Computational verification of the birch and swinnerton-dyer conjecture for individual elliptic curves

Grigor Grigorov; Andrei Jorza; Stefan Patrikis; William A. Stein; Corina E. Tarnita

doi:10.1090/S0025-5718-09-02253-4

Computational verification of the birch and swinnerton-dyer conjecture for individual elliptic curves

Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein, Corina E. Tarnita

Princeton University

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specic elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the E-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely veries the full conjecture for these curves up to the primes excluded above.

Original language	American English
Pages (from-to)	2397-2425
Number of pages	29
Journal	Mathematics of Computation
Volume	78
Issue number	268
DOIs	https://doi.org/10.1090/S0025-5718-09-02253-4
State	Published - Oct 2009

ASJC Scopus subject areas

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

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abstract = "We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specic elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the E-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely veries the full conjecture for these curves up to the primes excluded above.",

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AU - Grigorov, Grigor

AU - Jorza, Andrei

AU - Patrikis, Stefan

AU - Stein, William A.

AU - Tarnita, Corina E.

PY - 2009/10

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N2 - We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specic elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the E-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely veries the full conjecture for these curves up to the primes excluded above.

AB - We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specic elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the E-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely veries the full conjecture for these curves up to the primes excluded above.

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Computational verification of the birch and swinnerton-dyer conjecture for individual elliptic curves

Abstract

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