% $Header: /home/was/papers/thesis/RCS/preface.tex,v 1.4 2000/05/10 20:32:33 was Exp $12% intro.tex3\chapter*{Preface}4\markboth{}{}56\addcontentsline{toc}{chapter}{Preface}78\begin{quote}9The object of numerical computation is theoretical advance.\\10\hfill --{\em A.\thinspace{}O.\thinspace{}L. Atkin, see~\cite{birch:atkin}}11\end{quote}1213The definition of the spaces of modular14forms as functions on the upper half plane satisfying15a certain equation is very abstract. The definition16of the Hecke operators\index{Hecke operators} even more so.17Nevertheless, one wishes to carry out explicit18investigations into these objects.1920We are fortunate that we now have methods available that allow us to21transform the vector space of cusp forms of given weight and level22into a concrete object, which can be explicitly computed. We have the23work of Atkin-Lehner, Birch, Swinnerton-Dyer, Manin, Merel, and many24others to thank for this (see \cite{antwerpiv, cremona:algs,25merel:1585}). For example, the Eichler-Selberg trace formula, as26extended in \cite{hijikata:trace}, can be used to compute27characteristic polynomials of Hecke operators.\index{Hecke operators} One can compute Hecke operators\index{Hecke operators}28using Brandt matrices and quaternion algebras29\cite{kohel:hecke, pizer:alg}; another closely related method involves30the module of enhanced supersingular elliptic31curves~\cite{mestre:graphs}. In the course of computing large tables32of invariants of elliptic curves in \cite{cremona:algs}, Cremona33demonstrated the power of systematic computation using modular34symbols.3536Various methods often must be used in concert to obtain information37about the package of invariants attached to a modular form. For38example, computing orders of component groups of optimal quotients of39$J_0(N)$ involves computations on the module of supersingular elliptic40curves combined with modular symbols techniques (see41Chapter~\ref{chap:compgroups}).4243Chapter~\ref{chap:bsd} is an attempt to systematically prove the Birch44and Swinnerton-Dyer conjecture for a certain finite list of rank-$0$45quotients of $J_0(N)$ that have nontrivial Shafarevich-Tate groups.46The key idea is to use Barry Mazur's notion of visibility, coupled47with explicit computations, to produce lower bounds on the48Shafarevich-Tate group. I have not finished the proof of the49conjecture in these examples; this would require computing explicit50upper bounds on the order of this group. However, I obtain explicit51formulas and data that will be helpful in further investigations.5253The following three chapters describe the algorithms used in54Chapter~\ref{chap:bsd}, along with generalizations to eigenforms on55$\Gamma_1(N)$ of integral weight greater than two. I have used these56algorithms to investigate the Artin57Conjecture~\cite{buzzard-stein:artin}, Serre's conjecture, and many58other problems not described in this thesis. I have implemented59most of the algorithms60that are described in Chapters~\ref{chap:modsym}--\ref{chap:compgroups}61in both \magma{} and {\sf C++}; this implementation should be available62in the standard release of \magma{} in versions 2.7 and greater.63\vspace{1ex}6465\par\noindent66William A. Stein\\67University of California, Berkeley686970717273747576