Surjectivity of Galois representations in rational families of abelian varieties

Abstract

In this article, we show that for any nonisotrivial 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 $g\ge 3$, there are infinitely many abelian varieties over $\mathbb{Q}$ with adelic Galois representation having image equal to all of ${GSp}_{2g}\left(\hat{\mathbb{Z}}\right)$.