# FINITE RAMIFICATION FOR PREIMAGE FIELDS OF POSTCRITICALLY FINITE MORPHISMS

@article{Bridy2015FINITERF,
title={FINITE RAMIFICATION FOR PREIMAGE FIELDS OF POSTCRITICALLY FINITE MORPHISMS},
author={Andrew Bridy and Patrick Ingram and Rafe Jones and Jamie Juul and A. Levy and M. Manes and Simon Rubinstein-Salzedo and J. Silverman},
journal={arXiv: Number Theory},
year={2015}
}
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… Expand
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