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

TL;DR

This paper explicitly parameterizes genus 2 curves with full -structure, constructs abelian surfaces with nontrivial Tate-Shafarevich groups for many quadratic twists, and provides bounds on ranks and rational points.

Contribution

It offers an explicit rational model for genus 2 curves with -structure and analyzes the Tate-Shafarevich groups and ranks of associated Jacobians across quadratic twists.

Findings

-structure on Jacobians is explicitly parameterized.

Nontrivial -torsion in Tate-Shafarevich groups for a positive proportion of twists.

Bounds on average ranks and rational point sizes for quadratic twists.

Abstract

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 their Jacobian $J$. We use this model to construct abelian surfaces $A$ with the property that $\mathrm{Sha}(A_d)[3] \neq 0$ for a positive proportion of quadratic twists $A_d$. In fact, for $100\%$ of $x \in \mathcal{H}_3(\mathbb{Q})$, this holds for the surface $A = \mathrm{Jac}(C_x)/\langle P \rangle$, where $P$ is the marked point of order $3$. Our methods also give an explicit bound on the average rank of $J_d(\mathbb{Q})$, as well as statistical results on the size of $\#C_d(\mathbb{Q})$, as $d$ varies through squarefree integers.