On the Brauer–Siegel ratio for abelian varieties over function fields

Ulmer, Douglas

doi:10.2140/ant.2019.13.1069

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On the Brauer–Siegel ratio for abelian varieties over function fields

Douglas Ulmer

Vol. 13 (2019), No. 5, 1069–1120

DOI: 10.2140/ant.2019.13.1069

Abstract

Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

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Keywords

abelian variety, Tate–Shafarevich group, regulator, height, Brauer–Siegel ratio, function field

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G10, 11G40

Milestones

Received: 11 June 2018

Revised: 27 February 2019

Accepted: 2 April 2019

Published: 12 July 2019

Authors

Douglas Ulmer
Department of Mathematics University of Arizona Tucson, AZ United States

Vol. 13, No. 5, 2019