Compact Riemann surfaces : prime Galois coverings of P¹

Abstract

The uniqueness of the hyperelliptic involution is well known in the theory of Riemann surfaces. More precisely, we know that if X is a hyperelliptic compact Riemann surface, there is a unique automorphism τ of order 2 such that X/〈τ〉 ≅ ℙ¹ . We wish to generalize the situation slightly. We say X is a prime Galois covering of ℙ¹ if there exists an automorphism τ of (odd) prime order p such that X/〈T〉 ≅ ℙ¹. This leads us to ask the question: When is this automorphism τ unique? We begin by building the necessary background to understand prime Galois coverings of ℙ¹. We then prove a theorem due to Gonzlez-Diez that answers our question about uniqueness. The proof given here follows his proof (given in [G-D]) quite closely, though we elaborate and modify certain details to make it more self contained.