Semi-direct Galois covers of the affine line

TL;DR

This paper investigates Galois covers of the projective line over an algebraically closed field of characteristic p, focusing on minimal genus curves with specific semi-direct product Galois groups and their enumeration.

Contribution

It determines the minimal genus of curves admitting certain Galois covers and bounds the number of such curves, depending only on group parameters.

Findings

Minimal genus depends only on , p, and the order of  modulo p.

Number of minimal genus curves with such covers is at most (p-1)/a.

Provides explicit construction and classification of these covers.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $\psi:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.