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

Abstract

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)/Ω 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 ≤ 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 (A f ). We find that there are at least 168 such A f for which the Birch and Swinnerton-Dyer conjecture implies that (A f ) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of (A f ) really divides # (A f ) by constructing nontrivial elements of (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.

Modular Forms, a Computational Approach

“…It is also important for computing special values of the L-function L(f, s) at integer points in the critical strip. discusses higher-weight modular symbols and applies modular symbols to study Shafarevich-Tate groups (see also [Aga00]). Martin's thesis [Mar01] is about an attempt to study an analogue of analytic modular symbols for weight 1.…”

Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves

“…(1) In the case of 681b, one can alternatively use [CM00] and [AS05,App. ] instead of an explicit 3-descent to see that 9 | #X(E).…”

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

“…For 3185c1, the algorithm of Stein and Wuthrich [54] provides the upper bound of 2. In all twelve cases [17] (and the appendix of [2]) found visible non-trivial parts of X(Q, E) [5]. Since the order must be a square, #X(Q, E) must be exactly 25 in each case.…”