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

Jennifer S. Balakrishnan, J. Steffen Müller, William A. Stein
2015 Mathematics of Computation  
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
more » ... comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, and Teitelbaum in dimension 1 and the classical Birch and 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. Conjecture 1.1 (BSD conjecture for abelian varieties). Let A be an abelian variety of dimension g over a number field K, and let A ∨ be its dual. Then the Mordell-Weil rank r of A(K) is equal to the analytic rank ord s=1 L(A, s) of A and lim s→1
doi:10.1090/mcom/3029 fatcat:7udy5hw2rvdytbjhug7p6of2qm