I 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 Barry Mazur's notion of visibility, coupled with explicit computations, to produce lower bounds on the Shafarevich-Tate group. I have not finished the proof of the conjecture in these examples; this would require computing explicit upper bounds on the order of this group.
I next describe how to compute in
spaces of modular forms of weight at least two.
I give 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 measure. Taken together, these
algorithms are frequently enough to compute the odd part of the
conjectural order of the Shafarevich-Tate group of an 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.
Click on one of the following links to obtain a typeset version of my thesis.