Abelian varieties over large algebraic fields with infinite torsion

2010/12/11 by David Zywina, Zywina, David
Computer Science · Mathematics · #14K15 (Primary) 11F80 (Secondary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1012.2477

openalex publication_date 2010/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in Kbar. We shall prove that the torsion subgroup of A(Kbar(sigma)) is infinite for all sigma in Gal(Kbar/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of Geyer and Jarden from 1978.

Related

• Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion
• Torsion bounds for a fixed abelian variety and varying number field
• Finiteness properties of torsion fields of abelian varieties
• Arithmetic of abelian varieties with constrained torsion
• A Mordell-Weil theorem for abelian varieties over fields generated by torsion points