\documentclass[12pt]{article}12\newcommand{\doc}{Research Summary}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://www.math.berkeley.edu/\~{\mbox{}}was}}7\newcommand{\address}{2041 Francisco Street, \#14\\8Berkeley, CA 94709\\9USA}101112\hoffset=-.1\textwidth13\textwidth=1.2\textwidth14\voffset=-.1\textheight15\textheight=1.2\textheight1617\usepackage{fancyhdr,ifthen}18\pagestyle{fancy}19\cfoot{} % no footers (in pagestyle fancy)20% running left heading21\lhead{\bfseries\LARGE\em \noindent{}\hspace{-.2em}\myname{}22\hfill \doc \vspace{-.2ex}\\}23% running right heading24\newcommand{\spc}{.33ex}25%\rhead{\em {\small{\phone{}}} \hspace{\spc}$\bullet$\hspace{\spc} \email{}26% \hspace{\spc}$\bullet$\hspace{\spc} \www{}}27% adjust, because the header is now taller than usual.28\rhead{}29\setlength{\headheight}{7ex}303132\usepackage{amsmath}33\usepackage{amsthm}34\usepackage{amssymb}3536\newcommand{\Adual}{A^{\vee}}37\newcommand{\isom}{\cong}38\newcommand{\mysection}[1]{\section{#1}\vspace{-1.5ex}}39\newcommand{\mysubsection}[1]{\subsection{#1}\vspace{-1ex}}40\newcommand{\Q}{\mathbf{Q}}41\newcommand{\Z}{\mathbf{Z}}42\newcommand{\R}{\mathbf{R}}43\newcommand{\C}{\mathbf{C}}44\newcommand{\PGL}{\mbox{\rm PGL}}45\newcommand{\GL}{\mbox{\rm GL}}46\newcommand{\Div}{\mbox{\rm Div}}47\newcommand{\GQ}{G_\Q}48\newcommand{\ra}{\rightarrow}49\newcommand{\tensor}{\otimes}50\DeclareMathOperator{\tor}{tor}51\font\cyr=wncyr10 scaled \magstep 152\newcommand{\Sha}{\mbox{\cyr X}}53\font\cyrbig=wncyr10 scaled \magstep 254\newcommand{\BigSha}{\mbox{\cyrbig X}}55\newcommand{\intersect}{\cap}56\newcommand{\comment}[1]{}57\theoremstyle{plain}58\newtheorem{theorem}{Theorem}59\newtheorem{corollary}[theorem]{Corollary}60\newtheorem{conjecture}[theorem]{Conjecture}6162\begin{document}63\begin{center}64\LARGE\bf \doc65\end{center}66676869\mysection{Introduction}70My research program reflects the essential interplay between abstract71theory and explicit machine computation during72the latter half of the twentieth century;73it sits at the intersection of recent work of74B.~Mazur, K.~Ribet, R.~Taylor, and A.~Wiles75on Galois representations76with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre77on explicit computations involving modular abelian varieties.78My work on the Birch and Swinnerton-Dyer79conjecture for modular abelian varieties and search for new80examples of modular icosahedral Galois representations81has led me to discover and implement algorithms for82explicitly computing with modular forms.8384\mysection{Objectives}85The main outstanding problem in my field86is the conjecture of Birch and Swinnerton-Dyer (\nobreak{BSD} conjecture),87which ties together the constellation of88arithmetic invariants of an89elliptic curve.90There is still no general class of elliptic curves for which91the full BSD conjecture is known to hold.92Approaches to the BSD conjecture that rely on congruences93between modular forms are likely to94require a deeper understanding of the analogous95conjecture for modular abelian varieties, which are higher96dimensional analogues of elliptic curves.9798As a first step, I have obtained theorems that make possible99computation of some of the arithmetic invariants of100modular abelian varieties.101My objective is to find ways to explicitly compute all of102the arithmetic invariants.103Cremona has enumerated these invariants for the first104few thousand elliptic curves, and I am working to do the same105for abelian varieties.106It is hoped that this work will continue to107yield theoretical results.108I am also writing modular forms software that I hope will be used109by many mathematicians and have practical applications in110the development of elliptic curve cryptosystems.111112My long-range goal is to give a general hypothesis, valid for113infinitely many abelian varieties,114under which the full BSD conjecture holds.115My approach involves combining Euler system techniques of K.~Kato and116K.~Rubin with visibility and congruence ideas of Mazur and Ribet.117118119\mysection{Modular abelian varieties}120My primary objective is to verify the BSD conjecture for121specific modular abelian varieties, by using the122rich theory of their arithmetic.123124The BSD conjecture asserts that if~$A$ is a modular abelian125variety with $L(A,1)\neq 0$, then126$$\frac{L(A,1)}{\Omega_A} =127\frac{\#\Sha(A)\cdot\prod c_p}128{\# A(\Q)_{\tor}\cdot\#\Adual(\Q)_{\tor}}.$$129Here $A(\Q)_{\tor}$ is the group of rational torsion points on~$A$;130the Shafarevich-Tate group $\Sha(A)$ is a measure of the failure131of the local-to-global principle; the Tamagawa numbers~$c_p$ are the132orders of certain component groups associated to~$A$;133the real number~$\Omega_A$ is134the volume of~$A(\R)$ with respect to a basis of differentials having135everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.136137\mysubsection{The ratio $L(A,1)/\Omega_A$}138Extending Manin's work on elliptic curves, A.~Agash\'{e} and I139found a computable formula for the rational number $L(A,1)/\Omega_A$.140Using similar techniques, I hope to find computable formulas141for rational parts of special values of twists, and of $L$-functions142attached to forms of weight greater than two. I have143already computed $L(A,1)/\Omega_A$ for several thousand abelian144varieties, and hope to extend these computations.145146\mysubsection{The Tamagawa numbers $c_p$}147When~$A$ has semistable reduction at~$p$, I have148found a way to explicitly compute149the number~$c_p$, up to a power of~$2$.150I hope to find a way to compute~$c_p$ in the remaining cases.151152\mysubsection{Bounding $\#\BigSha$}153V.~Kolyvagin and K.~Kato154obtained upper bounds on~$\#\Sha(A)$.155To verify the full BSD conjecture for certain abelian156varieties, it is necessary is to make these bounds explicit.157Kolyvagin's bounds involve computations with Heegner points,158and Kato's involve a study of the Galois representations159associated to~$A$. I plan to carry out such160computations in many specific cases.161162One approach to showing that~$\Sha(A)$ is as large as predicted163by the BSD conjecture is suggested by Mazur's notion of164the visible part of~$\Sha(A)$.165Consider an abelian variety~$A$ that sits naturally166in the Jacobian $J_0(N)$ of the modular curve $X_0(N)$.167The {\em visible part} of $\Sha(A)$ is the collection of those168elements of $\Sha(A)$169that go to~$0$ under the natural map to $\Sha(J_0(N))$.170Cremona and Mazur observed that if an element of order~$p$171in~$\Sha(A)$ is visible,172then it is explained by a jump in the rank of Mordell-Weil,173in the sense that there is another abelian subvariety $B\subset J_0(N)$174such that $p \mid \#(A\intersect B)$ and~$B$ has many rational points.175I am trying to find the precise degree to which this observation176can be turned around: if there is177another abelian variety~$B$ with many rational points and178$p\mid \#( A\intersect B)$, then under what hypotheses179is there an element of~$\Sha(A)$ of order~$p$?180181182\mysection{Icosahedral Galois representations}183E.~Artin conjectured184that the $L$-series associated to any continuous irreducible185representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.186Recent exciting work of Taylor and others suggests that a complete proof187of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,188is on the horizon.189190By combining the main result of191a recent paper of K.~Buzzard and Taylor192with a computer computation,193Buzzard and I recently194proved that the icosahedral Artin representations of195conductor~$1376=2^5\cdot 43$ are modular.196If I can extend a congruence result of J.~Sturm, then197our method will yield several more examples.198These ongoing computations are laying a part of the199foundation necessary for a full proof of the Artin conjecture200for odd two-dimensional~$\rho$, as well as stimulating201the development of new algorithms for computing with modular forms202in characteristic~$\ell$.203204\end{document}205206