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

Research output: Contribution to journalArticlepeer-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 languageEnglish (US)
Pages (from-to)2397-2425
Number of pages29
JournalMathematics of Computation
Volume78
Issue number268
DOIs
StatePublished - Oct 2009

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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