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Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

Authors: E. Victor Flynn, Franck Leprévost, Edward F. Schaefer, William A. Stein, Michael Stoll and Joseph L. Wetherell
Journal: Math. Comp. 70 (2001), 1675-1697
MSC (2000): Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40, 14H45
Published electronically: May 11, 2001
MathSciNet review: 1836926
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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.

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Additional Information

E. Victor Flynn
Affiliation: Department of Mathematical Sciences, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, England

Franck Leprévost
Affiliation: Université Grenoble I, Institut Fourier, BP 74, F-38402 Saint Martin d’Hères Cedex, France

Edward F. Schaefer
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053

William A. Stein
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138

Michael Stoll
Affiliation: Mathematisches Institut der Heinrich-Heine-Universität, Universitätsstr. 1, 40225 Düsseldorf, Germany

Joseph L. Wetherell
Affiliation: Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089-1113

Keywords: Birch and Swinnerton-Dyer conjecture, genus~2, Jacobian, modular abelian variety
Received by editor(s): August 16, 1999
Published electronically: May 11, 2001
Additional Notes: The first author thanks the Nuffield Foundation (Grant SCI/180/96/71/G) for financial support.
The second author did some of the research at the Max-Planck Institut für Mathematik and the Technische Universität Berlin.
The third author thanks the National Security Agency (Grant MDA904-99-1-0013).
The fourth author was supported by a Sarah M. Hallam fellowship.
The fifth author did some of the research at the Max-Planck-Institut für Mathematik.
The sixth author thanks the National Science Foundation (Grant DMS-9705959). The authors had useful conversations with John Cremona, Qing Liu, Karl Rubin and Peter Swinnerton-Dyer. The authors are grateful to Xiangdong Wang and Michael Müller for making data available to them and to the referee for helpful suggestions.
Article copyright: © Copyright 2001 American Mathematical Society