1I investigate the Birch and Swinnerton-Dyer conjecture, which ties
2together the constellation of invariants attached to an abelian variety.
3I attempt to verify this conjecture for certain specific modular abelian
4varieties of dimension greater than one. The key idea is to use
5Barry Mazur's notion of visibility, coupled with explicit computations,
6to produce lower bounds on the Shafarevich-Tate group. I have
7not finished the
8proof of the conjecture in these examples; this would
9require computing explicit upper bounds on the order of this group.
11I next describe how to compute in
12spaces of modular forms of weight at least two.
13I give an integrated package for computing, in many
14cases, the following invariants of a modular abelian variety: the
15modular degree, the rational part of the special value of the
16-function, the order of the component group at primes of
17multiplicative reduction, the period lattice, upper and lower bounds
18on the torsion subgroup, and the real volume. Taken together, these
19algorithms are frequently sufficient to compute the odd part of the
20conjectural order of the Shafarevich-Tate group of an analytic
21rank optimal quotient of , with square-free. I have
22not determined the exact structure of the component
23group, the order of the component group at primes whose square divides
24the level, or the exact order of the torsion subgroup in all cases.
25However, I do provide generalizations of some of the above algorithms
26to higher weight forms with nontrivial character.