Math. Ann. 308, 31– 45 (1997) On symmetries of p-hyperelliptic Riemann surfaces E. Bujalance ? , A.F. Costa ? Departamento de Matem aticas Fundamentales, Facultade Ciencias, U.N.E.D., E-28040 Madrid, Spain Received: 2 August 1995 / Revised version: 22 February 1996 Mathematics Subject Classication (1991): 30F10, 14H55, 20H10 1 Introduction 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 − 4 p points (and this number determines the topological type of a conformal in- volution). 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, 0 5 k 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 sup- pose 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¿ 4 p + 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) conju- gate 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 ? Partially supported by DGICYT PB 92-D716 and CEE CHRX-CT93-0408