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Register a new userFinite ramification for preimage fields of postcritically finite morphisms
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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.
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