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# Mathematical Research Letters

## Volume 24 (2017)

### Number 6

### Finite ramification for preimage fields of post-critically finite morphisms

Pages: 1633 – 1647

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n6.a3

#### Authors

#### 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 étale, 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.

The authors would like to thank AIM and the organizers of the March 2014 AIM workshop on Postcritically finite maps in complex and arithmetic dynamics at which this research was started. Bridy’s research partially supported by NSF Grant #EMSW21-RTG. Ingram’s research partially supported by Simons Collaboration Grant #283120. Juul’s research partially supported by DMS-1200749. Levy’s research partially supported by the Göran Gustafsson Foundation. Manes’s research partially supported by NSF DMS #1102858. Silverman’s research partially supported by Simons Collaboration Grant #241309.

Received 9 December 2015

Accepted 3 December 2016

Published 29 January 2018