On the brauer–siegel ratio for abelian varieties over function fields

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1069-1120
Number of pages52
JournalAlgebra and Number Theory
Volume13
Issue number5
DOIs
StatePublished - Jan 1 2019

Keywords

  • Abelian variety
  • Brauer
  • Function field
  • Height
  • Regulator
  • Shafarevich group
  • Siegel ratio
  • Tate

ASJC Scopus subject areas

  • Algebra and Number Theory

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