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\begin{document}
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Visibility of the Shafarevich-Tate Group at Higher Level
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Dimitar P. Jetchev \\
William A. Stein
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We study visibility of Shafarevich-Tate groups of modular abelian
varieties in Jacobians of modular curves of higher level. We prove a theorem
about the existence of visible elements at a specific higher level under
hypotheses that can be verified explicitely. We also provide a table
of examples of visible subgroups at higher level and state conjectures inspired
by our data.
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Dimitar P. Jetchev
Department of Mathematics
University of California
Berkeley, CA 94720-3840
{\tt\small jetchev@math.berkeley.edu}
\Address
William A. Stein
Department of Mathematics
University of Washington
Seattle, WA 98195-4350
{\tt\small wstein@u.washington.edu}
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\section{Introduction}\label{sec:intro}
\subsection{Motivation}
Mazur suggested that the Shafarevich-Tate group $\Sha(K, E)$ of an abelian variety
$A$ over a number field $K$ could be studied via a collection of finite subgroups (the \emph{visible
subgroups}) corresponding to different embeddings of the variety into larger abelian varieties
$C$ over $K$ (see~\cite{mazur:visord3} and~\cite{cremona-mazur}). The advantage of this approach is
that the isomorphism classes of principal homogeneous spaces, for which one has \emph{\`a priori} little
geometric information, can be given a much more explicit description as $K$-rational points on the quotient
abelian variety $C / A$ (the reason why they are called \emph{visible elements}).
Agashe, Cremona, Klenke and the second author built upon the ideas of Mazur and developed a systematic theory
of visibility of Shafarevich-Tate groups of abelian varieties over number fields
(see~\cite{agashe:invisible, agashe-stein:visibility, agashe-stein:bsd, cremona-mazur, klenke:phd, stein:phd}).
More precisely, Agashe and Stein provided sufficient conditions for the existence of visible sugroups of
certain order in the Shafarevich-Tate group and applied their general theory to the case of newform subvarieties
${A_f}_{/\Q}$ of the Jacobian $J_0(N)_{/\Q}$ of the modular curve $X_0(N)_{/\Q}$ (here, $f$ is a newform of level
$N$ and weight 2 which is an eigenform for the Hecke operators acting on the space $S_2(\Gamma_0(N))$ of cuspforms
of level $N$ and weight 2). Unfortunately, there is no guarantee that a non-trivial element of
$\Sha(\Q, A_f)$ is visible for the embedding $A_f \hra J_0(N)$.
In this paper we consider the case of modular abelian varieties over
$\Q$ and make use of the algebraic and arithmetic properties of the
corresponding newforms to provide sufficient conditions for the
existence of visible elements of $\Sha(\Q, A_f)$ in modular Jacobians of level a multiple of the base level
$N$. More precisely, we consider morphism of the form $A_f \hra J_0(N) \xra{\phi} J_0(MN)$, where $\phi$ is
a suitable linear combination of degeneracy maps which makes the kernel of the composition morphism almost
trivial (i.e., trivial away from the 2-part). For specific examples, the sufficient conditions can be verified
explicitely. We also provide a table of examples where certain elements of $\Sha(\Q, A_f)$ which are invisible
in $J_0(N)$ become visible at a suitably chosen higher level. At the end, we state some general conjectures
inspired by our results.
\subsection{Organization of the paper}
Section~\ref{sec:notation} discusses the basic definitions and notation for modular abelian varieties, modular forms,
Hecke algebras, the Shimura construction and modular degrees. Section~\ref{sec:visdef} is a brief introduction to
visibility theory for Shafarevich-Tate groups. In Section~\ref{sec:evis} we state and
prove an equivariant version of a theorem of Agashe-Stein (see~\cite[Thm 3.1]{agashe-stein:bsd})
which guarantees existence of visible elements. The theorem is more general because it makes use
of the action of the Hecke algebra on the modular Jacobian.
In Section~\ref{sec:strongvis} we introduce the notion of \emph{strong visibility} which
is relevant for visualizing cohomology classes in Jacobians of modular
curves whose level is a multiple of the level of the original abelian variety.
Theorem~\ref{thm:strongvis} guarantees
existence of strongly visible elements of the Shafarevich-Tate group
under some hypotheses on the component groups, a congruence condition
between modular forms, and irreducibility of the Galois representation.
In Section~\ref{subsec:strongvis-simpler} we prove a variant of the same theorem
(Theorem~\ref{thm:strongestvis}) with more stringent hypotheses that are easier
to verify in specific cases.
Section~\ref{sec:example} discusses in detail two computational examples for
which strongly visible elements of certain order exist which provides evidence
for the Birch and Swinnerton-Dyer conjecture. We state a general conjecture
(Conjecture~\ref{conj1}) in Section~\ref{sec:conj} according to which every element of
the Shafarevich-Tate group of a modular abelian variety becomes visible at higher level.
We provide evidence for the the conjecture in Section~\ref{subsec:evidence} and tables of
computational data in Section~\ref{subsec:data}.
\vspace{2ex}
\noindent{\bf Acknowledgement:} The authors would like to thank David Helm, Ben Howard, Barry Mazur,
Bjorn Poonen and Ken Ribet for discussions and comments on the paper.
\section{Notation}\label{sec:notation}
\noindent \emph{1. Abelian varieties.} For a number field $K$, $A_{/K}$ denotes an abelian variety over $K$.
We denote the dual of $A$ by $A^\vee_{/K}$. If $\vphi: A\to B$ is an isogeny of degree~$n$, we denote the
\emph{complementary isogeny} by $\vphi'$; this is the isogeny $\phi' : B \ra A$, such that
$\vphi\circ \vphi' = \vphi' \circ \vphi = [n]$, the multiplication-by-$n$ map on $A$. Unless otherwise specified,
N\'eron models of abelian varieties will be denoted by the corresponding caligraphic letters, e.g., $\cA$ denotes
the N\'eron model of $A$.
\vspace{0.1in}
\noindent \emph{2. Galois cohomology.}
For a fixed algebraic closure $\Kbar$ of $K$, $G_K$ will be the Galois group
$\Gal(\Kbar/K)$. If $v$ is any non-archimedean place of $K$, $K_v$ and $k_v$ will always mean the completion
and the residue field of $K$ at $v$, respectively. By $K_v^{\ur}$ we always mean the maximal unramified extension
of the completion $K_v$. Given a $G_K$-module $M$, we let $\H^1(K, M)$ denote the
Galois cohomology group $\H^1(G_K, M)$.
\vspace{0.1in}
\noindent \emph{3. Component groups.} The \emph{component group} of $A$ at $v$ is
the finite group $\Phi_{A,v} = \cA_{k_v} / \cA_{k_v}^0$ which also has a structure of a finite
group scheme over $k_v$. The \emph{Tamagawa number} of $A$ at $v$ is
$c_{A,v} = \#\Phi_{A,v}(k_v)$, and the \emph{component group order} of $A$ at $v$ is
$\cbar_{A,v} = \#\Phi_{A,v}(\kbar_v)$.
\vspace{0.1in}
\noindent \emph{4. Modular abelian varieties.} Let $h = 0$ or $1$. A \emph{$J_h$-modular abelian variety}
is an abelian variety $A_{/K}$ which is a quotient of $J_h(N)$ for some $N$, i.e., there exists a surjective
morphism $J_h(N) \twoheadrightarrow A$ defined over $K$. We define the~\emph{level} of a modular abelian variety
$A$ to be the minimal $N$, such that $A$ is a quotient of $J_h(N)$. The modularity theorem of Wiles et al.
(see~\cite{breuil-conrad-diamond-taylor}) implies that all elliptic curves over~$\Q$ are modular.
Serre's modularity conjecture implies that the modular abelian varieties over $\Q$ are precisely
the abelian varieties over~$\Q$ of $\GL_2$-type (see \cite[\S4]{ribet:abvars}).
\vspace{0.1in}
\noindent \emph{5. Shimura construction.}
Let $\displaystyle f=\sum_{n=1}^\infty a_n q^n\in S_2(\Gamma_0(N))$ be a newform of level $N$ and weight 2 for
$\Gamma_0(N)$ which is an eigenform for all Hecke operators in the Hecke algebra $\T(N)$. Shimura
(see~\cite[Thm.~7.14]{shimura:intro}) associated to $f$ an abelian subvariety ${A_f}_{/\Q}$ of $J_0(N)$, simple
over $\Q$, of dimension $d = [K : \Q]$, where $K = \Q(\dots, a_n, \dots)$ is the Hecke eigenvalue field. More
precisely, if $I_f = \Ann_{\T(N)}(f)$ then $A_f$ is the connected component containing the identity of the $I_f$-torsion
subgroup of $J_0(N)$, i.e., $A_f = J_0(N)[I_f]^0 \subset J_0(N)$. The quotient $\T(N)/I_f$ of the Hecke algebra
$\T(N)$ is a subalgebra of the
endomorphism ring $\End_\Q(A_{/\Q})$. Also $\displaystyle L(A_f,s) = \prod_{i = 1}^d L(f_i,s)$, where the $f_i$
are the $G_\Q$-conjugates of~$f$. We also consider the dual abelian variety $A_f^\vee$ which is a quotient variety
of $J_0(N)$.
\comment{
\begin{remark} In this paper $A_f$ always denotes an abelian
subvariety of $J_0(N)$. By abuse of notation, it is also common to denote by $A_f$ the dual of the subvariety $A_f$,
which is a quotient of $J_0(N)$ (see e.g.~\cite{shimura:factors}).
\end{remark}
}
\vspace{0.1in}
\noindent \emph{6. $I$-torsion submodules.} If $M$ is a module over a commutative ring $R$ and $I$
is an ideal of~$R$, let
$$
M[I] = \{ x \in M : mx = 0 \text{ all } m \in I\}
$$
be the \emph{$I$-torsion submodule} of $M$.
\vspace{0.1in}
\noindent \emph{7. Hecke algebras.} Let $S_2(\Gamma)$ denote the space of cusp forms of weight~$2$ for any
congruence subgroup~$\Gamma$ of $\SL_2(\Z)$. Let
$$
\T(N) = \Z[\ldots, T_n, \ldots]\subseteq \End_{\Q}(J_0(N))
$$
be the Hecke algebra, where $T_n$ is the $n$th Hecke operator. $\T(N)$ also acts on $S_2(\Gamma_0(N))$
and the integral homology $H_1(X_0(N),\Z)$.
\vspace{0.1in}
\noindent \emph{8. Modular degree.} If $A$ is an abelian subvariety of $J_0(N)$, let
$$
\theta:A\to J_0(N) \isom J_0(N)^{\vee} \to A^{\vee}
$$
be the induced polarization.
The \emph{modular degree} of~$A$ is
$$
m_A = \sqrt{\#\Ker(A\xra{\theta} A^{\vee})}.
$$
See~\cite{agashe-stein:visibility} for why $m_A$ is an integer and for an algorithm to compute it.
\section{Visible Subgroups of Shafarevich-Tate Groups}\label{sec:visdef}
Let $K$ be a number field and $\iota : A_{/K} \hra C_{/K}$ be an embedding of an abelian variety into another
abelian variety over $K$.
\begin{definition}\label{def:vish1}
The {\em visible subgroup} of $\H^1(K,A)$ relative to~$\iota$ is
\[
\Vis_{C} \H^1(K,A) = \Ker\left(\iota_* : \H^1(K,A)\to \H^1(K,C)\right).
\]
The visible subgroup of $\Sha(K, A)$ relative to the embedding $\iota$
is
\begin{align*}
\Vis_{C} \Sha(K, A) &= \Sha(K, A)\cap \Vis_C \H^1(K,A) \\
&= \Ker\left(\Sha(K, A) \to \Sha(K, C)\right)
\end{align*}
\end{definition}
\vspace{0.1in}
Let $Q$ be the abelian variety $C/\iota(A)$, which is defined over $K$. The long exact sequence of Galois cohomology
corresponding to the short exact sequence $0\to{} A\to{}C\to{}Q \to{}0$ gives rise to the following
exact sequence
\[
0 \to A(K) \to C(K) \to Q(K) \to \Vis_{C} \H^1(K,A)\to 0.
\]
The last map being surjective means that the cohomology classes of
$\Vis_C \H^1(K,A)$ are images of $K$-rational points on $Q$, which explains the meaning of the word \emph{visible}
in the definition. The group $\Vis_C \H^1(K,A)$ is finite since it is torsion and since the Mordell-Weil group
$Q(K)$ is finitely generated.
\begin{remark}\label{rem:vis}
If~$A_{/K}$ is an abelian variety and $c \in \H^1(K,A)$ is any cohomology class,
there exists an abelian variety $C_{/K}$ and an embedding $\iota : A \hra C$ defined over
$K$, such that $c\in\Vis_C \H^1(K,A)$, i.e., $c$ is visible in $C$
(see~\cite[Prop.~1.3]{agashe-stein:visibility}).
The $C$ of \cite[Prop.~1.3]{agashe-stein:visibility} is
the restriction of scalars of $A_L = A \times_K L$ down to~$K$,
where~$L$ is any finite extension of~$K$ such that~$c$ has trivial image in $\H^1(L,A)$.
\end{remark}
\section{Equivariant Visibility}\label{sec:evis}
Let $K$ be a number field, let $A_{/K}$ and $B_{/K}$ be abelian
subvarieties of an abelian variety $C_{/K}$, such that
$C=A+B$ and $A \cap B$ is finite. Let $Q_{/K}$ denotes the quotient $C/B$.
Let $N$ be a positive integer divisible by all primes of bad reduction
for $C$.
Let $\ell$ be a prime such that $B[\ell]\subset A$
and $e < \ell-1$, where~$e$ is the largest ramification index of any
prime of $K$ lying over~$\ell$. Suppose that $$\ell
\nmid N \cdot \#B(K)_{\tor} \cdot \#Q(K)_{\tor} \cdot
\prod_{v\mid N} c_{A,v} c_{B,v}.$$
Under those conditions, Agashe and Stein (see~\cite[Thm.~3.1]{agashe-stein:visibility})
construct a homomorphism $B(K)/\ell B(K) \ra \Sha(K, A)[\ell]$ whose kernel has
$\F_\ell$-dimension bounded by the Mordell-Weil rank of $A(K)$.
In this paper, we refine \cite[Prop.~1.3]{agashe-stein:visibility} by taking into account the algebraic
structure coming from the endomorphism ring $\End_K(C)$.
In particular, when we apply the theory to modular abelian varieties, we would like to use the
additional structure coming from the Hecke algebra. There are numerous example (see~\cite{agashe-stein:bsd})
where \cite[Prop.~1.3]{agashe-stein:visibility} does not apply, but nevertheless, we can use our refinement
to prove existence of visible elements of $\Sha(\Q, A_f)$ at higher level (e.g.,
see Propositions~\ref{prop:767vis} and~\ref{prop:959vis} below).
\subsection{The main theorem}
Let $A_{/K}$, $B_{/K}$, $C_{/K}$, $Q_{/K}$, $N$ and $\ell$ be as above. Let~$R$ be a commutative subring of
$\End_K(C)$ that leaves $A$ and $B$ stable and let $\m$
be a maximal ideal of~$R$ of residue characteristic~$\ell$. By the N\'eron mapping property,
the subgroups $\Phi_{A, v}(k_v)$ and $\Phi_{B, v}(k_v)$ of $k_v$-points of the corresponding component groups
can be viewed as $R$-modules.
\begin{theorem}[Equivariant Visibility Theorem]\label{thm:evis}
Suppose that $A(K)$ has rank zero and that the groups $Q(K)[\m]$, $B(K)[\m]$,
$\Phi_{A,v}(k_v)[\m]$ and $\Phi_{B,v}(k_v)[\ell]$ are all trivial
for all nonarchimedean places $v$ of $K$. Then there is an injective homomorphism of
$R/\m$-vector spaces
\begin{equation}\label{eqn:evis}
(B(K)/\ell{}B(K))[\m]\hra \Vis_C(\Sha(K, A))[\m].
\end{equation}
\end{theorem}
\begin{remark}
Applying the above result for $R=\Z$, we recover the result of Agashe and Stein in the case when $A(K)$ has
Mordell-Weil rank zero. We could relax the hypothesis that $A(K)$ is finite and instead give a bound on
the dimension of the kernel of (\ref{eqn:evis}) in terms of the rank of $A(K)$ similar to the bound
in~\cite[Thm.~3.1]{agashe-stein:visibility}. We will not need this stronger result in our paper.
\end{remark}
\subsection{Some commutative algebra}
Before proving Theorem~\ref{thm:evis} we recall
some well-known lemmas from commutative algebra.
Let $M$ be a module over a commutative
ring $R$ and let~$\m$ be a finitely
generated prime ideal of~$R$.
%\edit{Ken says:
%Lemma 3.4 is very well known. I wonder if you can find a reference
%and not have to give a proof. Did you look in Bourbaki Algebre
%Commutative I and II?}
\begin{lemma}\label{lem:mtor}
If $M_\m$ is Artinian, then $M_\m \neq 0\iff M[\m] \neq 0.$
\end{lemma}
\begin{proof}
$(\Longleftarrow)$
We first prove that $M_\m=0$ implies $M[\m]=0$
by a slight modification of the proof of \cite[Prop.~I.3.8]{am}.
Suppose $M_\m=0$, yet there is a nonzero $x\in M[\m]$.
Let $I=\Ann_R(x)$. Then $I\neq (1)$ is an ideal that
contains $\m$, so $I=\m$. Consider $\frac{x}{1} \in M_\m$.
Since $M_\m=0$, we have $x/1=0$, hence by definition
of localization, $x$ is killed by
some element of $R-\m$ (set-theoretic difference).
But this is impossible since $\Ann_R(x)=\m$.
$(\Longrightarrow)$
Next we prove that $M_\m \neq 0$ implies
$M[\m]\neq 0$.
Since $M_\m$ is an Artinian module over the (local)
ring $R_\m$, by \cite[Prop.~6.8]{am}, $M_\m$
has a composition series:
$$
M_\m = M_0 \supset M_1 \supset \cdots \supset
M_{n-1} \supset M_n = 0,
$$
where by definition each quotient $M_i/M_{i+1}$ is a simple
$R_\m$-module. In particular, $M_{n-1}$ is a simple
$R_\m$-module. Suppose $x\in M_{n-1}$ is nonzero, and
let $I=\Ann_{R_\m}(x)$. Then $$R_\m/I \isom R_\m\cdot x \subset M_{n-1},$$
so by simplicity $R_\m /I \isom M_{n-1}$ is simple. Thus
$I=\m$, otherwise $R_\m/I$ would have $\m/I$ as a
proper submodule. Thus $x\in M_{n-1}[\m]$ is nonzero.
Write $x=[y,a]$ with $y\in M$ and $a\in R-\m$, where $[y/a]$ means the
class of $y/a$ in the localization (same as $(y,a)$ on page 36 of
\cite{am}). Since $a\in R - \m$, the element $a$ acts as a unit on
$M_\m$, hence $ax = [y/1] \in M_{n-1}$ is nonzero and also still
annihilated by $\m$ (by commutativity).
To say that $[y/1]$ is annihilated by $\m$ means that for all
$\alpha\in \m$ there exists $t\in R-\m$ such that $t\alpha y = 0$ in
$M$. Since $\m$ is finitely generated, we can write
$\m=(\alpha_1,\ldots, \alpha_n)$ and for each $\alpha_i$ we get
corresponding elements $t_1,\ldots, t_n$ and a product $t=t_1\cdots
t_n$. Also $t\not\in\m$ since $\m$ is a prime ideal and each
$t_i\not\in\m$. Let $z=ty$. Then for all $\alpha\in\m$ we have
$\alpha z = t\alpha y = 0$. Also $z\neq 0$ since $t$ acts as a unit
on $M_{n-1}$. Thus $z\in M[\m]$, and is nonzero, which completes the
proof of the lemma.
\end{proof}
\begin{lemma}\label{lem:ex_art}
Suppose
$
0 \to M_1 \to N \to M_2 \to 0
$
is an exact sequence of $R$-modules each of
whose localization at $\m$ is Artinian.
Then
$N[\m]\neq 0 \iff (M_1\oplus M_2) [\m]\neq 0.$
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:mtor} we have $N[\m]\neq 0$
if and only if $N_\m\neq 0$.
By Proposition 3.3 on page 39 of \cite{am}, the localized sequence
$$
0 \to (M_1)_\m \to N_\m \to (M_2)_\m \to 0
$$
is exact.
Thus $N_\m\neq 0$ if and only if at least one
of $(M_1)_\m$ or $(M_2)_\m$ is nonzero.
Again by Lemma~\ref{lem:mtor}, at least one
of $(M_1)_\m$ or $(M_2)_\m$ is nonzero if and
only if at least one of $M_1[\m]$ or $M_2[\m]$
is nonzero. The latter is the case
if and only if $(M_1\oplus M_2)[\m]\neq 0$.
\end{proof}
\begin{remark}
One could also prove the lemmas using the isomorphism $M[\m]\isom
\Hom_R(R/\m, M)$ and exactness properties of $\Hom$, but even with
this approach many of the details in Lemma~\ref{lem:mtor} still have
to be checked.
\end{remark}
\begin{remark}
In Theorem~\ref{thm:evis}, we
have $R\subset \End(C)$, hence $R$
is finitely generated as a $\Z$-module, so $R$ is noetherian.
\end{remark}
\begin{lemma}\label{lem:h0h1}
Let $G$ be a finite cyclic group,
$M$ be a finite $G$-module that
is also a module over a commutative ring~$R$ such
that the action of $G$ and $R$ commute (i.e.,
$M$ is an $R[G]$-module).
Suppose $\p$ is a finitely-generated prime ideal of $R$,
and $H^0(G,M)[\p]=0$.
Then $H^1(G, M)[\p]= 0$.
\end{lemma}
\begin{proof}
Argue as in \cite[Prop. VIII.4.8]{serre:localfields}, but
noting that all modules are modules over~$R$ and maps are
morphisms of $R$-modules.
\end{proof}
\comment{
\begin{proof}
This argument was inspired by the proof
of \cite[Prop. VIII.4.8]{serre:localfields}.
Let $s$ be a generator for $G$, and let $D=s-1$.
There is an exact sequence
\begin{equation}\label{eqn:agex}
0 \to M^G \to M \xra{D} M \to M_G \to 0,
\end{equation}
and each map is a homomorphism of $R$-modules\np{ (here
we use that the action of~$R$ and~$G$ commutes, so
that the middle map is an $R$-module homomorphism)}.
By \cite[Prop.~3.3]{am},
the localization
$$
0 \to (M^G)_\p \to M_\p \xra{D} M_\p \to (M_G)_\p \to 0
$$
of (\ref{eqn:agex}) is exact. By hypothesis
$H^0(G,M)[\p]=0$, and by definition $M^G = H^0(G,M)$,
so by Lemma~\ref{lem:mtor}, $(M^G)_\p = 0$,
and the following sequence is exact:
$$
0 \to M_\p \xra{D} M_\p \to (M_G)_\p \to 0.
$$
But $D:M_\p \to M_\p$ is an injective map
of finite sets, so it is a bijection, hence $(M_G)_\p=0$.
Again using Lemma~\ref{lem:mtor} it follows that $M_G[\p]=0$.
It follows from \cite[Ch. VIII, \S4]{serre:localfields} that
$$H^1(G,M) \isom M[N]/DM \qquad\text{(as functors in $M$)},$$
where $M[N]$ is the kernel
of the map $M \xra{N} M$, and $N = \sum_{g \in G}g$. Since $M_G M/DM$, we have an $R$-module inclusion
$H^1(G,M) \hra M_G$. We showed above that $M_G[\p]=0$, so the lemma follows.
\end{proof}
}
\comment{\begin{remark}
If $\p$ were replaced by a prime number $p\in\Z$ then the result
would be immediate since using Herbrand quotients one shows that
$\#H^0(G,M) = \#H^1(G,M)$ (see
\cite[Prop.~VIII.4.8]{serre:localfields}). It is unclear to the
authors if the result is true in general, i.e., if~$G$ is replaced
by an arbitrary group.
\end{remark}}
%\subsection{A Lemma About Divisible Points}
%\edit{Ribet's remark about this lemma: I find Lemma 3.10 very
% strangely stated. Why not say simply that a certain group is
% divisible by every integer n prime to the residue characteristic;
% the group might be denoted $A^0(K)$. I imagine that this lemma is in
% SGA 7I Expose IX (by Grothendieck). Is it really true that x can be
% divided by n only in $A(K)$ (and not in $A^0(K)$)? }
%The following proposition will be necessary for proving that certain
%cohomology clases are locally trivial. For proofs of this result,
%see~\cite[\S 3.2]{agashe-stein:visibility}.
\comment{
\begin{lemma}[Grothendieck]\label{lem:divisible}
Let $A$ be an abelian variety over the fraction field $K$ of a
strictly Henselian discrete valuation ring $R$ (e.g. the maximal
unramified extension of local field).
Let $n$ be an integer coprime to the residue characteristic of $K$. Let $x \in A(K)$ be
a point whose reduction lands in the identity component of the
closed fiber of the N\'eron model of $A$. Then $x \in nA(K)$.
\end{lemma}
}
\comment{
\begin{proposition}
Let $A$ be an abelian variety over the fraction field $K$ of a
strictly Henselian discrete valuation ring $R$. Let $\cA$ be the
N\'eron model of $A$ and let $\cA^0$ be the connected component of
the identity element of $\cA$. Then the group $\cA^0(R)$ of
$R$-points on $\cA^0$ is $n$-divisible for every $n$ that is
relatively prime to the characteristic.
\end{proposition}
\begin{remark}
Suppose that $P \in A(K)$ is a point whose reduction lies in the
identity component of the special fiber of the N\'eron model of
$\cA$. Since $A(K) \simeq \cA(R)$ because of the N\'eron mapping
property, the proposition implies that $P$ is $n$-divisible in
$A(K)$. Furthermore, $P = nP'$ for a point $P' \in A(K)$ whose
reduction also lies in the identity component of the special fiber
of $\cA$ (although we will not need this last property).
\end{remark}
}
\subsection{Proof of Theorem~\ref{thm:evis}}
\begin{proof}[Proof of Theorem~\ref{thm:evis}]
We argue as in the proof of
\cite[Thm.~3.1]{agashe-stein:visibility}.
The construction of the map (\ref{eqn:evis}) is similar to the one in
the proof of~\cite[Lem.~3.6]{agashe-stein:visibility}.
We have the commutative diagram
$$\xymatrix{
0 \ar[r] & B[\ell] \ar[r]\ar[d] & B \ar[d]\ar[r]^{\ell}\ar[dr]^{\psi} &
B \ar[d]^{\pi}\ar[r] & 0 \\
0 \ar[r] & A \ar[r] & C \ar[r] & Q \ar[r] & 0,
}
$$
where $\psi : B \ra Q$ is the composition of the inclusion
$B \hra C$ with the quotient map $C \ra Q$, and the existence of the
morphism $\pi : B \ra Q$ follows from the inclusion
$B[\ell] \subset \Ker(\psi) = A \cap B$. By naturality for the long
exact sequence of Galois cohomology we obtain the following commutative
diagram with exact rows and columns
$$\xymatrix{
& M_0\ar[d] & M_1\ar[d]& M_2\ar[d]\\
0 \ar[r] & B(K)/(B(K)[\ell]) \ar[r]^{\qquad \ell}\ar[d] & B(K)\ar[dr]^{\pi} \ar[r]\ar[d]
& B(K)/\ell{}B(K)\ar[r]\ar[d]^{\vphi} & 0\\
0 \ar[r] & C(K)/A(K)\ar[r]\ar[d] & Q(K) \ar[r] & {\Vis_C(\H^1(K,A))} \ar[r] & 0\\
& M_3.
}
$$
Here, $M_0$, $M_1$ and $M_2$ denote the kernels of the corresponding
vertical maps and $M_3$ denotes the cokernel of the first map.
Since $R$ preserves $A$, $B$, and $B[\ell]$, all objects in the diagram
are $R$-module and the morphisms of abelian varieties are also
$R$-module homomorphisms.
The snake lemma yields an exact sequence
$$
0 \to M_0\ra M_1 \ra M_2 \ra M_3.
$$
By hypothesis, $B(K)[\m]=0$, so $N_0=\Ker(B(K) \to C(K)/A(K))$ has
no~$\m$ torsion. Noting that $B(K)[\ell] \subset N_0$, it follows
that $M_0=N_0/(B(K)[\ell])$ has no~$\m$ torsion either,
by Lemma~\ref{lem:ex_art}.
Also, $M_1[\m]=0$ again since $B(K)[\m]=0$.
By the long exact sequence on Galois cohomology,
the quotient $C(K)/B(K)$ is isomorphic to a subgroup of $Q(K)$
and by hypothesis $Q(K)[\m]=0$, so $(C(K)/B(K))[\m]=0$.
Since $Q$ is isogenous to $A$ and $A(K)$ is finite
and $C(K)/B(K) \hra Q(K)$, we see that
$C(K)/B(K)$ is finite. Thus $M_3$ is a quotient
of the finite $R$-module $C(K)/B(K)$, which has no
$\m$-torsion, so Lemma~\ref{lem:ex_art} implies
that $M_3[\m]=0$. The same lemma
implies that $M_1/M_0$ has no $\m$-torsion,
since it is a quotient of the finite module
$M_1$, which has no $\m$-torsion. Thus, we
have an exact sequence
$$
0 \to M_1/M_0 \to M_2 \to M_3 \to 0,
$$
and both of $M_1/M_0$ and $M_3$ have trivial
$\m$-torsion. It follows by Lemma~\ref{lem:ex_art},
that $M_2[\m]=0$. Therefore, we have an injective morphism
of $R/\m$-vector spaces
$$
\vphi: (B(K)/\ell{}B(K))[\m]\hra \Vis_C(H^1(K,A))[\m].
$$
It remains to show that for any $x \in B(K)$, we have
$\vphi(x) \in \Vis_C(\Sha(K, A))$, i.e., that $\vphi(x)$
is locally trivial.
\comment{For real archimedian places $v$ the cohomology group
$\H^1(\overline{K}_v/K_v,A)$ is trivial. For complex archimedian
places, every cohomology class has order 2 since
$\Gal(\overline{K}_v/K) \cong \Gal(\C/\R) \cong \Z/2\Z$ and the order
of any cohomology class divides the order of the group
\cite{serre:local}. Since $\res_v(\vphi(\pi(x)))$ is also
$\ell$-torsion and $\ell$ is odd (since $1\leq e < \ell - 1$), then
$\res_v(\vphi(\pi(x))) = 0$.
Let $v$ be a non-archimedian place for which char$(v) \ne \ell$. If
$m = c_{B,v}$ denotes the Tamagawa number at $v$ for $B$, then the
reduction of $mx$ lands in the identity component of the closed fiber
of the N\'eron model of $B$. The field $K_v^{\ur}$ is
the fraction field of a strictly Henselian discrete valuation ring, so
we can apply Proposition~\ref{lem:divisible} to obtain a point
$z \in B(K^{\ur}_v)$, such that $mx = \ell z$. The cohomology class
$\res_v(\pi(mx))$ is represented by the 1-cocycle
$\xi : \Gal(\overline{K}_v/K_v) \ra A(K_v^{\ur} )$, given by
$\sigma \mapsto \sigma(z)-z \in A(K_v^{\ur})$. It follows that $[\xi]$
is an unramified cohomology class, i.e., $\left [\xi \right ] \in
H^1(K_v^{\ur} / K_v ,A(K_v^{\ur}))$, i.e., $\res_v(\pi(mx))$ is unramified.
}
We proceed exactly as in Section~3.5 of \cite{agashe-stein:bsd}.
In both cases $\text{char}(v)\neq \ell$ and $\text{char}(v)=\ell$ we
arrive at the conclusion that the restriction
of $\vphi(x)$ to $\H^1(K_v,A)$ is an element
$c\in \H^1(K_v^{\ur}/K_v, A(K_v^{\ur}))$.
(Note that in the case $\text{char}(v)\neq \ell$ the proof
uses our hypothesis that $\ell\nmid \#\Phi_{B,v}(k_v)$.)
By \cite[Prop I.3.8]{milne:duality}, there is an
isomorphism
\begin{equation}\label{eqn:prop38}
\H^1(K_v^{\ur}/K_v, A(K_v^{\ur}))
\isom \H^1(\kbar_v/k_v, \Phi_{A,v}(\kbar_v)).
\end{equation}
We will use our hypothesis that
$$\Phi_{A,v}(k_v)[\m] = \Phi_{B,v}(k_v)[\ell] = 0$$
for all $v$ of bad reduction to deduce that
the image of $\vphi$ lies in $\Vis_C(\Sha(K, A))[\m]$.
Let $d$ denote the image of $c$ in $\H^1(\kbar_v/k_v, \Phi_{A,v}(\kbar_v))$.
The construction of $d$ is compatible with the
action of~$R$ on Galois cohomology, since (as is explained in the
proof of \cite[Prop.~I.3.8]{milne:duality})
the isomorphism (\ref{eqn:prop38})
is induced from the exact sequence of
$\Gal(K_v^{\ur}/K_v)$-modules
$$
0 \to \cA^0(K_v^{\ur}) \to \cA(K_v^{\ur}) \to \Phi_{A,v}(\kbar_v)\to 0,
$$
where $\cA$ is the N\'eron model of $A$ and $\cA^0$ is the subgroup scheme
whose generic fiber is $A$ and whose closed fiber is the
identity component of $\cA_{k_v}$.
Since $\vphi(x)\in \H^1(K,A)[\m]$, it follows that
$$
d \in \H^1(\kbar_v/k_v, \Phi_{A,v}(\kbar_v))[\m].
$$
Lemma~\ref{lem:h0h1}, our hypothesis that
$\Phi_{A,v}(k_v)[\m]= 0$,
and that
$$
\H^1(\kbar_v/k_v, \Phi_{A,v}(\kbar_v)) = \varinjlim
\H^1(\Gal(k_v'/k_v),\Phi_{A,v}(k_v'))),
$$
together imply that $\H^1(\kbar_v/k_v, \Phi_{A,v}(\kbar_v))[\m]= 0$,
hence $d=0$. Thus $c=0$, so $\vphi(x)$ is locally
trivial, which completes the proof.
\end{proof}
\section{Strong Visibility at Higher Level}\label{sec:strongvis}
\subsection{Strongly visible subgroups}
Let $A_{/\Q}$ be an abelian subvariety of $J_0(N)_{/\Q}$ and let $p\nmid N$ be a prime. Let
\begin{equation}\label{eqn:phi}
\vphi = \delta_1^* + \delta_p^* : J_0(N) \to J_0(pN),
\end{equation}
where $\delta_1^*$ and $\delta_p^*$ are the pullback maps on equivalence classes of degree-zero
divisors of the degeneracy maps $\delta_1, \delta_p : X_0(pN) \ra X_0(N)$. Let $\H^1(\Q,A)^{\odd}$ be
the prime-to-2-part
of the group $\H^1(\Q,A)$.
\begin{definition}[Strongly Visibility]\label{defn:strongvis}
The \emph{strongly visible} subgroup of $\H^1(\Q,A)$ for $J_0(pN)$
is
$$
\Vis_{pN}\H^1(\Q,A) = \Ker\left(\H^1(\Q,A)^{\odd} \xra{\vphi_*} \H^1(\Q,J_0(pN))\right)
\subset \H^1(\Q,A).
$$
Also,
$$
\Vis_{pN}\Sha(\Q, A) = \Sha(\Q, A) \cap \Vis_{pN}\H^1(\Q,A).
$$
The reason we replace $\H^1(\Q,A)$ by $\H^1(\Q,A)^{\odd}$ is that
the kernel of $\vphi$ is a $2$-group (see \cite{ribet:raising}).
\end{definition}
\begin{remark}
We could obtain more visible subgroups by considering the map
$\delta_1^* - \delta_p^*$ in Definition~\ref{defn:strongvis}. However, the methods of this paper do not
apply to this map.
\end{remark}
For a positive integer~$N$, let
$$
\nu(N) = \frac{1}{6} \cdot \prod_{q^r\| N} (q^r+q^{r-1}) = \frac{1}{6} \cdot [\SL_2(\Z) : \Gamma_0(N)].
$$
We call the number $\nu(N)$ the \emph{Sturm bound} (see~\cite{sturm:cong}).
\begin{theorem}\label{thm:strongvis}
Let $A_{/\Q}=A_f$ be a newform abelian subvariety of $J_0(N)$ for which
$L(A_{/\Q},1)\neq 0$ and let $p\nmid N$ be a prime. Suppose that there is a
maximal ideal $\lambda \subset \T(N)$ and an elliptic curve~$E_{/\Q}$ of conductor $pN$ such that:
\begin{enumerate}
\item{}[Nondivisibility]\label{hypo:nondiv}
The residue characteristic $\ell$ of $\lambda$ satisfies
$$\ell\nmid 2 \cdot N \cdot p \cdot \prod_{q\mid N} c_{E,q}.$$
\item{}[Component Groups]\label{hypo:comp}
For each prime $q\mid N$,
$$
\Phi_{A,q}(\F_q)[\lambda] = 0.
$$
\item{}[Fourier Coefficients]\label{hypo:cong}
Let $a_n(E)$ be the $n$-th Fourier coefficient of the modular
form attached to~$E$, and $a_n(f)$ the $n$-th Fourier coefficient of~$f$.
Assume that $a_p(E) = -1$,
$$
a_p(f) \con -(p+1)\pmod{\lambda}\quad\text{ and }\quad a_q(f) \con a_q(E) \pmod{\lambda},
$$
for all primes $q\neq p$ with
$q\leq \nu(pN)$.
% [[Idea: put $\ell\mid\ord_p(\Delta(E))$ so get level $N$,
% then put $\nu(N)$ instead of $\nu(pN)$. Then get two
% newforms, so can use enhanced Sturm at level~$N$. Also,
% with mod deg hypothesis, just need to show that no other
% congruence.
% ]]
\item{}[Irreducibility]\label{hypo:isog}
The mod~$\ell$ representation $\rhobar_{E,\ell}$ is irreducible.
\end{enumerate}
Then there is an injective homomorphism
$$
E(\Q)/\ell E(\Q) \hra \Vis_{pN}(\Sha(\Q, A_f))[\lambda].
$$
\end{theorem}
\begin{remark}
In fact, we have
$$
E(\Q)/\ell E(\Q) \hra \Ker(\Sha(\Q, A_f) \to \Sha(\Q, C))[\lambda]
\subset \Vis_{pN}(\Sha(\Q, A_f))[\lambda],
$$
where $C\subset J_0(pN)$ is isogenous to $A_f\times E$.
\end{remark}
\subsection{Some auxiliary lemmas}
We will use the following lemmas in the proof
of Theorem~\ref{thm:strongvis}. The notation is as
in the previous section. In addition, if $f \in S_2(\Gamma_0(N))$, we denote by $a_n(f)$
the $n$-th Fourier coefficient of $f$ and by $K_f$ and $\cO_f$ the Hecke eigenvalue field and its ring of integers,
respectively.
\begin{lemma}\label{lem:getint}
Suppose $A_f\subset J_0(N)$ and $A_g \subset J_0(pN)$ are attached
to newforms~$f$ and $g$ of level $N$ and $pN$, respectively, with $p\nmid N$.
Suppose that there is a prime ideal~$\lambda$ of residue characteristic $\ell\nmid 2pN$
in an integrally closed subring $\cO$ of $\Qbar$ that contains
the ring of integers of the composite field $K = K_fK_g$
such that for $q\leq \nu(pN)$,
$$
a_q(f) \con
\begin{cases} a_q(g) \pmod{\lambda} & \text{if $q\neq p$,}\\
(p+1)a_p(g)\pmod{\lambda} & \text{if $q=p$.}
\end{cases}
$$
Assume that $a_p(g) = -1$.
Let $\lambda_f = \cO_f \cap \lambda$ and
$\lambda_g = \cO_g \cap \lambda$ and assume that
$A_f[\lambda_f]$ is an irreducible $G_\Q$-module.
Then we have an equality
$$
\vphi(A_f[\lambda_f]) = A_g[\lambda_g]
$$
of subgroups of $J_0(pN)$, where $\vphi$ is as in (\ref{eqn:phi}).
\end{lemma}
\begin{proof}
Our hypothesis that
$a_p(f) \con -(p+1) \pmod{\lambda_f}$
implies, by the proofs in \cite{ribet:raising},
that
$$
\vphi(A_f[\lambda_f]) \subset \vphi(A_f) \cap J_0(pN)_{\pnew},
$$
where $J_0(pN)_{\pnew}$ is the $\pnew$ abelian subvariety of $J_0(N)$.
By \cite[Lem.~1]{ribet:raising}, the operator
$U_p=T_p$ on $J_0(pN)$ acts as $-1$ on
$\vphi(A_f[\lambda_f])$.
Consider the action of $U_p$ on the 2-dimensional vector space spanned by $\{f(q), f(q^p)\}$. The matrix
of $U_p$ with respect to this basis is
$$
U_p = \mtwo{a_p(f)}{p}{-1}{0}.
$$
In particular, neither of $f(q)$ and $f(q^p)$ is an
eigenvector for $U_p$.
The characteristic polynomial of $U_p$
acting on the span of $f(q)$ and $f(q^p)$ is
$x^2-a_p(f)x+p$.
Using our hypothesis on $a_p(f)$
again, we have
$$
x^2 - a_p(f)x+p \con x^2 + (p+1) x + p \con
(x + 1) (x + p)\pmod{\lambda}.
$$
Thus we can choose an algebraic integer $\alpha$ such that
$$
f_1(q) = f(q) + \alpha f(q^p)
$$
is an eigenvector of $U_p$ with eigenvalue congruent
to $-1$ modulo~$\lambda$.
(It does not matter for our purposes whether $x^2+a_p(f)x+p$ has distinct
roots; nonetheless, since $p\nmid N$, \cite[Thm.~2.1]{coleman-voloch}
implies that it does have distinct roots.)
The cusp form $f_1$ has the same prime-indexed
Fourier coefficients as~$f$ at primes other than~$p$.
Enlarge $\cO$ if necessary so that $\alpha\in\cO$.
The $p$-th coefficient of $f_1$
is congruent modulo~$\lambda$ to~$-1$ and $f_1$ is an
eigenvector for the full Hecke algebra.
It follows from the recurrence relation for coefficients of the eigenforms
that
$$
a_n(g) \con a_n(f_1) \pmod{\lambda}
$$
for all integers $n\leq \nu(pN)$.
By \cite{sturm:cong}, we have $g\con f_1\pmod{\lambda}$, so $a_q(g)
\con a_q(f)\pmod{\lambda}$ for all primes $q\neq p$. Thus by the
Brauer-Nesbitt theorem \cite{curtis-reiner}, the 2-dimensional $G_\Q$-representations
$\vphi(A_f[\lambda_f])$ and $A_g[\lambda_g]$ are isomorphic.
Let $\m$ be a maximal ideal of the Hecke algebra $\T(pN)$ that annihilates the module $A_g[\lambda_g]$.
Note that $A_g[\m]=A_g[\lambda_g]$ since $A_g[\m] \subset A_g[\lambda_g]$ and
$A_g[\lambda_g]\isom \vphi(A_f[\lambda_f])$ is irreducible as a $G_\Q$-module.
The maximal ideal $\m$ gives rise to a Galois representation $\rhobar_\m : G_\Q \ra \GL_2(\T(pN)/\m)$ isomorphic to $A_g[\lambda_g]$,
which is irreducible since the Galois module $A_f[\lambda_f]$ is irreducible. Finally, we apply
\cite[Thm.~2.1(i)]{wiles:fermat} for $H = (\Z / N\Z)^\times$ (i.e., $J_H = J_0(N)$) to conclude
that $J_0(N)(\Qbar)[\m] \isom (\T(pN)/\m)^2$, i.e., the representation $\rhobar_\m$ occurs with multiplicity
one in $J_0(pN)$.
Thus
$$
A_g[\lambda_g] = \vphi(A_f[\lambda_f]).
$$
\end{proof}
%\begin{remark}
% The bounds in \cite{sturm:cong} are sometimes much better than
% $\nu(pN)$ when comparing eigenforms (see also
% \cite{buzzard-stein:artin}). In the proof of Lemma~\ref{lem:getint}
% we are comparing eigenforms, so these better bounds apply. They are
% more complicated to state so we do not include them here.
%\footnote{Should we just state them? Ribet says so.}
%\end{remark}
\begin{lemma}\label{lem:pushsha}
Suppose $\vphi:A \to B$ and $\psi:B \to C$ are
homomorphisms of abelian varieties over a number field $K$, with
$\vphi$ an isogeny and $\psi$ injective.
Suppose~$n$ is an integer that is relatively
prime to the degree of~$\vphi$.
If $G=\Vis_C(\Sha(\Q, B))[n^{\infty}]$,
then there is some injective homomorphism
$$
f: G \hra \Ker\left\{(\psi\circ\vphi)_*:\Sha(\Q, A) \lra \Sha(\Q, C)\right\},
$$
such that $\vphi_*(f(G)) = G$.
\end{lemma}
\begin{proof}
Let $m$ be the degree of the isogeny $\vphi:A \to B$.
Consider the complementary isogeny $\vphi':B\to A$,
which satisfies
$\vphi\circ \vphi' = \vphi' \circ \vphi = [m]$.
By hypothesis~$m$ is coprime to~$n$, so
$\gcd(m,\#G)=\gcd(m,n^{\infty})=1$, hence
$$
\vphi_* (\vphi'_*(G)) = [m]G = G.
$$
Thus $\vphi'_*(G)$ maps, via $\vphi_*$, to
$G\subset \Sha(\Q, B)$, which in turn maps to~$0$ in $\Sha(\Q, C)$.
\end{proof}
\begin{lemma}\label{lem:heckegen}
Let $M$ be an odd integer coprime to $N$ and
let~$R$ be the subring of $\T(N)$ generated by all Hecke operators
$T_n$ with $\gcd(n,M) = 1$. Then $R = \T(N)$.
\end{lemma}
\begin{proof}
See the lemma on page 491 of \cite{wiles:fermat}. (The condition
that $M$ is odd is necessary, as there is a counterexample when
$N=23$ and $M=2$.)
\end{proof}
\comment{\begin{proof}
Let $J = J_0(N)$ and $\T=\T(N)$.
Suppose $p\mid M$ is a prime.
We first show that there is a prime~$q\neq p$ such that $T_q$ and $T_p$
act in the same way on $J[\ell]$. Consider the Galois
representation
$$
\rhobar_{\ell}: G_\Q \to \Aut(\Tate_\ell(J)).
$$
View $\Tate_\ell(J)$ as a $2$-dimensional
$\T\tensor\Q_\ell$-module, and the
elements of $\Aut(\Tate_\ell(J))$ as $2\times 2$-matrices
with entries in $\T\tensor \Q_\ell$.
For each prime~$q\nmid N$,
we have
$$
\Tr(\rhobar_{\ell}(\Frob_q)) = T_q \in \T \tensor \Z_\ell.
$$
Since $J[\ell]$ is finite,
the representation
$$
\rhobar_{\ell}: G_\Q \to \Aut(J[\ell])
$$
factors through the Galois group of a finite extension
of~$\Q$.
The Chebotarev density theorem
implies that there exists a prime $q\neq p$ such that
$$
\Tr(\rhobar_{\ell}(\Frob_p)) = \Tr(\rhobar_{\ell}(\Frob_q)).
$$
Thus
$$
T_p \con T_q \pmod{\ell},
$$
where congruence modulo~$\ell$ means their difference is
in $\ell \T$.
Hence $T_q$ acts on $J[\ell]$ in the same way as $T_p$.
Define $S$ by the exact sequence
$$
0 \to R \to \T(N) \to S \to 0.
$$
Let $\ell$ be any prime.
Then we have an exact sequence
$$
R\tensor\F_\ell \to \T(N)\tensor\F_\ell \to S\tensor \F_\ell \to 0.
$$
Using what we did above,for
each prime $p\mid M$ we find a prime $q\nmid M$ such
that $T_q \equiv T_p$ on $J[\ell]$.
Thus $R\tensor\F_\ell \to \T\tensor\F_\ell$
is surjective, hence
$S \tensor \F_\ell=0$.
Since $\T$ is a finitely generated abelian group,
so is~$S$, so we must have $S=0$.
\end{proof}
}
\begin{lemma}\label{lem:twolam}
Suppose~$\lambda$ is a maximal ideal of $\T(N)$ with generators a prime~$\ell$
and $T_n - a_n$ (for all $n\in\Z$), with $a_n \in \Z$. For each integer $p\nmid
N$, let $\lambda_p$ be the ideal in $\T(N)$ generated by
$\ell$ and all $T_n - a_n$, where $n$ varies over integers
coprime to $p$. Then $\lambda = \lambda_p$.
\end{lemma}
\begin{proof}
Since $\lambda_p\subset\lambda$ and $\lambda$ is maximal, it
suffices to prove that $\lambda_p$ is maximal. Let $R$ be the
subring of $\T(N)$ generated by Hecke operators $T_n$ with $p\nmid
n$. The quotient $R/\lambda_p$ is a quotient of $\Z$ since each
generator $T_n$ is equivalent to an integer. Also,
$\ell \in \lambda_p$, so $R/\lambda_p = \F_\ell$.
But by Lemma~\ref{lem:heckegen},
$R = \T(N)$, so $\T(N)/\lambda_p = \F_\ell$,
hence $\lambda_p$ is a maximal ideal.
\end{proof}
\begin{lemma}\label{lem:am2}
Suppose that $A$ is an abelian variety over a field~$K$. Let $R$ be
a commutative subring of $\End(A)$ and $I$ an ideal of $R$.
Then
$$
(A/A[I])[I] \cong A[I^2]/A[I],
$$
where the isomorphism is an
isomorphism of $R[G_K]$-modules.
\end{lemma}
\begin{proof}
Let $a + A[I]$, for some $a \in A$, be an $I$-torsion element of
$A/A[I]$. Then by
definition, $xa \in A[I]$ for each $x \in I$.
Therefore, $a \in A[I^2]$. Thus $a + A[I] \mapsto a + A[I]$
determines a well-defined homomorphism of $R[G_K]$-modules
$$
\varphi : (A/A[I])[I] \ra A[I^2]/A[I].
$$
Clearly this homomorphism is injective. It is also surjective as every
element $a + A[I] \in A[I^2]/A[I]$ is
$I$-torsion as an element of $A/A[I]$, as $I{}a \in
A[I]$. Therefore, $\varphi$ is an isomorphism of
$R[G_K]$-modules.
\end{proof}
\begin{lemma}\label{lem:tam-l-div}
Let $\ell$ be a prime and let $\phi : E \ra E'$ be an isogeny of elliptic curves
of degree coprime to $\ell$ defined over a number field $K$. If $v$ is any place of $K$ then
$\ell \mid c_{E, v}$ if and only if $\ell \mid c_{E', v}$.
\end{lemma}
\begin{proof}
Consider the complementary isogeny $\phi' : E' \ra E$. Both $\phi$ and $\phi'$ induce homomorphisms
$\phi : \Phi_{E, v}(k_v) \ra \Phi_{E', v}(k_v)$ and $\phi' : \Phi_{E',v}(k_v) \ra \Phi_{E, v}(k_v)$ and
$\phi \circ \phi'$ and $\phi' \circ \phi$ are multiplication-by-$n$ maps. Since $(n, \ell) = 1$ then
$\# \ker \phi$ and $\# \ker \phi'$ must be coprime to $\ell$ which implies the statement.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:strongvis}}
\begin{proof}[Proof of Theorem~\ref{thm:strongvis}]
By \cite{breuil-conrad-diamond-taylor} $E$ is modular, so there
is a rational newform $f \in S_2^{\new}(pN)$ which is an eigenform for the
Hecke operators and an isogeny $E \to E_f$ defined over~$\Q$,
which by Hypothesis~\ref{hypo:isog} can be chosen to have degree
coprime to~$\ell$. Indeed, every cyclic rational isogeny is a composition
of rational isogenies of prime degree, and~$E$ admits no rational $\ell$-isogeny
since $\rhobar_{E,\ell}$ is irreducible.
By Hypothesis~\ref{hypo:nondiv} the Tamagawa numbers of $E$ are coprime
to~$\ell$. Since~$E$ and~$E_f$ are related by an isogeny of degree coprime
to~$\ell$, the Tamagawa numbers of~$E_f$ are also not divisible by~$\ell$
by Lemma~\ref{lem:tam-l-div}. Moreover, note that
$$E(\Q)\tensor \F_\ell \isom E_f(\Q) \tensor \F_\ell.$$
Let $\m$ be the ideal of $\T(pN)$ generated by $\ell$
and $T_n-a_n(E)$ for all integers $n$ coprime to $p$.
Note that $\m$ is maximal by Lemma~\ref{lem:twolam}.
Let $\vphi$ be as in (\ref{eqn:phi}), and let $A=\vphi(A_f)$.
Note that if $T_n\in \T(pN)$ then $T_n(E_f) \subset E_f$ since
$E_f$ is attached to a newform, and if, moreover $p\nmid n$, then
$T_n(A) \subset A$ since the Hecke operators with index coprime to~$p$
commute with the degeneracy maps.
Lemma~\ref{lem:getint} implies that
$$
E_f[\ell] = E_f[\m] = \vphi(A_f[\lambda]) \subset A,
$$
so $\Psi = E_f[\ell]$ is a subgroup of~$A$
as a $G_{\Q}$-module.
Let
$$
C= (A\times E_f)/\Psi,
$$
where we embed $\Psi$ in $A\times E_f$ anti-diagonally, i.e., by the map
$x\mapsto (x,-x)$.
The antidiagonal map $\Psi \to A\times E_f$ commutes with
the Hecke operators $T_n$ for $p\nmid n$, so
$(A\times E_f)/\Psi$ is preserved by the $T_n$
with $p\nmid n$. Let $R$ be the subring of $\End(C)$
generated by the action of all Hecke operators~$T_n$, with $p\nmid n$.
Also note that $T_p \in \End(J_0(pN))$ acts by Hypothesis~\ref{hypo:cong} as $-1$ on
$E_f$, but $T_p$ need {\em not} preserve $A$.
% (Proof: If $T_p=1$ on $B$, then
% the Tamagawa number $c_{B,p}$ would equal the order of the component
% group $\Phi_{B,p}(\Fpbar)$, which is divisible by $\ell$, since
% the mod $\ell$ representation $\rhobar_{B,p}$ is unramified at $p$ by
% Hypothesis~\ref{hypo:cong}.)
Suppose for the moment that we have verified that the
hypothesis of Theorem~\ref{thm:evis} are satisfied with $A$, $B = E_f$, $C$, $Q = C/B$, $R$ as above and
$K=\Q$.
Then we obtain an injective homomorphism
$$
E(\Q)/\ell E(\Q) \isom E_f(\Q)/\ell E_f(\Q) \hra \Ker(\Sha(\Q, A) \to \Sha(\Q, C))[\m].
$$
We then apply Lemma~\ref{lem:pushsha} with $n=\ell$,
$A_f$, $A$, and $C$, respectively, to see that
$$
E_f(\Q)/\ell E_f(\Q) \subset \Ker(\Sha(\Q, A_f) \to \Sha(\Q, C))[\lambda].
$$
That $E_f(\Q)/\ell E_f(\Q)$ lands in the~$\lambda$-torsion is because
the subgroup of $\Vis_C(\Sha(\Q, E_f))$ that we constructed
is $\m$-torsion.
Finally, consider
$A\times E_f \to J_0(pN)$ given by $(x,y)\mapsto x+y$.
Note that $\Psi$ maps to $0$, since $(x,-x)\mapsto 0$
and the elements of $\Psi$ are of the form $(x,-x)$.
We have a (not-exact!) sequence of maps
$$
\Sha(\Q, A_f) \to \Sha(\Q, C) \to \Sha(\Q, J_0(pN)),
$$
hence inclusions
\begin{eqnarray*}
E_f(\Q)/\ell E_f(\Q) &\subseteq& \Ker(\Sha(\Q, A_f) \to \Sha(\Q, C)) \\
&\subseteq& \Ker(\Sha(\Q, A_f) \to \Sha(\Q, J_0(pN))),
\end{eqnarray*}
which gives the conclusion of the theorem.
It remains to verify the hypotheses of Theorem~\ref{thm:evis}.
That $C=A+B$ is clear from the definition of~$C$. Also,
$A\cap E_f = E_f[\ell]$, which is finite.
We explained above when defining~$R$ that each of~$A$ and~$E_f$ is preserved by~$R$.
Since $K=\Q$ and~$\ell$ is odd the condition $1=e<\ell-1$ is satisfied.
That $A(\Q)$ is finite follows from our hypothesis
that $L(A_f,1)\neq 0$ (by \cite{kolyvagin-logachev:finiteness}).
It remains is to verify that the groups
$$
Q(\Q)[\m] ,\quad E_f(\Q)[\m] ,\quad \Phi_{A,q}(\F_q)[\m] ,\quad \text{ and } \Phi_{E_f,q}(\F_q)[\ell] ,
$$
are $0$ for all primes $q\mid pN$.
Since $\ell\in \m$, we have by Hypothesis~\ref{hypo:isog} that
$$
E_f(\Q)[\m] = E_f(\Q)[\ell] = 0.
$$
We will now verify that $Q(\Q)[\m]=0$.
From the definition of~$C$ and~$\Psi$ we have $Q \isom A/\Psi.$
Let $\lambda_p$ be as in Lemma~\ref{lem:twolam} with $a_n = a_n(E)$.
The map~$\vphi$ induces an isogeny of $2$-power degree
$$
A_f/(A_f[\lambda]) \to A/\Psi.
$$
Thus there is $\lambda_p$-torsion in $(A_f/(A_f[\lambda]))(\Q)$
if and only if there is $\m$-torsion in
$(A/\Psi)(\Q)$.
%(Note that $\lambda_p$ and $\m$
%are both ideals generated by $\ell$ and $T_n-a_n$
%for all~$n$ coprime to~$p$, but for $\lambda_p$
%the $T_n\in \T(N)$, and for $\m$ they are
%in $\T(pN)$.)
Thus it suffices to prove that
$(A_f/A_f[\lambda])(\Q)[\lambda_p] = 0$.
By Lemma~\ref{lem:twolam}, we have
$\lambda_p = \lambda$, and
by Lemma~\ref{lem:am2},
$$
(A_f/A_f[\lambda])[\lambda] \isom A_f[\lambda^2]/A_f[\lambda].
$$
By \cite[\S
II.14]{mazur:eisenstein}, the quotient $A_f[\lambda^2]/A_f[\lambda]$
injects into a direct sum of copies of $A_f[\lambda]$ as Galois
modules. But $A_f[\lambda] \isom E[\ell]$ is irreducible, so
$(A_f[\lambda^2]/A_f[\lambda])(\Q) = 0$, as required.
By Hypothesis~\ref{hypo:comp}, we have $\Phi_{A_f,q}(\F_q)[\lambda] = 0$ for each prime
divisor $q\mid N$.%, so $\Phi_{A_f,q}(\F_q)[\m]=0$.
Since $A$ is $2$-power isogenous to $A_f$ and $\ell$ is odd, this
verifies the Tamagawa number hypothesis for $A$. Our hypothesis that
$a_p(E)=-1$ implies that $\Frob_p$ acts on $\Phi_{E_f,p}(\Fpbar)$ as
$-1$. Thus $\Phi_{E_f,p}(\F_p)[\ell] = 0$ since $\ell$ is odd.
This completes the proof.
\end{proof}
\begin{remark}
An essential ingrediant in the proof of the above theorem is the
multiplicity one result used in the paper of Wiles
(see~\cite[Thm.~2.1.]{wiles:fermat}). Since this result holds for Jacobians
$J_H$ of the curves $X_H(N)$ that are intermediate
covers for the covering $X_1(N) \ra X_0(N)$ corresponding to subgroups
$H \subseteq (\Z/N\Z)^\times$ (i.e., the Galois group of $X_1(N) \ra X_H$
is $H$), one should be able to give a generalization of Theorem~\ref{thm:strongvis}
which holds for newform subvarieties of $J_H$. This
requires generalizing some results from \cite{ribet:raising} to arbitrary $H$.
\end{remark}
\subsection{A Variant of Theorem~\ref{thm:strongvis}
with Simpler Hypothesis}\label{subsec:strongvis-simpler}
\begin{proposition}\label{prop:kill_comp_grp}
Suppose $A=A_f\subset J_0(N)$ is a newform abelian variety
and $q$ is a prime that exactly divides~$N$.
Suppose $\m\subset\T(N)$ is a non-Eisenstein maximal ideal
of residue characteristic~$\ell$ and that $\ell\nmid m_A$,
where $m_A$ is the modular degree of~$A$.
Then $\Phi_{A,q}(\Fbar_q)[\m]=0$.
\end{proposition}
\begin{proof}
The component group of $\Phi_{J_0(N),q}(\Fbar_q)$ is Eisenstein
by~\cite{ribet:comp}, so $$\Phi_{J_0(N),q}(\Fbar_q)[\m]=0.$$
By Lemma~\ref{lem:ex_art}, the image of
$\Phi_{J_0(N),q}(\Fbar_q)$ in $\Phi_{A^{\vee},q}(\Fbar_q)$ has
no~$\m$ torsion.
By the main theorem of \cite{conrad-stein:compgroup},
the cokernel $\Phi_{J_0(N),q}(\Fbar_q)$ in $\Phi_{A^{\vee},q}(\Fbar_q)$
has order that divides $m_A$. Since $\ell\nmid m_A$,
it follows that the cokernel also has no~$\m$ torsion.
Thus Lemma~\ref{lem:ex_art} implies
that $\Phi_{A^{\vee},q}(\Fbar_q)[\m]=0$.
Finally, the modular polarization $A\to A^{\vee}$
has degree $m_A$, which is coprime to~$\ell$, so
the induced map $\Phi_{A,q}(\Fbar_q)\to \Phi_{A^{\vee},q}(\Fbar_q)$
is an isomorphism on $\ell$ primary parts. In particular,
that $\Phi_{A^{\vee},q}(\Fbar_q)[\m]=0$ implies that $\Phi_{A,q}(\Fbar_q)[\m]=0$.
\end{proof}
% RIBET SAYS THIS IS WRONG, and WE DON'T USE IT, so
% I'm commenting it out!!
% \begin{lemma}\label{prop:nochange}
% Suppose $A=A_f \subset J_0(N)$ is a newform abelian
% variety and $\m$ is a non-Eisenstein maximal
% ideal of $\T(N)$. If $\psi:A_f \to A'$ is
% an isogeny with $\T(N)$-invariant kernel, then
% $A_f[\m]\isom A'[\m]$.
% \end{lemma}
% \begin{proof}
% This is a consequence of the construction of
% mod~$\ell$-representations attached to modular forms and
% Brauer-Nesbitt theorem, which says that the traces and determinant
% of an irreducible two-dimensional representation over a finite field
% determine the representation. When we construct $\rhobar_{f,\m}$, we
% first construct the $\Q_{\ell}$-adic Tate module, break it up using
% the Hecke algebra, then choose a lattice, and reduce to get the
% mod~$\ell$ representation. The semisimplification of the
% representation we obtain does not depend on that choice. Replacing
% $A$ by $A'$ is the same as making a different choice of lattice.
% \end{proof}
If $E$ is a semistable elliptic curve over~$\Q$ with discriminant
$\Delta$, then
we see using Tate curves that
$
\cbar_p = \ord_p(\Delta).
$
\begin{theorem}\label{thm:strongestvis}
Suppose $A=A_f\subset J_0(N)$ is a newform abelian variety with
$L(A_{/\Q},1)\neq 0$ and~$N$ square free, and let~$\ell$ be a prime.
Suppose that $p\nmid N$ is a prime, and that there is an elliptic
curve~$E$ of conductor $pN$ such that:
\begin{enumerate}
\item{}[Rank] The Mordell-Weil rank of $E(\Q)$ is positive.
\item{}[Divisibility]\label{hypo:2:nondiv}
We have $a_p(E)=-1$,
$\displaystyle \ell \mid \cbar_{E,p}$, and $$
\ell\nmid 2 \cdot N \cdot p \cdot c_{E,p} \cdot
\prod_{q\mid N} \cbar_{E,q}.
$$
\item{}[Irreducibility]\label{hypo:2:isog}
The mod~$\ell$ representation $\rhobar_{E,\ell}$ is irreducible.
\item{}[Noncongruence]\label{hypo:2:cong}
The representation $\rhobar_{E,\ell}$ is not isomorphic
to any representation $\rhobar_{g,\lambda}$ where $g\in S_2(\Gamma_0(N))$
is a newform of level dividing $N$ that is not conjugate to~$f$.
\end{enumerate}
Then there is an element of
order~$\ell$ in $\Sha(\Q, A_f)$ that is not visible in $J_0(N)$
but is strongly visible in $J_0(pN)$. More precisely,
there is an inclusion
$$
E(\Q)/\ell E(\Q) \hra \Ker(\Sha(\Q, A_f) \to \Sha(\Q, C))[\lambda]
\subset \Vis_{pN}(\Sha(\Q, A_f))[\lambda],
$$
where $C\subset J_0(pN)$ is isogenous to $A_f \times E$,
the homomorphism $A_f \to C$ has degree a power of~$2$,
and~$\lambda$ is the maximal ideal of $\T(N)$ corresponding
to $\rhobar_{E,\ell}$.
\end{theorem}
\begin{proof}
The divisibility assumptions of Hypothesis~\ref{hypo:2:nondiv}
on the $\cbar_{E,q}$ imply that the
Serre level of $\rhobar_{E,\ell}$ is~$N$ and since $\ell\nmid N$,
the Serre weight is $2$ (see \cite[Thm.~2.10]{ribet-stein:serre}).
Since $\ell$ is odd, Ribet's level lowering theorem~\cite{ribet:lowering}
implies that there
is {\em some} newform $h =\sum b_n q^n\in S_2(\Gamma_0(N))$ and a maximal
ideal $\lambda$ over~$\ell$
such that $a_q(E) \con b_q \pmod{\lambda}$ for all primes $q\neq p$.
By our non-congruence hypothesis, the only possibility is that $h$
is a $G_{\Q}$-conjugate of~$f$.
Since we can replace $f$ by any Galois conjugate of $f$ without changing
$A_f$, we may assume that $f=h$.
Also $a_p(f) \con -(p+1)\pmod{\lambda}$,
as explained in \cite[pg.~506]{ribet:congrel}.
Hypothesis~\ref{hypo:2:isog} implies that $\lambda$
is not Eisenstein, and by assumption $\ell\nmid m_A$,
so Proposition~\ref{prop:kill_comp_grp} implies
that $\Phi_{A,q}(\Fbar_q)[\lambda]=0$ for each $q\mid N$.
The theorem now follows from Theorem~\ref{thm:strongvis}.
\end{proof}
\begin{remark}
The condition $a_p(E) = -1$ is redundant. Indeed,
we have $\cbar_{E,p} \neq c_{E,p}$ since $\cbar_{E,p}$ is divisible by $\ell$
and $c_{E,p}$ is not. By studying the action of Frobenius on
the component group at $p$ one can show that this implies
that~$E$ has nonsplit multiplicative reduction,
so $a_p(E) = -1$.
\end{remark}
\begin{remark}
The non-congruence hypothesis of Theorem~\ref{thm:strongestvis}
can be verified using modular symbols as follows.
Let $W \subset H_1(X_0(N),\Z)_{\new}$ be the saturated submodule of
$H_1(X_0(N),\Z)$ that corresponds to all newforms in
$S_2(\Gamma_0(N))$ that are not Galois conjugate to~$f$. Let
$\overline{W} = W\tensor \F_{\ell}$. We require that the
intersection of the kernels of $T_q|_{\overline{W}} - a_q(E)$, for
$q\neq p$, has dimension~$0$.
\end{remark}
% We have $B[3] \subset A$, so $B[\m]=B[3][\m]\subset A[\m]$. Also,
% $B[3]$ has dimension 2 as a vector space over $\F_3$. But $A[\m]
% \subseteq J_0(2 \cdot 767)[\m]$ and by~\cite[Theorem
% 9.2]{edixhoven:weight}, we see that $J_0(2 \cdot 767)[\m]$ has
% dimension~$2$ over $\F_3$. Hence, $A[\m]$ has dimension at most 2, so
% $B[\m] = B[3] = A[\m]$.
% We have isomorphisms $C/B = (A+B)/B \isom A/(A \cap B)$. By using the inclusion
% $A[\m] = B[3] \subseteq A \cap B$ and Lemma~\ref{lemma:quotient} we obtain
% $$
% \left ( C/B \right)[\m] \isom \left ( A/(A \cap B) \right )[\m] \subseteq (A/A[\m])[\m] \simeq A[\m^2] / A[\m].
% $$
% According to \cite[\S II.14]{mazur:eisenstein}, the quotient $A[\m^2]/A[\m]$ injects into a direct sum of copies of
% $A[\m]$ (as Galois modules). But $A(\Q)[\m] = B(\Q)[\m] = 0$ as we verified in the previous step, so
% $(A[\m^2]/A[\m])(\Q) = 0$. Therefore $(C/B)(\Q)[\m] = 0$.
\section{Computational Examples}\label{sec:example}
In this section we give examples that illustrate how to use Theorem~\ref{thm:strongestvis}
to prove existence of elements of the Shafarevich-Tate group of a newform subvariety of $J_0(N)$ (for $767$
and $959$) which are invisible at the base level, but become visible in a modular Jacobian of higher
level.
\begin{hypothesis}
The statements in this section all make the hypothesis that certain
commands of the computer algebra system Magma \cite{Magma} produce correct output.
\end{hypothesis}
\subsection{Level $767$}
Consider the modular Jacobian $J_0(767)$. Using the modular symbols package in Magma,
one decomposes $J_0(767)$ (up to isogeny) into a product of six optimal quotients of dimensions 2, 3, 4,
10, 17 and 23. The duals of these quotients are subvarieties ${A_2}, {A_3}, {A_4},
A_{10}, {A_{17}}$ and ${A_{23}}$ defined over $\Q$, where $A_d$ has dimension $d$. Consider the subvariety
$A_{23}$.
We first show that the Birch and Swinnerton-Dyer conjectural formula predicts that the orders of
the groups $\Sha(\Q, A_{23})$ and $\Sha(\Q, A_{23}^\vee)$ are both divisible by 9.
\begin{proposition}\label{prop:767}
Assume~\cite[Conj.~2.2]{agashe-stein:bsd}.
Then
$$
3^2 \mid \#\Sha(\Q, A_{23})\quad\text{ and }\quad 3^2\mid \#\Sha(\Q, A^{\vee}_{23}).
$$
\end{proposition}
\begin{proof}
Let $A = A^\vee_{23}$. We use~\cite[\S 3.5 and \S3.6]{agashe-stein:bsd} (see also \cite{katz:torsion})
to compute a multiple of the order of the torsion subgroup $A(\Q)_{\tor}$. This multiple
is obtained by injecting the torsion subgroup into the group of $\F_p$-rational points on the reduction
of $A$ for odd primes~$p$ of good reduction and then computing the order of that group. Hence, the multiple
is an isogeny invariant, so one gets the same multiple for $A^\vee(\Q)_{\tor}$. For producing a divisor of
$\#A(\Q)_{\tor}$, we use the injection of the subgroup of rational cuspidal divisor classes of degree~$0$ into
$A(\Q)_{\tor}$.
Using the implementation in Magma we obtain $120 \mid \#A(\Q)_{\tor} \mid 240$.
To compute a divisor of $A^\vee(\Q)_{\tor}$, we use the algorithm described
in~\cite[\S 3.3]{agashe-stein:bsd} to find that the modular degree $m_A = 2^{34}$, which
is not divisible by any odd primes, hence $15 \mid \#A^\vee(\Q)_{\tor} \mid 240$.
Next, we use~\cite[\S4]{agashe-stein:bsd} to compute the ratio of the special value of the
$L$-function of $A_{/\Q}$ at 1 over the real N\'eron period $\Omega_{A}$. We obtain \linebreak
$\displaystyle \frac{L(A_{/\Q}, 1)}{\Omega_{A}} =
c_{A} \cdot \frac{2^9 \cdot 3}{5}$, where $c_{A}\in\Z$ is the Manin constant.
Since $c_{A} \mid 2^{\textrm{dim}(A)}$ by~\cite{agashe-ribet-stein} then
$$
\frac{L(A_{/\Q}, 1)}{\Omega_{A}} = \frac{2^{n+2} \cdot 3}{5},
$$
for some $0 \leq n \leq 23$. In particular, the modular abelian variety $A_{/\Q}$ has rank
zero over $\Q$.
Next, using the algorithms from \cite{conrad-stein:compgroup, kohel-stein:ants4} we compute the Tamagawa
number $c_{A,13} = 1920 = 2^3 \cdot 3 \cdot 5$. We also find that $2 \mid c_{A,59}$ is a power
of $2$ because $W_{59}$ acts as~$1$ on $A$, and on the component
group $\Frob_{59}=-W_{59}$, so the fixed subgroup $\Phi_{A,59}(\F_{59})$ of Frobenius
is a $2$-group (for more details, see \cite[Prop.~3.7--8]{ribet:modreps}).
% In fact, we can get only a multiple of this order by
% computing the order of the component group (i.e., the group of
% $\overline{\F}_{59}$-rational points). We obtain .
Finally, the Birch and Swinnerton-Dyer conjectural formula for abelian varieties of Mordell-Weil rank zero
(see \cite[Conj.~2.2]{agashe-stein:bsd}) asserts that
$$
\frac{L(A_{/\Q},1)}{\Omega_{A}} = \frac{\#\Sha(\Q, A) \cdot c_{A,13} \cdot
c_{A,59}}{\# A(\Q)_{\tor} \cdot \# A^\vee(\Q)_{\tor}}.
$$
By substituting what we computed above, we obtain $3^2 \mid \#\Sha(\Q, A)$.
Since $L(A_{/\Q}, 1) \ne 0$, \cite{kolyvagin-logachev:finiteness} implies
that $\Sha(\Q, A)$ is finite. By the nondegeneracy of the Cassels-Tate pairing,
$\# \Sha(\Q, A) = \# \Sha(A^\vee/\Q)$. Thus, if the BSD conjectural
formula is true then $3^2 \mid \# \Sha(\Q, A) = \# \Sha(\Q, A^\vee)$.
\end{proof}
We next observe that there are no visible elements of odd order for the embedding
${A_{23}}_{/\Q} \hookrightarrow {J_0(767)}_{/\Q}$.
\begin{lemma}
Any element of $\Sha(\Q, A_{23})$ which is visible in $J_0(767)$ has order a power
of~$2$.
\end{lemma}
\begin{proof}
Since $m_{A_{23}} = 2^{34}$, \cite[Prop. 3.15]{agashe-stein:bsd} implies that any element
of $\Sha(\Q,A_{23})$ that is visible in $J_0(767)$ has order a power of $2$.
\end{proof}
Finally, we use Theorem~\ref{thm:strongestvis} to prove the existence of non-trivial elements of order 3
in $\Sha(\Q, A_{23})$ which are invisible at level $767$, but become visible at higher level.
In particular, we prove unconditionally that $3 \mid \#\Sha(\Q, A_{23})$ which provides evidence for
the Birch and Swinnerton-Dyer conjectural formula.
\begin{proposition}\label{prop:767vis}
There is an element of order 3 in $\Sha(\Q, A_{23})$ which is not
visible in $J_0(767)$ but is strongly visible in $J_0(2 \cdot 767)$.
\end{proposition}
\begin{proof}
Let $A=A_{23}$, and note that $A$ has rank~$0$, since $L(A_{/\Q},1)\neq 0$.
Using Cremona's database~\cite{cremona:onlinetables} we find that the
elliptic curve
$$
E:\qquad y^2 + xy = x^3 - x^2 + 5x + 37
$$
has conductor $2\cdot 767$ and Mordell-Weil group $E(\Q) = \Z \oplus \Z$.
Also
$$
c_2 = 2, c_{13} = 2, c_{59} = 1,
\cbar_2 = 6, \cbar_{13} = 2, \cbar_{59}=1.
$$
We apply Theorem~\ref{thm:strongestvis} with $\ell=3$ and $p=2$.
Since $E$ does not admit any rational $3$-isogeny (by
\cite{cremona:onlinetables}),
Hypothesis~\ref{hypo:2:isog} is satisfied.
The level is square free and the modular degree of $A$ is a power of~$2$, so
Hypothesis~\ref{hypo:2:nondiv} is satisfied.
% We have $a_2=-1$, and
% $$
% \nu(2\cdot 767) = (2+1)\cdot (13+1)\cdot (59+1) = 2520,
% $$
% so to finish verifying the hypothesis of Theorem~\ref{thm:strongvis},
% we check that $a_q(E) \con a_q(f) \pmod{\lambda}$,
% for the $367$ odd primes $q< 2520$. Using Magma we find
% that this congruence condition is satisfied, which completes the proof.
% Finally we can verify
% that $a_q(E) \con a_q(f) \pmod{\lambda}$,
% for all odd primes $q$ vastly more efficiently as follows.
% The discriminant of $E$ is $-638144=2^6 \cdot 13^2 \cdot 59$,
% so the Serre level of the mod~$3$ representation $\rho_{E,3}$
% is $13\cdot 59$ (cite Krauss's paper...). By Ribet's theorem \cite{},
% there is at least one newform $h\in S_2(\Gamma_0(13\cdot 59))$
% such that $a_q(E) \con a_q(h) \pmod{\wp}$, where $\wp\mid 3$.
We have $a_3(E) = -3$. Using Magma we find
$$
\det(T_3|_{\Wbar} - (-3)) \con 1\pmod{3},
$$
which verifies the noncongruence hypothesis and completes the proof.
% \begin{verbatim}
% > M := ModularSymbols(767,2,+1);
% > S := CS(NS(M));
% > D := ND(S);
% > A := D[6];
% > B := Complement(A) meet S;
% > f :=CharacteristicPolynomial(HeckeOperator(B,3));
% > Evaluate(f,-3);
% 9952742800
% > (Integers()!Evaluate(f,-3)) mod 3;
% 1
% \end{verbatim}
% When we have the modular degree hypothesis, i.e., $\ell\nmid m_A$,
% there will always be at most one factor that is congruent
% to $\sum a_n(E) q^n$, so the above much more efficient algorithm
% will always work in this case.
\end{proof}
\subsection{Level 959}
We do similar computations for a $24$-dimensional
abelian subvariety of $J_0(959)$.
We have $959=7\cdot 137$, which is square free.
There are five newform abelian subvarieties of the
Jacobian, $A_2, A_7, A_{10}, A_{24}$ and $A_{26}$, whose dimensions
are the corresponding subscripts.
Let $A_f = A_{24}$ be the 24-dimensional newform abelian subvariety.
\begin{proposition}\label{prop:959vis}
There is an element of order 3 in $\Sha(A_f/\Q)$ which is not
visible in $J_0(959)$ but is strongly visible in $J_0(2 \cdot 959)$.
\end{proposition}
\begin{proof}
Using Magma we find that $m_A = 2^{32} \cdot 583673$, which is coprime
to~$3$. Thus we apply Theorem~\ref{thm:strongestvis} with $\ell=3$ and
$p=2$. Consulting \cite{cremona:onlinetables} we find the curve
E=1918C1, with Weierstrass equation
$$
y^2 +xy + y = x^3 - 22x - 24,
$$
with Mordell-Weil group $E(\Q)\isom \Z \oplus \Z \oplus (\Z/2\Z)$,
and
$$
c_2 = 2, c_7 = 2, c_{137}=1, \cbar_2 = 6, \cbar_7 = 2, \cbar_{137}=1.
$$
Using \cite{cremona:onlinetables} we find that $E$ has no rational $3$-isogeny.
The modular form attached to $E$ is
$$
g = q - q^2 - 2q^3 + q^4 - 2q^5 + \cdots,
$$
and we have
$$
\det(T_2|_{\overline{W}} - (-2)) = 2177734400 \con 2 \pmod{3},
$$
which completes the verification.
\end{proof}
\section{Conjecture, evidence and more computational data}\label{sec:conj}
We state several conjectures, provide some evidence and finally, provide a table that
we computed using similar techniques to those in Section~\ref{sec:example}
\subsection{The conjecture}
The two examples computed in Section~\ref{sec:example} show that for an abelian subvariety
$A$ of $J_0(N)$ an invisible element of $\Sha(\Q, A)$ at the base level $N$ might become visible
at a multiple level $NM$. We state a general conjecture according to which any element of
$\Sha(\Q, A)$ should have such a property.
\comment{
\begin{conjecture}
Let $h = 0$ or $1$. Suppose that $A$ is an abelian subvariety of $J_h(N)$
and $c\in\Sha(\Q, A)$. Then there is a positive integer~$M$, and a
homomorphism of abelian varieties $\iota :A \to J_h(NM)$ of finite degree coprime
to the order of~$c$ such that $\iota _*c=0$.
\end{conjecture}
}
\begin{conjecture}\label{conj1}
Let $h=0$ or $1$. Suppose $A$ is a $J_h$-modular abelian variety and
$c \in \Sha(\Q,A)$. Then there is a $J_h$-modular abelian variety $C$
and an inclusion $\iota : A \to C$ such that $\iota_* c = 0$.
\end{conjecture}
\begin{remark}
For any prime $\ell$, the Jacobian $J_h(N)$ comes
equipped with two morphisms $\alpha^*, \beta^* : J_h(N) \ra J_h(N \ell)$
induced by the two degeneracy maps $\alpha, \beta : X_h(\ell N) \ra X_h(N)$
between the modular curves of levels $\ell N$ and $N$, and it is natural to
consider visibility of $\Sha(\Q, A)$ in $J_h(N\ell)$ via morphisms $\iota$
constructed from these degeneracy maps.
\end{remark}
\begin{remark}
It would be interesting to understand the set of all levels $N$ of $J_h$-modular abelian varieties
$C$ that satisfy the conclusion of the conjecture.
\end{remark}
\subsection{Theoretical Evidence for the Conjectures}\label{subsec:evidence}
The first piece of theoretical evidence for Conjecture~\ref{conj1} is Remark~\ref{rem:vis}, according
to which any cohomology class $c\in \H^1(K,A)$ is visible in some abelian variety~$C_{/K}$.
The next proposition gives evidence for elements of $\Sha(\Q, E)$ for an elliptic curve $E$ and elements
of order 2 or 3.
\begin{proposition}
Suppose $E$ is an elliptic curve over $\Q$.
Then Conjecture~\ref{conj1} for $h = 0$ is true for all
elements of order $2$ and $3$ in $\Sha(\Q, E)$.
\end{proposition}
\begin{proof}
We first show that there is an abelian
variety $C$ of dimension $2$ and an injective homomorphism
$i:E\hra C$ such that $c\in \Vis_C(\Sha(\Q, E))$.
If $c$ has order~$2$, this follows from
\cite[Prop.~2.4]{agashe-stein:visibility} or
\cite{klenke:phd},
and if $c$ has order~$3$, this follows from
\cite[Cor.~pg.~224]{mazur:visord3}.
The quotient $C/E$ is an elliptic curve, so $C$ is isogenous to a product
of two elliptic curves. Thus by
\cite{breuil-conrad-diamond-taylor}, $C$ is a quotient
of $J_0(N)$, for some~$N$.
\end{proof}
We also prove that Conjecture~\ref{conj1} is true with $h=1$
for all elements of $\Sha(\Q, A)$ which split over abelian extensions.
\begin{proposition}
Suppose $A_{/\Q}$ is a $J_1$-modular abelian variety over~$\Q$ and
$c\in\Sha(\Q, A)$ splits over an abelian extension of~$\Q$. Then
Conjecture~\ref{conj1} is true for~$c$ with $h = 1$.
\end{proposition}
\begin{proof}
Suppose $K$ is an abelian extension such that $\res_K(c)=0$ and let $C=\Res_{K/\Q}(A_K)$.
Then $c$ is visible in $C$ (see Section~\ref{rem:vis}). It remains to verify that $C$
is modular. As discussed in \cite[pg.~178]{milne:bsdres}, for any abelian variety $B$ over $K$,
we have an isomorphism of Tate modules
$$
\Tate_\ell(\Res_{K/\Q}(B_K)) \isom \Ind^{G_\Q}_{G_K} \Tate_\ell(B_K),
$$
and by Faltings's isogeny theorem~\cite{faltings:finiteness}, the Tate module determines an
abelian variety up to isogeny. Thus if $B=A_f$ is an abelian variety attached to a
newform, then $\Res_{K/\Q}(B_K)$ is isogenous to a product of
abelian varieties $A_{f^{\chi}}$, where $\chi$ runs through
Dirichlet characters attached to the abelian extension $K/\Q$.
Since $A$ is isogenous to a product of abelian varieties of the form
$A_f$ (for various $f$), it follows that the restriction of scalars $C$ is modular.
\end{proof}
\begin{remark}
Suppose that $E$ is an elliptic curve and $c\in \Sha(\Q, E)$. Is there
an abelian extension $K/\Q$ such that $\res_K(c)=0$? The answer is ``yes'' if and only if there is a
$K$-rational point (with $K$-abelian) on the locally trivial principal homogeneous space corresponding to $c$
(this homogenous space is a genus one curve). Recently, M. Ciperiani and A. Wiles proved that any genus one curve over $\Q$ which
has local points everywhere and whose Jacobian is a semistable elliptic curve admits a point over a solvable
extension of $\Q$ (see~\cite{ciperiani-wiles}). Unfortunately, this paper does not answer our question about
the existence of abelian points.
\end{remark}
\begin{remark}
As explained in \cite{stein:nonsquaresha}, if $K/\Q$ is an abelian extension of
prime degree then there is an exact sequence
$$
0 \to A \to \Res_{K/\Q}(E_K) \xra{\Tr} E \to 0,
$$
where $A$ is an abelian variety with $L(A_{/\Q},s) = \prod L(f_i,s)$ (here, the $f_i$'s
are the $G_\Q$-conjugates of the twist of the newform $f_E$ attached to~$E$
by the Dirichlet character associated to $K/\Q$). Thus one could approach
the question in the previous remark by investigating whether or not
$L(f_E,\chi,1)=0$ which one could do using modular symbols (see~\cite{chantal:vanish}).
The authors expect that $L$-functions of twists of degree larger than three are very
unlikely to vanish at $s=1$ (see \cite{chantal:vanish}), which suggests that in general,
the question might have a negative answer for cohomology classes of order larger than
$3$.
\end{remark}
\subsection{Visibility of Kolyvagin cohomology classes}
It would also be interesting to study visibility at higher
level of Kolyvagin cohomology classes. The following is a first
``test question'' in this direction.
\begin{question}
Suppose $E\subset J_0(N)$ is an elliptic curve with conductor~$N$,
and fix a prime~$\ell$ such that $\rhobar_{E,\ell}$ is surjective.
Fix a quadratic imaginary field $K$ that satisfies the Heegner
hypothesis for~$E$. For any prime $p$ satisfying the
conditions of~\cite[Prop.~5]{rubin:kolyvagin}, let
$c_{p} \in \H^1(\Q,E)[\ell]$ be the corresponding Kolyvagin
cohomology class.
There are two natural homomorphisms
$\delta_1^*, \delta_p^*:E \to J_0(Np)$.
When is
$$
(\delta_1^* \pm \delta_\ell^*)_*(c_\ell) = 0 \in \H^1(\Q,J_0(Np))?
$$
When is
$$
\res_v((\delta_1^* \pm \delta_\ell^*)_*(c_\ell)) = 0 \in \H^1(\Q_v,J_0(Np))?
$$
\end{question}
\subsection{Table of Strong Visibility at Higher Level}\label{subsec:data}
The following is a table that gives the known examples of ${A_f}_{/\Q}$
with square free conductor $N \leq 1339$,
such that the Birch and Swinnerton-Dyer conjectural formula predicts an
odd prime divisor $\ell$ of $\Sha(\Q, A_f)$, but $\ell$ does
not divide the modular degree of $A_f$.
These were taken from \cite{agashe-stein:bsd}. If there is an entry in the
fourth column, this means we have verified the hypothesis of
Theorem~\ref{thm:strongestvis}, hence there really is a nonzero element in $\Sha(\Q, A_f)$
that is not visible in $J_0(N)$, but is strongly visible in $J_0(pN)$.
The notation in the fourth column is $(p,E,q)$, where $p$ is the
prime used in Theorem~\ref{thm:strongestvis}, $E$ is an elliptic
curve, denoted using a Cremona label,
and $q\neq p$ is a prime such that
$$
\bigcap_{q'\leq q} \Ker(T_q'|_{\Wbar} - a_{q'}(E)) = 0.
$$
\vspace{0.2in}
{\small
\begin{center}
\begin{tabular}{|l|c|c|c|l|}\hline
$A_f$ & dim & $\ell\mid\Sha(A_f)_?$ & moddeg & $(p,E,q)$'s\\\hline
{\bf 551H} & 18 & 3 & $2^{?} \cdot 13^2$ & (2, 1102A1, -) \\\hline
{\bf 767E} & 23 & 3 & $2^{34}$ & (2, 1534B1, 3)\\\hline
{\bf 959D} & 24 & 3 & $2^{32}\cdot 583673$ & (2, 1918C1, 5), (7, 5369A1,2)\\\hline
{\bf 1337E} & 33 & 3 & $2^{59} \cdot 71$ & (2, 2674A1, 5) \\\hline
{\bf 1339G} & 30 & 3 & $2^{48} \cdot 5776049$ & (2, 2678B1, 3), (11, 14729A1,2) \\\hline
\end{tabular}
\end{center}}
\bibliography{biblio}
%\bibliographystyle{amsalpha} \bibliography{biblio}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{BCDT01}
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\end{thebibliography}
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\Addresses
\end{document}