^2\mid{}N$, it is
possible to bound $c_p$; see, e.g., \cite[Cor.~15.2.1]{silverman:aec}
where one finds that when $\dim A=1$ and $p^2\mid N$, we have $c_p\leq 4$.
In Section~\ref{sec:bsdratio}, we use modular symbols to compute
the rational number $L(A,1)/\Omega_A$, up to a bounded Manin constant.

\subsubsection{Torsion subgroup}\label{sec:torsion}
We obtain an upper bound on $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$
as follows.
The characteristic polynomial $\chi_p(X)$ of the Hecke operator~$T_p$
acting on~$A$ is a monic polynomial having integer coefficients
and degree equal to the dimension of~$A$.
\begin{proposition}
Both $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$ divide
$$\gcd \{ \chi_p(p+1) : (p,2N)=1,\, \text{\rm p prime} \}.$$
\end{proposition}
\begin{proof}
Use the Eichler-Shimura relation and that for primes~$p$ for which $p\nmid 2N$ the maps
$A(\Q)_{\tor} \ra \tilde{A}(\F_p)$ and
$\Adual(\Q)_{\tor} \ra \tilde{A}^{\vee}(\F_p)$
are both injective, and that
$\#\Adual(\F_p)_{\tor}=\#\tilde{A}^{\vee}(\F_p)$.
\end{proof}

The difference of two cusps $\alp,\beta \in{} X_0(N)$ defines
a point $(\alp)-(\beta) \in J_0(N)(\C)$.  Manin observed
in \cite{manin:parabolic} that $(0)-(\infty)$ is rational.
The order of the image of $(0)-(\infty)$ in  $A(\Q)$ can be computed as follows.
Let
$$V=\Hom(S_2(\Gamma_0(N)),\C),$$
and $V_I = \Hom(S_2(\Gamma_0(N))[I],\C)$.
The integration pairing $\langle f, \gamma \rangle = 2\pi i \int_\gamma f(z)dz$
between homology and cusp forms gives rise to a map $P: H_1(X_0(N),\Q)\ra V_I$.
By the Abel-Jacobi theory (see, e.g., \cite[Thm~IV.2.2]{lang:modular}),
$A(\C) \isom V_I/P(H_1(X_0(N),\Z))$.
\begin{proposition}
The order of the image of $(\alp)-(\beta)$ in $A(\C)$
equals the order of the image of the modular symbol $\{\alp,\beta\}$
in $P(H_1(X_0(N),\Q))/P(H_1(X_0(N),\Z)).$
\end{proposition}
The quotient appearing in the proposition can be computed algebraically
by replacing~$P$ by a map with the same kernel as~$P$.  Such a map can
be computed using the Hecke operators (see \cite[\S3.7]{stein:phd}).

\subsubsection{Tamagawa numbers}\label{sec:tamagawa}
Suppose~$p$ is a prime that exactly divides~$N$ and let $\Phi_{A,p}$
denote the component group of~$A$ at~$p$.
We have an exact sequence,
$$0\ra \cA_{\Fp}^0\ra \cA_{\Fp} \ra \Phi_{A,p}\ra 0,$$
where $\cA_{\Fp}$ is the closed fiber of the N\'eron model of~$A$ over $\Z_p$ and $\cA_{\Fp}^0$
is the component of $\cA_{\Fp}$ that contains the identity.
A formula for $\#\Phi_{A,p}(\Fbar_p)$ and, up to a power of $2$,
for $\#\Phi_{A,p}(\F_p)$,
is given in \cite{kohel-stein:ants4} and \cite{stein:compgroup}.

\subsubsection{Rational part of the special value}\label{sec:bsdratio}
As in Section~\ref{sec:torsion},
let $P : H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N))[I],\C)$
be the map induced by integration.
Let $P(H_1(X_0(N),\Z))^+$ denote the $+1$-eigenspace for the action
of the involution induced by complex conjugation on the image of~$P$.

\begin{theorem}\label{thm:ratpart}
$$\frac{L(A,1)}{\Omega_{A}} = [P(H_1(X_0(N),\Z))^+ : P(\T\{0,\infty\})]/(c_\infty\cdot c_A),$$
where $c_\infty$ is the number of components of $A(\R)$ and $c_A$
is the Manin constant of~$A$, as defined below.
\end{theorem}
In order to define the Manin constant of~$A$,
let $\cA$ denote the N\'eron model of~$A$ over~$\Z$.
\begin{definition}[Manin constant]
The \defn{Manin constant}~$c_A$ of~$A$ is the index
$$c_A := [S_2(\Gamma_0(N);\Z)[I]:H^0(\cA,\Omega_{\cA/\Z})].$$
\end{definition}
In the definition, we have implicitly mapped $H^0(\cA,\Omega_{\cA/\Z})$ into
$S_2(\Gamma_0(N);\Q)$ using the composition of the following maps:
$$H^0(\cA,\Omega_{\cA/\Z}) \ra H^0(\cJ,\Omega_{\cJ/\Z})[I] \ra H^0(J,\Omega_{J/\Q})[I] \ra S_2(\Gamma_0(N);\Q)[I].$$
For a discussion of why $H^0(\cA,\Omega_{\cA/\Z})$ is in fact
contained in $S_2(\Gamma_0(N);\Z)[I]$, see \cite{agashe-stein:manin}.
\begin{theorem}
If $\ell \mid c_A$ then $\ell^2 \mid 4N$.
\end{theorem}
\begin{proof}
See~\cite[\S4]{mazur:rational} when~$A$ has dimension~$1$,
and \cite{agashe-stein:manin} in general.
\end{proof}

We now give the proof of Theorem~\ref{thm:ratpart}.
\begin{proof}[Proof of Theorem~\ref{thm:ratpart}]
Let $H=H_1(X_0(N),\Z)$ and $S=S_2(\Gamma_0(N))$.
There is a perfect pairing $\T \cross S \ra \Z$ given by
$\langle T_n, f\rangle = a_n(f)$, which
induces a canonical isomorphism of rings $\T\isom \Hom_\Z(S,\Z)$,
where $\Hom_\Z(S,\Z)$ is a ring under multiplication of functions.
The subring $W=\Hom_\Z(S[I],\Z)$ of $\Hom_\Z(S[I],\R)$ is
isomorphic to $\T/I$, since $S[I]$ is saturated in~$S$.
Thus
\begin{eqnarray*}
[W : P(\{0,\infty\})W] &=& [W:P(\T \{0,\infty\})]\\
&=& [W:P(H)^+] \cdot [P(H)^+ : P(\T \{0,\infty\})].
\end{eqnarray*}
To complete the proof, observe that
that $\Omega_A = [W:P(H)^+] \cdot c_\infty\cdot c_A$ and observe
that multiplication by $P(\{0,\infty\})$ has determinant
$\prod_{i=1}^d 2\pi i \int_{\{0,\infty\}} f^{(i)} = \pm L(A,1)$.
\end{proof}

\subsection{Emerton's work}
When~$N$ is prime, M.~Emerton has proved in \cite{emerton:myconj}
that $\#A_f(\Q)$ and $c_p(A_f)$ divide the numerator of $(N-1)/12$.

\section{Visibility tables}\label{sec:tables}
The tables in this section guide and motivate the conjectures
and questions of Section~\ref{sec:conj}.

In {\bf Table~\ref{table:invisible}}, we list each of the $8$ invisible odd
Shafarevich-Tate groups found in \cite{cremona-mazur}, and
prove\footnote{This computation is currently only partially complete.} that
they are visible in some $J_0(Nq)$.

{\bf Table~\ref{table:prime}} lists every quotient $A_f$ of $J_0(p)$
with $p\leq 2593$ and $L(A_f,1)\neq 0$ such that the BSD conjecture
predicts that $\#\Sha(\Adual_f)$ is divisible by an odd
prime.  In addition, the table contains data that can frequently be
used in conjuction with Theorem~\ref{thm:shaexists} to deduce that
there are visible elements of $\Sha(\Adual_f)$.  When the {\bf B}
column is labeled NONE then there is definitely nothing in
$\Sha(\Adual_f)$ of the predicted order.  When the {\bf B} column
contains an elliptic curve, its rank has been computed and is~$2$, so
there are visible elements of $\Sha(\Adual_f)$.  When the {\bf B}
column contains an abelian variety of dimension greater than~$1$, we
have verified that $L(B,1)=0$, so the BSD conjecture predicts that
$B(\Q)$ is infinite; however, we have not proved that $B(\Q)$ is
infinite.  If we assume that $B(\Q)$ is infinite, it follows in these cases that
$\Sha(A)$ is visible in $J_0(p)$.  Note that~$B$ has rank~$2$ over the
Hecke algebra here, so the results of \cite{gross-zagier} say
nothing about $B(\Q)$.

{\bf Table~\ref{table:prime2}} continues the computations of Table~\ref{table:prime}
up to level $5647$.  For each prime~$p$ between $2609$ and $5674$, we computed each
factor $A$ such that $L(A,1)\neq 0$ and the odd part of
$\#\Shaan(A)$ is nontrivial.  We then found all factors~$B$ such that
$L(B,1)=0$ and there is a mod~$\ell$ congruence between~$A$ and~$B$,
where $\ell\mid \#\Shaan(A)$.  The column labeled~$N$ gives the level,
the column labeled $d(A)$ gives the dimension of~$A$, the column labeled
$d(B)$ gives the dimension of~$B$, and the column labeled cong'' gives
the odd part of $\gcd(\# A\intersect B, \#\Shaan(A))$.

{\bf Table~\ref{table:mordell-weil}}
lists every quotient $A_f$ of $J_0(N)$ with $N\leq 1642$ such
that $L(A_f,1)=0$ but the sign in the functional equation for~$f$ is $+1$.
For each such $A_f$, we looked for an abelian variety~$B$ such that~$B$
has rank~$0$ and $A_f^{\vee}$ probably gives rise to odd visible
elements of $\Sha(B)$.  This table contains initial data towards the idea of constructing
points on high-rank abelian varieties by constructing visible elements
of Shafarevich-Tate groups using, e.g., Euler system methods.
For example, to prove that $A=\mathbf{1061B}$ really has positive rank,
we consider the variety $B=\mathbf{1061D}$.
To prove that $A(\Q)\neq 0$, it suffices to construct an appropriate element of $\Sha(B)$
and show that this element is visible in $A+B\subset J_0(1061)$.

{\bf Table~\ref{table:motive}} suggests a first tenuous step towards a
computational theory of motives attached to modular forms of weight
greater than two.  This table is organized like
Table~\ref{table:prime}, except that the abelian varieties are
replaced by motives attached to weight~$4$ modular forms.  For
example, at prime level~$127$ there is a $17$-dimensional motive $\cM$
such that $\Sha(\cM)(2)$ seems to contain elements of order~$43$.  The
computations used to suggest this conclusion were carried out using
algorithms for higher weight modular symbols as described in
\cite{merel:1585}, \cite{stein:phd}, and \cite{stein-verrill:periods}.

\subsection{Odd invisible $\BigSha$ in \cite{cremona-mazur}}
\label{table:invisible}
$$\begin{array}{lccll} \mbox{\rm\bf E}&\#\Sha(E)& \text{mod deg}(E) & \mbox{\rm\bf F} & \text{Where \Sha(E) is visible}\\ & & & & \vspace{-3ex} \\ \mbox{\rm\bf 2849A}& 3^2 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 3\cdot 2849}\\ \mbox{\rm\bf 3364C}& 7^2 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& \text{visible using a 3-dimensional F at level 3364}\\ \mbox{\rm\bf 4229A}& 3^2 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& \text{not visible at level 4299,}\\ &&&&\text{???}\\ \mbox{\rm\bf 4343B}& 3^2 &2^4\cdot1583 &\mbox{\rm\bf NONE}& ???\\ \mbox{\rm\bf 4914N}& 3^2 &2^4\cdot 3^5 &\mbox{\rm\bf none}& ??? \\ \mbox{\rm\bf 5054C}& 3^2 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& ???\\ \mbox{\rm\bf 5073D}& 3^2 &2^5\cdot 3\cdot 5\cdot7\cdot23 &\mbox{\rm\bf none}& ???\\ \mbox{\rm\bf 5389A}& 3^2 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 7\cdot 5389}\\ \end{array}$$

\comment{\subsubsection*{Remarks}
The elliptic curve~$E$ denoted {\bf 3364C} is  labeled none'' because there is no
{\em elliptic curve} that satisfies an appropriate $7$-congruence with
{\bf 3364C}.  However, the modular degree is divisible by~$7$, so there
must be some abelian subvariety that satisfies a $7$-congruence with~$E$.
Computing, we find a $3$-dimensional abelian variety~$A$ such that $T_2$, $T_3$,
and $T_5$ satisfies the polynomials $x^3$, $x^3 + 5x^2 + 6x + 1$, and
$x^3 + 5x^2 + 6x + 1$ on~$A$, respectively.  Furthermore, $L(A,1)=0$, so
the BSD conjecture strongly suggests that there are elements of $\Sha(E)$
of order~$7$ that are visible in $E+A \subset J_0(3364)$.
}

\newpage
\subsection{Visibility of $\BigSha$ at prime level}\label{table:prime}
The entries in the columns mod deg'' and $\Shaan$'' are only really
the odd parts of mod deg'' and $\Shaan$''.  Theorem~\ref{thm:shaexists} does not
apply to the two entries marked with a $*$.\vspace{-.25ex}
{\small
$$\begin{array}{lccclcc} \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)\\ & & & & & & \vspace{-3ex} \\ \mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ \mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ \mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ \mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ \mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ \mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ \mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ \mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ \mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ \mbox{\rm\bf 1531D}*& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ \mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ \mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ \mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ \mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ \mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ \mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ \mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ \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\\ \mbox{\rm\bf 1933C}*& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ \mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ \mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ \mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ \mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2213C}& 101 & 3^4 & ? & \mbox{\rm NONE} & & \\ \mbox{\rm\bf 2239B}& 110 & 11^4 & 1 & \mbox{\rm NONE} & & \\ \mbox{\rm\bf 2251E}& 99 & 37^2 & 37 & \mbox{\rm\bf 2251A} & 1 & 37\\ \mbox{\rm\bf 2273C}& 105 & 7^2 & ? & \mbox{\rm NONE}& & \\ \mbox{\rm\bf 2287B}& 109 & 71^2 & 1 & \mbox{\rm NONE}& & \\ \mbox{\rm\bf 2293C}& 96 & 479^2& 479 & \mbox{\rm\bf 2293A} & 2 & 479\\ \mbox{\rm\bf 2311B}& 110 & 5^2 & 1 & \mbox{\rm NONE}& & \\ \mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ \mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2593C}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213&\mbox {\bf 2593A}& 4 & 67 \cdot 2213\\ \end{array}$$
}

\subsection{More $\Sha$ at prime level}\label{table:prime2}
Only odd parts of $\Shaan$ and congruences are given.
Observe that $\Shaan$ is only visible roughly 10 percent of the time!
As the level gets large, we find that there is almost always some
nontrivial $\Sha$ in a large-dimensional factor of $J_0(p)$, and that
this $\Sha$ is invisible.
(Warning: In making this table, $53$ primes below $5647$ were not analyzed.)
{\tiny
$$\hspace{-6em}\begin{array}{lcccc|} N & d(A) & \Shaan & d(B) & cong\\ \mathbf{2609} & 127 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ \mathbf{2617} & 114 & 11^{2}\cdot19^{2} & 2 & 11\cdot19 \\ \mathbf{2647} & 117 & 13^{2} & \text{NONE} & \\ \mathbf{2659} & 123 & 53^{2} & \text{NONE} & \\ \mathbf{2663} & 132 & 43^{2} & \text{NONE} & \\ \mathbf{2671} & 122 & 37^{2} & \text{NONE} & \\ \mathbf{2677} & 115 & 3^{2} & 1 & 3 \\ \mathbf{2693} & 122 & 3^{4} & \text{NONE} & \\ \mathbf{2699} & 125 & 19^{2} & \text{NONE} & \\ \mathbf{2707} & 119 & 5^{2} & \text{NONE} & \\ \mathbf{2713} & 118 & 19^{2} & \text{NONE} & \\ \mathbf{2731} & 124 & 53^{2} & \text{NONE} & \\ \mathbf{2749} & 124 & 7^{2} & \text{NONE} & \\ \mathbf{2767} & 125 & 5^{2} & \text{NONE} & \\ \mathbf{2789} & 136 & 83^{2} & \text{NONE} & \\ \mathbf{2791} & 135 & 29^{2} & \text{NONE} & \\ \mathbf{2797} & 119 & 11^{2} & 1 & 11 \\ \mathbf{2819} & 138 & 13^{2} & \text{NONE} & \\ \mathbf{2837} & 128 & 23^{2} & 1 & 23 \\ \mathbf{2843} & 129 & 3^{6}\cdot587^{2} & \text{NONE} & \\ \mathbf{2851} & 129 & 7^{2} & \text{NONE} & \\ \mathbf{2861} & 133 & 11^{4}\cdot61^{2} & 2 & 11\cdot61 \\ \mathbf{2879} & 148 & 97^{2} & \text{NONE} & \\ \mathbf{2903} & 150 & 643^{2} & \text{NONE} & \\ \mathbf{2939} & 150 & 17^{2}\cdot19^{2} & \text{NONE} & \\ \mathbf{2953} & 127 & 29^{2} & 1 & 29 \\ \mathbf{2963} & 134 & 5^{2}\cdot31^{2}\cdot61^{2} & 2 & 31\cdot61 \\ \mathbf{2969} & 136 & 103^{2} & \text{NONE} & \\ \mathbf{2999} & 161 & 1459^{2} & \text{NONE} & \\ \mathbf{3001} & 132 & 3^{4} & \text{NONE} & \\ \mathbf{3011} & 146 & 5^{2}\cdot101^{2} & \text{NONE} & \\ \mathbf{3019} & 130 & 3259^{2} & 2 & 3259 \\ \mathbf{3041} & 147 & 103^{2} & \text{NONE} & \\ \mathbf{3067} & 134 & 5^{4} & \text{NONE} & \\ \mathbf{3079} & 148 & 131^{2} & \text{NONE} & \\ \mathbf{3083} & 141 & 179^{2} & \text{NONE} & \\ \mathbf{3089} & 135 & 5^{2}\cdot131^{2} & 2 & 5\cdot131 \\ \mathbf{3109} & 136 & 5^{2} & \text{NONE} & \\ \mathbf{3119} & 164 & 11^{2}\cdot59^{2} & \text{NONE} & \\ \mathbf{3181} & 144 & 43^{2} & \text{NONE} & \\ \mathbf{3187} & 139 & 3^{4} & \text{NONE} & \\ \mathbf{3191} & 167 & 53^{2} & \text{NONE} & \\ \mathbf{3203} & 143 & 13^{2} & \text{NONE} & \\ \mathbf{3221} & 149 & 7^{2}\cdot41^{2} & \text{NONE} & \\ \mathbf{3229} & 142 & 3^{2} & \text{NONE} & \\ \mathbf{3251} & 166 & 3^{4} & \text{NONE} & \\ \mathbf{3257} & 143 & 13^{2} & \text{NONE} & \\ \mathbf{3271} & 146 & 7^{4}\cdot43^{2}\cdot71^{2} & 3 & 7\cdot43\cdot71 \\ \mathbf{3299} & 164 & 6131^{2} & \text{NONE} & \\ \mathbf{3301} & 145 & 5^{2} & \text{NONE} & \\ \mathbf{3319} & 158 & 5^{4} & \text{NONE} & \\ \mathbf{3323} & 155 & 179^{2} & \text{NONE} & \\ \mathbf{3329} & 157 & 83^{2} & \text{NONE} & \\ \mathbf{3331} & 152 & 937^{2} & \text{NONE} & \\ \mathbf{3343} & 148 & 7^{2}\cdot53^{2} & \text{NONE} & \\ \mathbf{3347} & 150 & 139^{2} & \text{NONE} & \\ \mathbf{3359} & 174 & 67^{4} & \text{NONE} & \\ \mathbf{3371} & 159 & 1259^{2} & \text{NONE} & \\ \mathbf{3391} & 159 & 29^{2} & \text{NONE} & \\ \mathbf{3407} & 170 & 499^{2} & \text{NONE} & \\ \mathbf{3433} & 148 & 5^{4}\cdot7^{2} & \text{NONE} & \\ \mathbf{3449} & 168 & 107^{2} & \text{NONE} & \\ \mathbf{3461} & 167 & 83^{2} & \text{NONE} & \\ \mathbf{3463} & 151 & 199^{2} & 2 & 199 \\ \mathbf{3467} & 162 & 5^{4} & \text{NONE} & \\ \mathbf{3469} & 151 & 47^{2} & \text{NONE} & \\ \mathbf{3491} & 168 & 67^{2} & \text{NONE} & \\ \mathbf{3511} & 166 & 37^{2} & \text{NONE} & \\ \mathbf{3527} & 179 & 659^{2} & \text{NONE} & \\ \mathbf{3529} & 153 & 79^{2} & \text{NONE} & \\ \mathbf{3533} & 164 & 3^{4} & \text{NONE} & \\ \mathbf{3539} & 170 & 1871^{2} & \text{NONE} & \\ \mathbf{3541} & 156 & 5^{4} & \text{NONE} & \\ \mathbf{3557} & 156 & 229^{2} & \text{NONE} & \\ \mathbf{3559} & 170 & 1109^{2} & \text{NONE} & \\ \end{array} \begin{array}{|lcccc|} N & d(A) & \Shaan & d(B) & cong\\ \mathbf{3571} & 163 & 67^{2} & \text{NONE} & \\ \mathbf{3583} & 161 & 3319^{2} & 2 & 3319 \\ \mathbf{3607} & 159 & 7^{4}\cdot19^{2} & \text{NONE} & \\ \mathbf{3613} & 156 & 7^{2} & \text{NONE} & \\ \mathbf{3617} & 165 & 3^{2} & \text{NONE} & \\ \mathbf{3623} & 172 & 3^{6} & \text{NONE} & \\ \mathbf{3631} & 172 & 433^{2} & \text{NONE} & \\ \mathbf{3643} & 160 & 5^{2} & \text{NONE} & \\ \mathbf{3659} & 181 & 3^{2}\cdot11^{4} & \text{NONE} & \\ \mathbf{3671} & 193 & 509^{2} & \text{NONE} & \\ \mathbf{3691} & 166 & 353^{2} & \text{NONE} & \\ \mathbf{3701} & 174 & 3^{4}\cdot281^{2} & 2 & 3^{2}\cdot281 \\ \mathbf{3709} & 164 & 3^{12} & \text{NONE} & \\ \mathbf{3719} & 188 & 13^{2}\cdot977^{2} & \text{NONE} & \\ \mathbf{3739} & 166 & 83^{2} & \text{NONE} & \\ \mathbf{3761} & 176 & 677^{2} & \text{NONE} & \\ \mathbf{3769} & 168 & 13^{2} & \text{NONE} & \\ \mathbf{3779} & 187 & 73^{2}\cdot149^{2} & 1 & 73 \\ \mathbf{3797} & 172 & 19^{2} & \text{NONE} & \\ \mathbf{3803} & 171 & 2531^{2} & \text{NONE} & \\ \mathbf{3821} & 182 & 307^{2} & \text{NONE} & \\ \mathbf{3823} & 173 & 7^{2} & \text{NONE} & \\ \mathbf{3863} & 191 & 11^{2}\cdot23^{2}\cdot311^{2} & \text{NONE} & \\ \mathbf{3907} & 168 & 3^{4} & \text{NONE} & \\ \mathbf{3919} & 182 & 71^{2} & \text{NONE} & \\ \mathbf{3929} & 185 & 877^{2} & \text{NONE} & \\ \mathbf{3931} & 174 & 31^{2} & \text{NONE} & \\ \mathbf{3943} & 173 & 2479319^{2} & 4 & 2479319 \\ \mathbf{3967} & 180 & 3^{6}\cdot13^{2} & 1 & 3\cdot13 \\ \mathbf{4007} & 195 & 7321^{2} & \text{NONE} & \\ \mathbf{4013} & 176 & 61^{2} & \text{NONE} & \\ \mathbf{4019} & 186 & 3^{4}\cdot5^{2}\cdot7^{4} & \text{NONE} & \\ \mathbf{4021} & 182 & 5^{4}\cdot71^{2} & \text{NONE} & \\ \mathbf{4027} & 174 & 29^{2}\cdot79^{2} & 2 & 29\cdot79 \\ \mathbf{4049} & 186 & 5^{2}\cdot3491^{2} & \text{NONE} & \\ \mathbf{4057} & 173 & 103^{2} & \text{NONE} & \\ \mathbf{4079} & 212 & 5^{2}\cdot157^{2}\cdot179^{2} & \text{NONE} & \\ \mathbf{4091} & 203 & 7^{4} & \text{NONE} & \\ \mathbf{4093} & 174 & 3^{2}\cdot89^{4} & 2 & 89^{2} \\ \mathbf{4099} & 185 & 3^{4}\cdot19^{2} & \text{NONE} & \\ \mathbf{4111} & 190 & 229^{2} & \text{NONE} & \\ \mathbf{4139} & 188 & 29^{2}\cdot67^{2} & 1 & 67 \\ \mathbf{4153} & 177 & 7^{2} & \text{NONE} & \\ \mathbf{4157} & 193 & 373^{2} & \text{NONE} & \\ \mathbf{4159} & 188 & 997^{2} & \text{NONE} & \\ \mathbf{4177} & 183 & 3^{2}\cdot17^{2} & \text{NONE} & \\ \mathbf{4217} & 186 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ \mathbf{4219} & 190 & 71^{2} & \text{NONE} & \\ \mathbf{4229} & 1 & 3^{2} & \text{NONE} & \\ \mathbf{4229} & 194 & 3^{4} & \text{NONE} & \\ \mathbf{4231} & 201 & 3^{6} & \text{NONE} & \\ \mathbf{4253} & 184 & 3^{6}\cdot2843^{2} & 3 & 3^{3}\cdot2843 \\ \mathbf{4261} & 185 & 5^{2} & \text{NONE} & \\ \mathbf{4271} & 210 & 163^{2}\cdot853^{2} & \text{NONE} & \\ \mathbf{4273} & 183 & 181^{2} & \text{NONE} & \\ \mathbf{4283} & 198 & 683^{2} & \text{NONE} & \\ \mathbf{4289} & 205 & 8807^{2} & \text{NONE} & \\ \mathbf{4339} & 196 & 17^{2} & \text{NONE} & \\ \mathbf{4349} & 191 & 127^{2} & \text{NONE} & \\ \mathbf{4357} & 187 & 7^{2}\cdot13^{2}\cdot17^{2} & 1 & 7\cdot13 \\ \mathbf{4373} & 199 & 3^{12}\cdot29^{2} & \text{NONE} & \\ \mathbf{4391} & 222 & 5^{4}\cdot372037^{2} & \text{NONE} & \\ \mathbf{4409} & 200 & 157^{2} & \text{NONE} & \\ \mathbf{4421} & 206 & 1523^{2} & \text{NONE} & \\ \mathbf{4423} & 200 & 3^{6}\cdot587^{2} & \text{NONE} & \\ \mathbf{4441} & 198 & 59^{2}\cdot101^{2} & \text{NONE} & \\ \mathbf{4451} & 213 & 809^{2} & \text{NONE} & \\ \mathbf{4457} & 199 & 337^{2} & \text{NONE} & \\ \mathbf{4463} & 213 & 8951^{2} & \text{NONE} & \\ \mathbf{4483} & 193 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ \mathbf{4517} & 201 & 181^{2} & \text{NONE} & \\ \mathbf{4519} & 202 & 2503^{2} & \text{NONE} & \\ \mathbf{4547} & 205 & 73^{2} & 1 & 73 \\ \mathbf{4549} & 203 & 19^{2}\cdot53^{2} & \text{NONE} & \\ \mathbf{4591} & 215 & 6317^{2} & \text{NONE} & \\ \end{array} \begin{array}{|lcccc} N & d(A) & \Shaan & d(B) & cong\\ \mathbf{4597} & 200 & 7^{2}\cdot17^{2} & \text{NONE} & \\ \mathbf{4603} & 198 & 829^{2} & \text{NONE} & \\ \mathbf{4621} & 196 & 13^{2} & \text{NONE} & \\ \mathbf{4639} & 218 & 89^{2} & \text{NONE} & \\ \mathbf{4649} & 215 & 751^{2} & \text{NONE} & \\ \mathbf{4651} & 210 & 13^{4} & \text{NONE} & \\ \mathbf{4673} & 207 & 11^{2}\cdot197^{2} & \text{NONE} & \\ \mathbf{4691} & 216 & 43^{2} & \text{NONE} & \\ \mathbf{4729} & 204 & 673^{2} & \text{NONE} & \\ \mathbf{4733} & 210 & 17^{2} & \text{NONE} & \\ \mathbf{4783} & 210 & 797^{2} & \text{NONE} & \\ \mathbf{4789} & 206 & 13^{4} & \text{NONE} & \\ \mathbf{4799} & 230 & 3^{2}\cdot7^{2}\cdot12203^{2} & 1 & 3\cdot7 \\ \mathbf{4801} & 213 & 60271^{2} & \text{NONE} & \\ \mathbf{4813} & 207 & 3^{2}\cdot6883^{2} & \text{NONE} & \\ \mathbf{4817} & 214 & 283^{2} & \text{NONE} & \\ \mathbf{4831} & 217 & 1151^{2} & \text{NONE} & \\ \mathbf{4861} & 216 & 204749^{2} & \text{NONE} & \\ \mathbf{4877} & 219 & 3^{4}\cdot103^{2} & \text{NONE} & \\ \mathbf{4931} & 240 & 17^{2}\cdot37^{2}\cdot43^{2} & \text{NONE} & \\ \mathbf{4933} & 211 & 239^{2} & \text{NONE} & \\ \mathbf{4957} & 212 & 5^{2} & \text{NONE} & \\ \mathbf{4967} & 236 & 7^{2}\cdot53881^{2} & \text{NONE} & \\ \mathbf{4969} & 220 & 11^{4} & \text{NONE} & \\ \mathbf{4973} & 223 & 5^{2}\cdot11^{2} & \text{NONE} & \\ \mathbf{4993} & 215 & 4013^{2} & \text{NONE} & \\ \mathbf{4999} & 224 & 985121^{2} & \text{NONE} & \\ \mathbf{5003} & 220 & 97^{2}\cdot1861^{2} & 3 & 97\cdot1861 \\ \mathbf{5009} & 223 & 23^{2}\cdot977^{2} & \text{NONE} & \\ \mathbf{5011} & 229 & 11^{4} & \text{NONE} & \\ \mathbf{5021} & 225 & 1609^{2} & \text{NONE} & \\ \mathbf{5023} & 221 & 51431^{2} & \text{NONE} & \\ \mathbf{5039} & 251 & 166363^{2} & \text{NONE} & \\ \mathbf{5051} & 239 & 13^{2}\cdot2633^{2} & \text{NONE} & \\ \mathbf{5059} & 229 & 5^{2}\cdot13^{2}\cdot31^{2} & \text{NONE} & \\ \mathbf{5077} & 216 & 283^{2} & \text{NONE} & \\ \mathbf{5081} & 240 & 19^{2}\cdot149^{2} & \text{NONE} & \\ \mathbf{5099} & 251 & 7^{4}\cdot11^{2}\cdot461^{2} & \text{NONE} & \\ \mathbf{5113} & 223 & 19^{2}\cdot61^{2} & \text{NONE} & \\ \mathbf{5119} & 232 & 53^{2}\cdot103^{2} & \text{NONE} & \\ \mathbf{5153} & 223 & 3^{4}\cdot41^{2} & \text{NONE} & \\ \mathbf{5167} & 231 & 367^{2} & \text{NONE} & \\ \mathbf{5171} & 249 & 73^{2}\cdot773^{2} & 1 & 73 \\ \mathbf{5179} & 226 & 7^{2}\cdot13^{2} & \text{NONE} & \\ \mathbf{5189} & 240 & 83^{2} & \text{NONE} & \\ \mathbf{5197} & 223 & 37^{2} & \text{NONE} & \\ \mathbf{5209} & 227 & 181^{2}\cdot1471^{2} & \text{NONE} & \\ \mathbf{5227} & 232 & 3^{2}\cdot7717^{2} & \text{NONE} & \\ \mathbf{5231} & 255 & 4507^{2} & \text{NONE} & \\ \mathbf{5233} & 223 & 163^{2} & \text{NONE} & \\ \mathbf{5237} & 229 & 7^{2} & \text{NONE} & \\ \mathbf{5261} & 239 & 24103^{2} & \text{NONE} & \\ \mathbf{5273} & 227 & 17389^{2} & \text{NONE} & \\ \mathbf{5279} & 263 & 120431^{2} & \text{NONE} & \\ \mathbf{5281} & 232 & 67^{2} & \text{NONE} & \\ \mathbf{5297} & 238 & 397^{2} & \text{NONE} & \\ \mathbf{5303} & 247 & 13^{2}\cdot73^{2}\cdot15467^{2} & \text{NONE} & \\ \mathbf{5309} & 247 & 1822693^{2} & \text{NONE} & \\ \mathbf{5323} & 233 & 3^{4}\cdot120563^{2} & 3 & 120563 \\ \mathbf{5333} & 237 & 967^{2} & \text{NONE} & \\ \mathbf{5347} & 231 & 3643^{2} & \text{NONE} & \\ \mathbf{5501} & 250 & 163^{2} & \text{NONE} & \\ \mathbf{5503} & 241 & 7^{2}\cdot17^{2} & \text{NONE} & \\ \mathbf{5507} & 252 & 103^{2}\cdot233^{2} & \text{NONE} & \\ \mathbf{5519} & 278 & 61^{2}\cdot211469^{2} & \text{NONE} & \\ \mathbf{5521} & 244 & 5^{4} & \text{NONE} & \\ \mathbf{5531} & 253 & 977^{2} & \text{NONE} & \\ \mathbf{5563} & 246 & 3^{4}\cdot1213^{2} & \text{NONE} & \\ \mathbf{5569} & 239 & 3^{4}\cdot5^{2}\cdot13^{2} & \text{NONE} & \\ \mathbf{5573} & 247 & 9901^{2} & \text{NONE} & \\ \mathbf{5581} & 242 & 28927^{2} & \text{NONE} & \\ \mathbf{5591} & 282 & 3^{2}\cdot13^{4}\cdot1061^{2} & \text{NONE} & \\ \mathbf{5639} & 278 & 229717^{2} & \text{NONE} & \\ \mathbf{5641} & 244 & 41^{2}\cdot431^{2} & \text{NONE} & \\ \mathbf{5647} & 245 & 4463^{2} & \text{NONE} & \\ \end{array}$$
}

\subsection{Mordell-Weil groups of positive even rank and the $\BigSha$ they probably induce}
\label{table:mordell-weil}

\noindent
$$\begin{array}{lclccc} L(1)=0 & d & \hspace{-.5em}L(1)\neq 0 & \hspace{-.6em}d & \hspace{-.8em}\text{cong} & \hspace{-.7em} L(1)/\Omega \cdot 2^*\\ & & & & & \vspace{-2ex}\\ \mbox{\bf 389A} & 1 & \mbox{\bf 389E} & 20 & 5 & 25/97\\ \mbox{\bf 433A} & 1 & \mbox{\bf 433D} & 16 & 7 & 49/9\\ \mbox{\bf 446B} & 1 & \mbox{\bf 446F} & 8 & 11 & 121/3\\ \mbox{\bf 563A} & 1 & \mbox{\bf 563E} & 31 & 13 & 169/281\\ \mbox{\bf 571B} & 1 & \mbox{\bf 571D} & 2 & 3 & 9\\ \mbox{\bf 643A} & 1 & \text{NONE} & & & \\ \mbox{\bf 655A} & 1 & \mbox{\bf 655D} & 13 & 9 & 81\\ \mbox{\bf 664A} & 1 & \mbox{\bf 664F} & 8 & 5 & 25\\ \mbox{\bf 681C} & 1 & \mbox{\bf 681B} & 1 & 3 & 9\\ \mbox{\bf 707A} & 1 & \mbox{\bf 707G} & 15 & 13 & 169\\ \mbox{\bf 718B} & 1 & \mbox{\bf 718F} & 7 & 7 & 49\\ \mbox{\bf 794A} & 1 & \mbox{\bf 794G} & 12 & 11 & 121/3\\ \mbox{\bf 817A} & 1 & \mbox{\bf 817E} & 15 & 7 & 49/5\\ \mbox{\bf 916C} & 1 & \mbox{\bf 916G} & 9 & 11 & 121\\ \mbox{\bf 944E} & 1 & \mbox{\bf 944O} & 6 & 7 & 49\\ \mbox{\bf 997B} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ \mbox{\bf 997C} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ \mbox{\bf 1001C} & 1 & \mbox{\bf 1001F} & 3 & 3 & 27\\ \mbox{\bf 1001C} & 1 & \mbox{\bf 1001L} & 7 & 7 & 49\\ \mbox{\bf 1028A} & 1 & \mbox{\bf 1028E} & 14 & 11 & 3267\\ \mbox{\bf 1034A} & 1 & \text{NONE} & & & \\ \mbox{\bf 1041B} & 2 & \mbox{\bf 1041E} & 4 & 5 & 25\\ \mbox{\bf 1041B} & 2 & \mbox{\bf 1041J} & 13 & 25 & 625\\ \mbox{\bf 1058C} & 1 & \mbox{\bf 1058D} & 1 & 5 & 25\\ \mbox{\bf 1061B} & 2 & \mbox{\bf 1061D} & 46 & 151 & 22801/265\\ \mbox{\bf 1070A} & 1 & \mbox{\bf 1070M} & 7 & 15 & 75\\ \mbox{\bf 1073A} & 1 & \text{NONE} & & & \\ \mbox{\bf 1077A} & 1 & \mbox{\bf 1077J} & 15 & 9 & 81\\ \mbox{\bf 1088J} & 1 & \mbox{\bf 1088R} & 2 & 3 & 9\\ \mbox{\bf 1094A} & 1 & \mbox{\bf 1094F} & 13 & 11 & 121/3\\ \mbox{\bf 1102A} & 1 & \mbox{\bf 1102K} & 4 & 3 & 9\\ \mbox{\bf 1126A} & 1 & \mbox{\bf 1126F} & 11 & 11 & 121\\ \mbox{\bf 1132A} & 1 & \mbox{\bf 1132F} & 12 & 5 & 225\\ \mbox{\bf 1137A} & 1 & \mbox{\bf 1137C} & 14 & 9 & 81\\ \mbox{\bf 1141A} & 1 & \mbox{\bf 1141I} & 22 & 7 & 1524537/41\\ \end{array} \begin{array}{lclccc} & & & & & \\ & & & & & \vspace{-2ex}\\ \mbox{\bf 1143C} & 1 & \mbox{\bf 1143J} & 9 & 11 & 121\\ \mbox{\bf 1147A} & 1 & \mbox{\bf 1147H} & 23 & 5 & 225/19\\ \mbox{\bf 1171A} & 1 & \mbox{\bf 1171D} & 53 & 11 & 121/195\\ \mbox{\bf 1246C} & 1 & \mbox{\bf 1246B} & 1 & 5 & 25\\ \mbox{\bf 1309B} & 1 & \text{NONE} & & & \\ \mbox{\bf 1324A} & 1 & \mbox{\bf 1324E} & 14 & 9 & 6561\\ \mbox{\bf 1325E} & 1 & \mbox{\bf 1325T} & 11 & 9 & 2187\\ \mbox{\bf 1363B} & 2 & \mbox{\bf 1363F} & 25 & 31 & 961/5\\ \mbox{\bf 1431A} & 1 & \mbox{\bf 1431L} & 14 & 3 & 9\\ \mbox{\bf 1436A} & 1 & \text{NONE} & & & \\ \mbox{\bf 1443C} & 1 & \mbox{\bf 1443G} & 5 & 7 & 49\\ \mbox{\bf 1446A} & 1 & \mbox{\bf 1446N} & 7 & 3 & 9\\ \mbox{\bf 1466B} & 1 & \mbox{\bf 1466H} & 23 & 13 & \hspace{-2em}\mbox{{\tiny 4331806939187/367}}\\ \mbox{\bf 1477A} & 1 & \mbox{\bf 1477C} & 24 & 13 & 169\\ \mbox{\bf 1480A} & 1 & \mbox{\bf 1480G} & 5 & 7 & 49\\ \mbox{\bf 1483A} & 1 & \mbox{\bf 1483D} & 67 & 15 & 225/247\\ \mbox{\bf 1525C} & 1 & \mbox{\bf 1525O} & 16 & 7 & 49\\ \mbox{\bf 1531A} & 1 & \mbox{\bf 1531D} & 73 & 3 & 3/85\\ \mbox{\bf 1534B} & 1 & \mbox{\bf 1534J} & 6 & 3 & 3\\ \mbox{\bf 1567B} & 3 & \mbox{\bf 1567D} & 69 & 287 & 82369/261\\ \mbox{\bf 1570B} & 1 & \mbox{\bf 1570J} & 6 & 11 & 121\\ \mbox{\bf 1576A} & 1 & \mbox{\bf 1576E} & 14 & 11 & 121\\ \mbox{\bf 1591A} & 1 & \mbox{\bf 1591F} & 35 & 31 & 6727/19\\ \mbox{\bf 1594A} & 1 & \mbox{\bf 1594J} & 17 & 3 & 3370648239/19\\ \mbox{\bf 1608A} & 1 & \mbox{\bf 1608J} & 6 & 13 & 169\\ \mbox{\bf 1611D} & 1 & \mbox{\bf 1611O} & 11 & 9 & 81\\ \mbox{\bf 1613A} & 1 & \mbox{\bf 1613D} & 75 & 5 & 25/403\\ \mbox{\bf 1615A} & 1 & \mbox{\bf 1615J} & 13 & 9 & 102141\\ \mbox{\bf 1621A} & 1 & \mbox{\bf 1621C} & 70 & 17 & 289/135\\ \mbox{\bf 1627A} & 1 & \mbox{\bf 1627C} & 73 & 9 & 81/271\\ \mbox{\bf 1633A} & 3 & \mbox{\bf 1633D} & 27 & 189 & 35721\\ \mbox{\bf 1639B} & 1 & \mbox{\bf 1639G} & 34 & 17 & 680017/25\\ \mbox{\bf 1641B} & 1 & \mbox{\bf 1641J} & 24 & 23 & 529\\ \mbox{\bf 1642A} & 1 & \mbox{\bf 1642D} & 14 & 7 & 49\\ & & & & & \\ \end{array}$$

\mbox{}\par\noindent
{{\bf 643A}, {\bf 1034A}, {\bf 1073A}, {\bf 1309B} all have modular degree
a power of~$2$; {\bf 1436A} has modular degree divisible by~$3$.

\newpage
\subsection{Conjecturally visible $\BigSha$ of modular motives of weight~$4$}
\label{table:motive}
Suppose $f$ and $g$ are elements of $S_4(\Gamma_0(N))$ such that $p^2 \mid L(\sM_f,2)/\Omega$
and $L(\sM_g,2)=0$.   If~$f$ and~$g$ satisfy a $p$-congruence'',
does~$p$ then divide the visible part'' of $\Sha(\sM_f(2))$?

$$\begin{array}{lcclcl} %\hspace{2em}\chi\hspace{2em}\mbox{} & \sM_f \hspace{1em}\mbox{}& \text{dim} & \hspace{2em}p^2\hspace{2em}\mbox{} & \sM_g & \text{dim}\\ & & & & & \vspace{-2ex} \\ %(1,1) & \mbox{\bf 99C} & 8 & 19^2 & \mbox{\bf 99B} & 2 \\ \mbox{\bf 127k4C} & 17 & 43^2 & \mbox{\bf 127k4A} & 1\\ \mbox{\bf 159k4E} & 8 & 23^2 & \mbox{\bf 159k4B} & 1\\ %(0,0,1) & \mbox{\bf 200E} & 8 & 7^2 & \mbox{\bf 200G} & 2\\ \mbox{\bf 365k4E} & 18 & 29^2 & \mbox{\bf 365k4A} & 1\\ \mbox{\bf 369k4I} & 9 & 13^2 & \mbox{\bf 369k4A} & 1\\ \mbox{\bf 453k4E} & 23 & 17^2 & \mbox{\bf 453k4A} & 1\\ \mbox{\bf 465k4H} & 7 & 11^2 & \mbox{\bf 465k4A} & 1\\ \mbox{\bf 477k4L} & 12 & 73^2 & \mbox{\bf 477k4A} & 1\\ \mbox{\bf 567k4G} & 8 & 13^2, 23^2 & \mbox{\bf 567k4A} & 1\\ \mbox{\bf 581k4E} & 34 & 19^2 & \mbox{\bf 581k4A} & 1 \end{array}$$
%Here~$\chi$ is the common nebentypus character of~$f$ and~$g$.

\section{Questions and conjectures}\label{sec:conj}
The following questions and conjectures were motivated by the
tables above and the computations that went into creating them.
The first conjecture suggests a generalization of a result of
Ribet on level raising.  The second conjecture asserts that
$\Sha$ is always visible in an appropriate modular Jacobian.
The third question suggests a new approach to the long-standing
open problem of constructing points on abelian varieties of
analytic rank greater than~$1$ over the Hecke algebra.

\subsection{Level raising nonvanishing conjecture}
Let $f\in S_2(\Gamma_0(N))$ be a newform such that the sign
of the functional equation of $L(A_f,s)$ is equal to $+1$,
and fix a prime~$\lambda$ such
that the associated Galois representation $\rho_{f,\lambda}=A_f[\lambda]$
is irreducible.
For each prime~$q$ not dividing~$N$, let $\delta: J_0(N)\ra J_0(Nq)$ be
the injection obtained from the sum of the two degeneracy maps.
Ribet's construction in \cite{ribet:raising}
produces infinitely many primes~$q$ and newforms
$g\in S_2(\Gamma_0(qN))$ such that
$$\delta(\Adual_f[\lambda]) \subset \delta(\Adual_f)\intersect \Adual_g$$
and the Tamagawa number $c_q$ of $\Adual_g$ is a power of~$2$.
\begin{conjecture}
Fix~$f$ and~$\lambda$.
\begin{enumerate}
\item Then there is a~$g$ among those constructed by Ribet
such that $L(A_g,1)\neq 0$.
\item If~$\lambda$ is in the support of the $\T$-module
$[P(H_1(X_0(N),\Z))^+: P(\T\{0,\infty\})]$
(see Theorem~\ref{thm:ratpart}),
then there is a~$g$ as above such that $L(A_g,1)=0$.
\end{enumerate}
\end{conjecture}

\subsection{Eventual visibility conjecture}
Let~$S$ be the set of all square-free positive integers.
If $M, N\in S$ with $M\mid N$ then there is
a natural injection $J_0(M)\hookrightarrow J_0(N)$, and hence a map
$\Sha(J_0(M))\ra \Sha(J_0(N))$.
These maps are compatible, so the collection of groups $\Sha(J_0(N))$, with
$N\in S$, forms a directed system.  Let
$\lim_{N\in S} \Sha(J_0(N))$
be the direct limit of the $\Sha(J_0(N))$.
\begin{conjecture}
$\lim_{N\in S} \Sha(J_0(N))=0$
\end{conjecture}
If true, this would imply that if $A\subset J_0(N)$, then
each element of $\Sha(A)$ is visible in some $J_0(N')$,
for some multiple~$N'$ of~$N$.

\subsection{Euler systems}
\label{sec:constructing}
In \cite{kolyvagin:structureofsha} and \cite{mccallum:kolyvagin}
one finds a construction using the Heegner point Euler system of
Kolyvagin of the Shafarevich-Tate groups of certain abelian varieties.
Under an unverified hypothesis on Heegner points, the construction
gives much of $\Sha(A_f/K)$, where~$K$ is a suitable imaginary
quadratic field.  Is it possible to verify the unverified hypothesis,
construct $\Vis_J(\Sha(A_f/K))$, and thus prove that $\Sha(\Adual_f/K)$
contains visible elements, when the BSD conjecture suggests that it should?  If so,
it would follow that there is a congruent $B_f^{\vee}$ having positive
algebraic rank, as predicted by the BSD conjecture.  Thus a construction of
visible elements of $\Sha(\Adual_f)$ also leads to a construction of points
on abelian varieties of positive analytic rank.

\bibliography{biblio}

\end{document}