Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The effect of twisting on the 2-Selmer group

SIR PETER SWINNERTON–DYERa1

a1 DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. e-mail: hpfs100@dpmms.cam.ac.uk

Abstract

Let Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb 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 Γb 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.

(Received June 01 2007)

(Revised February 09 2008)