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

TL;DR

This paper proves an algebraic analogue of the Brauer-Siegel theorem for abelian varieties over function fields, extending previous results from elliptic curves to higher-dimensional cases using direct algebraic methods.

Contribution

It introduces an algebraic approach to establish the Brauer-Siegel ratio for abelian varieties, broadening the scope beyond elliptic curves to higher dimensions.

Findings

Confirmed the Brauer-Siegel ratio for certain families of abelian varieties.

Extended results to higher-dimensional abelian varieties.

Provided a new algebraic proof technique for these theorems.

Abstract

Hindry has proposed an analogue 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 similar 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.