An open image theorem for a general class of abelian varieties

TL;DR

This paper proves that for a broad class of polarized abelian varieties over number fields, the Galois group of the torsion points' splitting field is as large as possible for almost all primes, under certain fiber conditions.

Contribution

It establishes an open image theorem for the Galois action on torsion points of abelian varieties with specific fiber properties, extending previous results to a more general setting.

Findings

Galois group is GSp_{2g}(Z/ell) for almost all primes ell

Condition on the Neron model's fiber with potential toric dimension one

Applicable to a broad class of polarized abelian varieties

Abstract

Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ell, the Galois group of the splitting field of the ell-torsion of A is GSp_{2g}(Z/ell).