Finite ramification for preimage fields of postcritically finite morphisms

2015/11/01 by Bridy, Andrew, Ingram, Patrick, Jones, Rafe +5 · 4 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1511.00194

Abstract

Given a finite endomorphism φ of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K(φ^-∞(α)) : = \bigcup_{n ≥ 1} K(φ^-n(α)) generated by the preimages of α under all iterates of φ. In particular when φ is post-critically finite, i.e., there exists a non-empty, Zariski-open W ⊆ X such that φ^-1(W) ⊆ W and φ: W → X is étale, we prove that K(φ^-∞(α)) 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 = \mathbbA¹ and Cullinan-Hajir, Jones-Manes in the case X = ℙ¹. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = ℙ¹. The proof relies on Faltings' theorem and a local argument.

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