Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

TL;DR

This paper proves the Geyer-Jarden conjecture regarding torsion in the Mordell-Weil group for a broad class of abelian varieties over finitely generated fields, focusing on those with large Galois monodromy.

Contribution

It establishes the conjecture for abelian varieties with big monodromy over finitely generated fields of any characteristic.

Findings

Proves the Geyer-Jarden conjecture for a large class of abelian varieties.

Shows that the Galois representation on l-torsion points contains the full symplectic group for almost all primes l.

Extends the conjecture's validity to arbitrary characteristic fields.

Abstract

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on l-torsion points, for almost all primes l, contains the full symplectic group.