Finite ramification for preimage fields of postcritically finite morphisms

TL;DR

This paper proves that for post-critically finite morphisms, the field generated by all preimages of a point has ramification only at finitely many primes, extending previous results and proposing a conjecture for characterization.

Contribution

It establishes finite ramification for preimage fields of post-critically finite morphisms and conjectures this property characterizes such morphisms, with a new proof for the projective line case.

Findings

Preimage fields are ramified at finitely many primes for post-critically finite morphisms.

The result generalizes earlier work on affine and projective lines.

A conjecture links finite ramification to post-critical finiteness, supported by a new proof for .

Abstract

Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$ generated by the preimages of $\alpha$ under all iterates of $\varphi$. In particular when $\varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$ and $\varphi : W \to X$ is \'etale, we prove that $K(\varphi^{-\infty}(\alpha))$ 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.