On projecting symmetries by unbranched regular coverings of Riemann surfaces (Q1024628)

An antiholomorphic involution of a compact Riemann surface \(X\) of genus \(g\) is called an \(M\)-symmetry if its fixed point set consists of \(g+1\) closed Jordan curves, and an \(M-1\) symmetry if its fixed point set consists of \(g\) Jordan curves. The authors prove that if \(X\) is a surface admitting an \(M\)-symmetry, then any surface \(Y\) covered by \(X\) admits an \(M\) or \(M-1\) symmetry. Given a surface \(Y\), they count the coverings \(p: X \rightarrow Y\) with \(X\) hyperelliptic and admitting an \(M\)-symmetry. Their results were previously obtained by \textit{S. N. Natanzon} [Sel. Math. Sov. 1, 81--99 (1981; Zbl 0462.14009)]. The main tools in the article are direct calculations using the Fuchsian groups associated to \(X\) and \(Y\).