Numerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecture

Abstract

Let be an elliptic curve and a finite Galois extension with group G. We write E_K for the base change of E and consider the equivariant Tamagawa number conjecture for the pair (h ¹(E_K )(1),). This conjecture is an equivariant refinement of the Birch and Swinnerton-Dyer conjecture for E/K. For almost all primes l, we derive an explicit formulation of the conjecture that makes it amenable to numerical verifications. We use this to provide convincing numerical evidence in favor of the conjecture.

2000 AMS Subject Classification:

• 11G40
• 14G10
• 11G05

Keywords:

• Birch-Swinnerton-Dyer conjecture
• Equivariant Tamagawa Number conjecture

ACKNOWLEDGMENTS

I am very grateful to David Burns and Tom Fisher for their interest and helpful discussions. I also would like to thank the referee for his careful reading of the manuscript and many valuable comments.

Notes

¹Available from http://www.mathematik.uni-muenchen.de/ bley/pub.html.

²Available online at http://www.mathematik.uni-muenchen.de/ bley/pub.html

³Available online at http://www.mathematik.uni-muenchen.de/ bley/pub.html