Minimal $a$-numbers of Artin--Schreier covers of ordinary curves

TL;DR

This paper computes the $a$-numbers of certain Artin-Schreier covers of ordinary curves over perfect fields, confirming a known lower bound is tight for a generic subset of polynomials.

Contribution

It explicitly determines the $a$-numbers for a generic family of Artin-Schreier covers, establishing the tightness of a previously known lower bound.

Findings

The $a$-number attains the lower bound for generic polynomials.

The result confirms the bound of Booher and Cais is tight.

Provides explicit computation of $a$-numbers for a family of covers.

Abstract

Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight.