An explicit Abelian surface with maximal Galois action

Abstract

We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\mathrm{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image ${\mathrm{GSp}}_4(\hat{\mathbb{Z}})$.