An Infinite Family of Artin-Schreier Curves with Minimal a-number

TL;DR

This paper constructs infinite families of Artin-Schreier curves over algebraically closed fields of odd characteristic that achieve the minimal possible a-number, confirming the optimality of a known lower bound related to ramification.

Contribution

It introduces a method using formal patching to generate infinite families of Artin-Schreier curves with minimal a-number in any characteristic, demonstrating the bound's sharpness.

Findings

Existence of Artin-Schreier curves with minimal a-number in all characteristics

Use of formal patching to construct infinite families

Validation of the lower bound's optimality

Abstract

Let $p$ be an odd prime and $k$ be an algebraically closed field with characteristic $p$. Booher and Cais showed that the $a$-number of a $\mathbb Z/p \mathbb Z$-Galois cover of curves $\phi: Y \to X$ must be greater than a lower bound determined by the ramification of $\phi$. In this paper, we provide evidence that the lower bound is optimal by finding examples of Artin-Schreier curves that have $a$-number equal to its lower bound for all $p$. Furthermore we use formal patching to generate infinite families of Artin-Schreier curves with $a$-number equal to the lower bound in any characteristic.