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\newcommand{\doc}{Research Summary}
\newcommand{\myname}{William A.\ Stein}
\newcommand{\phone}{(510) 883-9938}
\newcommand{\email}{{\tt was@math.berkeley.edu}}
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Berkeley, CA  94709\\
USA}

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\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, R.~Taylor, and A.~Wiles
on Galois representations
with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
on explicit computations involving modular abelian varieties.
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.

\mysection{Objectives}
The main outstanding problem in my field
is the conjecture of Birch and Swinnerton-Dyer (\nobreak{BSD} conjecture),
which ties together the constellation of
arithmetic invariants of an
elliptic curve.
There is still no general class of elliptic curves for which
the full BSD conjecture is known to hold.
Approaches to the BSD conjecture that rely on congruences
between modular forms are likely to
require a deeper understanding of the analogous
conjecture for modular abelian varieties, which are higher
dimensional analogues of elliptic curves.

As a first step, I have obtained theorems that make possible
computation of some of the arithmetic invariants of
modular abelian varieties.
My objective is to find ways to explicitly compute all of
the arithmetic invariants.
Cremona has enumerated these invariants for the first
few thousand elliptic curves, and I am working to do the same
for abelian varieties.
It is hoped that this work will continue to
yield theoretical results.
I am also writing modular forms software that I hope will be used
by many mathematicians and have practical applications in
the development of elliptic curve cryptosystems.

My long-range goal is to give a general hypothesis, valid for
infinitely many abelian varieties,
under which the full BSD conjecture holds.
My approach involves combining Euler system techniques of K.~Kato and
K.~Rubin with visibility and congruence ideas of Mazur and Ribet.

\mysection{Modular abelian varieties}
My primary objective is to verify the BSD conjecture for
specific modular abelian varieties, by using the
rich theory of their arithmetic.

The BSD conjecture asserts that if~$A$ is a modular abelian
variety with $L(A,1)\neq 0$, then
$$\frac{L(A,1)}{\Omega_A} = \frac{\#\Sha(A)\cdot\prod c_p} {\# A(\Q)_{\tor}\cdot\#\Adual(\Q)_{\tor}}.$$
Here $A(\Q)_{\tor}$ is the group of rational torsion points on~$A$;
the Shafarevich-Tate group $\Sha(A)$ is a measure of the failure
of the local-to-global principle; the Tamagawa numbers~$c_p$ are the
orders of certain component groups associated to~$A$;
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~$A$.

\mysubsection{The ratio $L(A,1)/\Omega_A$}
Extending Manin's work on elliptic curves, A.~Agash\'{e} and I
found a computable formula for the rational number $L(A,1)/\Omega_A$.
Using similar techniques, I hope to find computable formulas
for rational parts of special values of twists, and of $L$-functions
attached to forms of weight greater than two.   I have
already computed $L(A,1)/\Omega_A$ for several thousand abelian
varieties, and hope to extend these computations.

\mysubsection{The Tamagawa numbers $c_p$}
When~$A$ has semistable reduction at~$p$, I have
found a way to explicitly compute
the number~$c_p$, up to a power of~$2$.
I hope to find a way to compute~$c_p$ in the remaining cases.

\mysubsection{Bounding $\#\BigSha$}
V.~Kolyvagin and K.~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.

One approach to showing that~$\Sha(A)$ is as large as predicted
by the BSD conjecture is suggested by Mazur's notion of
the visible part of~$\Sha(A)$.
Consider an abelian variety~$A$ that sits naturally
in the Jacobian $J_0(N)$ of the modular curve $X_0(N)$.
The {\em visible part} of $\Sha(A)$ is the collection of those
elements of $\Sha(A)$
that go to~$0$ under the natural map to $\Sha(J_0(N))$.
Cremona and Mazur observed that if an element of order~$p$
in~$\Sha(A)$ 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 $p \mid \#(A\intersect B)$ and~$B$ has many rational points.
I am trying to find the precise degree  to which this observation
can be turned around: if there is
another abelian variety~$B$ with many rational points and
$p\mid \#( A\intersect B)$, then under what hypotheses
is there an element of~$\Sha(A)$ of order~$p$?

\mysection{Icosahedral Galois representations}
E.~Artin conjectured
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.

By combining the main result of
a recent paper of K.~Buzzard and Taylor
with a computer computation,
Buzzard and I recently
proved that the icosahedral Artin representations of
conductor~$1376=2^5\cdot 43$ are modular.
If I can extend a congruence result of J.~Sturm, then
our method will yield several more examples.
These ongoing computations are laying a part of the
foundation 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
in characteristic~$\ell$.

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