Empirical evidence for the Birch and
Swinnerton-Dyer conjectures for
modular Jacobians of genus 2 curves
E. Victor Flynn
and Franck Leprevost
and Edward F. Schaefer
and William A. Stein
and Michael Stoll
and Joeseph L. Wetherell
Abstract
This paper provides empirical evidence for the Birch and
Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves.
The second of these conjectures relates six quantities associated to
a Jacobian over the rational numbers. One of these
six quantities is
the size of the Shafarevich-Tate group.
Unable to compute that, we
computed the five other quantities and solved for the last one. In
all 32 cases, the result is very close to an integer that is a power
of 2. In addition, this power of 2 agrees with the size of the
2-torsion of the Shafarevich-Tate group, which we could compute.
My local copy of the paper is evidence3.pdf (or evidence3.dvi).
A nicer version of this paper has appeared in
Mathematics of Computation.