Families of Abelian Varieties and Large Galois Images

Abstract

Abstract Associated with an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho _A\colon \operatorname {Gal}({\overline {K}}/K)\to \operatorname {GL}_{2g}(\widehat {{\mathbb {Z}}})$. 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 non-empty 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 that assumed that the family of abelian varieties has “big monodromy”. We also give a version for families of abelian varieties with a more general base.