Galois covers of type (p,...,p), vanishing cycles formula, and the existence of torsor structures

TL;DR

This paper establishes a local Riemann-Hurwitz formula for Galois covers of type (p,...,p) over p-adic curves, generalizing previous results, and explores conditions for torsor structures in such covers.

Contribution

It introduces a generalized vanishing cycles formula for higher-degree Galois covers and investigates the existence of torsor structures in this context.

Findings

Proved a local Riemann-Hurwitz formula for type (p,...,p) covers.

Extended vanishing cycles comparison to higher-degree Galois covers.

Analyzed criteria for torsor structure existence in p-adic Galois covers.

Abstract

In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type (p,p,...,p) between formal germs of p-adic curves and which generalises the formula proven by the first author in the case of Galois covers of degree p. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type (p,p,...,p) between p-adic schemes.