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\begin{center}
\Large \bf Abstract
\end{center}
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\begin{center}
{\bf Explicit approaches to modular abelian varieties}\\
by\vspace{1ex}\\
William Arthur Stein\vspace{1ex}\\
Doctor of Philosophy in Mathematics\vspace{1ex}\\
University of California at Berkeley\vspace{1ex}\\
Professor Hendrik Lenstra, Chair\vspace{1ex}\\
\end{center}
I investigate the Birch and Swinnerton-Dyer conjecture, which ties
together the constellation of invariants attached to an abelian variety.
I attempt to verify this conjecture for certain specific modular abelian
varieties of dimension greater than one. The key idea is to use
Barry Mazur's notion of visibility, coupled with explicit computations,
to produce lower bounds on the Shafarevich-Tate group. I have
not finished the
proof of the conjecture in these examples; this would
require computing explicit upper bounds on the order of this group.
I next describe how to compute in
spaces of modular forms of weight at least two.
I give an integrated package for computing, in many
cases, the following invariants of a modular abelian variety: the
modular degree, the rational part of the special value of the
$L$-function, the order of the component group at primes of
multiplicative reduction, the period lattice, upper and lower bounds
on the torsion subgroup, and the real volume. Taken together, these
algorithms are frequently sufficient to compute the odd part of the
conjectural order of the Shafarevich-Tate group of an analytic
rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free. I have
not determined the exact structure of the component
group, the order of the component group at primes whose square divides
the level, or the exact order of the torsion subgroup in all cases.
However, I do provide generalizations of some of the above algorithms
to higher weight forms with nontrivial character.
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\noindent{}Professor Hendrik Lenstra\\
Dissertation Committee Chair
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