Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
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- by Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein and Corina Tarniţǎ PDF
- Math. Comp. 78 (2009), 2397-2425 Request permission
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.References
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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
- MR Author ID: 876071
- Stefan Patrikis
- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544-1000
- MR Author ID: 876004
- William A. Stein
- Affiliation: Department of Mathematics, University of Washington, Seattle, Box 354350, Seattle, Washington 98195-4350
- MR Author ID: 679996
- Corina Tarniţǎ
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- 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.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 2397-2425
- MSC (2000): Primary 11Y99
- DOI: https://doi.org/10.1090/S0025-5718-09-02253-4
- MathSciNet review: 2521294