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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Selmer companion curves


Authors: Barry Mazur and Karl Rubin
Journal: Trans. Amer. Math. Soc. 367 (2015), 401-421
MSC (2010): Primary 11G056; Secondary 11G40, 11G07
DOI: https://doi.org/10.1090/S0002-9947-2014-06114-X
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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Additional Information

Barry Mazur
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
MR Author ID: 121915
ORCID: 0000-0002-1748-2953
Email: mazur@math.harvard.edu

Karl Rubin
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697
MR Author ID: 151435
Email: krubin@math.uci.edu

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.