Realizing Artin–Schreier covers with minimal a-numbers in positive characteristic

Abstract

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B\subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $\mathbb{Z}∕p\mathbb{Z}$-cover of $X$ with branch locus $B$. For odd primes $p$, in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$-numbers of $\mathbb{Z}∕p\mathbb{Z}$-covers of the affine line. As an application, when $p=3$ or $p=5$, for any ordinary curve $X$ and any choice of $B$, we prove that the lower bound is realized for Artin–Schreier covers of $X$ with branch locus $B$.