Selmer groups of elliptic curves

Recently, Poonen and Rains proposed heuristics for the distributions of Selmer groups of elliptic curves. Work dating back to Heath-Brown in 1994 suggests that these heuristics should be satisfied within quadratic twist families of elliptic curves as well. I will be presenting results of Mazur, Rubin, and myself showing that, after a parity adjustment, these heuristics are satisfied within quadratic twist families of elliptic curves with an S_3 2-division field. I will also explain how our work challenges the conventional wisdom of Goldfeld’s conjecture about how ranks are distributed within quadratic twist families of elliptic curves over general number fields.