On the ranks of the $2$-Selmer groups of twists of a given elliptic curve

Abstract

Swinnerton-Dyer considered the proportion of twists of an elliptic curve with full $2$-torsion that have $2$-Selmer group of a particular dimension. Swinnerton-Dyer obtained asymptotic results on the number of such twists using an unusual notion of asymptotic density. We build on this work to obtain similar results on the density of twists with particular rank of $2$-Selmer group using the natural notion of density.