An explicit Abelian surface with maximal Galois action

TL;DR

This paper constructs an explicit genus 2 curve over a number field whose Jacobian's Galois representation has maximal possible image, demonstrating a case of maximal Galois action on the Jacobian's adelic points.

Contribution

It provides the first explicit example of a genus 2 curve with Jacobian exhibiting maximal Galois image in the symplectic group.

Findings

Constructed an explicit genus 2 curve with maximal Galois image.

Demonstrated the Jacobian's Galois representation is surjective onto GSp_4(ℤ̂).

Contributed to understanding of Galois actions on abelian surfaces.

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 $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.