Resumen de Genus two curves with full √3-level structure and Tate–Shafarevich groups

Nils Bruin, E. Victor Flynn, Ari Shnidman

• 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 X(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.