Roots of unity and higher ramification in iterated extensions

TL;DR

This paper investigates the ramification properties of iterated field extensions generated by preimages of points under rational functions, revealing new infinite ramification phenomena and proposing conjectures linking dynamics and ramification theory.

Contribution

It introduces new classes of rational functions with infinite higher ramification groups and proposes a conjecture connecting PCF maps with subextensions exhibiting arithmetic profiniteness.

Findings

Several families of rational functions generate extensions with infinite higher ramification groups.

New examples of iterated extensions with subextensions satisfying arithmetic profiniteness.

Conjecture that all PCF map extensions contain subextensions with strong ramification properties.

Abstract

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the $n$th iterate of $\phi$. We ask when a finite extension of $K(\phi^{-\infty}(b))$ can contain all $m$-power roots of unity for some $m \geq 2$, and prove that several families of rational functions do so. A motivating application is to understand the higher ramification filtration when $K$ is a finite extension of $\mathbb{Q}_p$ and $p$ divides the degree of $\phi$, especially when $\phi$ is post-critically finite (PCF). We show that all higher ramification groups are infinite for new families of iterated extensions, for example those given by bicritical rational functions with periodic critical points. We also give new examples of iterated extensions with subextensions satisfying an even stronger ramification-theoretic condition called arithmetic profiniteness. We conjecture that every iterated extension arising from a PCF map should have a subextension with this stronger property, which would give a dynamical analogue of Sen's theorem for PCF maps.