Number Theory Seminar: Karl Rubin

Abstract: I will discuss some recent results (joint with Zev Klagsbrun and Barry Mazur) on the distribution of 2-Selmer ranks in families of quadratic twists of elliptic curves over arbitrary number fields. We study the density of twists with a given 2-Selmer rank, and obtain some surprising results on the fraction of twists with 2-Selmer rank of given parity. Since the 2-Selmer rank is an upper bound for the Mordell-Weil rank, this work has consequences for Mordell-Weil ranks in families of quadratic twists.