  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 F_p-representations of the absolute Galois group of
K by characters of order p.
