| Affiliation: | Department of Mathematical Sciences, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, England ; Université Grenoble I, Institut Fourier, BP 74, F-38402 Saint Martin d'Hères Cedex, France ; Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053 ; Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138 ; Mathematisches Institut der Heinrich-Heine-Universität, Universitätsstr.~1, 40225 Düsseldorf, Germany ; Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089-1113 |
| 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. |
