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Roots of unity and higher ramification in iterated extensions

Spencer Hamblen, Rafe Jones

Abstract

Given a field $K$, a rational function $φ\in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(φ^{-\infty}(b))$ generated by the union over $n$ of all solutions to $φ^n(x) = b$, where $φ^n$ is the $n$th iterate of $φ$. We ask when a finite extension of $K(φ^{-\infty}(b))$ 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 $φ$, especially when $φ$ 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.

Roots of unity and higher ramification in iterated extensions

Abstract

Given a field , a rational function , and a point , we study the extension generated by the union over of all solutions to , where is the th iterate of . We ask when a finite extension of can contain all -power roots of unity for some , and prove that several families of rational functions do so. A motivating application is to understand the higher ramification filtration when is a finite extension of and divides the degree of , especially when 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.
Paper Structure (1 section, 1 theorem)

This paper contains 1 section, 1 theorem.

Table of Contents

  1. Introduction

Key Result

Theorem 1.2

Let $K$ be a field and let $\phi \in K(x)$ have degree $d \geq 2$, where $d$ is not divisible by the characteristic of $K$. Then $K(\boldsymbol\mu_{m^\infty})$ is contained in a finite extension of $K(\phi^{-\infty}(b))$ for any non-exceptional $b$ if:

Theorems & Definitions (1)

  • Theorem 1.2