Selmer Ranks of Elliptic Curves in Families of Quadratic Twists


October 20, 2010


Karl Rubin


University of California at Irvine


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.