Quadratic twists of abelian varieties and disparity in Selmer ranks

Morgan, Adam

doi:10.2140/ant.2019.13.839

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Quadratic twists of abelian varieties and disparity in Selmer ranks

Adam Morgan

Vol. 13 (2019), No. 4, 839–899

DOI: 10.2140/ant.2019.13.839

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.

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

Vol. 13, No. 4, 2019