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 ${\text{GSp}}_4(\widehat{\mathbb{Z}}).$

Keywords:

• Galois representations
• hyperelliptic curves
• abelian varieties

Acknowledgements

We would like to thank Aaron Greicius and David Zureick-Brown for originally suggesting the problem and for providing comments on the manuscript. We also thank Brian Conrad and Jackson Morrow for helpful comments. We thank Samuele Anni, Vladimir Dokchitser, Zev Rosengarten, Jesse Silliman, Ashvin Swaminathan, James Tao, and Yujie Xu for beneficial conversations. Special thanks is due to the anonymous referee for their exceptionally attentive suggestions.

Declaration of Interest

No potential conflict of interest was reported by the authors.

Additional information

Funding

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656518.