Surjectivity of Galois Representations in Rational Families of Abelian Varieties

Aaron Landesman, Ashvin Swaminathan, James Tao, Yujie Xu

Published 2016-08-18Version 1

In this article, we show that for any family of abelian varieties over a rational base with big geometric 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 $\operatorname{GSp}_{2g}(\hat{\mathbb Z})$.