$p$-group Galois covers of curves in characteristic $p$

TL;DR

This paper investigates the cohomological properties of algebraic curves with finite p-group actions in characteristic p, introducing the concept of a 'magical element' to facilitate explicit computations of differentials and cohomology.

Contribution

It introduces the notion of a 'magical element' in the function field, enabling explicit calculations of equivariant differentials and cohomology for p-group covers of curves.

Findings

Generic p-group covers possess a 'magical element'

Explicit computation of de Rham cohomology for cyclic p-group actions

Provides a framework for understanding equivariant cohomology in characteristic p

Abstract

We study cohomologies of a curve with an action of a finite $p$-group over a field of characteristic $p$. Assuming the existence of a certain 'magical element' in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic $p$-group cover has a 'magical element'. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.