Morgan, A. (2019) Quadratic twists of abelian varieties and disparity in Selmer ranks. Algebra and Number Theory, 13(4), pp. 839-899. (doi: 10.2140/ant.2019.13.839)
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Abstract
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarized abelian variety over a number field. Specifically, we determine the proportion of twists having odd (respectively even) 2-Selmer rank. This generalizes work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree and we describe both.
| Item Type: | Articles |
|---|---|
| Additional Information: | This research is supported by EPSRC grant EP/M016846. |
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Morgan, Dr Adam |
| Authors: | Morgan, A. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | Algebra and Number Theory |
| Publisher: | Mathematical Sciences Publishers |
| ISSN: | 1937-0652 |
| ISSN (Online): | 1944-7833 |
| Published Online: | 06 May 2019 |
| Copyright Holders: | Copyright © 2019 Mathematical Sciences Publishers |
| First Published: | First published in Algebra and Number Theory 13(4);839-899 |
| Publisher Policy: | Reproduced in accordance with the publisher copyright policy |
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