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

Grigorov, Grigor; Jorza, Andrei; Patrikis, Stefan; Stein, William; Tarniţǎ, Corina

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

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.

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