Hilbertian fields and Galois representations

TL;DR

This paper introduces a new criterion for Hilbertian fields in towers with Galois steps of specific types and applies it to Galois representations, resolving a conjecture related to abelian varieties.

Contribution

The paper develops a novel Hilbertianity criterion for certain Galois towers and applies it to fields from Galois representations, settling Jarden's conjecture on abelian varieties.

Findings

Established a new Hilbertianity criterion for Galois towers.

Applied the criterion to fields from Galois representations.

Resolved Jarden's conjecture on abelian varieties.

Abstract

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties.