Vol. 13, No. 5, 2019

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

Douglas Ulmer

Vol. 13 (2019), No. 5, 1069–1120
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.

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