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

Abstract

For an elliptic curve with complex multiplication over a number field, the $p^{\infty }$-Selmer rank is even for all $p$. Česnavičius proved this using the fact that $E$ admits a $p$-isogeny whenever $p$ splits in the complex multiplication field, and invoking known cases of the $p$-parity conjecture. We give a direct proof, and generalise the result to abelian varieties.