Arbres de Hurwitz et automorphismes d'ordre p des disques et des couronnes p-adiques formels

TL;DR

This paper characterizes the minimal semi-stable models of order p-automorphisms of formal discs and annuli over mixed characteristic rings, using combinatorial Hurwitz trees to determine when such automorphisms can occur.

Contribution

It introduces a combinatorial criterion via Hurwitz trees for realizing order p-automorphisms on formal p-adic analytic spaces.

Findings

Describes the minimal semi-stable model for automorphisms

Defines Hurwitz trees as a combinatorial tool

Provides a necessary and sufficient condition for automorphisms to be represented by Hurwitz trees

Abstract

Let R be a complete discrete valuation ring of mixed characteristics (0,p). Given an order p-automorphism of a formal disc (or annulus) over R, we describe the minimal semi-stable model for which the specialisations of fixed points are distincts and lie in the smooth locus of the special fiber. The description leads to a combinatorial object called Hurwitz tree. Our main result is a necessary and sufficient condition for a Hurwitz tree to arise from an order p-automorphism.