%abstracts.tex1\documentclass[12pt]{article}2\newcommand{\myname}{William A.\ Stein}3\newcommand{\phone}{(510) 883-9938}4\newcommand{\email}{{\tt was@math.berkeley.edu}}5\newcommand{\www}{{\tt http://math.berkeley.edu/\~{\mbox{}}was}}6\newcommand{\address}{2041 Francisco Street, Apt.\ 14\\7Berkeley, CA 94709\\8USA}9\newcommand{\tfont}{\bf}10\newcommand{\Q}{\mathbf{Q}}11\newcommand{\C}{\mathbf{C}}12\newcommand{\GQ}{G_\Q}13\newcommand{\GL}{\mbox{\rm GL}}141516\hoffset=-.1\textwidth17\textwidth=1.2\textwidth18\voffset=-.1\textheight19\textheight=1.2\textheight2021\usepackage{fancyhdr,ifthen}22\pagestyle{fancy}23\cfoot{} % no footers (in pagestyle fancy)24% running left heading25\lhead{\bfseries\LARGE\em \noindent{}\hspace{-.2em}\myname{}26\hfill Abstracts\vspace{-.2ex}\\}27% running right heading28\newcommand{\spc}{.33ex}2930%\rhead{\em {\small{\phone{}}} \hspace{\spc}$\bullet$\hspace{\spc} \email{}31% \hspace{\spc}$\bullet$\hspace{\spc} \www{}}3233% adjust, because the header is now taller than usual.34\setlength{\headheight}{7ex}35363738\begin{document}39\begin{center}40{\LARGE\bf41Publications}\\42(published, accepted for publication, or submitted)43\end{center}4445\begin{enumerate}46\item {\tfont Lectures on modular forms and47Galois representations} (170 pages), with K.~Ribet: In 1996,48Ken Ribet taught an advanced49course on modular forms and Galois representations.50In collaboration with Ribet, I am51turning my course notes into a book that has been accepted for52publication in Springer-Verlag's Universitext series.53The chapters are: Overview; Modular representations; Modular forms;54Embedding Hecke operators in the dual; Rationality and integrality55questions; Modular curves; Higher weight modular forms; Newforms;56Some explicit genus computations; The field of moduli;57Hecke operators as correspondences; Abelian varieties from modular forms;58The Gorenstein property; Local properties of $\rho_\lambda$; Serre's59conjecture; Fermat's Last Theorem; Deformations; The Hecke60algebra $\mathbf{T}_{\Sigma}$.61Though a version of the notes has been circulating62for a few years, significant polishing remains to be done.6364\item {\tfont HECKE: The modular forms calculator:}65I wrote {\sc Hecke}, a popular {\tt C++} package for66computing with spaces of modular forms67and modular abelian varieties.68I have been invited to visit the {\sc Magma} group in69Sydney in order to make {\sc Hecke} a part of their computer70algebra system, and I have already ported my code to {\sc Magma}.717273\item {\tfont Explicit approaches to modular abelian74varieties} (130 pages), Ph.D. thesis75under H.\thinspace{}W.~Lenstra:76In the first part of my thesis, I give algorithms for77computing with modular abelian varieties.78These include algorithms for computing congruences between79modular eigenforms, the modular degree, the rational parts80of the special values of the $L$-function,81component groups at primes of multiplicative reduction, period82lattices, and the real volume.8384In the second part of my thesis, I use these algorithms to85investigate several open problems.86The first concerns the Birch and Swinnerton-Dyer conjecture,87which ties together the constellation of invariants attached88to an abelian variety; I verify this conjecture for certain specific89abelian varieties of dimension greater than one.90The key idea is to use B.~Mazur's notion of visibility, coupled with91explicit computations, to produce lower bounds on Shafarevich-Tate groups.92I also describe related computations that are used in an93upcoming joint paper with L.~Merel on94rational points of $X_0(N)$ over $\Q(\mu_p)$.9596E.~Artin conjectured in 1924 that the $L$-series associated to97a representation $\rho:\GQ\rightarrow\GL_2(\C)$ is entire.98K.~Buzzard and I verify this conjecture in eight new icosahedral cases;99the argument uses a theorem of K.~Buzzard and R.~Taylor along100with a computer computation, which is used to fill in a missing101mod-$5$ modularity result.102103Finally I turn to Serre's conjecture. I consider a family of104exceptional cases, and show that the conjecture appears to fail105as badly as possible. I also describe numerical experiments106towards a mod-$pq$ extension of Serre's conjecture.107108I expect to finish my thesis before May 2000.109\newpage110\item {\tfont Lectures on Serre's conjectures} (77 pages), with K.~Ribet:111This is an expository paper based on Ken Ribet's lectures at112the 1999 Park City Mathematics Institute; it will be113published in the IAS/Park City Mathematics Institute Lecture Series.114There are 9 lectures: Introduction to Serre's conjectures;115The weak and strong conjectures; The weight in Serre's conjecture;116Galois representations from modular forms; Introduction to level lowering;117Approaches to level lowering; Mazur's principle; Level lowering118without multiplicity one; Level lowering with multiplicity one;119Other directions. We have finished120all but the last three sections, and expect to finish121by early December, 1999.122123124\item {\tfont Empirical evidence for the Birch and125Swinnerton-Dyer conjecture for modular Jacobians126of genus~2 curves} (22 pages), with E.\thinspace{}V.~Flynn,127F.~Lepr\'{e}vost, E.\thinspace{}F.~Schaefer,128M.~Stoll, J.\thinspace{}L.~Wetherell:129We provide systematic numerical evidence for the BSD conjecture130in the case of dimension two.131This conjecture relates six quantities associated to132a Jacobian over the rational numbers. One of these133quantities is the size~$S$ of the Shafarevich-Tate group.134Unable to compute~$S$ directly, we compute the five other135quantities and solve for the conjectural value~$S_?$ of~$S$.136For all 32~curves considered, the real number~$S_?$137is very close to either~$1$,~$2$, or~$4$, and138agrees with the size of the 2-torsion of the139Shafarevich-Tate group, which we could compute.140This paper has been submitted.141142\item {\tfont Parity structures and generating143functions from Boolean rings \hspace{.2em}}(8 pages), with D.~Moulton:144Let~$S$ be a finite set and~$T$ be a subset of the power set of~$S$.145Call~$T$ a {\em parity structure} for~$S$ if, for146each subset~$b$ of~$S$ of odd size, the number of subsets of~$b$147that lie in~$T$ is even. We classify parity structures using generating148functions from a free boolean ring.149We also show that if~$T$ is a parity structure, then, for each150subset~$b$ of~$S$ of even size, the number of subsets of~$b$ of151odd size that lie in~$T$ is even. We then give several other152properties of parity structures and discuss a generalization.153This paper has been submitted.154155\item {\tfont Fallacies, Flaws, and156Flimflam \#92: An Inductive Fallacy}, (1 page)157with A.~Riskin: College Math.\ Journal 26:5 (1995), 382.158\end{enumerate}159160\vspace{1ex}161\begin{center}162\LARGE\bf163Work in progress164\end{center}165\begin{enumerate}166167\item {\tfont The modular forms database:}168This is an evolving collection of tables of modular eigenforms,169special values of $L$-functions, arithmetic invariants of170modular abelian varieties, and other data.171These tables are available at\\172{\tt http://shimura.math.berkeley.edu/\~{}was/Tables/}.173174175\item {\tfont Visibility of Shafarevich-Tate176groups of modular abelian varieties} (20 pages), with177A.~Agash\'{e}:178We study Mazur's notion of visibility179of Shafarevich-Tate of modular \nobreak{abelian} varieties,180and use it to verify the conjecture of Birch and Swinnerton-Dyer181for several specific abelian varieties. We also give evidence182that visibility is rare. The computations are done, but183we are still adding to the theoretical content of the paper184and polishing the presentation.185186\item {\tfont Component groups of optimal quotients of Jacobians} (16 pages):187Let~$A$ be an optimal quotient of~$J_0(N)$.188The main theorem of this paper gives a relationship between189the modular degree of~$A$ and the order of the component group190of~$A$. From this I deduce a computable formula for the191component group of any optimal \nobreak{quotient}192of~$J_0(N)$ at a prime of multiplicative193reduction. I then compute over one \nobreak{thousand}194examples leading me195to conjecture that the torsion and component groups of196quotients of $J_0(p)$, for~$p$ prime, are as simple as possible.197This paper is almost ready to be submitted.198199\item {\tfont Mod~$5$ approaches to modularity of icosahedral200Galois \nobreak{representations}} (16 pages), with K.~Buzzard:201Consider a continuous odd irreducible representation202$\rho:\mbox{\rm Gal}(\overline{\mathbf{Q}}/\mathbf{Q})203\rightarrow\mbox{\rm GL}_2(\mathbf{C}).$204A special case of a general conjecture of Artin205is that the $L$-function $L(\rho,s)$ associated to~$\rho$206is entire.207Buzzard and I give new examples of representations~$\rho$ that208satisfy this conjecture.209We obtained these examples by applying a recent210theorem of Buzzard and Taylor to a mod~$5$ reduction of~$\rho$,211combined with a computational verification of modularity of a212related mod~$5$ representation.213The computations of the paper are complete, and have been214mostly written up. We expect to finish the paper in time215for the Hot Topics workshop at MSRI in December, 1999.216217\item {\tfont Indexes of genus one curves} (10 pages):218I prove results about the complexity of genus one curves.219The greatest common divisor of the degrees of the fields in220which a curve of genus~$g$ over~$\mathbf{Q}$ has a rational221point divides $2g-2$; when $g=1$ this is no condition at all.222In the 1950s S.~Lang and J.~Tate asked whether, given a positive integer~$m$,223there exists a genus one curve so that the smallest number224field in which it has a rational point is of degree~$m$ over~$\mathbf{Q}$.225Using Euler systems, I prove this when~$m$ is not226divisible by~$4$. This paper is almost ready to be227submitted.228\end{enumerate}229230\end{document}231232233234235236