Finite ramification for preimage fields of postcritically finite morphisms

Abstract

Given a finite endomorphism $\phi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K({\phi }^{-\infty }(\alpha )):={\bigcup }_{n\ge 1}K({\phi }^{-n}(\alpha ))$ generated by the preimages of $\alpha$ under all iterates of $\phi$. In particular when $\phi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W\subseteq X$ such that ${\phi }^{-1}(W)\subseteq W$ and $\phi :W\to X$ is étale, we prove that $K({\phi }^{-\infty }(\alpha ))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X={\mathbb{A}}^1$ and Cullinan-Hajir, Jones-Manes in the case $X={\mathbb{P}}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X={\mathbb{P}}^1$. The proof relies on Faltings' theorem and a local argument.