Surjectivity of Galois Representations in Rational Families of Abelian Varieties

TL;DR

This paper proves that in certain rational families of abelian varieties with large monodromy, most members have maximal Galois representation images, leading to infinitely many varieties with full symplectic Galois images over ield.

Contribution

It establishes that for non-isotrivial families with big monodromy, the set of members with maximal adelic Galois image has density one, and constructs infinitely many abelian varieties with full GSp image.

Findings

Most members in these families have maximal Galois image.

There are infinitely many abelian varieties over ield with full GSp(ield) Galois image.

Results apply to families dominating moduli of hyperelliptic, trigonal, and plane curves.

Abstract

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation having image equal to all of $\operatorname{GSp}_{2g}(\widehat{\mathbb Z})$.