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Finite ramification for preimage fields of post-critically finite morphisms. (English) Zbl 1391.14004
Summary: 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 $$\varphi : W \rightarrow X$$ 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 W. Aitken et al. [Int. Math. Res. Not. 2005, No. 14, 855–880 (2005; Zbl 1160.11356)] in the case $$X = \mathbb{A}^1$$ and J. Cullinan and F. Hajir [Manuscr. Math. 137, No. 3–4, 273–286 (2012; Zbl 1235.14023)], the third and sixth authors [Comment. Math. Helv. 89, No. 1, 173–213 (2014; Zbl 1316.11104)] 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.

##### MSC:
 14A10 Varieties and morphisms 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 14G25 Global ground fields in algebraic geometry
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