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Mathematics of Computation

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Explicit isogeny descent on elliptic curves

Authors: Robert L. Miller and Michael Stoll
Journal: Math. Comp. 82 (2013), 513-529
MSC (2010): Primary 11G05; Secondary 14G05, 14G25, 14H52
Published electronically: June 11, 2012
MathSciNet review: 2983034
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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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Additional Information

Robert L. Miller
Affiliation: Warwick Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom – and – The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720-5070
Address at time of publication: Quid, Inc., 733 Front Street, C1A, San Francisco, California 94111

Michael Stoll
Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Received by editor(s): January 23, 2011
Received by editor(s) in revised form: August 2, 2011
Published electronically: June 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.