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\Large \bf Abstract
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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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First I describe how to compute spaces of modular forms of weight
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at least two.
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I then I describe 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 enough to compute the odd part of the
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conjectural order of the Shafarevich-Tate group of any 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 factor of~$2$, 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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Next, I use these algorithms to investigate the Birch and
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Swinnerton-Dyer conjecture, which ties together the constellation of
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invariants attached to an abelian variety. I attempt to verify this
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conjecture for certain specific modular abelian varieties of dimension
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greater than one. The key idea is to use B.~Mazur's notion of
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visibility, coupled with explicit computations, to produce lower
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bounds on the Shafarevich-Tate group. To finish the proof of the
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conjecture in these cases requires knowing upper bounds on the order
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of this group. K.~Kato has constructed an Euler system that
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is expected to give such a bound;
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however, this bound has only been made explicit in the
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case of elliptic curves.
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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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