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\begin{document}

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\begin{center}
{\Large\bf Research Plan}\\
William A.\ Stein\\
\today
\end{center}

\mysection{Introduction}
My research program reflects the essential interplay between abstract
theory and explicit machine computation during
the latter half of the twentieth century;
it sits at the intersection of recent work of
B.~Mazur, K.~Ribet, J-P.~Serre, R.~Taylor, and A.~Wiles
on Galois representations attached to modular abelian varieties
(see \cite{ribet:modreps, serre:conjectures,
taylor-wiles:fermat, wiles:fermat})
with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
on explicit computations involving modular forms
(see \cite{cremona:algs, elkies:ffield}).

In 1969 B.~Birch~\cite{birch:bsd} described computations that
led to the most fundamental open
conjecture in the theory of elliptic curves:
\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}
The rich tapestry of arithmetic conjectures and theory
we enjoy today would not exist without the ground-breaking
application of computing by Birch and Swinnerton-Dyer.
Computations in the 1980s by Mestre were key in convincing
Serre that his conjectures on modularity of odd
irreducible  Galois representations
were worthy of serious consideration (see~\cite{serre:conjectures}).
These conjectures have inspired much recent work;
for example, Ribet's proof of the
$\epsilon$-conjecture, which played an essential role in
Wiles's proof of Fermat's Last Theorem.

My work on the Birch and Swinnerton-Dyer
conjecture for modular abelian varieties and search for new
examples of modular icosahedral Galois representations
has led me to discover and implement algorithms for
explicitly computing with modular forms.
My research, which involves finding ways to compute
with modular forms and modular abelian varieties,
is driven by outstanding conjectures in number theory.

\mysection{Invariants of modular abelian varieties}
Now that the Shimura-Taniyama conjecture has been proved,
the main outstanding problem in the field
is the Birch and Swinnerton-Dyer conjecture (BSD conjecture),
which ties together the arithmetic invariants of an elliptic curve.
There is no general class of elliptic curves for which
the full BSD conjecture is known.
Approaches to the BSD conjecture that rely on congruences
between modular forms are likely to
require a deeper understanding of the analogous
conjecture for higher-dimensional abelian varieties.
As a first step, I have obtained theorems that make possible explicit
computation of some of the arithmetic invariants of modular
abelian varieties.

\mysubsection{The BSD conjecture}
that every elliptic curve over~$\Q$ is
a quotient of the curve~$X_0(N)$ whose complex points
are the isomorphism classes of pairs consisting of a
(generalized) elliptic curve and a cyclic subgroup of order~$N$.
Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian
variety of dimension equal to the genus of~$X_0(N)$ whose points
correspond to the degree~$0$ divisor classes on~$X_0(N)$.

An {\em optimal quotient} of $J_0(N)$ is a quotient by an abelian subvariety.
Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.
By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A/\Q)$
are both finite.
The BSD conjecture asserts that
$$\frac{L(A,1)}{\Omega_A} = \frac{\#\Sha(A/\Q)\cdot\prod_{p\mid N} c_p} {\# A(\Q)\cdot\#\Adual(\Q)}.$$
Here the Shafarevich-Tate group $\Sha(A/\Q)$ is a measure of the failure
of the local-to-global principle; the Tamagawa numbers~$c_p$ are the
orders of the component  groups of~$A$; the real number~$\Omega_A$ is
the volume of~$A(\R)$ with respect to a basis of differentials having
everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.
My goal is to verify the full
conjecture for many specific abelian varieties on a case-by-case basis.
This is the first step in a program to verify the above
conjecture for an infinite family of quotients of~$J_0(N)$.

\mysubsection{The ratio $L(A,1)/\Omega_A$}
Following Y.~Manin's work on elliptic curves,
A.~Agash\'e and I proved the following
theorem in~\cite{stein:vissha}.
\begin{theorem}\label{thm:ratpart}
Let~$m$ be the largest square dividing~$N$.
The ratio $L(A,1)/\Omega_A$ is a rational number that can be
explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.
\end{theorem}
The proof uses modular symbols combined with an extension of the argument
used by Mazur in~\cite{mazur:rational} to bound the Manin constant.
The ratio $L(A,1)/\Omega_A$ is expressed as the lattice index of
two modules over the Hecke algebra.
I expect the method to give similar results
for special values of twists, and of
$L$-functions attached to eigenforms of higher weight.
I have computed $L(A,1)/\Omega_A$ for all optimal
quotients of level $N\leq 1500$; this table continues to be
of value to number theorists.

\mysubsection{The torsion subgroup}
I can compute upper and lower bounds on $\#A(\Q)_{\tor}$, but I can not
determine $\#A(\Q)_{\tor}$ in all cases.  Experimentally, the deviation
between the upper and lower bound is reflected in congruences with
forms of lower level; I hope to exploit this in a precise way.
I also obtained the following
intriguing corollary that suggests cancellation between
torsion and~$c_p$; it generalizes to higher weight forms, thus
suggesting a geometric explanation for reducibility
of Galois representations.
\begin{corollary}
Let~$n$ be the order of the image of
$(0)-(\infty)$ in $A(\Q)$, and
let~$m$ be the largest square dividing~$N$.
Then $\displaystyle n\cdot L(A,1)/\Omega_A$ is an integer,
up to a unit in  $\Z[1/(2m)]$.
\end{corollary}

\mysubsection{Tamagawa numbers}
\begin{theorem}\label{thm:tamagawa}
When $p^2\nmid N$, the number~$c_p$ can be explicitly computed
(up to a power of~$2$).
\end{theorem}
I prove this in~\cite{stein:compgroup}.  Several related problems
remain: when $p^2 \mid N$ it may be possible to compute~$c_p$
using the Drinfeld-Katz-Mazur interpretation
of~$X_0(N)$; it should also be possible to use my
methods to treat optimal quotients of $J_1(N)$.

I was surprised to find that systematic computations using this
formula indicate the following conjectural refinement of a result of
Mazur~\cite{mazur:eisenstein}.
\begin{conjecture}
Suppose~$N$ is prime and~$A$
is an optimal quotient of $J_0(N)$.  Then $A(\Q)_{\tor}$ is
generated by the image of $(0)-(\infty)$
and $c_p = \#A(\Q)_{\tor}$.  Furthermore,
the product of the~$c_p$ over all optimal factors
equals the numerator of $(N-1)/12$.
\end{conjecture}
I have checked this conjecture for all $N\leq 997$ and,
up to a power of~$2$, for all $N\leq 2113$.
The first part is known when~$A$ is an elliptic
curve (see~\cite{mestre-oesterle:crelle}).
Upon hearing of this conjecture, Mazur proved it
when all $q$-Eisenstein quotients'' are simple.
There are three promising approaches to finding
a complete proof. One involves the explicit
formula of Theorem~\ref{thm:tamagawa};
another is based on Ribet's level lowering theorem,
and a third makes use of a simplicity result of Merel.

Theorem~\ref{thm:tamagawa}
also suggests a way to compute Tamagawa numbers of
motives attached to eigenforms of higher weight.
These numbers appear in the conjectures of Bloch and Kato,
which generalize the BSD conjecture to motives (see~\cite{bloch-kato}).

\mysubsection{Upper bounds on $\#\BigSha$}
V.~Kolyvagin and K.~Kato~\cite{kolyvagin:structureofsha, scholl:kato}
obtained upper bounds on~$\#\Sha(A)$.
To verify the full BSD conjecture for certain abelian
varieties, it is necessary is to make these bounds explicit.
Kolyvagin's bounds involve computations with Heegner points,
and Kato's involve a study of the Galois representations
associated to~$A$.  I plan to carry out such
computations in many specific cases.

\mysubsection{Lower bounds on $\#\BigSha$}
One approach to showing that~$\Sha$ is as large as predicted
by the BSD conjecture is suggested by Mazur's notion of
the visible part of~$\Sha$ (see~\cite{cremona-mazur,  mazur:visthree}).
Let~$\Adual$ be the dual of~$A$.
The visible part of $\Sha(\Adual/\Q)$ is the
kernel of $\Sha(\Adual/\Q)\ra \Sha(J_0(N))$.
Mazur observed that if an element of order~$p$
in~$\Sha(\Adual/\Q)$ is visible,
then it is explained by a jump in the rank of Mordell-Weil
in the sense that there is another abelian subvariety $B\subset J_0(N)$
such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.
I think that this observation can be turned around: if there is
another abelian
variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,
then, under mild hypotheses, there is an element of~$\Sha(\Adual/\Q)$
of order~$p$.  Thus the theory of congruences between modular
forms can be used to obtain a lower bound on~$\#\Sha(\Adual/\Q)$.
I am trying to use the cohomological methods of~\cite{mazur:tower}
and suggestions of B.~Conrad and Mazur to prove the following conjecture.
\begin{conjecture}
Let~$\Adual$ and~$B$ be abelian subvarieties of~$J_0(N)$.
Suppose that $p\mid \#(\Adual\intersect B)$, that~$p\nmid N$,
and that~$p$ does not divide the
order of any of the torsion subgroups
or component groups of~$A$ or~$B$.  Then
$(B(\Q) \oplus \Sha(B/\Q))\tensor\Z/p\Z \isom (\Adual(\Q)\oplus \Sha(\Adual/\Q) )\tensor\Z/p\Z$.
\end{conjecture}
Unfortunately, $\Sha(\Adual/\Q)$ can fail to be
visible inside~$J_0(N)$.
For example, I found that the BSD conjecture
predicts the existence of invisible elements of odd order in~$\Sha$
for at least~$15$ of the~$37$ optimal quotients of prime
level $\leq 2113$.
For every integer~$M$ (Ribet~\cite{ribet:raising} tells us which~$M$
to choose), we can consider the images of~$\Adual$ in $J_0(NM)$.
There is not yet enough evidence to conjecture the existence of
an integer~$M$ such that all of $\Sha(\Adual/\Q)$ is visible in
$J_0(NM)$.
I am gathering data to determine whether or not
to expect the existence of such~$M$.

\mysubsection{Motivation for considering abelian varieties}
If $A$ is an elliptic curve, then explaining~$\Sha(A/\Q)$ using
only congruences between elliptic curves is bound to fail.
This is because pairs of nonisogenous elliptic curves
with isomorphic $p$-torsion are, according to
E.~Kani's conjecture, extremely rare.
It is crucial to understand what happens in all dimensions.

Within the range accessible by computer,
abelian varieties exhibit more richly textured
structure than elliptic curves.
For example, I discovered a visible element of prime order $83341$
in the Shafarevich-Tate group of an abelian variety of
prime conductor~$2333$; in contrast, over all optimal elliptic
curves of conductor up to $5500$, it appears that
the largest order of an element of a Shafarevich-Tate
group is~$7$.

\mysection{Conjectures of Artin, Merel, and Serre}

\mysubsection{Icosahedral Galois representations}
E.~Artin conjectured in~\cite{artin:conjecture}
that the $L$-series associated to any continuous irreducible
representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.
Recent exciting work of Taylor and others suggests that a complete proof
of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,
is on the horizon.
This case of Artin's conjecture is known when the image
of~$\rho$ in $\PGL_2(\C)$ is solvable
(see~\cite{tunnell:artin}), and in infinitely
many cases when the image of~$\rho$ is not solvable (see~\cite{bdsbt}).

In 1998, K.~Buzzard suggested a way to combine the
main theorem of~\cite{buzzard-taylor}, along with
a computer computation, to deduce modularity of certain
icosahedral Galois representations.  Buzzard and I recently obtained
the following theorem.
\begin{theorem}
The icosahedral Artin representations of conductor~$1376=2^5\cdot 43$
are modular.
\end{theorem}
We expect our method to yield several more examples.
These ongoing computations are laying a small part of the technical
foundations necessary for a full proof of the Artin conjecture
for odd two dimensional~$\rho$, as well as stimulating
the development of new algorithms for computing with modular forms
using modular symbols in characteristic~$\ell$.

\mysubsection{Cyclotomic points on modular curves}
If~$E$ is an elliptic curve over~$\Q$
and~$p$ is an odd prime, then the $p$-torsion
on~$E$ can not all lie in~$\Q$; because of
the Weil pairing the $p$-torsion generates a field
that contains~$\Q(\mu_p)$.
For which primes~$p$ does
there exist an elliptic curve~$E$ over $\Q(\mu_p)$
with all of its $p$-torsion rational over $\Q(\mu_p)$?
When $p=2,3,5$ the corresponding moduli space has genus zero
and infinitely many examples exist.  Recent work of L.~Merel,
combined with computations he enlisted me to do,
suggest  that these are the only primes~$p$ for
which such elliptic curves exist.
In~\cite{merel:cyclo}, Merel exploits
cyclotomic analogues of the techniques used
in his proof of the uniform boundedness conjecture
to obtain an explicit criterion that can
be used to answer the above question for many primes~$p$,
on a case-by-case basis.
Theoretical work of Merel, combined with my computations of
twisted $L$-values and character groups of tori,
give the following result (see~\cite[\S3.2]{merel:cyclo}):
\begin{theorem}
Let $p \equiv 3\pmod{4}$ be a prime satisfying
$7 \leq p < 1000$. There are no elliptic curves
over $\Q(\mu_p)$ all of whose $p$-torsion is rational
over $\Q(\mu_p)$.
\end{theorem}
The case in which~$p$ is congruent to~$1$ modulo~$4$ presents
additional difficulties that involve showing that~$Y(p)$ has no
$\Q(\sqrt{p})$-rational points.  We hope to tackle these in
the near future.

\mysubsection{Genus one curves}
The index of an algebraic curve~$C$ over~$\Q$ is the
order of the cokernel of the degree map $\Div_\Q(C)\ra\Z$;
rationality of the canonical divisor implies that the index
divides $2g-2$, where~$g$ is the genus of~$C$.
When $g=1$ this is no condition at all; Artin conjectured, and
Lang and Tate~\cite{lang-tate} proved, that for every integer~$m$
there is a genus one curve of index~$m$ over some number field.
Their construction yields genus one curves over~$\Q$ only for a few
values of~$m$, and they ask whether one can find genus one curves
over~$\Q$ of every index.  I have answered
this question for odd~$m$.
\begin{theorem}
Let~$K$ be any number field.  There are genus one curves over~$K$
of every odd index.
\end{theorem}
The proof involves showing that enough cohomology classes in
Kolyvagin's Euler system of Heegner points do not vanish
combined with explicit Heegner point computations.
I hope to show that curves  of every index occur, and
to determine the consequences of my
nonvanishing result for Selmer groups.  This can be viewed
as a contribution to the problem of understanding $H^1(\Q,E)$.

\mysubsection{Serre's conjecture modulo $pq$}
Let~$p$ and~$q$ be primes, and consider a continuous
representation $\rho:\GQ\ra \GL(2,\Z/pq\Z)$
that is irreducible in the sense that its reductions
modulo~$p$ and modulo~$q$ are both irreducible.
Call~$\rho$ {\em modular} if there is a modular
form~$f$ such that a mod~$p$ representation attached to~$f$
is the mod~$p$ reduction of~$\rho$, and ditto for~$q$.
I have carried out specific computations suggested by Mazur
in hopes of determining when one should expect
that such mod~$pq$ representations are modular; the computation
suggests that the right conjectures are elusive.
Ribet's theorem (see~\cite{ribet:raising})
produces infinitely many levels $pq\ell$ at which there is
a form giving rise to $\rho$~mod~$p$ and another
giving rise to $\rho$~mod~$q$; we hope to determine if for some~$\ell$
there is a single form  giving rise to both reductions.

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\newcommand{\closer}{\vspace{-1.5ex}}
\begin{thebibliography}{10}

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\closer

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\closer

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\closer

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\closer

\end{thebibliography}

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