#
Visible Evidence
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

Mathematics of
Computation, Vol 74, Number 249, pages 455-484.

**Abstract**

This paper provides evidence for the Birch and Swinnerton-Dyer
conjecture for 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_{Af}, develop tools for computing
with A_{f}, and gather data about the arithmetic invariants
of the nearly 20000
abelian varieties A_{f} of level < 2334.
Over half
of these A_{f} have rank 0, and for these
we compute upper and lower bounds on the
conjectural order of Sha(A_{f}).
We find that there are 168 such that
Sha(A_{f}) 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(A_{f}) really divides Sha(A_{f}) by constructing nonzero
elements of Sha(A_{f}) 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
A(R)^0.