Abelian varieties over large algebraic fields with infinite torsion

TL;DR

This paper proves that for an abelian variety over a number field, the torsion subgroup over fixed fields of Galois automorphisms is infinite for almost all automorphisms, confirming a longstanding conjecture.

Contribution

It establishes the number field case of Geyer and Jarden's conjecture, showing infinite torsion in fixed fields of Galois automorphisms outside a measure zero set.

Findings

Torsion subgroup is infinite for almost all Galois automorphisms.

Confirms the Geyer-Jarden conjecture in the number field case.

Results hold for abelian varieties over number fields.

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.