On 2-selmer groups of twists after quadratic extension

Morgan, A. and Paterson, R. (2022) On 2-selmer groups of twists after quadratic extension. Journal of the London Mathematical Society, 105(2), pp. 1110-1166. (doi: 10.1112/jlms.12533)

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Abstract

Let E∕Q be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists Ed over a fixed quadratic extension K∕Q. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erdos–Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over Q, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich–Tate groups of the Ed over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of Gal(K∕Q) on the 2-Selmer group of Ed over K is trivial, and the Mordell–Weil group Ed(K) splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2- Selmer groups over K have non-trivial Gal(K∕Q)-action, illustrating that these previous results are genuinely statistical phenomena.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Paterson, Ross and Morgan, Dr Adam
Authors: Morgan, A., and Paterson, R.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of the London Mathematical Society
Publisher:Wiley
ISSN:0024-6107
ISSN (Online):1469-7750
Published Online:08 February 2022
Copyright Holders:Copyright © 2022 The Authors
First Published:First published in Journal of the London Mathematical Society 105(2): 1110-1166
Publisher Policy:Reproduced under a Creative Commons License

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