Selmer Ranks of Elliptic Curves in Families of Quadratic Twists

This talk will report on ongoing work with Barry Mazur that studies 2-Selmer ranks in the family of all quadratic twists of a fixed elliptic curve over a number field. Our goal is to compute the density of twists with a given 2-Selmer rank r, for every r. This has been done by Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over Q with all 2-torsion rational. Our methods are different and work best for curves with no rational points of order 2. So far we can prove under certain hypotheses that E has “many” twists of every 2-Selmer rank, but not that the set of such twists has positive density. In this talk I will describe these results and the methods involved, and discuss a basic question about algebraic number fields that arises in trying to improve our results.