p-Selmer ranks of CM abelian varieties

TL;DR

This paper proves that for CM abelian varieties, the p-infinity Selmer rank is always even, extending known results from elliptic curves to a broader class of abelian varieties.

Contribution

It provides a direct proof of the evenness of Selmer ranks for CM abelian varieties and generalizes previous elliptic curve results.

Findings

Selmer rank is even for all primes p in CM abelian varieties

Generalization from elliptic curves to higher-dimensional abelian varieties

Provides a new direct proof method

Abstract

For an elliptic curve with complex multiplication over a number field, the $p^{\infty}$--Selmer rank is even for all $p$. \v{C}esnavi\v{c}ius 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.