Families of abelian varieties and large Galois images

TL;DR

This paper proves that for most abelian varieties in a family over a number field, the associated Galois representations have large images, extending previous results by relaxing monodromy assumptions.

Contribution

It generalizes earlier results on Galois image size for abelian varieties by removing the need for big monodromy in families over various bases.

Findings

Galois images are large for most abelian varieties in the family.

The index of the Galois image in a certain group is uniformly bounded.

Results hold for families over different base varieties.

Abstract

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $\rho_A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $\rho_A$ will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results which assumed that the family of abelian varieties have "big monodromy". We also give a version for a family of abelian varieties with a more general base.