Selmer ranks in twists of CM abelian varieties

Abstract: We prove the distribution of Selmer ranks in the certain family of $p$-th twists of CM abelian varieties obeys the symplectic distribution $\sD_p^\sym$ or the unitary distribution $\sD_{p^2}^\uni$. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curve $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity is ``largely" unsolvable as $\delta$ varies.