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 $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. 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 $\mathrm{Sha}(E/K)[p^\infty]$ is infinite, it must contain $(\mathbb{Q}_p/\mathbb{Z}_p)^2$.