Vol. 13, No. 4, 2019

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Quadratic twists of abelian varieties and disparity in Selmer ranks

Adam Morgan

Vol. 13 (2019), No. 4, 839–899
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.

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Keywords
Abelian varieties, Selmer groups, quadratic twist, ranks, Shafarevich–Tate group
Mathematical Subject Classification 2010
Primary: 11G10
Milestones
Received: 1 December 2017
Revised: 2 November 2018
Accepted: 23 January 2019
Published: 6 May 2019
Authors
Adam Morgan
School of Mathematics and Statistics
University of Glasgow
United Kingdom