Now that the Shimura-Taniyama conjecture has been proved, the main outstanding problem in the field is the Birch and Swinnerton-Dyer conjecture (BSD conjecture), which ties together the arithmetic invariants of an elliptic curve. There is no general class of elliptic curves for which the full BSD conjecture is known. Approaches to the BSD conjecture that rely on congruences between modular forms are likely to require a deeper understanding of the analogous conjecture for higher-dimensional abelian varieties. As a first step, I have obtained theorems that make possible explicit computation of some of the arithmetic invariants of modular abelian varieties.

- The BSD conjecture
- The ratio
- The torsion subgroup
- Tamagawa numbers
- Upper bounds on
- Lower bounds on
- Motivation for considering abelian varieties