Automorphisms of Harbater-Katz-Gabber curves

TL;DR

This paper classifies automorphisms of formal power series over fields of characteristic p that have p-power order and can be explicitly described, using the framework of Harbater-Katz-Gabber curves to understand their structure.

Contribution

It provides explicit formulas for certain automorphisms of k[[t]] and criteria for when HKG G-curves extend to larger groups, advancing the understanding of automorphism groups in positive characteristic.

Findings

Explicit formulas for automorphisms of p-power order in k[[t]]

Criteria for extending HKG G-curves to larger groups

Classification of automorphisms via Artin-Schreier extensions

Abstract

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to classify the automorphisms of k[[t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a Z/pZ Artin-Schreier extension of k(t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve.