Explicit isogeny descent on elliptic curves

Miller, Robert; Stoll, Michael

doi:10.1090/S0025-5718-2012-02619-6

Explicit isogeny descent on elliptic curves
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by Robert L. Miller and Michael Stoll PDF

Math. Comp. 82 (2013), 513-529 Request permission

Abstract:

In this note, we consider an $\ell$-isogeny descent on a pair of elliptic curves over $\mathbb {Q}$. We assume that $\ell > 3$ is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite-dimensional $\mathbb {F}_\ell$-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the $\ell$-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.

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