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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 √©tale, 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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