Moscow Mathematical Journal

Volume 16, Issue 1, January–March 2016 pp. 45–93.

An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

Authors: Marc Hindry (1) and Amílcar Pacheco (2)
Author institution: (1) Université Paris Diderot, Institut de Mathématiques de Jussieu, UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris Cedex 13, FRANCE
(2) Universidade Federal do Rio de Janeiro, Instituto de Matemática. Mailing address: Rua Alzira Brandão 355/404, Tijuca, 20550-035 Rio de Janeiro, RJ, Brasil

Summary:

Consider a family of abelian varieties A_i of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevich–Tate group of A_i. We ask then when does the product of the order of the Shafarevich–Tate group by the regulator of A_i behave asymptotically like the exponential height of the abelian variety. We give examples of families of abelian varieties for which this analogue of the Brauer–Siegel theorem can be proved unconditionally, but also hint at other situations, where the behaviour is different. We also prove interesting inequalities between the degree of the conductor, the height and the number of components of the Néron model of an abelian variety.

2010 Math. Subj. Class. 11G05, 11G10, 11G40, 11G50, 11R58, 14G10, 14G25, 14G40, 14K15.

Keywords: Abelian varieties, global fields, function fields, L-function, Birch and Swinnerton-Dyer conjecture, heights, torsion points, Néron models, Brauer–Siegel theorem.

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