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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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
  • Email:
  • Michael Stoll
  • Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
  • Email:
  • Received by editor(s): January 23, 2011
  • Received by editor(s) in revised form: August 2, 2011
  • Published electronically: June 11, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 82 (2013), 513-529
  • MSC (2010): Primary 11G05; Secondary 14G05, 14G25, 14H52
  • DOI:
  • MathSciNet review: 2983034