Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af , 1)/ΩAf , develop tools for computing with Af , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of X(Af ). We find that there are at least 168 such Af for which the Birch and Swinnerton-Dyer conjecture implies that X(Af ) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of X(Af ) really divides #X(Af ) by constructing nontrivial elements of X(Af ) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.