Klagsbrun, ZevMazur, BarryRubin, Karl2014-12-032013Klagsbrun, Zev, Barry Charles Mazur, and Karl Rubin. 2013. "Disparity in Selmer Ranks of Quadratic Twists of Elliptic Curves." Annals of Mathematics 178 (1): 287–320.0003-486X1939-8980http://nrs.harvard.edu/urn-3:HUL.InstRepos:13456934We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0,1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual \(F_p\)-representations of the absolute Galois group of K by characters of order p.en-USarithmetic statisticselliptic curvesparityquadratic twistsSelmer groupsJournal Article2014-12-0310.4007/annals.2013.178.1.5