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 III (E/K) [p∞] for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the Z[subscript p]-corank of the p∞-Selmer group. We complete the proof of the p-parity conjecture for elliptic curves that have a p-isogeny for p > 3 (the cases p ≤ 3 were known). Tim and Vladimir Dokchitser have showed this in the case when E has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the p-parity conjecture holds for every E with complex multiplication defined over K. Consequently, if for such an elliptic curve III (E/K) [p∞] is infinite, it must contain (Q[subscript p]/Z[subscript p])².