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Author: William A. Stein
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Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

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.