Genus two curves with full √3-level structure and Tate-Shafarevich groups - Bruin, N; Flynn, EV; Shnidman, A (Springer, 2023)
We give an explicit rational parameterization of the surface H3 over ℚ whose points parameterize genus 2 curves C with full √3-level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that III (Ad) [3] ≠ 0 for a positive proportion of quadratic twists Ad. In fact, for 100% of x∈H3(ℚ), this holds for the surface A = Jac(Cx)/⟨P⟩, where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(ℚ), as well as statistical results on the size of #Cd(ℚ), as d varies through squarefree integers.