An Infinite Family of Artin-Schreier Curves with Minimal a-number
Iris Y. Shi
TL;DR
This work addresses the $a$-number of Artin–Schreier curves and the sharpness of Booher–Cais bounds for $ ext{Z}/p ext{Z}$-Galois covers. It first constructs explicit small Artin–Schreier curves where the $a$-number attains the lower bound $L(d)$, by analyzing equations of the form $y^p-y=f$ with carefully chosen poles (notably $d=p^2\, ext{±}\,1$). It then develops a formal patching framework to glue these curves and produce infinite families with increasing ramification breaks while preserving minimal $a$-numbers, valid in arbitrary characteristic. The main contribution is a concrete mechanism to realize $a$-numbers equal to $L(d)$ for a wide range of $d$ (often modulo $p^2$), establishing evidence for the optimality of the Booher–Cais bounds and providing a pathway to infinitely many examples with prescribed ramification. This deepens understanding of how the $a$-number behaves under ramification in $p$-group covers and demonstrates the robustness of the lower bound under controlled global constructions.
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 $φ: Y \to X$ 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.