CoCalc Public Fileswww / job / Abstracts.tex
Author: William A. Stein
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2\documentclass[12pt]{article}
3\newcommand{\myname}{William A.\ Stein}
4\newcommand{\phone}{(510) 883-9938}
5\newcommand{\email}{{\tt was@math.berkeley.edu}}
6\newcommand{\www}{{\tt http://math.berkeley.edu/\~{\mbox{}}was}}
8Berkeley, CA  94709\\
9USA}
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39\begin{document}
40\begin{center}
41{\LARGE\bf
42Publications}\\
43(published, accepted for publication, or submitted)
44\end{center}
45
46\begin{enumerate}
47\item  {\tfont Lectures on modular forms and
48Galois representations} (170 pages), with K.~Ribet: In 1996,
50course on modular forms and Galois representations.
51In collaboration with Ribet, I am
52turning my course notes into a book that has been accepted for
53publication in Springer-Verlag's Universitext series.
54The chapters are: Overview; Modular representations; Modular forms;
55Embedding Hecke operators in the dual; Rationality and integrality
56questions; Modular curves; Higher weight modular forms; Newforms;
57Some explicit genus computations; The field of moduli;
58Hecke operators as correspondences; Abelian varieties from modular forms;
59The Gorenstein property; Local properties of $\rho_\lambda$; Serre's
60conjecture; Fermat's Last Theorem; Deformations; The Hecke
61algebra $\mathbf{T}_{\Sigma}$.
62Though a version of the notes has been circulating
63for a few years, significant polishing remains to be done.
64
65\item {\tfont HECKE: The modular forms calculator:}
66I wrote {\sc Hecke}, a popular {\tt C++} package for
67computing with spaces of  modular forms
68and modular abelian varieties.
69I have been invited to visit the {\sc Magma} group in
70Sydney in order to make {\sc Hecke} a part of their computer
71algebra system, and I have already ported my code to {\sc Magma}.
72
73
74\item {\tfont Explicit approaches to modular abelian
75varieties} (130 pages), Ph.D. thesis
76under H.\thinspace{}W.~Lenstra:
77In the first part of my thesis, I give algorithms for
78computing with modular abelian varieties.
79These include algorithms for computing congruences between
80modular eigenforms, the modular degree, the rational parts
81of the special values of the $L$-function,
82component groups at primes of multiplicative reduction, period
83lattices, and the real volume.
84
85In the second part of my thesis, I use these algorithms to
86investigate several open problems.
87The first concerns the Birch and Swinnerton-Dyer conjecture,
88which ties together the constellation of invariants attached
89to an abelian variety; I verify this conjecture for certain specific
90abelian varieties of dimension greater than one.
91The key idea is to use B.~Mazur's notion of visibility, coupled with
92explicit computations, to produce lower bounds on Shafarevich-Tate groups.
93I also describe related computations that are used in an
94upcoming joint paper with L.~Merel on
95rational points of $X_0(N)$ over $\Q(\mu_p)$.
96
97E.~Artin conjectured in 1924 that the $L$-series associated to
98a representation $\rho:\GQ\rightarrow\GL_2(\C)$ is entire.
99K.~Buzzard and I verify this conjecture in eight new icosahedral cases;
100the argument uses a theorem of K.~Buzzard and R.~Taylor along
101with a computer computation, which is used to fill in a missing
102mod-$5$ modularity result.
103
104Finally I turn to Serre's conjecture. I consider a family of
105exceptional cases, and show that the conjecture appears to fail
106as badly as possible.  I also describe numerical experiments
107towards a mod-$pq$ extension of Serre's conjecture.
108
109I expect to finish my thesis before May 2000.
110\newpage
111\item  {\tfont Lectures on Serre's conjectures} (77 pages), with K.~Ribet:
112This is an expository paper based on Ken Ribet's lectures at
113the 1999 Park City Mathematics Institute; it will be
114published in the IAS/Park City Mathematics Institute Lecture Series.
115There are 9 lectures: Introduction to Serre's conjectures;
116The weak and strong conjectures; The weight in Serre's conjecture;
117Galois representations from modular forms; Introduction to level lowering;
118Approaches to level lowering; Mazur's principle; Level lowering
119without multiplicity one; Level lowering with multiplicity one;
120Other directions.  We have finished
121all but the last three sections, and expect to finish
122by early December, 1999.
123
124
125\item {\tfont Empirical evidence for the Birch and
126Swinnerton-Dyer conjecture for modular Jacobians
127of genus~2 curves} (22 pages), with E.\thinspace{}V.~Flynn,
128F.~Lepr\'{e}vost,  E.\thinspace{}F.~Schaefer,
129M.~Stoll, J.\thinspace{}L.~Wetherell:
130We provide systematic numerical evidence for the BSD conjecture
131in the case of dimension two.
132This conjecture relates six quantities associated to
133a Jacobian over the rational numbers.  One of these
134quantities is the size~$S$ of the Shafarevich-Tate group.
135Unable to compute~$S$ directly, we compute the five other
136quantities and solve for the conjectural value~$S_?$ of~$S$.
137For all 32~curves considered, the real number~$S_?$
138is very close to either~$1$,~$2$, or~$4$, and
139agrees with the size of the 2-torsion of the
140Shafarevich-Tate group, which we could compute.
141This paper has been submitted.
142
143\item {\tfont Parity structures and generating
144functions from Boolean rings \hspace{.2em}}(8 pages), with D.~Moulton:
145Let~$S$ be a finite set and~$T$ be a subset of the power set of~$S$.
146Call~$T$ a {\em parity structure} for~$S$ if, for
147each subset~$b$ of~$S$ of odd size, the number of subsets of~$b$
148that lie in~$T$ is even.  We classify parity structures using generating
149functions from a free boolean ring.
150We also show that if~$T$ is a parity structure, then, for each
151subset~$b$ of~$S$ of even size, the number of subsets of~$b$ of
152odd size that lie in~$T$ is even.  We then give several other
153properties of parity structures and discuss a generalization.
154This paper has been submitted.
155
156  \item {\tfont Fallacies, Flaws, and
157         Flimflam \#92:   An Inductive Fallacy}, (1 page)
158     with A.~Riskin:  College Math.\ Journal 26:5 (1995), 382.
159\end{enumerate}
160
161\vspace{1ex}
162\begin{center}
163\LARGE\bf
164Work in progress
165\end{center}
166\begin{enumerate}
167
168\item {\tfont The modular forms database:}
169This is an evolving collection of tables of modular eigenforms,
170special values of $L$-functions, arithmetic invariants of
171modular abelian varieties, and other data.
172These tables are available at\\
173{\tt http://shimura.math.berkeley.edu/\~{}was/Tables/}.
174
175
176\item {\tfont Visibility of Shafarevich-Tate
177       groups of modular abelian varieties} (20 pages), with
178A.~Agash\'{e}:
179We study Mazur's notion of visibility
180of Shafarevich-Tate of modular \nobreak{abelian} varieties,
181and use it to verify the conjecture of Birch and Swinnerton-Dyer
182for several specific abelian varieties.  We also give evidence
183that visibility is rare.   The computations are done, but
184we are still adding to the theoretical content of the paper
185and polishing the presentation.
186
187\item {\tfont Component groups of optimal quotients of Jacobians} (16 pages):
188Let~$A$ be an optimal quotient of~$J_0(N)$.
189The main theorem of this paper gives a relationship between
190the modular degree of~$A$ and the order of the component group
191of~$A$.  From this I deduce a computable formula for the
192component group of any optimal \nobreak{quotient}
193of~$J_0(N)$ at a prime of multiplicative
194reduction.  I then compute over one \nobreak{thousand}
196to conjecture that the torsion and component groups of
197quotients of $J_0(p)$, for~$p$ prime, are as simple as possible.
198This paper is almost ready to be submitted.
199
200\item {\tfont Mod~$5$ approaches to modularity of icosahedral
201Galois \nobreak{representations}} (16 pages), with K.~Buzzard:
202Consider a continuous odd irreducible representation
203$\rho:\mbox{\rm Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) 204 \rightarrow\mbox{\rm GL}_2(\mathbf{C}).$
205A special case of a general conjecture of Artin
206is that the $L$-function $L(\rho,s)$ associated to~$\rho$
207is entire.
208Buzzard and I give new examples of representations~$\rho$ that
209satisfy this conjecture.
210We obtained these examples by applying a recent
211theorem of Buzzard and Taylor to a mod~$5$ reduction of~$\rho$,
212combined with a computational verification of modularity of a
213related mod~$5$ representation.
214The computations of the paper are complete, and have been
215mostly written up.  We expect to finish the paper in time
216for the Hot Topics workshop at MSRI in December, 1999.
217
218\item {\tfont Indexes of genus one curves} (10 pages):
219I prove results about the complexity of genus one curves.
220The greatest common divisor of the degrees of the fields in
221which a curve of genus~$g$ over~$\mathbf{Q}$ has a rational
222point divides $2g-2$; when $g=1$ this is no condition at all.
223In the 1950s S.~Lang and J.~Tate asked whether, given a positive integer~$m$,
224there exists a genus one curve so that the smallest number
225field in which it has a rational point is of degree~$m$ over~$\mathbf{Q}$.
226Using Euler systems, I prove this when~$m$ is not
227divisible by~$4$.  This paper is almost ready to be
228submitted.
229\end{enumerate}
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