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\section{Introduction}
My research reflects the essential interplay of abstract theory
with explicit machine computation, which is illustrated by the
following quote of Bryan Birch~\cite{birch:bsd} about computations
that led to a central conjecture in number theory:
\begin{quote}
I want to describe some computations undertaken by myself and
Swinnerton-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}


I am primarily interested in abelian varieties attached to modular
forms via Shimura's construction \cite{shimura:factors}.
Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$.  We
may view~$f$ as 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 annihilator of~$f$ in $\T$, and
attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$
is an 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~$f$.

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

My investigations into modular abelian varieties are inspired by
the following special case of the Birch and Swinnerton-Dyer
conjecture (see \cite{tate:bsd, lang:nt3}):
\begin{conjecture}[BSD Conjecture]
\[
\frac{L(A_f,1)}{\Omega_{A_f}} =%
\frac{\prod c_p \cdot \#\Sha(A_f/\Q)}%
{\#A_f(\Q)\cdot \#A_f^{\vee}(\Q)}.
\]
\end{conjecture}
Here $L(A_f,s)$ is the canonical $L$-series attached to $A_f$, the
real volume $\Omega_{A_f}$ is the measure of $A_f(\R)$ with
respect to a basis of differentials for the N\'eron model of
$A_f$, the $c_p$ are the Tamagawa numbers of $A_f$, the dual of
$A_f$ is denoted $A_f^{\vee}$, and
\[
 \Sha(A_f/\Q) = \ker\left(\H^1(\Q,A_f) \to \bigoplus_{p\leq
 \infty} \H^1(\Q_p,A_f)\right)
\] is the
Shafarevich-Tate group of $A_f$.  When $A_f(\Q)$ is infinite, the
right hand side should be interpreted as~$0$, so, in particular,
the conjecture asserts that $L(A_f,1)=0$ if and only if $A_f(\Q)$
is infinite. Birch and Swinnerton-Dyer also conjectured that the
order of vanishing of $L(A_f,s)$ at $s=1$ equals the rank of
$A_f(\Q)$.


\section{Computing with modular abelian varieties}

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
the Birch and Swinnerton-Dyer for modular abelian varieties and
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, that is a much used resource for number theorists.  It is
analogous to Sloane's tables of integer sequences, and extends
Cremona's tables to dimension bigger than one and weight bigger
than two (\cite{cremona-tables} contain more precise data about
elliptic curves than \cite{mfd}).

The PI proposes to greatly expand the database.  The 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
better method for storing information about modular abelian so
that the database can grow larger, and to investigate methods used
by astronomers or the human genome project to see how they cope
with a torrent of data while making it available to their
colleagues.

The PI implemented the current database using PostgreSQL coupled
with a Python web interface.   To speed access and improve
efficiency, the PI is considering rewriting key portions of the
database using MySQL and PHP.  Also, the database currently runs
on a three-year-old 933Mhz Pentium III, so the PI is requesting
more modern hardware.

\subsubsection{M{\small AGMA} package for modular abelian varieties}%
{\sc Magma} is a nonprofit computer algebra system developed
primarily at the University of Sydney, which is supported mostly
by grant money from organizations such as the US National Security
Agency. {\sc Magma} is considered by many to be the most
comprehensive tool for research in number theory, finite group
theory, and cryptography, and it 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}.

An abelian variety $A$ over a number field 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 popular because most of them are easily described by giving a
level and the first few coefficients of a modular form, and the
$L$-functions of modular abelian varieties are particularly well
understood.   This is in sharp contrast to the case of general
abelian varieties which, in general, can only be described by
unwieldy systems of polynomial equations, and whose $L$-functions
are very mysterious.

The PI recently designed and partially implemented a general
package for computing with modular abelian varieties over number
fields. Several crucial algorithms still need to be developed or
refined.  When available, this package will likely be useful for
people working with modular abelian varieties.   The following
major problems arose in work on this package, and they must be
resolved in order to have a completely satisfactory system for
computing with modular abelian varieties:
\begin{itemize}
\item {\em Given a modular abelian variety $A$, efficiently
compute the endomorphism ring $\End(A)$ as a ring of matrices
acting $\H_1(A,\Z)$.} 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 number fields of given level.
Such an enumeration would be a major step towards the ultimate
possible generalization of Cremona's tables \cite{cremona:tables}
to modular abelian varieties. %
\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 mostly
well-studied norm equations.  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 of isomorphism testing is that it could be used to
prove that an abelian variety is not principally polarized.%
\end{itemize}

We finish be describing recent work with $J_1(p)$ that was
partially inspired by  PI's modular forms software. This
conjecture generalizes a conjecture of Ogg, which asserted 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}
Significant numerical evidence for this conjecture is given in
\cite{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 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?  If
so, 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.

In \cite{j1p}, it is proved that $J_1(p)$ has trivial component
group (component groups are closely related to the Tamagawa
numbers $c_p$, which appear in the Birch and Swinnerton-Dyer
conjecture).
\begin{question}
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 Birch and Swinnerton-Dyer conjecture,
extensive computations by the PI have not produced a conjectural
answer to this question.  However, he and Bjorn Poonen formulated
a strategy to answer this question in some interesting cases by
using an explicit decomposition of $\End({A_f}_{\Qbar})$ to obtain
a curve whose Jacobian is $A_f$.

\section{Visibility of Shafarevich-Tate and Mordell-Weil Groups}
\subsection{Computational problems}
 Before describing work the PI proposes to do on
visibility and its uses in obtaining evidence for the Birch and
Swinnerton-Dyer conjecture, we recall the precise definition of
visibility over~$\Q$ (the definition makes sense over any global
field, but we restrict to $\Q$ below for simplicity).

\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 $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to
\H^1(\Q,J)$.  The {\em visible subgroup} of $\Sha(A/\Q)$ is the
intersection of $\Sha(A/\Q)$ with the visible subgroup of
$\H^1(\Q,A)$; equivalently, it is the kernel of the induced map
$\Sha(A/\Q)\to \Sha(J/\Q)$.
\end{definition}

Before discussing theoretical questions about visibility, we
describe computational evidence for the Birch and Swinnerton-Dyer
conjecture for modular abelian varieties (and motives) that we
obtained using a theorem 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 large
rank to explicitly construct subgroups of Shafarevich-Tate groups
of 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 from $B(\Q)/p B(\Q)$ into the visible subgroup of
$\Sha(A/\Q)$.  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 is provably nontrivial.

In  \cite{agashe-stein:bsd}, the PI and Agashe describe algorithms
that we found and I implemented in {\sc Magma} for computing with
modular abelian varieties.  We then compute an odd divisor and
even multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the
over ten thousand quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq
0$. For over a hundred of these, our lower bound on the
conjectural order of $\Sha(A/\Q)$ is is divisible by an odd prime;
for a quarter of these we prove, using the main theorem of
\cite{agashe-stein:visibility}, that if $n$ is the conjectural
divisor of $\#\Sha(A/\Q)$, then there are at least~$n$ elements of
$\Sha(A/\Q)$ that are visible in $J_0(N)$.  \emph{The PI intends
to investigate the remaining 75\% of the examples with $n>1$ by
considering the image of $A$ in $J_0(NM)$ for small integer $M$.}
Information about which primes $\ell$ to choose can be extracted
from Ribet's level raising theorem (see \cite{ribet:raising}).  As
a test, the PI tried the first example not already done, which is
an $18$ dimension abelian variety of level $551$, whose
Shafarevich-Tate group conjecturally contains elements of order
$3$; these are not visible in $J_0(551)$, but he shows in
\cite{stein:bsdmagma} that they are visible in $J_(551\cdot 2)$.

One objective of the PI's past NSF supported research was to
generalize visibility to the context of modular motives.
Fortunately, Neil Dummigan, Mark Watkins, and the PI did
significant 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 very
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 $J_0(N M)$ for some integer~$M$.  It would be relatively
straightforward for the PI to do computations in this direction,
but none have been done yet.  The PI intends to do such
computations.

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{stein:compgroup}, Conrad
and the PI use Grothendieck's monodromy pairing to give a complete
proof of correctness of an algorithm the PI found for computing
the order of the component group of certain purely toric abelian
varieties.  This algorithm makes it practical to compute component
groups of quotients of $A_f$ of $J_0(N)$ at primes~$p$ that
exactly divide $N$; without such an algorithm it would probably be
difficult to get anywhere in computational investigations into the
Birch and Swinnerton-Dyer conjecture for abelian varieties;
indeed, the only other paper in this direction is \cite{evidence},
which restricts to the case of dimension~$2$ Jacobians.


\subsection{Theoretical problems}
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}\label{conj:allvis}
For each $x\in \Sha(A/\Q)$, there is an integer $M$ and a morphism
$f:A\to J_1(NM)$ of finite degree coprime to the order of~$x$ such
that the image of~$x$ in $\Sha(f(A)/\Q)$ 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 $M$.  The PI hopes to prove
Conjecture~\ref{conj:allvis} by understanding much more precisely
how $A$, $B$, and $J_1(NM)$ are related.

A more analytical approach to Conjecture~\ref{conj:allvis} is to
assume the rank statement of the Birch and Swinnerton-Dyer
conjecture and relate when elements of $\Sha(A/\Q)$ 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 wish 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 order
to give a good conjectural refinement of Ribet's theorem.


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.   Over fifteen years later, it still
seems mysterious to give an interpretation of points on elliptic
curves of rank higher than~$1$.  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~$C$.  Let $\delta : A(\Q)\to
\H^1(\Q,C)$ be the connecting homomorphism of Galois cohomology.
An element $x\in A(\Q)$ is \emph{$n$-visible in $\H^1$} (with
respect to $\pi$) if $\delta(x)$ has order divisible by~$n$, and
$x$ is \emph{$n$-visible in $\Sha$} if moreover $\delta(x)\in
\Sha(C/\Q)$.
\end{definition}

The following theorem is not difficult to prove by combining
Kato's powerful results towards the Birch and Swinnerton-Dyer
conjecture (see \cite{kato:secret,rubin:kato}) with a nonvanishing
theorem of Rohrlich \cite{rohrlich:cyclo}.
\begin{theorem}[Stein]\label{thm:allmwvis}
If~$A$ is a modular abelian variety and $x\in A(\Q)$ has infinite
order, then for every integer~$n$ there is a covering $J\to A$
with connected kernel such that $x$ is $n$-visible in $\H^1$.
\end{theorem}
%The key idea of the proof is that if $p$ is any prime and
%$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a
%nonvanishing theorem of Rohrlich  and Kato's theorem , the group
%$A(\Q_\infty)$ is finitely generated. From this we deduce that
%there is an (abelian) extension $K$ of $\Q$ such that $n$ divides
%the order of the image of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace
%defines a morphism from the restriction of scalars
%$\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with connected kernel.  Then $x$
%is $n$-visible with respect to $\pi$.

\begin{conjecture}[Stein]
If~$A$ be a modular abelian variety and $x\in A(\Q)$ has infinite
order, then for every integer~$n$, there is an abelian variety $B$
and a surjective morphism $\pi:B\to A$ with connected kernel such
that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$.
\end{conjecture}

The PI proved partial results towards this conjecture 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~$p\nmid N \prod c_p$ and such that the mod~$p$
Galois representation attached to $E$ is surjective. Suppose
$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character with
$\ell\nmid N$ such that
\[
  L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
  a_{\ell}(E) \not\con \ell+1 \pmod{p},
\]
and let $K$ be the corresponding abelian extension. In
\cite{stein:nonsquaresha}, the PI uses Kato's theorem and
restriction of scalars to construct abelian varieties $A$ and $J$
and an exact sequence
\[
0 \to E(\Q)/ p E(\Q) \to \Sha(A/\Q) \to \Sha(J/\Q) \to \Sha(E/\Q)
\to 0.
\]
Here $A$ is an abelian variety of rank $0$ with $L(A,1)\neq 0$.
Thus all of $E(\Q)$ is $p$-visible in $\Sha$ and one can interpret
$E(\Q)/ p E(\Q)$ as a visible subgroup of a rank~$0$ abelian
variety~$A$.

There are two problems with this picture.  First, the PI does not
know a proof that such a $\chi$ exists (he has verified existence
of $\chi$ in thousands of examples). Second, even if the existence
of $\chi$ were known it seems unlikely that visibility of
Mordell-Weil groups in an abstract restriction of scalars abelian
variety is likely to yield a satisfactory interpretation of
$E(\Q)$ when $E$ has large rank.  To answer these problems, the PI
intends to study the system of all possible abelian varieties $B$
in which $E(\Q)$ induces visible elements of Shafarevich-Tate
groups, and moreover to try to keep track of how these abelian
varieties $B$ are related to $J_1(NM)$ for various $M$.   This
will involve a combination of computer computation of examples
followed hopefully by the development of a concise notation for
keeping track of all relevant data, and eventually perhaps new
ideas that clarify our understanding of elliptic curves of rank
bigger than one.

\bibliographystyle{amsalpha}
\bibliography{biblio}
\end{document}



%%%%%%%%%%%



\begin{conjecture}[Stein]\label{conj:nonvanishchi}
Assume~$p$ is an odd prime and the mod~$p$ Galois representation
attached to $E$ is surjective.  Then there are infinitely many
Dirichlet characters $\chi$ as above that satisfies the following
hypothesis:
\[
  L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
  a_{\ell}(E) \not\con \ell+1 \pmod{p}.
\]
\end{conjecture}
I have verified this conjecture numerically in thousands of
examples, and I hope to prove something about it by assuming it is
false and constructing many relations between modular symbols.
Analytic methods involving averaging special values of
$L$-functions seem incapable of handling twists of high degree.

\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}
If $A$ is a modular abelian variety, and $n$ is an integer, we say
that the Mordell-Weil group of $A$ is {\em $n$-modular of level
$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected
kernel, such that $A'$ is isogenous to $A$ and%
$\pi(J_1(M))\subset n A'(\Q)$.  We say that the Mordell-Weil group
of $A$ is {\em modular} if it is $n$-modular for every
integer~$n$.
\end{definition}

I think the following conjecture is within reach.
\begin{conjecture}[Stein]\label{conj:mwallmodular}
The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
itself modular, in the sense of Definition~\ref{defn:modmw}.
\end{conjecture}
This is closely related to Theorem~\ref{thm:allmwvis} since the
restriction of scalars of a modular abelian variety is again
modular.  However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
and $J_1(M)$ all fit together, and it is essential to understand
{\em exactly} how they fit together in order to verify the
conjecture.   Also, for a given $n$, it would be interesting to
decide if the Mordell-Weil group is $n$-visible of level $M$ for
some naturally defined~$M$.






\section{OLD STUFF}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




[summary/intro/the point]


\begin{definition}[Visibility]
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 $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to
\H^1(\Q,J)$.  The visible subgroup of $\Sha(A/\Q)$ is the
intersection of $\Sha(A/\Q)$ with the visible subgroup of
$\H^1(\Q,A)$, or equivalently, the kernel of the induced map
$\Sha(A/\Q)\to \Sha(J/\Q)$.
\end{definition}

The paper \cite{agashe-stein:visibility} extends work of Cremona
and Mazur \cite{mazur:visord3, cremona-mazur} to lay the
foundations for studying visibility of Shafarevich-Tate groups of
abelian varieties.  In it, I use a restriction of scalars
construction to prove that if $A$ is an abelian variety and $x\in
\Sha(A/\Q)$ then there is an inclusion $A\hra B$ such that $x$ is
visible in $B$. We then prove 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 from $B(\Q)/p
B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$.  We apply this
theorem to prove that $25$ divides the order of the visible
subgroup of the Shafarevich-Tate group of an abelian variety of
dimension $20$ and level $389$. We also give the first explicit
example of an element of the Shafarevich-Tate group of an elliptic
curve that only becomes visible at higher level.

\subsubsection{Computational evidence for the BSD conjecture}
In \cite{agashe-stein:bsd}, Agashe and I
describe a number of algorithms that we found and I implemented in
{\sc Magma} for computing with modular abelian varieties.  We then
compute a divisor and multiple of the BSD conjectural order of
$\Sha(A/\Q)$ for the 10360 (optimal) abelian variety quotients
$A=A_f$ of $J_0(N)$ with $L(A,1)\neq 0$ and $f$ of level $N\leq
2333$. For $168$ of these $A$ our divisor is divisible by an odd
prime, and for $37$ of these $168$, we prove using visibility that
if $n$ is the prime--to-$2$ part of the conjectural order of
$\Sha(A/\Q)$, then there are $n$ elements of $\Sha(A/\Q)$ that are
visible in $J_0(N)$.   The challenge remains to use  other
techniques, e.g., visibility at higher level, to show that there
are $n$ elements in $\Sha(A/\Q)$ in the remaining $131$ cases. For
example, in \cite{stein:bsdmagma} we do this for the first
example, which has level $551$ and whose $\Sha$ becomes visible at
level $2\cdot 551$.

\subsubsection{Visibility for modular motives} One objective of my
past NSF supported research was to generalize visibility theory to
the context of modular motives. Fortunately, Neil Dummigan, Mark
Watkins, and I succeeded in carrying out such a program in
\cite{dummigan-stein-watkins:motives}. There we prove a general
theorem that can be used in many cases 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.  We give several pages of tables that suggest that
Shafarevich-Tate groups of modular motives of level~$N$ are very
rarely visible in the higher weight motivic analogue of $J_0(N)$,
much more rarely than for weight~$2$.


\subsubsection{Nonsquare Shafarevich-Tate groups} In the paper
\cite{stein:nonsquaresha} I give a surprising application of
visibility to understanding the possibilities for the orders of
Shafarevich-Tate groups. Before \cite{stein:nonsquaresha} no
examples of Shafarevich-Tate groups of order $p\cdot n^2$ were
known for any odd prime $p$, and the literature even suggested
such examples do not exist ([swinnerton-dyer]).  In
\cite{poonen-stoll}, Poonen and Stoll give the first examples of
Shafarevich-Tate groups of order $2\cdot n^2$, which inspired me
to look for examples of order $3\cdot n^2$.   Using an adaptation
of the ideas from visibility along with a deep theorem of Kato on
the Birch and Swinnerton-Dyer conjecture, I was able to construct,
under suitable hypothesis, an abelian variety $A$ with
$\#\Sha(A/\Q)=p\cdot n^2$; I then used a computer to verify the
hypothesis for all odd primes $p<25000$.   The $p$-part of
$\Sha(A/\Q)$ is visible in an abelian variety that is isogenous to
$A\times E$, where $E$ is an elliptic curve.  To remove the
hypothesis on~$p$ would require proving a nonvanishing result
about twists of~$L$ functions (see Section~\ref{conj:nonvanishchi}
below).

\subsection{The Arithmetic of modular abelian varieties}

\subsubsection{$J_1(p)$ has connected fibers} In the paper \cite{conrad-edixhoven-stein:j1p},
Conrad, S.~Edixhoven, and I prove a remarkable uniformity result
for the component group of the N\'eron model of $J_1(p)$: {\em It
has order~$1$ for all primes $p$!}  We do this by determining the
closed fiber at~$p$ of a model for $X_1(p)$, then do intersection
theory computations to find a regular model for $X_1(p)$ over
$\Z$.   In this paper, I use theorems of Mazur, Kato, and a
computation to determine the primes~$p$ such that $J_1(p)$ has
rank~$0$. I also do significant computations of many of the
invariants appearing in the BSD conjecture for each of the simple
factors of $J_1(p)$. The results of these numerical computations
combined with the main theoretical result of the paper on
component groups suggest several questions, which I intend to
address (see Section~\ref{sec:j1pques}).

\subsubsection{Component groups of purely toric abelian varieties}
In \cite{stein:compgroup}, Conrad and I use Grothendieck's
monodromy pairing to give a complete proof of correctness of an
algorithm I found for computing the order of the component group
of certain purely toric abelian varieties.  I found this algorithm
after reading a letter from Ribet to Mestre, which addressed
certain numerical relations for elliptic curves in terms of
Mestre's method of graphs.   In \cite{kohel-stein:ants4}, D.~Kohel
and I explain how to calculate Tamagawa numbers (the $c_p$ in the
BSD conjecture, which are orders of component groups) for purely
toric modular abelian varieties using this algorithm.  In general,
we only obtain the Tamagawa number up to a bounded power of $2$, a
shortcoming I intend to remedy with further work.   We also do not
determine the structure of the underlying component groups, which
is something I hope to do.

\subsubsection{The BSD conjecture for Jacobians of genus two
curves} The paper \cite{empirical} is about the BSD conjecture for
$32$ modular Jacobians of genus~$2$ curves. I learned of an early
version of \cite{empirical} before it was published, and was
shocked by the table of orders of Shafarevich-Tate groups that it
contained. I used the equation-free algorithms I developed in
\cite{stein:phd} to do the computations in a new way, and found
that the most striking example in the paper, a Shafarevich-Tate
group of order $49$, was incorrect.  I was made a coauthor and
wrote a section of the paper describing my methods.

\subsection{Other research}
\subsubsection{Application of Kolyvagin's Euler system}%
In \cite{stein:index} I give an innovative application of
Kolyvagin's Euler system to an old question of E.~Artin, S.~Lang,
and Tate (see \cite{lang-tate}).

 Let $X$ be a curve over $\Q$ (say) of
genus~$g$. The {\em index} of $X$ is the greatest common divisor
of the degrees of the extensions of $\Q$ in which $X$ has a
rational point. Then the canonical divisor has degree $2g-2$, so
the index of $X$ divides $2g-2$. When $g=1$ this is no condition
at all. {\bf Question:} {\em For every integer $n$, is there a
genus one curve with index exactly~$n$?}

In \cite{lang-tate}, Lang and Tate prove that for each $n$ there
is a genus one curve~$X$ over some number field~$K$ (which depends
on $n$) such that~$X$ has index~$n$.  In \cite{stein:index}, I
prove that if~$K$ is a fixed number field, then for any~$n$ not
divisible by~$8$ there is a genus one curve~$X$ over~$K$ of
index~$n$.  The proof involves reinterpreting genus one curves and
the notion of index in terms of Galois cohomology, then finding
nontrivial Galois cohomology classes with the requisite properties
in the Euler system of Heegner points on $X_0(17)$.

\subsubsection{Elliptic curves with full torsion} In \cite{merel-stein}, L.~Merel
and I investigate a natural question about fields of definition
that is connected with points on modular curves. Let $p$ be a
prime. Suppose $E$ is an elliptic curve over a number field~$K$
and all of the $p$ torsion on $E$ is defined over $K$. Properties
of the Weil pairing imply that the field $\Q(\zeta_p)$ of $p$th
roots of unity is contained in $K$. {\bf Question.} {\em Is there
an elliptic curve defined over $\Q(\zeta_p)$ all of whose
$p$-torsion is also defined over $\Q(\zeta_p)$?}   By combining
the significant theory developed in \cite{merel:cyclo} with a
nontrivial modular symbols computation, we show that the question
has a ``no'' answer for all $p<1000$, except $p=2,3,5,13$. (A
student of Merel showed that $13$ also has a ``no'' answer.)


\subsubsection{Modularity of icosahedral Galois representations} In \cite{buzzard-stein:artin},
K.~Buzzard and I prove $8$ new cases of the Artin conjecture about
modularity of icosahedral Galois representations, only $3$ of
which are covered by the subsequent landmark work of Taylor which
gave infinitely many new examples. Buzzard and I push through an
explicit application of \cite{buzzard-taylor} by combining various
theorems with significant modular symbols computations over the
finite field of order~$5$.


\subsubsection{Approximating $p$-adic modular forms} In
\cite{coleman-stein:padicapprox}, R.~Coleman and I consider from a
theoretical and computational point of view questions about
$p$-adic approximation of infinite slope modular eigenforms by
modular eigenforms of finite slope. The slope of an eigenform
$f=\sum a_n q^n$ is the $p$-adic valuation of $a_p$, so an
eigenform has infinite slope precisely when $a_p=0$.  When~$f$ is
an eigenform having infinite slope, Naomi Jochnowitz asked if for
every~$n$ there is an eigenform~$g$ of finite slope such that
$f\equiv g\pmod{p^n}$.  We show that the answer in general is no,
but prove that if~$f$ is a twist of a finite slope eigenform,
then~$f$ can be approximated. We also investigate computationally
which forms can be approximated and how the weight of~$g$ grows as
a function of~$n$. These computations lead to intriguing
unanswered questions.

\section{Project Proposal}

\subsection{Visibility of Shafarevich-Tate groups at higher level}

The following conjecture is the central open problem in visibility
theory.
\begin{conjecture}[Stein]\label{conj:allvis}
Let~$A$ be a modular abelian variety.%
\begin{enumerate}%
\item Then there is an integer~$N$ and a morphism $f:A\to J_1(N)$
such that every element of $\Sha(f(A))$ is visible in $J_1(N)$.%
\item The level $N$ should be determined in some natural way in
terms of properties of~$A$.  (Part of the conjecture is to give a
reasonable interpretation of natural.)
\end{enumerate}
\end{conjecture}

If true, Conjecture~\ref{conj:allvis} would imply finiteness of
the Shafarevich-Tate group of~$A$, which would massively
strengthen many current results towards the Birch and
Swinnerton-Dyer conjecture.  In \cite{agashe-stein:visibility}, I
proved that each element of the Shafarevich-Tate group of~$A$ is
visible in some modular abelian variety~$B$, but in this
construction~$B$ depends on the element. As a first step toward
Conjecture~\ref{conj:allvis}, I hope to use my result to prove
that if $\Sha(A)$ is finite then part 1 of the conjecture is true.
The main obstruction is that it is unclear  how $A$, $B$ and
$J_1(N)$ all fit together, and in order to prove the conjecture it
is essential to know {\em exactly} how these abelian varieties fit
together.  I strongly believe resolving this difficulty is within
reach and will lead to new ideas. (See
Conjecture~\ref{conj:mwallmodular} below for a similar situation.)

The first part of Conjecture~\ref{conj:allvis} for a single
element of $\Sha(A)$ is analogous to the easy-to-prove assertion
that each ideal class in the ring of integers of a number field
becomes principal in a suitable extension field, where the
extension depends on the ideal class. The second part of the
conjecture is reminiscent of the existence of the Hilbert class
field of a number field, and deeper investigation into it may
prove crucial to understanding Shafarevich-Tate groups.

I intend to revisit the computations of \cite{agashe-stein:bsd}
and see how far visibility at higher level goes toward
constructing the odd part of $\Sha(A/\Q)$ in the remaining $131$
cases not already covered.

Suppose $A=A_f$ with $f\in S_2(\Gamma_1(N))$ and $L(A_f,1)\neq 0$.
The following discussion illustrates one way in which ideas from
visibility have vague unexplored implications for the BSD
conjecture, namely for the assertion that if $p\mid \#\Sha(A)$
then $p$ divides the conjectural order of $\Sha(A)$.   This is
only one of many similar ideas.

Suppose $x\in \Sha(A/\Q)[p]$ is an element of prime order $p$ that
is visible in $J_1(NM)$ for some $M$. Then in most cases there
should be a factor $A_g$ of $J_1(NM)$ that has positive
Mordell-Weil rank such that $x$ is in the image of $A_g(\Q)$ under
some map. Usually this should imply that~$g$ that is congruent
to~$f$ modulo a prime of characteristic~$p$; then by Kato's
theorem \cite{kato:secret, rubin:kato} we must have $L(g,1)=0$,
since otherwise $A_g$ would have rank $0$.  Because congruences
between eigenforms usually induce congruences between special
values of $L$ functions, this will often imply that
\[
 L(A_f,1)/\Omega_{A_f} \con L(A_g,1)/\omega = 0 \pmod{p}.
 \]

%%%% I just commented this out because isn't it trivially true by switching parity in functional equation
%This discussion also motivates the following conjecture, which may
%be viewed as an analytic shadow of visibility:
%\begin{conjecture}[Stein]
%Suppose $f$ is a newform and $p$ is a prime  such that
%$L(A_f,1)/\Omega_{A_f}\con 0\pmod{p}$.  Then there exists a
%newform $g$ that is congruent to $f$ modulo~$p$ such that
%$L(g,1)=0$.
%\end{conjecture}
%Note that the conjecture is trivially true in case $L(f,1)=0$,
%since we just take $g=f$.



\subsection{Visibility of Mordell-Weil Groups of abelian varieties}
Turning Mazur's visibility idea on its head, I introduced the
notion of visibility of Mordell-Weil groups.

\begin{definition}[Visibility of Mordell-Weil]
Suppose that $\pi : J\to A$ is a surjective morphism of abelian
varieties with connected kernel~$C$.  Let $\delta : A(\Q)\to
\H^1(\Q,C)$ be the connecting homomorphism.  An element $x\in
A(\Q)$ is \emph{$n$-visible} with respect to $\pi$ if $\delta(x)$
has order divisible by~$n$, and $x$ is \emph{$n$-visible in
$\Sha$} if moreover $\delta(x)\in \Sha(C/\Q)$.
\end{definition}

\begin{theorem}\label{thm:allmwvis}
If~$A$ is a modular abelian variety and $x\in A(\Q)$, then for
every integer~$n$ there is a covering $J\to A$ with connected
kernel such that $x$ is $n$-visible in $\H^1(\Q,J)$.
\end{theorem}
The key idea of the proof is that if $p$ is any prime and
$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a
nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo} and Kato's
theorem \cite{kato:secret,rubin:kato}, the group $A(\Q_\infty)$ is
finitely generated. From this we deduce that there is an (abelian)
extension $K$ of $\Q$ such that $n$ divides the order of the image
of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace defines a morphism from
the restriction of scalars $\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with
connected kernel.  Then $x$ is $n$-visible with respect to $\pi$.

\begin{conjecture}[Stein]
Let $A$ be a modular abelian variety, let $x\in A(\Q)$, and let
$n$ be a positive integer.  Then there is a surjective morphism
$\pi:J\to A$ with connected kernel such that $x$ is $n$-visible in
$\Sha$ with respect to~$\pi$.
\end{conjecture}

My attempts so far to prove this conjecture led to the paper
\cite{stein:nonsquaresha}, the connection being as follows.
Suppose $E$ is an elliptic curve over $\Q$ with $E(\Q)=\Z{}x$, and
let $f$ be the newform attached to $f$.  Fix a prime~$p$.  Suppose
$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character that satisfies
several carefully chosen hypothesis, and let $K$ be the
corresponding abelian extension. By chasing the appropriate
diagrams and using results about \'etale cohomology and  N\'eron
models, I show that if $J=\Res_{K/\Q}(E)$ then $x$ is $p$-visible
in $\Sha$ with respect to $J\to E$.  This means that
\[
   E(\Q)/pE(\Q)\isom \Z/p\Z\subset \Sha(\ker(J\to E)),
\]
which is where the nonsquare part of $\Sha$ comes from.
\begin{conjecture}[Stein]\label{conj:nonvanishchi}
Assume~$p$ is an odd prime and the mod~$p$ Galois representation
attached to $E$ is surjective.  Then there are infinitely many
Dirichlet characters $\chi$ as above that satisfies the following
hypothesis:
\[
  L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
  a_{\ell}(E) \not\con \ell+1 \pmod{p}.
\]
\end{conjecture}
I have verified this conjecture numerically in thousands of
examples, and I hope to prove something about it by assuming it is
false and constructing many relations between modular symbols.
Analytic methods involving averaging special values of
$L$-functions seem incapable of handling twists of high degree.

\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}
If $A$ is a modular abelian variety, and $n$ is an integer, we say
that the Mordell-Weil group of $A$ is {\em $n$-modular of level
$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected
kernel, such that $A'$ is isogenous to $A$ and%
$\pi(J_1(M))\subset n A'(\Q)$.  We say that the Mordell-Weil group
of $A$ is {\em modular} if it is $n$-modular for every
integer~$n$.
\end{definition}

I think the following conjecture is within reach.
\begin{conjecture}[Stein]\label{conj:mwallmodular}
The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
itself modular, in the sense of Definition~\ref{defn:modmw}.
\end{conjecture}
This is closely related to Theorem~\ref{thm:allmwvis} since the
restriction of scalars of a modular abelian variety is again
modular.  However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
and $J_1(M)$ all fit together, and it is essential to understand
{\em exactly} how they fit together in order to verify the
conjecture.   Also, for a given $n$, it would be interesting to
decide if the Mordell-Weil group is $n$-visible of level $M$ for
some naturally defined~$M$.



\subsection{Computing with modular abelian varieties}
Bryan Birch once commented to me in reference to computation that
``It is always a good idea to try to prove true theorems.''  To
this end, the author 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 the Birch and Swinnerton-Dyer for modular abelian
varieties and the Bloch-Kato conjecture for modular motives.  The
results of this work should give an explicit picture of modular
abelian varieties that could never be obtained from general
theory.


\subsubsection{The Modular forms database} The modular forms
database \cite{mfd} contains a large collection of information
about objects attached to cuspidal eigenforms.  Though greatly
appreciated by the many mathematicians who use it, the database
currently only scratches the surface of what it should contain.

The database is stored using the database system PostgreSQL, and I
wrote the web user interface in Python.  During Summer 2003 the
Harvard undergraduate Dimitar Jetchev did extensive work on the
database, and this pointed out significant deficiencies in how it
is currently implemented.   It is more difficult than it should be
to modify the web interface to the database, the data is not
compressed well, and there is no way to submit new data to the
database using the web page.  I intend to completely rewrite the
database using MySQL and PHP, and investigate better algorithms
for storing $q$-expansions of modular forms much more efficiently.
Currently the limit on the database is not the difficulty of
computing modular forms, but the space and time used in storing
them.  This could be partially remedied by moving the database to
a more modern computer (it currently runs on a three year old
Pentium III), something I am requesting in this grant.

\subsubsection{Example database queries that have not yet been done}
\begin{itemize}
\item Suppose $d=2,3,4,5$, say.  Using the algorithm described in
\cite{agashe-stein:bsd}, compute a multiple of the order of the
torsion subgroup of $A_f(\Q)$ for each $d$-dimensional $A_f$ in
the database.  What is the maximum number that occurs?  After what
level do no new numbers appear?  For small $d$ such a computation
may suggest a conjectural generalization to modular abelian
varieties of Mazur's theorem on torsion points on elliptic curves.%

\item  Make a conjectural list of all number fields of degree~$d$
(for $d=2,3,4,5$, say) that arise as the field generated by the
eigenvalues of a newform in the database. Coleman has conjectured
that for each~$d$ only finitely many number fields of degree~$d$
appear. When do new $d$ seem to stop appearing?%
\end{itemize}

\subsubsection{M{\small AGMA} package for modular abelian varieties} I
wrote the modular forms and modular symbols packages that are part
of the {\sc Magma} computer algebra system \cite{magma}.  I spent
June 2003 in Sydney, Australia and did exciting work on designing
and implementing a very general package for computing with modular
abelian varieties over number fields.  Much work is left to be
done to finish this package, and several crucial algorithms still
need to be developed or refined.   When available this package
will likely be greatly appreciated by anybody working with modular
abelian varieties.

The following are problems that arose in work on this package,
which must be resolved in order to have a satisfactory system for
computing with modular abelian varieties.  I need to solve all of
these problems.
\begin{itemize}
\item {\bf Endomorphism ring over $\Qbar$:} {\em Giving a modular
abelian variety $A$, explicitly (and efficiently) compute the
endomorphism ring $\End(A)$ as a ring of matrices acting
$\H_1(A,\Z)$.}   I have a modular symbols solution that draw on
work of Ribet \cite{ribet:twistsendoalg} and Shimura
\cite{shimura:factors} but is too slow to be really useful in
practice; however, similar Manin symbols methods must
exist and be very efficient.%
\item {\bf Decomposition:} {\em Given the endomorphism ring of an
abelian variety $A$, compute its decomposition 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 a ``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 number
fields of given level.%
 \item {\bf Isomorphism testing:} {\em Given
two modular abelian varieties over a number field $K$, represented
as explicit quotients of Jacobians $J_1(N)$, decide whether there
is an isomorphism between them.}  I have solved this problem when
the two modular abelian varieties are simple.  There are analogues
of this problem in other categories, which I intend to investigate.%
\end{itemize}

\subsection{Some conjectures that were inspired by my computations}

\subsubsection{Questions about $J_1(p)$}\label{sec:j1pques} The following
conjecture generalizes a famous conjecture of Ogg 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]
The torsion subgroup of $J_1(p)(\Q)$ is exactly the group
generated by the cusps on $X_1(p)$ that lie over $\infty\in
X_0(p)$.  This is a group of order
\[
\frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps}
\]
where the product is over the nontrivial even Dirichlet
characters~$\varepsilon$ of conductor dividing~$p$, and
$B_{2,\eps}$ is a generalized Bernoulli number.
\end{conjecture}
Mazur's complete proof of the analogue of this statement for
$J_0(p)$ is quite deep, though the proof for the prime-to-$2$ part
of $J_0(p)(\Q)_{\tor}$ is much easier.  I hope to mimic Mazur's
method and prove the conjecture above for the prime-to-$2$ part of
$J_1(p)(\Q)_{\tor}$.

More generally, I would like to investigate the torsion in
quotients of $J_1(p)$.
\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?  If
so, 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}
 I conjectured that the analogous questions for $J_0(p)$ should have
``yes'' answers, and in \cite{emerton:optimal} M.~Emerton
subsequently proved this conjecture.   It is still not clear if
one should make this conjecture for $J_1(p)$.

\subsubsection{Congruences between modular forms of prime level}
Ken Ribet enticed me into studying modular forms as a graduate
student by asking me the following question: ``Is there a prime
$p$ so that $p$ is ramified in the Hecke algebra $\T$ attached to
$S_2(\Gamma_0(p))$?'' I answered his question by showing that
$p=389$ is the only prime less than $50000$ that ramifies in the
associated Hecke algebra of level~$p$.  The question remains
whether $p=389$ is the only such example, and this seems extremely
difficult to say anything useful about.   However a related
question exhibits a shockingly clear pattern, and this related
question is the question Ribet was really interested in for his
application to images of Galois representations
\cite{ribet:torsion}.

Let $p$ be a prime and $k$ a positive even integer.  Let $\T$ be
the Hecke algebra attached to $S_k(\Gamma_0(p))$ and let $d(k,p)$
be the valuation at $p$ of the index of $\T$ in its normalization.
The following conjecture is backed up by significant numerical
evidence, and was discovered by staring at tables and looking for
a pattern.
\begin{conjecture}[Stein, F.~Calegari]
Suppose $p\geq k-1$.  Then
  $$
d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot
  \binom{m}{2} + a(p,m),
  $$
  where the function $a$ only depends on $p$ modulo $12$ as
  follows:
  $$
  a(p,m) =
\begin{cases}
  0 & \text{if $p\equiv 1\pmod{12}$,}\\
  3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\
  2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\
  a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}
\end{cases}
$$
\end{conjecture}

The situation of interest to Ribet is $k=2$, in which case the
conjecture simply asserts that $\T\otimes\Z_p$ is normal, i.e.,
{\em there or no congruences in characteristic $p$ between
non-Galois conjugate newforms in $S_2(\Gamma_0(p))$.}  Calegari
has given a conjectural interpretation of some of the congruences
that the conjecture asserts must exist, which I intend to study
further.

\section{Summary}
This research proposal depicts an intricate network of ongoing
investigations into the arithmetic of modular abelian varieties,
which unite a theoretical and computational point of view.  The
basic foundations of visibility theory are nearly complete, but
solutions to the questions about visibility outlined in  this
proposal demand a new level of precision in our understanding of
the web of modular abelian varieties. I am determined to advance
our understanding in this direction.

My work has produced results and tools that are of use to other
mathematicians who are exploring the world of modular forms.   By
supporting my research, you will assure the sustained development
of this technology.





Let $p$ be a prime and $k$ a positive even integer.  Let $\T$ be
the Hecke algebra attached to the space $S_k(\Gamma_0(p))$ of cusp
forms for $\Gamma_0(p)$ and let $d(k,p)$ be the valuation at $p$
of the index of $\T$ in its normalization. The following
conjecture is backed up by significant numerical evidence, and was
discovered by staring at tables computed using the PIs {\sc Magma}
code and looking for a pattern.
\begin{conjecture}[Stein, F.~Calegari]
Suppose $p\geq k-1$.  Then
  $$
d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot
  \binom{m}{2} + a(p,m),
  $$
  where the function $a$ only depends on $p$ modulo $12$ as
  follows:
  $$
  a(p,m) =
\begin{cases}
  0 & \text{if $p\equiv 1\pmod{12}$,}\\
  3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\
  2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\
  a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}
\end{cases}
$$
\end{conjecture}
The conjecture is of interest to Ribet in the case $k=2$, because
it is a hypotheses to the main argument of \cite{ribet:torsion}.
For a long time it was unclear what to conjecture when $k=2$;
finally, investigation into what happens at higher weight
suggested the above conjectural formula, which specializes in
weight $2$ to the assertion that $\T\otimes \Z_p$ is normal. The
PI has no idea how to prove this conjecture when $k=2$, but
intends to at least find similar conjectures when $\Gamma_0(p)$ is
replaced by $\Gamma_1(p)$ and when~$p$ is replaced by a composite
number.



\newpage

\newcommand{\etalchar}[1]{$^{#1}$}
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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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\bibitem[DWS]{dummigan-stein-watkins:motives}
N.~Dummigan, M.~Watkins, and W.\thinspace{}A. Stein,
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  Swinnerton-Dyer proceedings.

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M.~Emerton, \emph{Optimal {Q}uotients of {M}odular {J}acobians},
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E.\thinspace{}V. Flynn, F.~\protect{Lepr\'{e}vost},
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  no.~236, 1675--1697 (electronic).

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\end{thebibliography}


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