On cyclic groups covers of the projective line

TL;DR

This paper generalizes the study of cyclic covers of the projective line, computing their fundamental groups and Galois modules using combinatorial and algebraic tools, including Smith Normal Form and Alexander modules.

Contribution

It extends previous work by considering arbitrary orders in the radicant, providing explicit computations of fundamental groups and Galois module structures for these covers.

Findings

Computed the fundamental group of general cyclic covers of the projective line.

Determined the Galois module structure of the first homology group.

Validated results using the Chevalley-Weil formula.

Abstract

This article extends the study of cyclic ramified covers of the projective line defined by Kummer equations. We consider the most general case of such covers, allowing arbitrary orders in the roots of the generating radicant. The primary goal is the computation of the fundamental group of both the open and complete curve. We employ tools of combinatorial group theory utilizing the Smith Normal Form. This result is further visualized through the theory of foldings and $S$-graphs. Finally, we apply the theory of Alexander modules and the Crowell exact sequence to compute the abelianization of the fundamental group, $H_{1}(X, \mathbb{Z})$, and determine its Galois~module~structure over a field $k$ confirming the result using the Chevalley-Weil formula.