The effect of twisting on the 2-Selmer group

Let Γ be an elliptic curve defined over Q , all of whose 2-division points are rational, and let Γ b be its quadratic twist by b . Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γ b as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.