Sharedwww / job / Abstracts.texOpen in CoCalc
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\newcommand{\myname}{William A.\ Stein}
\newcommand{\phone}{(510) 883-9938}
\newcommand{\email}{{\tt was@math.berkeley.edu}}
\newcommand{\www}{{\tt http://math.berkeley.edu/\~{\mbox{}}was}}
\newcommand{\address}{2041 Francisco Street, Apt.\ 14\\
Berkeley, CA  94709\\
USA}
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\begin{document}
\begin{center}
{\LARGE\bf
Publications}\\
(published, accepted for publication, or submitted)
\end{center}

\begin{enumerate}
\item  {\tfont Lectures on modular forms and
Galois representations} (170 pages), with K.~Ribet: In 1996,  
Ken Ribet taught an advanced  
course on modular forms and Galois representations.  
In collaboration with Ribet, I am 
turning my course notes into a book that has been accepted for
publication in Springer-Verlag's Universitext series. 
The chapters are: Overview; Modular representations; Modular forms;
Embedding Hecke operators in the dual; Rationality and integrality
questions; Modular curves; Higher weight modular forms; Newforms;
Some explicit genus computations; The field of moduli; 
Hecke operators as correspondences; Abelian varieties from modular forms;
The Gorenstein property; Local properties of $\rho_\lambda$; Serre's
conjecture; Fermat's Last Theorem; Deformations; The Hecke
algebra $\mathbf{T}_{\Sigma}$.
Though a version of the notes has been circulating 
for a few years, significant polishing remains to be done.

\item {\tfont HECKE: The modular forms calculator:}
I wrote {\sc Hecke}, a popular {\tt C++} package for 
computing with spaces of  modular forms
and modular abelian varieties.  
I have been invited to visit the {\sc Magma} group in 
Sydney in order to make {\sc Hecke} a part of their computer 
algebra system, and I have already ported my code to {\sc Magma}.


\item {\tfont Explicit approaches to modular abelian 
varieties} (130 pages), Ph.D. thesis 
under H.\thinspace{}W.~Lenstra:
In the first part of my thesis, I give algorithms for
computing with modular abelian varieties.
These include algorithms for computing congruences between 
modular eigenforms, the modular degree, the rational parts
of the special values of the $L$-function,
component groups at primes of multiplicative reduction, period 
lattices, and the real volume.

In the second part of my thesis, I use these algorithms to 
investigate several open problems.
The first concerns the Birch and Swinnerton-Dyer conjecture,
which ties together the constellation of invariants attached
to an abelian variety; I verify this conjecture for certain specific
abelian varieties of dimension greater than one.  
The key idea is to use B.~Mazur's notion of visibility, coupled with
explicit computations, to produce lower bounds on Shafarevich-Tate groups.
I also describe related computations that are used in an 
upcoming joint paper with L.~Merel on 
rational points of $X_0(N)$ over $\Q(\mu_p)$.

E.~Artin conjectured in 1924 that the $L$-series associated to
a representation $\rho:\GQ\rightarrow\GL_2(\C)$ is entire.
K.~Buzzard and I verify this conjecture in eight new icosahedral cases;
the argument uses a theorem of K.~Buzzard and R.~Taylor along 
with a computer computation, which is used to fill in a missing 
mod-$5$ modularity result.

Finally I turn to Serre's conjecture. I consider a family of 
exceptional cases, and show that the conjecture appears to fail 
as badly as possible.  I also describe numerical experiments 
towards a mod-$pq$ extension of Serre's conjecture.  

I expect to finish my thesis before May 2000.
\newpage
\item  {\tfont Lectures on Serre's conjectures} (77 pages), with K.~Ribet:
This is an expository paper based on Ken Ribet's lectures at 
the 1999 Park City Mathematics Institute; it will be 
published in the IAS/Park City Mathematics Institute Lecture Series.
There are 9 lectures: Introduction to Serre's conjectures;
The weak and strong conjectures; The weight in Serre's conjecture;
Galois representations from modular forms; Introduction to level lowering;
Approaches to level lowering; Mazur's principle; Level lowering
without multiplicity one; Level lowering with multiplicity one;
Other directions.  We have finished 
all but the last three sections, and expect to finish 
by early December, 1999.


\item {\tfont Empirical evidence for the Birch and 
Swinnerton-Dyer conjecture for modular Jacobians 
of genus~2 curves} (22 pages), with E.\thinspace{}V.~Flynn, 
F.~Lepr\'{e}vost,  E.\thinspace{}F.~Schaefer,
M.~Stoll, J.\thinspace{}L.~Wetherell:
We provide systematic numerical evidence for the BSD conjecture
in the case of dimension two.
This conjecture relates six quantities associated to
a Jacobian over the rational numbers.  One of these 
quantities is the size~$S$ of the Shafarevich-Tate group. 
Unable to compute~$S$ directly, we compute the five other 
quantities and solve for the conjectural value~$S_?$ of~$S$.
For all 32~curves considered, the real number~$S_?$
is very close to either~$1$,~$2$, or~$4$, and
agrees with the size of the 2-torsion of the 
Shafarevich-Tate group, which we could compute.
This paper has been submitted.

\item {\tfont Parity structures and generating 
functions from Boolean rings \hspace{.2em}}(8 pages), with D.~Moulton:
Let~$S$ be a finite set and~$T$ be a subset of the power set of~$S$.
Call~$T$ a {\em parity structure} for~$S$ if, for
each subset~$b$ of~$S$ of odd size, the number of subsets of~$b$
that lie in~$T$ is even.  We classify parity structures using generating
functions from a free boolean ring.
We also show that if~$T$ is a parity structure, then, for each
subset~$b$ of~$S$ of even size, the number of subsets of~$b$ of
odd size that lie in~$T$ is even.  We then give several other
properties of parity structures and discuss a generalization.                 
This paper has been submitted.

  \item {\tfont Fallacies, Flaws, and 
         Flimflam \#92:   An Inductive Fallacy}, (1 page)
     with A.~Riskin:  College Math.\ Journal 26:5 (1995), 382.
\end{enumerate}

\vspace{1ex}
\begin{center}
\LARGE\bf
Work in progress
\end{center}
\begin{enumerate}

\item {\tfont The modular forms database:}
This is an evolving collection of tables of modular eigenforms,
special values of $L$-functions, arithmetic invariants of
modular abelian varieties, and other data. 
These tables are available at\\
{\tt http://shimura.math.berkeley.edu/\~{}was/Tables/}.


\item {\tfont Visibility of Shafarevich-Tate 
       groups of modular abelian varieties} (20 pages), with
A.~Agash\'{e}:
We study Mazur's notion of visibility 
of Shafarevich-Tate of modular \nobreak{abelian} varieties,
and use it to verify the conjecture of Birch and Swinnerton-Dyer
for several specific abelian varieties.  We also give evidence
that visibility is rare.   The computations are done, but 
we are still adding to the theoretical content of the paper
and polishing the presentation.

\item {\tfont Component groups of optimal quotients of Jacobians} (16 pages):
Let~$A$ be an optimal quotient of~$J_0(N)$.
The main theorem of this paper gives a relationship between
the modular degree of~$A$ and the order of the component group
of~$A$.  From this I deduce a computable formula for the
component group of any optimal \nobreak{quotient} 
of~$J_0(N)$ at a prime of multiplicative 
reduction.  I then compute over one \nobreak{thousand} 
examples leading me
to conjecture that the torsion and component groups of 
quotients of $J_0(p)$, for~$p$ prime, are as simple as possible.
This paper is almost ready to be submitted.

\item {\tfont Mod~$5$ approaches to modularity of icosahedral
Galois \nobreak{representations}} (16 pages), with K.~Buzzard:
Consider a continuous odd irreducible representation
$\rho:\mbox{\rm Gal}(\overline{\mathbf{Q}}/\mathbf{Q})
    \rightarrow\mbox{\rm GL}_2(\mathbf{C}).$
A special case of a general conjecture of Artin
is that the $L$-function $L(\rho,s)$ associated to~$\rho$
is entire.  
Buzzard and I give new examples of representations~$\rho$ that 
satisfy this conjecture.  
We obtained these examples by applying a recent 
theorem of Buzzard and Taylor to a mod~$5$ reduction of~$\rho$,
combined with a computational verification of modularity of a 
related mod~$5$ representation.  
The computations of the paper are complete, and have been
mostly written up.  We expect to finish the paper in time 
for the Hot Topics workshop at MSRI in December, 1999.

\item {\tfont Indexes of genus one curves} (10 pages): 
I prove results about the complexity of genus one curves.  
The greatest common divisor of the degrees of the fields in 
which a curve of genus~$g$ over~$\mathbf{Q}$ has a rational 
point divides $2g-2$; when $g=1$ this is no condition at all.
In the 1950s S.~Lang and J.~Tate asked whether, given a positive integer~$m$, 
there exists a genus one curve so that the smallest number
field in which it has a rational point is of degree~$m$ over~$\mathbf{Q}$.
Using Euler systems, I prove this when~$m$ is not 
divisible by~$4$.  This paper is almost ready to be 
submitted.
\end{enumerate}

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