Disparity in Selmer ranks of quadratic twists of elliptic curves

Abstract

We 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 $\mathbf{F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.