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Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

Part of: Number theory Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 November 2011

Robert L. Miller
Robert L. Miller*
Affiliation:
Warwick Mathematics Institute Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley CA 94720-5070, USA

Abstract

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We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 11-04: Explicit machine computation and programs (not the theory of computation or programming)

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 327 - 350
Copyright
Copyright © London Mathematical Society 2011

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