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3\Large \bf Abstract
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7{\bf Explicit approaches to modular abelian varieties}\\
8by\vspace{1ex}\\
9William Arthur Stein\vspace{1ex}\\
10Doctor of Philosophy in Mathematics\vspace{1ex}\\
11University of California at Berkeley\vspace{1ex}\\
12Professor Hendrik Lenstra, Chair\vspace{1ex}\\
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15First I describe how to compute spaces of modular forms of weight
16at least two.
17I then I describe an integrated package for computing, in many
18cases, the following invariants of a modular abelian variety: the
19modular degree, the rational part of the special value of the
20$L$-function, the order of the component group at primes of
21multiplicative reduction, the period lattice, upper and lower bounds
22on the torsion subgroup, and the real volume.  Taken together, these
23algorithms are frequently enough to compute the odd part of the
24conjectural order of the Shafarevich-Tate group of any analytic
25rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free.  I have
26not determined the factor of~$2$, the exact structure of the component
27group, the order of the component group at primes whose square divides
28the level, or the exact order of the torsion subgroup in all cases.
29However, I do provide generalizations of some of the above algorithms
30to higher weight forms with nontrivial character.
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32Next, I use these algorithms to investigate the Birch and
33Swinnerton-Dyer conjecture, which ties together the constellation of
34invariants attached to an abelian variety.  I attempt to verify this
35conjecture for certain specific modular abelian varieties of dimension
36greater than one.  The key idea is to use B.~Mazur's notion of
37visibility, coupled with explicit computations, to produce lower
38bounds on the Shafarevich-Tate group.  To finish the proof of the
39conjecture in these cases requires knowing upper bounds on the order
40of this group. K.~Kato has constructed an Euler system that
41is expected to give such a bound;
42however, this bound has only been made explicit in the
43case of elliptic curves.
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48\noindent{}Professor Hendrik Lenstra\\
49Dissertation Committee Chair
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