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

Abstract

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank $0$ abelian varieties $A_f$ that are optimal quotients of $J_0(N)$ attached to newforms. We prove theorems about the ratio $L(A_f,1)/\Omega _{A_f}$, develop tools for computing with $A_f$, and gather data about certain arithmetic invariants of the nearly $20,000$ abelian varieties $A_f$ of level $\leq 2333$. Over half of these $A_f$ have analytic rank $0$, and for these we compute upper and lower bounds on the conjectural order of $\Sha (A_f)$. We find that there are at least $168$ such $A_f$ for which the Birch and Swinnerton-Dyer conjecture implies that $\Sha (A_f)$ is divisible by an odd prime, and we prove for $37$ of these that the odd part of the conjectural order of $\Sha (A_f)$ really divides $\#\Sha (A_f)$ by constructing nontrivial elements of $\Sha (A_f)$ 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.

Locations

• Mathematics of Computation - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF