Algebra & Number Theory

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

Douglas Ulmer

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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.

Article information

Algebra Number Theory, Volume 13, Number 5 (2019), 1069-1120.

Received: 11 June 2018
Revised: 27 February 2019
Accepted: 2 April 2019
First available in Project Euclid: 17 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

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


Ulmer, Douglas. On the Brauer–Siegel ratio for abelian varieties over function fields. Algebra Number Theory 13 (2019), no. 5, 1069--1120. doi:10.2140/ant.2019.13.1069.

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