p∞$p^{\infty }$‐Selmer ranks of CM abelian varieties

Abstract

AbstractFor an elliptic curve with complex multiplication over a number field, the ‐Selmer rank is even for all . Česnavičius proved this using the fact that admits a ‐isogeny whenever splits in the complex multiplication field, and invoking known cases of the ‐parity conjecture. We give a direct proof, and generalise the result to abelian varieties.