$a$-Numbers of Cyclic Degree $p^2$ Covers of the Projective Line

TL;DR

This paper studies the $a$-numbers of cyclic covers of the projective line with group $Z/p^2Z$ in characteristic $p>2$, extending existing techniques to derive these invariants from branching data.

Contribution

It extends a method for computing $a$-numbers from $Z/pZ$-covers to $Z/p^2Z$-covers, enabling explicit calculations from branching data.

Findings

$a$-numbers of minimal $Z/9Z$-covers can be determined from branching data.

The technique is generalized to covers with group $Z/p^2Z$ in characteristic $p>2$.

The approach simplifies the computation of $a$-numbers for these covers.

Abstract

We investigate the $a$-numbers of $\mathbb{Z}/p^2\mathbb{Z}$-covers in characteristic $p>2$ and extend a technique originally introduced by Farnell and Pries for $\mathbb{Z}/p\mathbb{Z}$-covers. As an application of our approach, we demonstrate that the $a$-numbers of ``minimal'' $\mathbb{Z}/9\mathbb{Z}$-covers can be deduced from the associated branching datum.