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24\begin{center}
25\Large \bf Abstract
26\end{center}
27\vspace{-1ex}
28\begin{center}
29{\bf Explicit approaches to modular abelian varieties}\\
30by\vspace{1ex}\\
31William Arthur Stein\vspace{1ex}\\
32Doctor of Philosophy in Mathematics\vspace{1ex}\\
33University of California at Berkeley\vspace{1ex}\\
34Professor Hendrik Lenstra, Chair\vspace{1ex}\\
35\end{center}
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38I investigate the Birch and Swinnerton-Dyer conjecture, which ties
39together the constellation of invariants attached to an abelian variety.
40I attempt to verify this conjecture for certain specific modular abelian
41varieties of dimension greater than one.  The key idea is to use
42Barry Mazur's notion of visibility, coupled with explicit computations,
43to produce lower bounds on the Shafarevich-Tate group.  I have
44not finished the
45proof of the conjecture in these examples; this would
46require computing explicit upper bounds on the order of this group.
47
48I next describe how to compute in
49spaces of modular forms of weight at least two.
50I give an integrated package for computing, in many
51cases, the following invariants of a modular abelian variety: the
52modular degree, the rational part of the special value of the
53$L$-function, the order of the component group at primes of
54multiplicative reduction, the period lattice, upper and lower bounds
55on the torsion subgroup, and the real volume.  Taken together, these
56algorithms are frequently sufficient to compute the odd part of the
57conjectural order of the Shafarevich-Tate group of an analytic
58rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free.  I have
59not determined the exact structure of the component
60group, the order of the component group at primes whose square divides
61the level, or the exact order of the torsion subgroup in all cases.
62However, I do provide generalizations of some of the above algorithms
63to higher weight forms with nontrivial character.
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67\hfill\begin{minipage}[l]{3in}
68\underline{\mbox{}\hspace{3in}\mbox{}}\\
69\noindent{}Professor Hendrik Lenstra\\
70Dissertation Committee Chair
71\end{minipage}
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