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Selmer companion curves

Authors: Barry Mazur and Karl Rubin
Journal: Trans. Amer. Math. Soc. 367 (2015), 401-421
Published electronically: September 4, 2014
MathSciNet review: 3271266
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Abstract: We say that two elliptic curves $ E_1, E_2$ over a number field $ K$ are $ n$-Selmer companions for a positive integer $ n$ if for every quadratic character $ \chi $ of $ K$, there is an isomorphism $ \operatorname {Sel}_n(E_1^\chi /K) \cong \operatorname {Sel}_n(E_2^\chi /K)$ between the $ n$-Selmer groups of the quadratic twists $ E_1^\chi $, $ E_2^\chi $. We give sufficient conditions for two elliptic curves to be $ n$-Selmer companions, and give a number of examples of non-isogenous pairs of companions.

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Barry Mazur
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Karl Rubin
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697

Received by editor(s): August 18, 2012
Received by editor(s) in revised form: February 28, 2013
Published electronically: September 4, 2014
Additional Notes: This material is based upon work supported by the National Science Foundation under grants DMS-1065904 and DMS-0968831
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.