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

Abstract. This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quo-tients of J0(N) attached to newforms. We prove theorems about the ratio L(Af; 1)=Af, develop tools for computing with Af, and gather data about certain arithmetic invariants of the nearly 20000 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 ¯nd that there are at least 168 such that the Birch and Swinnerton-Dyer Conjecture implies that X(Af) is divisible by an odd prime, and we prove for 39 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, ¯lls in some gaps in the theoretical discussion in their paper on visibility o