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2019 Quadratic twists of abelian varieties and disparity in Selmer ranks
Adam Morgan
Algebra Number Theory 13(4): 839-899 (2019). 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.

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Adam Morgan. "Quadratic twists of abelian varieties and disparity in Selmer ranks." Algebra Number Theory 13 (4) 839 - 899, 2019. https://doi.org/10.2140/ant.2019.13.839

Information

Received: 1 December 2017; Revised: 2 November 2018; Accepted: 23 January 2019; Published: 2019
First available in Project Euclid: 18 May 2019

zbMATH: 07059758
MathSciNet: MR3951582
Digital Object Identifier: 10.2140/ant.2019.13.839

Subjects:
Primary: 11G10

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.13 • No. 4 • 2019
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