Counting Non-abelian Coverings of Algebraic Curve

TL;DR

This paper investigates the enumeration of non-abelian Galois coverings of algebraic curves, specifically focusing on coverings with semi-direct product Galois groups and providing counts under certain coprimality conditions.

Contribution

It provides a method to count non-abelian coverings of algebraic curves with specific Galois groups, extending understanding of their structure and enumeration.

Findings

Determined the number of non-abelian coverings with given cyclic covers.

Established conditions for the existence of such coverings when gcd(m,n)=1.

Extended classical results on abelian coverings to non-abelian cases.

Abstract

In this article, we study the etale coverings of an algebraic curve $C$ with Galois group a semi-direct product $\mathbb{Z}/m\mathbb{Z} \rtimes \mathbb{Z}/n\mathbb{Z}$. Especially, for a given etale cyclic $n$-covering $D \to C$, we determine how many curves $E$ are there, satisfying $E \to D$ is an etale cyclic $m$-covering and $E \to C$ is Galois with non-abelian Galois group, under the assumption $gcd(m,n)=1$.