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\begin{center}
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\Large \bf Abstract
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\vspace{-1ex}
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\begin{center}
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{\bf Explicit approaches to modular abelian varieties}\\
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by\vspace{1ex}\\
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William Arthur Stein\vspace{1ex}\\
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Doctor of Philosophy in Mathematics\vspace{1ex}\\
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University of California at Berkeley\vspace{1ex}\\
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Professor Hendrik Lenstra, Chair\vspace{1ex}\\
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I investigate the Birch and Swinnerton-Dyer conjecture, which ties
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together the constellation of invariants attached to an abelian variety.
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I attempt to verify this conjecture for certain specific modular abelian
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varieties of dimension greater than one. The key idea is to use
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Barry Mazur's notion of visibility, coupled with explicit computations,
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to produce lower bounds on the Shafarevich-Tate group. I have
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not finished the
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proof of the conjecture in these examples; this would
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require computing explicit upper bounds on the order of this group.
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I next describe how to compute in
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spaces of modular forms of weight at least two.
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I give an integrated package for computing, in many
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cases, the following invariants of a modular abelian variety: the
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modular degree, the rational part of the special value of the
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$L$-function, the order of the component group at primes of
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multiplicative reduction, the period lattice, upper and lower bounds
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on the torsion subgroup, and the real volume. Taken together, these
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algorithms are frequently sufficient to compute the odd part of the
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conjectural order of the Shafarevich-Tate group of an analytic
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rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free. I have
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not determined the exact structure of the component
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group, the order of the component group at primes whose square divides
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the level, or the exact order of the torsion subgroup in all cases.
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However, I do provide generalizations of some of the above algorithms
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to higher weight forms with nontrivial character.
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\vfill
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\hfill\begin{minipage}[l]{3in}
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\underline{\mbox{}\hspace{3in}\mbox{}}\\
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\noindent{}Professor Hendrik Lenstra\\
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Dissertation Committee Chair
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\end{minipage}
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