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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
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, Engineering and Medicine > Science > Mathematics | ||||
| Journal or Publication Title: | Quarterly Journal of Mathematics | ||||
| Publisher: | Oxford University Press | ||||
| ISSN: | 0033-5606 | ||||
| Official Date: | 8 November 2012 | ||||
| Dates: |
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| 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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