Summary: MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000--000
S 0025­5718(XX)0000­0
VISIBLE EVIDENCE FOR THE BIRCH AND
SWINNERTON­DYER CONJECTURE FOR MODULAR ABELIAN
VARIETIES OF ANALYTIC RANK ZERO
(WITH AN APPENDIX BY J. CREMONA AND B. MAZUR)
AMOD AGASHE AND WILLIAM STEIN
Abstract. This paper provides evidence for the Birch and Swinnerton­Dyer
conjecture for analytic rank 0 abelian varieties A f that are optimal quo­
tients of J 0 (N) attached to newforms. We prove theorems about the ratio
L(A f ,
1)/# A f , develop tools for computing with A f , and gather data about
certain arithmetic invariants of the nearly 20000 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 X(A f ). We find that
there are at least 168 such that the Birch and Swinnerton­Dyer Conjecture
implies that X(A f ) is divisible by an odd prime, and we prove for 37 of these
that the odd part of the conjectural order of X(A f ) really divides #X(A f )
by constructing nontrivial elements of X(A f ) using visibility theory. We also

Source: Agashe, Amod - Department of Mathematics, Florida State University

Collections: Mathematics