On the numerology of ramification data for power series in characteristic $p$

TL;DR

This paper investigates the ramification behavior of automorphisms of power series fields over perfect fields of characteristic p, utilizing the field of norms construction to analyze ramification data.

Contribution

It introduces a new approach to understanding ramification in power series automorphisms using the field of norms, providing deeper insights into their numerology.

Findings

Characterization of ramification properties of automorphisms

Application of Fontaine-Wintenberger field of norms construction

Enhanced understanding of ramification data in characteristic p

Abstract

Let $k$ be a perfect field of characteristic $p$ and set $K=k((t))$. In this paper we study the ramification properties of elements of Aut$_k(K)$. By choosing a uniformizer for $K$ we may interpret our theorems in terms of power series over $k$. The most important tool that we use is the field of norms construction of Fontaine and Wintenberger.