(joint with Vladimir Dokchitser) For an elliptic curve E over the rational numbers, there are various modulo 2 versions of the Birch-Swinnerton-Dyer Conjecture, each sometimes called the Parity Conjecture. One asserts that the Mordell-Weil rank of E has the same parity as the analytic rank. Another
one is the same statement for the p-infinity Selmer rank for some prime p. I will explain the background behind the conjectures and the ideas behind the proof of the second one. (This completes earlier work by Birch, Stephens, Greenberg, Guo, Monsky, Nekovar and Kim.)