Big monodromy theorem for abelian varieties over finitely generated fields

TL;DR

This paper proves that certain abelian varieties over finitely generated fields have large Galois image on torsion points, extending previous results without restrictions on the field's characteristic or transcendence degree.

Contribution

It establishes big monodromy for all abelian varieties with trivial endomorphism ring and semistable reduction of toric dimension one over finitely generated fields, generalizing prior work.

Findings

Abelian varieties with specified reduction have big monodromy.

No restrictions on characteristic or transcendence degree.

Generalizes recent results of Chris Hall.

Abstract

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall.