Genus two curves with full $\sqrt{3}$-level structure and Tate-Shafarevich groups

Nils Bruin, E. Victor Flynn, Ari Shnidman

We give an explicit rational parameterization of the surface ${\mathcal{H}}_3$ over $\mathbb{Q}$ whose points parameterize genus 2 curves~$C$ with full $\sqrt{3}$-level structure on…