Some remarks on cyclic Galois coverings of the projective line over finite fields

TL;DR

This paper investigates cyclic Galois extensions of the rational function field over finite fields, focusing on quotient curves by subgroups of PGL(2,q), and provides enumeration formulas using Stirling numbers.

Contribution

It introduces a method to count cyclic Galois coverings of the projective line over finite fields via subgroup analysis of PGL(2,q).

Findings

Derived enumeration formulas involving Stirling numbers.

Analyzed quotient curves under subgroup actions.

Enhanced understanding of Galois coverings over finite fields.

Abstract

We study cyclic finite Galois extensions of the rational function field of the projective line P^{1}(F_q) over a finite field F_q with q elements defined by considering quotient curves by finite subgroups of the projective linear group PGL(2,q), and we enumerate them expressing the count in terms of Stirling numbers.