Elliptic curves with p-Selmer growth for all p

Bartel, A. (2013) Elliptic curves with p-Selmer growth for all p. Quarterly Journal of Mathematics, 64(4), pp. 947-954. (doi: 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: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bartel, Professor Alex
• Authors: Bartel, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Quarterly Journal of Mathematics
• Publisher: Oxford University Press
• ISSN: 0033-5606
• ISSN (Online): 1464-3847
• Published Online: 08 November 2012
• Copyright Holders: Copyright © 2012 Oxford University Press
• First Published: First published in Quarterly Journal of Mathematics 64(4):947-954
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher