Powers of the Cartier operator on Artin-Schreier covers

TL;DR

This paper generalizes the analysis of the Cartier operator on Artin-Schreier covers to arbitrary powers, providing new bounds on the kernel dimension and restrictions on their Ekedahl-Oort types in positive characteristic.

Contribution

It extends previous sheaf-theoretic methods to higher powers of the Cartier operator, offering novel bounds and insights into the structure of Artin-Schreier covers.

Findings

Derived bounds for the kernel dimension of higher powers of the Cartier operator.

Established new restrictions on the Ekedahl-Oort types of Artin-Schreier covers.

Generalized existing methods to a broader class of Cartier operator powers.

Abstract

Curves in positive characteristic have a Cartier operator acting on their space of regular differentials. The $a$-number of a curve is defined to be the dimension of the kernel of the Cartier operator. In \cite{BoCaASc}, Booher and Cais used a sheaf-theoretic approach to give bounds on the $a$-numbers of Artin-Schreier covers. In this paper, that approach is generalized to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin-Schreier covers.