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

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 $\mathrm{Sha}(A_d)[3]\ne 0$ 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 $A=\mathrm{Jac}(C_x)/⟨P⟩$, 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.

25 pages, small updates to presentation

References in corpus (1)

• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes