Representations of cyclic groups in positive characteristic and Weierstrass semigroups

TL;DR

This paper investigates the structure of holomorphic differentials on algebraic curves with cyclic group actions in positive characteristic, linking group representations to Weierstrass semigroups at Galois points.

Contribution

It provides new insights into the module structure of differentials under cyclic group actions and explores their connection to Weierstrass semigroups in positive characteristic.

Findings

Characterization of the $k[G]$-module structure of holomorphic differentials.

Analysis of the relation between group actions and Weierstrass semigroups.

Results applicable to Galois Weierstrass points in algebraic geometry.

Abstract

We study the $k[G]$-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group $G$ of order $p^\ell n$. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points.