a-Numbers of Curves in Artin-Schreier Covers

TL;DR

This paper explores how the a-number of Artin-Schreier covers relates to ramification and the base curve, providing bounds and examples, and analyzing the Cartier operator's kernel.

Contribution

It establishes bounds on the a-number of covers in Artin-Schreier extensions and analyzes the Cartier operator's kernel to understand their relationship.

Findings

Bounds on the a-number of Y in terms of X and ramification.

Examples demonstrating the sharpness of these bounds.

Analysis of the Cartier operator's kernel in this context.

Abstract

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.