Ramification of Wild Automorphisms of Laurent Series Fields

TL;DR

This paper studies wild automorphisms of Laurent series fields in positive characteristic, deriving conditions for specific ramification behaviors and generalizing previous results in arithmetic dynamics.

Contribution

It computes the first nontrivial coefficient of the pth iterate of certain power series and characterizes $b$-ramified automorphisms, extending prior theorems in the field.

Findings

Derived necessary and sufficient conditions for $b$-ramified automorphisms.

Generalized Nordqvist's 2017 theorem on 2-ramified power series.

Progress towards bounding norms of periodic points in nonarchimedean dynamics.

Abstract

Let $K$ be a complete discrete valuation field with residue class field $k$, where both are of positive characteristic $p$. Then the group of wild automorphisms of $K$ can be identified with the group under composition of formal power series over $k$ with no constant term and $X$-coefficient $1$. Under the hypothesis that $p > b^2$, we compute the first nontrivial coefficient of the $p$th iterate of a power series over $k$ of the form $f = X + \sum_{i \geq 1} a_iX^{b+i}$. As a result, we obtain a necessary and sufficient condition for an automorphism to be ``$b$-ramified,'' having lower ramification numbers of the form $i_n(f) = b(1 + \cdots + p^n)$. This is a vast generalization of Nordqvist's 2017 theorem on $2$-ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in $\mathbf{C}_p$ and a 2015 result of Lindahl--Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl--Nordqvist's 2018 theorem bounding the norm of periodic points of $2$-ramified power series.