For elliptic curve E over a number field K, there are various 'modulo 2' versions of the Birch–Swinnerton-Dyer Conjecture, each sometimes called the Parity Conjecture. One sserts that the algebraic rank of E/K has the same parity as the analytic rank (as given by the root number). Another one is the same statement for the p-infinity Selmer rank for some prime p. I will explain the proof of the second conjecture for all elliptic curves E/Q and all p (this completes earlier work by Greenberg, Guo, Monsky, Nekovar and Kim), and a weaker result over general number fields.

This is joint work with Vladimir Dokchitser.