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Elliptic curves with p-Selmer growth for all p

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Bartel, Alex (2012) Elliptic curves with p-Selmer growth for all p. Quarterly Journal of Mathematics, Volume 64 (Number 4). pp. 947-954. doi:10.1093/qmath/has030

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Official URL: http://dx.doi.org/10.1093/qmath/has030

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Abstract

It is known that, for every elliptic curve over ℚ, there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the size of the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves E/ℚ whose 2-Selmer group grows in size in every bi-quadratic extension, and such that, moreover, for any odd prime p, the size of the p-Selmer group grows in every D2p-extension and every elementary abelian p-extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit this behaviour. We also discuss generalizations to other Galois groups.

Item Type: Journal Article
Divisions: Faculty of Science > Mathematics
Journal or Publication Title: Quarterly Journal of Mathematics
Publisher: Oxford University Press
ISSN: 0033-5606
Official Date: 8 November 2012
Dates:
DateEvent
8 November 2012Published
Volume: Volume 64
Number: Number 4
Page Range: pp. 947-954
DOI: 10.1093/qmath/has030
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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