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Mathematics of Computation

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Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves


Authors: Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein and Corina Tarnita
Journal: Math. Comp. 78 (2009), 2397-2425
MSC (2000): Primary 11Y99
DOI: https://doi.org/10.1090/S0025-5718-09-02253-4
Published electronically: June 8, 2009
MathSciNet review: 2521294
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Abstract: We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over  $ \mathbb{Q}$ of analytic ranks 0 and $ 1$. We apply our techniques to show that if $ E$ is a non-CM elliptic curve over  $ \mathbb{Q}$ of conductor $ \leq 1000$ and rank 0 or $ 1$, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the $ L$-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 verifies the full conjecture for these curves up to the primes excluded above.


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Additional Information

Grigor Grigorov
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Andrei Jorza
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544-1000

Stefan Patrikis
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544-1000

William A. Stein
Affiliation: Department of Mathematics, University of Washington, Seattle, Box 354350, Seattle, Washington 98195-4350

Corina Tarnita
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

DOI: https://doi.org/10.1090/S0025-5718-09-02253-4
Received by editor(s): June 30, 2005
Received by editor(s) in revised form: October 30, 2008
Published electronically: June 8, 2009
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. 0400386.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.