On the number of ovals of a symmetry of a compact Riemann surface

Abstract

Let $X$ be a symmetric compact Riemann surface whose full group of conformal automorphisms is cyclic. We derive a formula for counting the number of ovals of the symmetries of $X$ in terms of few data of the monodromy of the covering $X\to X/G$, where $G={\mathrm{Aut}}^{\pm }X$ is the full group of conformal and anticonformal automorphisms of $X$.