Exceptional maps and abelian points in backward orbits

Abstract

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $\alpha$ for $f$. We show that the extension $K(f^{-\infty}(\alpha))/K$ generated by the backward orbit of $\alpha$ is virtually abelian if and only if $f$ is an exceptional map with CM and $\alpha$ is $f$-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over $\mathbb{C}$ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.