Selmer ranks in twists of CM abelian varieties

Abstract

We prove the Selmer ranks in certain families of $p$-th twists of CM abelian varieties obey the symplectic or unitary distributions. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curves $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity are ``largely" unsolvable as $\delta$ varies. We also discuss the rank growth in cyclic extensions of prime degree for CM abelian varieties.