On symmetries of p-hyperelliptic Riemann surfaces

Abstract

Let X be a compact Riemann surface of genus g? 1. The surface X is p-hyperelliptic with p= 0 if X admits a conformal involution p such that X= p has genus p (if p=0 then X is hyperelliptic and if p = 1 then X is elliptic hyperelliptic). The involution p is called a p-hyperelliptic involution. The topological type of a p-hyperelliptic involution p is determined by the number p because the xed point set of p, F( p) consists of 2g + 2 − 4p points (and this number determines the topological type of a conformal involution). A symmetry of X is an anticonformal involution : X → X of X . The topological type of a symmetry is determined by properties of its xed point set F( ). The set F( ) consists of k disjoint Jordan curves, 05k 5 g + 1 (Harnack’s theorem [H]). The space X − F( ) has either one or two components. It consists of one component if X= is nonorientable and of two components if X= is orientable. Let be a symmetry of X and suppose that F( ) has k disjoint curves, then we shall say that the species of is +k or −k according to whether X − F( ) has two or one component respectively. In this work we shall study the species of symmetries of p-hyperelliptic Riemann surfaces. If X is p-hyperelliptic having genus g, g? 4p + 1 and is a symmetry of X then the p-hyperelliptic involution p of X is unique and therefore p is another symmetry in general not (conformally) conjugate to . F. Klein in [K] (see also [BS]) studied the species of these pairs , p in the case p = 0, the hyperelliptic case. We shall study such species in the general case p= 0 in Sect. 3. We shall de ne M − r (with r = 0) Riemann surface as a Riemann surface admitting a symmetry with g + 1 − r xed curves. S.M. Natanzon [N1, N2, N3, N4] has studied the automorphisms