for the Birch and Swinnerton-Dyer
Conjecture for Rank 0 Modular Abelian Varieties
With an Appendix by Barry Mazur and John Cremona
This paper has appeared in
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.
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.
shacomp_v13.dvi shacomp_v13.tex mcom-l.cls
shacomp_v13.ps shacomp_v13.pdf html
Note that the above version below fixes some typos that (will?) appear in
the published version:
- We actually do prove nontriviality of Sha for 794G and 817E, so those entries
should be starred in the big table.
- In section 4.2 of our visible evidence for BSD paper,
in the second paragraph and Lem 4.4, the A(R) should really be