On symmetries of compact Riemann surfaces with cyclic groups of automorphisms

Abstract

A Riemann surface X is said to be of type ( n , m ) if its full automorphism group Aut X is cyclic of order n and the quotient surface X / Aut X has genus m. In this paper we determine necessary and sufficient conditions on the integers n , m , g and γ, where n is odd, for the existence of a Riemann surface of genus g and type ( n , m ) admitting a symmetry with γ ovals.