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\begin{center}
{\bf\large American Mathematical Society}\vspace{2ex}
{\bf \LARGE Centennial Fellowship for 2007-2008}
\vspace{2ex}
{\bf \large Application Form}
\end{center}
\noindent{}Membership and Programs Department\\
American Mathematical Society\\
201 Charles Street\\
Providence, RI 02904-2294\\
\i{Date} November 28, 2006
\i{Name in Full (First, Middle, Last)} William Arthur Stein
\i{Present Address}\\
University of Washington \\
Department of Mathematics \\
Padelford C-138\\
Seattle, WA 98195-4350
\i{Email Address} wstein@gmail.com
\i{Telephone Number} (858) 220-6876
\i{Fax Number} (206) 543-0397
\i{Present Position} Associate Professor of Mathematics
\i{All Positions and Fellowships since Ph.D.}
(see attached)
\vspace{1ex}
\noindent{}I am currently employed full time with a tenured
position at the University of
Washington, which is a North American institution.
\i{Graduate Education}
UC Berkeley, Ph.D., 05/01/1995--05/01/2000
\i{Doctoral Thesis Advisor} Hendrik Lenstra
\i{Title of Thesis} Explicit Approaches to Modular Abelian Varieties
\i{Honors Received} I received the Cal@SiliconValley fellowship and
the Vice Chancellor Research Grant when I was a
Berkeley graduate student.
\i{Publications} (see attached)
\i{References}\\
Name and institution: {\bf Barry Mazur, Harvard University}\\
Name and institution: {\bf Kenneth Ribet, UC Berkeley}\\
Name and institution: {\bf Karl Rubin, UC Irvine}
\i{Research} (see attached)
\vspace{3ex}
\i{Signature of Applicant}
\verb+_________________________________________________+
\newpage
\begin{center}
{\bf\Large Positions Held}
\end{center}
\begin{itemize}\setlength{\itemsep}{0ex}
\item Associate Professor of Mathematics (with tenure),
University of Washington, September 2006--present.
\item Associate Professor of Mathematics (with tenure),
UC San Diego, July 2005--June 2006.
\item Benjamin Peirce Assistant Professor of Mathematics,
Harvard University, July 2001--May 2005.
\item NSF Postdoctoral Research Fellowship
under Barry Mazur at Harvard University, August 2000--May 2004.
\item Clay Mathematics Institute Liftoff Fellow, Summer 2000.
\end{itemize}
\newpage
\newcommand{\ptitle}{\em}
\begin{center}
{\bf\Large Publication List}
\begin{enumerate}
\item {\ptitle Modular forms, a computational approach}, (284 pages),
Graduate Studies in Mathematics (AMS), Volume 79, 2007, with
an appendix by Paul Gunnells.
\item {\ptitle The Manin Constant},
with A. Agashe and K. Ribet (22 pages), 2006, to appear in
World Scientific's Coate's Volume.
\item {\ptitle The Modular Degree, Congruence Primes and Multiplicity One},
with A. Agashe and K. Ribet (16 pages), 2006, submitted.
%, to appear in Documenta
%Mathematica's Coate's Volume.
\item {\ptitle Computation of $p$-Adic Heights and Log Convergence},
with B. Mazur and J. Tate (36 pages), 2005, to appear in Documenta
Mathematica's Coate's Volume.
\item {\ptitle Verification of the Birch and Swinnerton-Dyer
Conjecture for Specific Elliptic Curves}, with G. Grigorov, A.
Jorza, S. Patrikis, and C. Patrascu (26 pages), 2005, submitted.
\item {\ptitle Visibility of Mordell-Weil Groups} (20 pages), 2005,
to appear in Documenta Mathematica.
\item {\ptitle SAGE: System for Algebra and Geometry Experimentation}
with D. Joyner,
Communications in Computer Algebra, vol 39, June 2005, pages 61--64.
\item {\ptitle Modular Parametrizations of Neumann-Setzer
Elliptic Curves}, with M. Watkins, in IMRN 2004, no. 27, 1395--1405.
\item {\ptitle Studying the Birch and Swinnerton-Dyer Conjecture for
Modular Abelian Varieties Using} MAGMA (23 pages), to appear in
a Springer-Verlag book edited by J.~Cannon and W.~Bosma.
\item {\ptitle Conjectures about Discriminants of Hecke Algebras of
Prime Level} (16 pages), with F.~Calegari, in ANTS VI, Vermont, 2004.
\item {\ptitle Constructing Elements in Shafarevich-Tate Groups of Modular Motives},
with N.~Dummigan and M.~Watkins, in ``Number theory and algebraic geometry---to Peter Swinnerton-Dyer on his 75th birthday'', Ed.
M. Reid and A. Skorobogatov, pages 91--118.
\item {\ptitle Approximation of Infinite Slope Modular Eigenforms
By Finite Slope Eigenforms} (13 pages), with R.~Coleman, in the
Dwork Proceedings.
\item {\ptitle $J_1(p)$ has connected fibers}, with B.~Conrad and
B.~Edixhoven, Documenta Mathematica, {\bf 8} (2003), 331--408.
\item {\ptitle Shafarevich-Tate Groups of Nonsquare Order},
in Progress in Math., {\bf 224} (2004), 277--289, Birkhauser.
\item {\ptitle Visible Evidence for the Birch and Swinnerton-Dyer
Conjecture for Rank $0$ Modular Abelian Varieties} (30 pages),
with A.~Agashe, appeared in Mathematics of Computation.
\item {\ptitle A Database of Elliptic Curves--First Report}
(10 pages) with M.~Watkins, in ANTS V proceedings, Sydney, Australia, 2002.
\item {\ptitle Visibility of Shafarevich-Tate Groups
of Abelian Varieties}, with A.~Agashe, J. Number Theory, {\bf 97}
(2002), no. 1, 171--185.
\item {\ptitle Cuspidal Modular Symbols are Transportable}, with H.~Verrill,
LMS J.\ Comput.\ Math., {\bf 4} (2001), 170--181.
\item Appendix to Lario and Schoof's {\ptitle Some computations
with Hecke rings and deformation rings}, with A.~Agashe,
Experiment. Math. {\bf 11} (2002), no.~2, 303--311.
\item {\ptitle There are genus one curves over $\mathbf{Q}$
of every odd index}, J. Reine Angew. Math. {\bf 547}
(2002), 139--147.
\item {\ptitle Component groups of purely toric quotients of
semistable Jacobians}, with B.~Conrad, Math. Res. Lett., {\bf
8} (2001), no. 5--6, 745--766.
\item {\ptitle The field generated by the points of small prime
order on an elliptic curve}, with L.~Merel,
Int.\ Math.\ Res.\ Notices, 2001, no.~20, 1075--1082.
\item {\ptitle An introduction to computing modular forms using
modular symbols} (10 pages), to appear in an MSRI proceedings volume.
\item {\ptitle A mod five approach to modularity of icosahedral Galois
representations}, with K.~Buzzard, Pac. J.
Math., {\bf 203} (2002), no. 2, 265--282.%
\item {\ptitle Lectures on Serre's conjectures},
with K.\thinspace{}A.~Ribet,
in Arithmetic Algebraic Geometry, IAS/Park City Math.
Inst. Series, Vol.~9, 143--232.
\item {\ptitle Component groups of quotients of $J_0(N)$},
with D.~Kohel, Proceedings of the 4th International
Symposium (ANTS-IV), 2000, 405--412.
\item {\ptitle Empirical evidence for
the Birch and Swinnerton-Dyer conjectures for modular Jacobians of
genus 2 curves},
with E.\thinspace{}V.~Flynn, F.~Lepr\'{e}vost,
E.\thinspace{}F. Schaefer,
M.~Stoll, J.\thinspace{}L.~Wetherell,
Math.\ of Comp. {\bf 70} (2001), no. 236, 1675--1697.
\end{enumerate}
\end{center}
\newpage
\begin{center}
{\bf\Large Research Statement}
\end{center}
\par{}\noindent{}My research reflects the rewarding interplay of theory
with explicit computation, as illustrated by
Bryan Birch \cite{birch:bsd}:
\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}
My main research goal is to carry out a wide range of
computational and theoretical investigations into elliptic
curves and abelian varieties motivated by
the Birch and
Swinnerton-Dyer conjecture (BSD conjecture).
This will hopefully
improve our practical
computational capabilities, extend the data that
researchers have available for formulating
conjectures, and deepen our understanding of
theorems about the BSD conjecture.
I am one of the more sought after people by the worldwide
community of number
theorists, for computational confirmation of conjectures, for modular
forms algorithms, for data, and for
ways of formulating
problems so as to make them more accessible to algorithms.
I have also been successful at involving numerous
undergraduate and graduate students at all levels in my
research, and have started a major project (SAGE) to
improve the quality and accessibility of open source
mathematics software.
\section*{My Ph.D. Research}
In my Ph.D. thesis, I investigate the Birch and Swinnerton-Dyer conjecture, which ties
together the constellation of invariants attached to an abelian variety.
I attempt to verify this conjecture for certain specific modular abelian
varieties of dimension greater than one. The key idea is to use
Barry Mazur's notion of visibility, coupled with explicit computations,
to produce lower bounds on the Shafarevich-Tate group. Nobody
has yet proved the full conjecture in these examples; this would
require computing explicit upper bounds.
I next describe how to compute in
spaces of modular forms of weight at least two.
I give an integrated package for computing, in many
cases, the following invariants of a modular abelian variety: the
modular degree, the rational part of the special value of the
$L$-function, the order of the component group at primes of
multiplicative reduction, the period lattice, upper and lower bounds
on the torsion subgroup, and the real volume. Taken together, these
algorithms are frequently sufficient to compute the odd part of the
conjectural order of the Shafarevich-Tate group of an analytic
rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free, and
to give tight bounds in many other cases.
I also provide generalizations of some of the above algorithms
to higher weight forms with nontrivial character.
\section*{My Computational Research}
%It would also provide new insight into the Birch and Swinnerton-Dyer
%conjecture.
\subsection*{SAGE: Software for Algebra and Geometry\\Experimentation}
I am the director of SAGE---Software for Algebra and
Geometry Experimentation \cite{sage},
a project I started in January 2005.
The goal of SAGE is to create quality free open source
software for
research and teaching in number theory, algebra, geometry,
cryptography and numerical computation. SAGE
does not reinvent the wheel, but instead builds upon and
unifies decades of work on mathematical software. It
provides an environment in
which to use all of your favorite mathematical software (free or commercial) in a
better way. My work so far on SAGE has led to publications and
numerous collaborations with undergraduate and graduate students,
and there are now over 30 contributors to SAGE.
A Centennial Fellowship would increase the amount of energy I
could focus on SAGE development and student research projects that
involve SAGE.
\begin{itemize}
\item SAGE is free open source software for
research in {\bf algebra}, {\bf geometry}, {\bf number theory},
{\bf cryptography}, and {\bf numerical computation}.
\item SAGE is an {\bf environment for
rigorous mathematical computation}
built using Python, GAP, Maxima, Singular,
PARI, etc.,
and provides a {\bf unified interface} to
Mathematica, Maple, Magma, MATLAB, etc.
\item I have organized {\bf several successful SAGE workshops}, and
there are many active SAGE developers.
\item The primary goal of SAGE is to make
{\bf modern research-level algorithms}
available in an integrated package with
a graphical interface.
\end{itemize}
\subsection*{The Modular Forms Database}%
The modular forms database (see \cite{mfd})
is a freely-available collection of data about objects attached to
modular forms. It is analogous to Neil Sloane's tables of integer
sequences, and generalizes John Cremona's tables of elliptic curves \cite{cremona:onlinetables}
to
dimension bigger than one and weight bigger than two. The database is
used by many prominent number theorists.
I hope to greatly expand the databases with more information
about modular forms, elliptic curves, and modular abelian
varieties.
Support from a Centennial Fellowship would help me to rework
and expand the
database and make it easier to use.
\subsection*{Theory and Computation}
The PI, 3 undergraduates and a graduate student
proved the following in \cite{bsdalg1}:
\begin{theorem}\label{thm:main}
Suppose that $E$ is a non-CM elliptic curve of rank $\leq 1$,
conductor $\leq 1000$ and that $p$ is a prime. If $p$ is odd,
assume further that the mod~$p$ representation $\overline{\rho}_{E,p}$ is
irreducible and~$p$ does not divide any Tamagawa number of~$E$. Then
the Birch and Swinnerton-Dyer conjecture for $E$ at $p$ is true.
\end{theorem}
The proof involves an application of results of Kato and Kolyvagin, new
refinements of Kolyvagin's theorem, explicit 2-descent and 3-descent
and much explicit calculation. This is a first step toward
the following goals:
\begin{goal}\label{prob:bsd01}
Verify the full Birch and Swinnerton-Dyer Conjecture for every
elliptic curve over $\Q$ of conductor $<1000$, except
for the $18$ curves of rank $2$.
\end{goal}
\begin{goal}\label{prob:bsd2}
For each curve $E$ over $\Q$ of conductor $<1000$
and rank $2$, prove that $\Sha(E)[p]=0$ for all
$p<1000$.
\end{goal}
\section*{Travel Plans}
I would use funds from a Centennial Fellowship to visit Harvard
University where I would work mainly with Barry Mazur to
{\bf finish a widely
accessible book} we are writing on the Riemann Hypothesis, and
do work with Mazur on $p$-adic heights
and Heegner points (continuing the work initiated
in \cite{mazur-stein-tate:padic}). Also, one of the main SAGE
developers, David Harvey, is a Harvard graduate student so
I would work with him on state-of-the-art algorithms for
polynomial arithmetic and computation of $p$-adic heights.
I would also visit
Ken Ribet in Berkeley in order to {\bf finish a book} on
modular forms, Hecke operators, and modular curves that we are writing for the
Springer-Verlag GTM series. Our book may be viewed as
a sequel to \cite{diamond-shurman} that complements my new
AMS
book \cite{stein:modform}.
Finally, I would visit John Cremona to do joint work
on computing with modular forms.
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