The p-parity conjecture for elliptic curves with a p-isogeny

Abstract: For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of the p-primary part of the Shafarevich-Tate group Sha(E/K) for a prime p, this is equivalent to the p-parity conjecture: the global root number matches the parity of the Z_p-corank of the p-infinity Selmer group. We prove the latter unconditionally for E that have a K-rational p-isogeny. We deduce that the p-parity conjecture holds for every E with complex multiplication defined over K, and that for such E, if the p-primary part of Sha(E/K) is infinite, it must contain (Q_p/Z_p)^2.