Objectives

The main outstanding problem in my field is the conjecture of Birch and Swinnerton-Dyer (BSD conjecture), which ties together the constellation of arithmetic invariants of an elliptic curve. There is still no general class of elliptic curves for which the full BSD conjecture is known to hold. Approaches to the BSD conjecture that rely on congruences between modular forms are likely to require a deeper understanding of the analogous conjecture for modular abelian varieties, which are higher dimensional analogues of elliptic curves.

As a first step, I have obtained theorems that make possible computation of some of the arithmetic invariants of modular abelian varieties. My objective is to find ways to explicitly compute all of the arithmetic invariants. Cremona has enumerated these invariants for the first few thousand elliptic curves, and I am working to do the same for abelian varieties. I am also writing modular forms software that I hope will be used by many mathematicians and have practical applications in the development of elliptic curve cryptosystems.

My long-range theoretical goal is to give a general hypothesis, valid for infinitely many abelian varieties, under which the full BSD conjecture holds. My approach involves combining Euler system techniques of K. Kato and K. Rubin with visibility and congruence ideas of Mazur and Ribet.