A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for
modular abelian varieties

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and
Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of
their conjecture in the good ordinary case to higher dimensional modular
abelian varieties over the rationals by constructing the p-adic L-function of a
modular abelian variety and showing that it satisfies the appropriate
interpolation property. This relies on a careful normalization of the p-adic
L-function, which we achieve by a comparison of periods. Our generalization
agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the
classical Birch Swinnerton-Dyer conjecture formulated by Tate in rank 0. We
describe the theoretical techniques used to formulate the conjecture and give
numerical evidence supporting the conjecture in the case when the modular
abelian variety is of dimension 2.