\comment{1$Header: /home/was/papers/thesis/RCS/abstract.tex,v 1.6 2000/05/11 04:41:11 was Exp $23$Log: abstract.tex,v $4Revision 1.6 2000/05/11 04:41:11 was5to make hendrik's ver.67Revision 1.5 2000/05/10 18:38:53 was8touched up wording910Revision 1.4 2000/05/10 18:26:51 was11Added $Log: abstract.tex,v $12Added Revision 1.6 2000/05/11 04:41:11 was13Added to make hendrik's ver.14Added15Added Revision 1.5 2000/05/10 18:38:53 was16Added touched up wording17Added181920}2122\setcounter{page}{1} % as per thesis guidelines23\begin{center}24\Large \bf Abstract25\end{center}26\vspace{-1ex}27\begin{center}28{\bf Explicit approaches to modular abelian varieties}\\29by\vspace{1ex}\\30William Arthur Stein\vspace{1ex}\\31Doctor of Philosophy in Mathematics\vspace{1ex}\\32University of California at Berkeley\vspace{1ex}\\33Professor Hendrik Lenstra, Chair\vspace{1ex}\\34\end{center}353637I investigate the Birch and Swinnerton-Dyer conjecture, which ties38together the constellation of invariants attached to an abelian variety.39I attempt to verify this conjecture for certain specific modular abelian40varieties of dimension greater than one. The key idea is to use41Barry Mazur's notion of visibility, coupled with explicit computations,42to produce lower bounds on the Shafarevich-Tate group. I have43not finished the44proof of the conjecture in these examples; this would45require computing explicit upper bounds on the order of this group.4647I next describe how to compute in48spaces of modular forms of weight at least two.49I give an integrated package for computing, in many50cases, the following invariants of a modular abelian variety: the51modular degree, the rational part of the special value of the52$L$-function, the order of the component group at primes of53multiplicative reduction, the period lattice, upper and lower bounds54on the torsion subgroup, and the real volume. Taken together, these55algorithms are frequently sufficient to compute the odd part of the56conjectural order of the Shafarevich-Tate group of an analytic57rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free. I have58not determined the exact structure of the component59group, the order of the component group at primes whose square divides60the level, or the exact order of the torsion subgroup in all cases.61However, I do provide generalizations of some of the above algorithms62to higher weight forms with nontrivial character.6364\vfill6566\hfill\begin{minipage}[l]{3in}67\underline{\mbox{}\hspace{3in}\mbox{}}\\68\noindent{}Professor Hendrik Lenstra\\69Dissertation Committee Chair70\end{minipage}71727374757677