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\Large \bf Abstract
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{\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}\\
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First I describe how to compute spaces of modular forms of weight
at least two.
I then I describe 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 enough to compute the odd part of the
conjectural order of the Shafarevich-Tate group of any analytic
rank~$0$ optimal quotient of $J_0(N)$, with~$N$ square-free. I have
not determined the factor of~$2$, 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.
Next, I use these algorithms to 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 B.~Mazur's notion of
visibility, coupled with explicit computations, to produce lower
bounds on the Shafarevich-Tate group. To finish the proof of the
conjecture in these cases requires knowing upper bounds on the order
of this group. K.~Kato has constructed an Euler system that
is expected to give such a bound;
however, this bound has only been made explicit in the
case of elliptic curves.
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\noindent{}Professor Hendrik Lenstra\\
Dissertation Committee Chair
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