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

2024/11/17 by Shi, Iris Y.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.11201

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 ϕ: Y → X must be greater than a lower bound determined by the ramification of ϕ. 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.

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