Visible Evidence for the Birch and Swinnerton-Dyer Conjecture for Rank 0 Modular Abelian Varieties

William A. Stein

Amod Agashe

With an Appendix by Barry Mazur and John Cremona

This paper has appeared in
Mathematics of Computation, Vol 74, Number 249, pages 455-484.


This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af,1)/OmegaAf, develop tools for computing with Af, and gather data about the arithmetic invariants of the nearly 20000 abelian varieties Af of level < 2334. Over half of these Af have rank 0, and for these we compute upper and lower bounds on the conjectural order of Sha(Af). We find that there are 168 such that Sha(Af) should be divisible by an odd prime, and we prove for 39 of these 168 that the odd part of the conjectural order of Sha(Af) really divides Sha(Af) by constructing nonzero elements of Sha(Af) using visibility theory.

The appendix, by John Cremona and Barry Mazur, fills in gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.

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Note that the above version below fixes some typos that (will?) appear in the published version: