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\section{Background}
The proposed project reflects the interplay of abstract theory
with explicit machine computation, as illustrated by the following
quote of Bryan Birch~\cite{birch:bsd}:
\begin{quote}
I want to describe some computations undertaken by myself and
Swin-nerton-Dyer on EDSAC by which we have calculated the
zeta-functions of certain elliptic curves. As a result of these
computations we have found an analogue for an elliptic curve of
the Tamagawa number of an algebraic group; and conjectures (due to
ourselves, due to Tate, and due to others) have proliferated.
\end{quote}


The PI is primarily interested in abelian varieties attached to
modular forms via Shimura's construction \cite{shimura:factors},
which we now recall. Let~$f=\sum a_n q^n$ be a weight~$2$ newform
on $\Gamma_1(N)$. Then~$f$ corresponds to a differential on the
modular curve $X_1(N)$, which is a curve whose affine points
over~$\C$ correspond to isomorphism classes of pairs $(E,P)$,
where~$E$ is an elliptic curve and $P\in E$ is a point of
order~$N$.  We view the Hecke algebra
\[\T=\Z[T_1,T_2,T_3,\ldots]\]
 as a subring of the endomorphism ring of the Jacobian $J_1(N)$
of $X_1(N)$. Let $I_f$ be the kernel of the homomorphism $\T\to
\Z[a_1,a_2,a_3]$ that sends $T_n$ to $a_n$, and attach to~$f$ the
quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple abelian
variety over~$\Q$ of dimension equal to the degree of the field
$\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$.  We
also sometimes consider a similar construction with $J_1(N)$
replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$
that parametrizes isomorphism classes of pairs $(E,C)$, where $C$
is a cyclic subgroup of $E$ of order~$N$.

\begin{definition}[Modular abelian variety]
A {\em newform abelian variety} is an abelian variety over~$\Q$ of
the form $A_f$.  An abelian variety over a number field is a {\em
modular abelian varieties} if it is a quotient of $J_1(N)$ for
some~$N$.
\end{definition}
Over $\Q$, newform abelian varieties are simple and every modular
abelian variety is isogenous to a product of copies of newform
abelian varieties.  Newform abelian varieties are typically not
absolutely simple.

Newform abelian varieties $A_f$ are important. For example,the
celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond,
R.~Taylor, and A.~Wiles \cite{breuil-conrad-diamond-taylor}
asserts that every elliptic curve over~$\Q$ is isogenous to some
$A_f$.  Also, J-P.~Serre conjectures that, up to twist, every
two-dimensional odd irreducible mod~$p$ Galois representation
appears in the torsion points on some $A_f$.

Much of this research proposal is inspired by the following
special case of the Birch and Swinnerton-Dyer conjecture:
\begin{conjecture}[BSD Conjecture (special case)]\label{conj:bsd}
Let $A$ be a modular abelian variety over~$\Q$.
\begin{enumerate}%
\item $L(A,1)=0$ if and
only if $A(\Q)$ is infinite.%
\item If $L(A,1)\neq 0$, then
\[
\frac{L(A,1)}{\Omega_{A}} =%
\frac{\prod c_p \cdot \#\Sha(A)}%
{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}},
\]
where the objects and notation in this formula are discussed
below.
\end{enumerate}
\end{conjecture}
Here $L(A,s)$ is the $L$-series attached to $A$, which is entire
because~$A$ is modular, so $L(A,1)$ makes sense.  The real volume
$\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of
differentials for the N\'eron model of $A$. For each prime $p\mid
N$, the integer $c_p=\#\Phi_{A,p}(\F_p)$ is the {\em Tamagawa
number} of~$A$ at~$p$, where $\Phi_{A,p}$ denotes the component
group of the N\'eron model of~$A$ at $p$.  The dual of $A$ is
denoted $A^{\vee}$, and in the conjecture $A(\Q)_{\tor}$ and
$A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em
Shafarevich-Tate group} of $A$ is
\[
 \Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq
 \infty} \H^1(\Q_p,A)\right),
\]
which is a group that measures the failure of a local-to-global
principle.  When $L(A,1)\neq 0$, Kato proved in \cite{kato:secret}
that $\Sha(A)$ and $A(\Q)$ are finite, so $\#\Sha(A)$ makes sense
and one implication of part 1 of the conjecture is known.

\begin{remark}
The general Birch and Swinnerton-Dyer conjecture (see
\cite{tate:bsd, lang:nt3}) is a conjecture about any abelian
variety $A$ over a global field~$K$. It asserts that the order of
vanishing of $L(A,s)$ at $s=1$ equals the free rank of $A(K)$, and
gives a formula for the leading coefficient of the Taylor
expansion of $L(A,s)$ about $s=1$.
\end{remark}

The rest of this proposal is divided into two parts.  The first is
about computing with modular forms and abelian varieties, and
making the results of these computations available to the
mathematical community.
 The second is about visibility of Mordell-Weil and Shafarevich-Tate groups, the
ultimate goal being to obtain relationships between parts 1 and 2
of Conjecture~\ref{conj:bsd}.

\section{Computing with modular forms}

The PI proposes to continue developing algorithms and making
available tools for computing with modular forms, modular abelian
varieties, and motives attached to modular forms. This includes
finishing a major new {\sc Magma} \cite{magma} package for
computing directly with modular abelian varieties over number
fields, extending the Modular Forms Database \cite{mfd}, and
searching for algorithms for computing the quantities appearing in
Conjecture~\ref{conj:bsd} and in the Bloch-Kato conjecture for
modular motives.



\subsection{The Modular Forms Database}%
The Modular Forms Database \cite{mfd} is a freely-available
collection of data about objects attached to cuspidal modular
forms. It is analogous to Sloane's tables of integer sequences,
and extends Cremona's tables \cite{cremona:onlinetables} to
dimension bigger than one and weight bigger than two. Cremona's
tables contain more refined data about elliptic curves than
\cite{mfd}, but the PI intends to work with Cremona to make the
modular forms database a superset of \cite{cremona:onlinetables}.

The database is used world-wide by prominent number theorists,
including Noam Elkies, Matthias Flach, Dorian Goldfeld, Benedict
Gross, Ken Ono, and Don Zagier.

The PI proposes to greatly expand the database.  A major challenge
is that data about modular abelian varieties of large dimension
takes a huge amount of space to store.  For example, the database
currently occupies 40GB disk space.   He proposes to find and
implement a better method for storing information about modular
abelian varieties so that the database will be more useful.  He
has found a method whereby a certain eigenvector is computed by
the database server, which may (or may not!) enable storing
coefficients of modular forms far more efficiently; however, he
has not yet tried to implement it and study its properties.

The PI proposes to improve the usability of the database.  It is
currently implemented using a PostgreSQL database coupled with a
Python web interface. To speed access and improve efficiency, he
is considering rewriting key portions of the database using MySQL
and PHP.  He hopes to rewrite key portions of the database in
response to user feedback that he has received.   The database
currently runs on a three-year-old 933Mhz Pentium III, which has
unduly limited disk space and no offsite backup, so the PI is
requesting a powerful modern computer with a large hard drive
array and external hard drives for offsite backups.

\subsubsection{M{\small AGMA} package for modular abelian varieties}\label{sec:magma}%
The PI's software is published as part of the non-commercial {\sc
Magma} computer algebra system. The core of {\sc Magma} is
developed by a group of academics at the University of Sydney, who
are supported mostly by grant money. {\sc Magma} is considered by
many to be the most comprehensive tool for research in number
theory, finite group theory, and cryptography, and is widely
distributed.  The PI has already written over 400 pages (26000
lines) of modular forms code and extensive documentation that is
distributed with {\sc Magma}, and intends to ``publish'' future
work in {\sc Magma}.

As mentioned above, an abelian variety $A$ over a number field~$K$
is {\em modular} if it is a quotient of $J_1(N)$ for some $N$.
Modular abelian varieties were studied intensively by Ken Ribet,
Barry Mazur, and others during recent decades, and studying them
is popular because results about them often yield surprising
insight into number theoretic questions.  Computation with modular
abelian varieties is attractive because they are much easier to
describe than arbitrary abelian varieties, and their $L$-functions
are reasonably well understood when~$K$ is an abelian extension
of~$\Q$.

The PI recently designed and partially implemented a general
system for computing with modular abelian varieties over number
fields.  He hopes to develop and refine several crucial components
of the system.  For example, three major problems arose, and the
PI intends to resolve them in order to have a completely
satisfactory system for computing with modular abelian varieties.
\begin{enumerate}
\item {\em Given a modular abelian variety $A$, efficiently
compute the endomorphism ring $\End(A)$ as a ring of matrices
acting on $\H_1(A,\Z)$.} The PI has found a modular symbols
solution that draws on work of Ribet \cite{ribet:twistsendoalg}
and Shimura \cite{shimura:factors}, but it is too slow to be
really useful in practice.  In \cite{merel:1585}, Merel uses
Herbrand matrices and Manin symbols to give efficient algorithms
for computing with Hecke operators. The PI intends to carry over
Merel's method to
give an efficient algorithm to compute $\End(A)$.%
\item {\em Given $\End(A)\otimes\Q$, compute an isogeny
decomposition of $A$ as a product of simple abelian varieties.}
This is a standard and difficult problem in general, but it might
be possible to combine work of Allan Steel on his
``characteristic~$0$ Meataxe'' with special features of modular
abelian varieties to solve it in practice.  It is absolutely {\em
essential} to solve this problem in order to explicitly enumerate
all modular abelian varieties over $\Qbar$ of given level~$N$.
Such an enumeration would be a major step towards the ultimate
possible generalization of Cremona's tables
\cite{cremona:onlinetables} to modular abelian varieties.
Computation of a decomposition is also crucial to other
algorithms, e.g., computing
complements and duals of abelian subvarieties.%
\item {\em Given two modular abelian varieties over a number field
$K$, decide whether there is an isomorphism between them.} When
the endomorphism ring of each abelian variety is known and both
are simple, it is possible to reduce this problem to the solution
of a norm equation, which has been studied extensively in many
cases. This problem is analogous to the problem of testing
isomorphism for modules over a fixed ring, which has been solved
with much effort for many classes of rings. One application is
to proving that specific abelian varieties can not be principally polarized.%
\end{enumerate}

\subsection{An Example: The Arithmetic of $J_1(p)$}
We finish be describing recent work of the PI on the modular
Jacobian $J_1(p)$, where~$p$ is a prime, that was partly inspired
by computation. The following conjecture generalizes a conjecture
of Ogg, which asserts that $J_0(p)(\Q)_{\tor}$ is cyclic of order
the numerator of $(p-1)/12$, a fact that Mazur proved in
\cite{mazur:eisenstein}.
\begin{conjecture}[Stein]\label{conj:tor}
Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the
group generated by the cusps on $X_1(p)$ that lie over $\infty\in
X_0(p)$.
\end{conjecture}
The PI gives significant numerical evidence for this conjecture in
\cite{conrad-edixhoven-stein:j1p}, and cuspidal subgroups of
$J_1(p)$ are considered in detail in \cite{kubert-lang}, where,
e.g.,  they compute orders of such groups in terms of Bernoulli
numbers.

Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the
proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier.  The
PI intends to explore whether or not it is possible to build on
Mazur's method and prove results towards
Conjecture~\ref{conj:tor}.  The PI also intends to develop his
computational methods for computing torsion subgroups in order to
answer, at least conjecturally, the following question.
\begin{question}
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? Is
the product of the orders of all $A_f(\Q)_{\tor}$ over all classes
of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
\end{question}
The PI conjectured that the analogous questions for $J_0(p)$
should have ``yes'' answers, and in \cite{emerton:optimal}
M.~Emerton proved this conjecture.  There he also proved that the
natural map from the component group of $J_0(p)$ to that of $A_f$
is surjective.  By \cite{conrad-edixhoven-stein:j1p}, the
component group of $J_1(p)$ is trivial, which suggests the
following question.
\begin{question}\label{ques:comp}
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
component group of $A_f$ trivial?
\end{question}
Even assuming the full BSD conjecture (and a conjecture about a
Manin constant), the PI has not yet produced enough data to give a
conjectural answer to this question.   He has many examples in
which the conjecture predicts that either $\Sha(A_f)$ is
nontrivial or the component group of $A_f$ is nontrivial.  He and
B.~Poonen formulated, and hope to carry out, a strategy to decide
which of these two is nontrivial by using an explicit description
of $\End({A_f/\Qbar})$  to obtain a curve whose Jacobian is $A_f$.
Note that computing $\End(A_f/\Qbar)$ in general is the second
problem in Section~\ref{sec:magma} above.

\section{Visibility}
The underlying motivation for this part of the proposal is to
prove implications between the two parts of
Conjecture~\ref{conj:bsd}, in examples and eventually in some
generality.  That is, we link information about the first part of
the BSD conjecture for an abelian variety~$B$ to information about
the second part of the conjecture for a related abelian
variety~$A$.  Visibility provides a conceptual framework in which
to organize our ideas.

\subsection{Computational problems}\label{sec:compprob}
Barry Mazur introduced visibility in order to unify various
constructions of Shafarevich-Tate groups.
\begin{definition}[Visibility of Shafarevich-Tate Groups]
Suppose that $$\iota:A\hra J$$ is an inclusion of abelian
varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$
with respect to $J$ is
\[
\Vis_J\H^1(\Q,A) := \Ker(\H^1(\Q,A)\to \H^1(\Q,J)).
\]
The {\em visible subgroup} of $\Sha(A)$ in~$J$ is the intersection
of $\Sha(A)$ with $\Vis_J\H^1(\Q,A)$; equivalently,
\[
 \Vis_J\Sha(A) := \Ker(\Sha(A)\to \Sha(J)).
\]
\end{definition}
The terminology ``visible'' arises from the fact that if
$x\in\Sha(A)$ is visible in~$J$, then a principal homogenous
space~$X$ corresponding to~$x$ can be realized as a subvariety
of~$J$.

Before discussing theoretical questions about visibility, we
describe computational evidence for the Birch and Swinnerton-Dyer
conjecture for modular abelian varieties (and motives) that the PI
and A.~Agashe obtained using theorems inspired by the definition
of visibility. In \cite{agashe-stein:visibility}, the PI and
Agashe prove a theorem that makes it possible to use abelian
varieties of positive rank to explicitly construct subgroups of
Shafarevich-Tate groups of other abelian varieties.   The main
theorem is that if $A$ and $B$ are abelian subvarieties of an
abelian variety $J$, and $B[p]\subset A$, then, under certain
hypothesis, there is an injection
\[ B(\Q)/p B(\Q) \hra \Vis_J\Sha(A).\]
The paper concludes with the first ever example of an abelian
variety $A_f$ attached to a newform, of large dimension ($20$),
whose Shafarevich-Tate group has order that is provably divisible
by an odd prime ($5$).

The PI has used the result described above to give evidence for
the BSD conjecture for many $A\subset J_0(N)$, where $A$ is
attached to a newform of level~$N\leq 2333$.  The PI proposes to
give similar evidence using visibility in $J_0(NM)$ for small~$M$.
More precisely, \cite{agashe-stein:bsd} describes the computation
of an odd divisor of the BSD conjectural order of $\Sha(A)$ for
over ten thousand $A\subset J_0(N)$ with $L(A,1)\neq 0$ (these are
{\em all} simple $A$ with $N\leq 2333$ and $L(A,1)\neq 0$). For
over a hundred of these, the divisor of the conjectural order of
$\Sha(A)$ is divisible by an odd prime; for a quarter of these the
PI and Agashe prove that if~$n$ is the conjectural divisor of the
order of $\Sha(A)$, then there are at least~$n$ elements of
$\Sha(A)$ that are visible in $J_0(N)$.

The PI intends to investigate the remaining 75\% of the $A$ with
$n>1$ by considering the image of $A$ in $J_0(NM)$ for small
integers~$M$. Information about which~$M$ to choose can be
extracted from Ribet's level raising theorem (see
\cite{ribet:raising}).  As a test, the PI recently tried the first
example with conjectural odd $\Sha(A)$ that is not visible in
$J_0(N)$ (this is an $18$ dimensional abelian variety $A$ of level
$551$ such that $9\mid \#\Sha(A)$). He showed in
\cite{stein:bsdmagma} that there are elements of order~$3$ in
$\Sha(A)$ that are visible in $J_0(551\cdot 2)$.   Since the
dimension of $J_0(NM)$ grows very quickly, a huge amount of
computer memory will be required to investigate visibility at
higher level.  Fortunately, the PI recently received a grant from
Sun Microsystems for a \$67,000 computer that contains 22GB of
contiguously addressable RAM (the processors are relatively slow
and the hard drive is small, making this computer less suitable as
a platform for the modular forms database, which requires a large
hard drive but not so much RAM).

Some of these ideas generalize to the context of Grothendieck
motives, which A.~Scholl attached to newforms of weight greater
than two. N.~Dummigan, M.~Watkins, and the PI did work in this
direction in \cite{dummigan-stein-watkins:motives}. There we prove
a theorem that can sometimes be used to deduce the existence of
visible Shafarevich-Tate groups in motives attached to modular
forms, assuming a conjecture of Beilinson about ranks of Chow
groups. However, we give several pages of tables that suggest that
Shafarevich-Tate groups of modular motives of level~$N$ are rarely
visible in the higher-weight motivic analogue of $J_0(N)$, much
more rarely than for weight~$2$.  Just as above, the question
remains to decide whether one expects these groups to be visible
in the analogue of $J_0(N M)$ for some integer~$M$.  It would be
relatively straightforward for the PI to do computations in this
direction, and he intends to do so.

Before moving on to theoretical questions about visibility, we
pause to emphasize that the above computational investigations
into the Birch and Swinnerton-Dyer conjecture motivated the PI and
others to develop new algorithms for computing with modular
abelian varieties.  For example, in \cite{conrad-stein:compgroup},
B.~Conrad and the PI use Grothendieck's monodromy pairing to give
an algorithm for computing orders of component groups of certain
purely toric abelian varieties.  This algorithm makes it practical
to compute component groups of quotients $A_f$ of $J_0(N)$ at
primes~$p$ that exactly divide $N$.  Without such an algorithm it
would probably be difficult to get very far in computational
investigations into the Birch and Swinnerton-Dyer conjecture for
abelian varieties; indeed, the only other paper in this direction
is \cite{empirical}, which restricts to the case of Jacobians of
genus~$2$ curves.


\subsection{Theoretical problems}
\subsubsection{Visibility at higher level}
Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and
let $A=A_f^{\vee}\subset J_1(N)$ be its dual.  One expects that
most of $\Sha(A)$ is {\em not} visible in $J_1(N)$.  The following
conjecture then arises.
\begin{conjecture}[Stein]\label{conj:allvis} For each $x\in \Sha(A)$, there is an integer
$M$ and a morphism $f:A\to J_1(NM)$, of finite degree and coprime
to the order of~$x$, such that the image of $x$ in $\Sha(f(A))$ is
visible in $J_1(NM)$.
\end{conjecture}
In \cite{agashe-stein:visibility}, the PI proved that if $x\in
\H^1(\Q,A)$, then there is an abelian variety~$B$ and an inclusion
$\iota:A\to B$ such that~$x$ is visible in~$B$; moreover,~$B$ is a
quotient of $J_1(NM)$ for some~$M$.  This theorem is the main
reason why the PI makes Conjecture~\ref{conj:allvis}. The PI hopes
to prove Conjecture~\ref{conj:allvis} by understanding the precise
relationship between  $A$, $B$, and $J_1(NM)$.  First he will
investigate explicitly the example with $N=551$ described in
Section~\ref{sec:compprob} above.

A more analytical, and possibly deeper, approach to
Conjecture~\ref{conj:allvis} is to assume the rank statement of
the Birch and Swinnerton-Dyer conjecture and relate when elements
of $\Sha(A)$ becoming visible at level $NM$ to when there is a
congruence between $f$ and a newform $g$ of level $NM$ with
$L(g,1)=0$.  Such an approach leads one to try to formulate a
refinement of Ribet's level raising theorem that includes a
statement about the behavior of the value at $1$ of the
$L$-function attached to the form at higher level. The PI intends
to do further computations in the hopes of finding a satisfactory
conjectural refinement of Ribet's theorem, which he then hopes to
subsequently prove.

The PI also proposes to investigate whether there is an~$M$ that
is minimal with respect to some property, such that every element
of  $\Sha(A)$ is simultaneously visible in $J_1(NM)$. This is well
worth looking into, since the payoffs could be huge---the
existence of such an~$M$ would imply finiteness of $\Sha(A)$,
since $\Vis_J(\Sha(A))$ is always finite. Finiteness of $\Sha(A)$
is a mysterious open problem when $L(A,1)=0$ and $A$ is not a
quotient of $J_0(N)$ with $\ord_{s=1}L(A,s)=\dim A$.

\subsubsection{Visibility of Mordell-Weil groups}
The Gross-Zagier theorem asserts that points on elliptic curves of
rank $1$ come from Heegner points, and that points on curves of
rank bigger than one do not.   It seems difficult to describe
where points on elliptic curves of rank bigger than~$1$ ``come
from''. The PI introduced the following definition, in hopes of
eventually creating a framework for giving a conjectural
explanation.

\begin{definition}[Visibility of Mordell-Weil Groups]
Suppose that $\pi : J\to A$ is a surjective morphism of abelian
varieties with connected kernel. The {\em visible quotient of
$A(\Q)$} with respect to~$J$ (and $\pi$) is
\[
  \Vis^J(A(\Q)) := \Coker(J(\Q)\to A(\Q)).
\]
\end{definition}

Visibility of Mordell-Weil groups is closely connected to
visibility of Shafarevich-Tate groups.  If $C$ is the kernel
of~$\pi$ and $\delta : A(\Q)\to \H^1(\Q,C)$ is the connecting
homomorphism of Galois cohomology, then $\delta$ induces an
isomorphism
\[
  \tilde{\delta}: \Vis^J(A(\Q)) \isom \Vis_J(\H^1(\Q,C)).
\]
Note that this implies $\Vis^J(A(\Q))$ is finite.  Let
\[
  \Vis^J_{\scriptsize\Sha}(A(\Q)) := \tilde{\delta}^{-1}(\Vis_J(\Sha(C))).
\]


Though we have introduced nothing fundamentally new, this
different point of view suggested questions that seemed unnatural
before,  which inspired the following theorem and conjecture (the
proof of the theorem relies on \cite{kato:secret,rubin:kato} and
\cite{rohrlich:cyclo}):
\begin{theorem}[Stein]\label{thm:allmwvis}
Let $A$ be an elliptic curve. If $x\in A(\Q)$ has order~$n$ (set
$n=0$ if $x$ has infinite order), then for every divisor $d$ of
$n$, there is surjective morphism $J\to A$, with connected kernel,
such that the image of~$x$ in $\Vis^J(A(\Q))$ has order~$d$.
\end{theorem}

\begin{conjecture}[Stein]
Suppose~$A$ is a modular abelian variety and $x\in A(\Q)$ has
order~$n$. For every divisor~$d$ of $\,n$ there is a surjective
morphism $J\to A$, with connected kernel, such that the image of
$\,x$ in $\Vis^J(A(\Q))$ lies in $\Vis_{\scriptsize \Sha}^J(A(\Q))$ and
has order~$d$.
\end{conjecture}

We now describe partial results about this conjecture that the PI
proved in \cite{stein:nonsquaresha}. Suppose $E$ is an elliptic
curve over $\Q$ with conductor $N$, and let $f$ be the newform
attached to~$E$. Fix a prime~$p\nmid 2 N \prod c_p$ such that the
Galois representation $\Gal(\Qbar)\to \Aut(E[p])$ is surjective.
\begin{conjecture}[Stein]\label{conj:nonvanishtwist}
There is a prime~$\ell\nmid N$ and a surjective Dirichlet
character $\chi:(\Z/\ell\Z)^*\to\mu_p$  such that
\[
  L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
  a_{\ell}(E) \not\con \ell+1 \pmod{p}.
\]
\end{conjecture}
According to Sarnak and Kowalski, this conjecture does not seem
amenable to standard analytic averaging arguments.  The PI has
verified this conjecture for the elliptic curve of rank~$1$ and
conductor~$37$ and all $p\leq 25000$.  In almost all cases, the
smallest $\ell\nmid N$ such that $a_{\ell}(E) \not \con
\ell+1\pmod{p}$ and $\ell\con 1\pmod{p}$ satisfies the conjecture.

The PI proved the following theorem in \cite{stein:nonsquaresha}.
\begin{theorem}[Stein]\label{thm:exact}
Let $E$ be an elliptic curve over~$\Q$ and suppose~$p$ and $\chi$
are as in Conjecture~\ref{conj:nonvanishtwist} above.  Then there
is an exact sequence $0\to A\to J\to E\to 0$ that induces an exact
sequence
\[
0 \to E(\Q)/ p E(\Q) \to \Sha(A) \to \Sha(J) \to \Sha(E) \to 0.
\]
In particular,
\[
  E(\Q)/p E(\Q) \isom \Vis_{\scriptsize\Sha}^J(E(\Q))  \isom \Vis_J(\Sha(A)).
\]
\end{theorem}
%When the hypothesis of the theorem are satisfied, the conclusion
%explains $E(\Q)\otimes\F_p$ in terms of the Shafarevich-Tate group
%of an abelian variety with analytic rank~$0$.  It thus links parts
%1 and 2 of Conjecture~\ref{conj:bsd}.  The PI proposes to explore
%the significance of this further.

%There are two problems with this picture.  First, the PI does not
%know a proof of Conjecture~\ref{conj:nonvanishtwist}, and is
%having difficulty finding one. Second, even if
%Conjecture~\ref{conj:nonvanishtwist} were known, it is unclear
%what implications the isomorphism
%\[
%E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A))
%\]
%has for Conjecture~\ref{conj:bsd}.
%

We finish this research proposal by explaining how
Theorem~\ref{thm:exact} may lead to a link between the two parts
of the BSD Conjecture (Conjecture~\ref{conj:bsd}). Suppose $E$ is
an elliptic curve over~$\Q$ and $L(E,1)=0$. Then part 1 of
Conjecture~\ref{conj:bsd} asserts that $E(\Q)$ is infinite. Under
our hypothesis that $L(E,1)=0$, a standard argument shows that
$$\frac{L(A,1)}{\Omega_A} \con 0\pmod{p},$$ where $A$ is as in
Theorem~\ref{thm:exact}. If part 2 of Conjecture~\ref{conj:bsd}
were true, there would be an element $x\in \Sha(A)$ of order~$p$
(the proof of Theorem~\ref{thm:exact} rules out the possibility
that~$p$ divides a Tamagawa number). If, in addition,~$x$ were
visible in $J$, then $E(\Q)$ would be infinite, since $E(\Q)$ has
no elements of order~$p$. Part 2 of Conjecture~\ref{conj:bsd} does
not assert that $x$ is visible in~$J$, so one can only hope that a
close examination of an eventual proof of part 2 of
Conjecture~\ref{conj:bsd} would yield some insight into whether or
not $x$ is visible. Alternatively, one could try to replace the
isomorphism $E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A)) $ by an
isomorphism
\[
\mbox{\rm Sel}^{(p)}(E)\isom \Sha(A)[I]
\]
where $I$ is an appropriate ideal in the ring $\Z[\mu_p]$ of
endomorphism of~$A$.  Then an appropriate refinement of part 2 of
Conjecture~\ref{conj:bsd} might imply that $\Sha(A)[I]$ contains
an element of order~$p$, which would imply that either $E(\Q)$ is
infinite or $\Sha(E/\Q)[p]$ is nonzero.

One can also work orthogonally to the above approach by
investigating similar situations coming from level raising, where
isomorphisms like the ones above may arise.  The PI intends to
investigate this cluster of ideas from various directions in hopes
of finding a new perspective on where points on elliptic curves of
rank bigger than one come from it.

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