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Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch andSwinnerton-Dyer conjecture for elliptic curves. We provide a generalization oftheir conjecture in the good ordinary case to higher dimensional modularabelian varieties over the rationals by constructing the p-adic L-function of amodular abelian variety and showing that it satisfies the appropriateinterpolation property. This relies on a careful normalization of the p-adicL-function, which we achieve by a comparison of periods. Our generalizationagrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and theclassical Birch Swinnerton-Dyer conjecture formulated by Tate in rank 0. Wedescribe the theoretical techniques used to formulate the conjecture and givenumerical evidence supporting the conjecture in the case when the modularabelian variety is of dimension 2.

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