Fixed points of automorphisms of compact Riemann surfaces and higher-order Weierstrass points

Abstract

A sufficient condition for fixed points of an automorphism of prime order on a compact Riemann surface to be higher-order Weierstrass points is given. This leads us to a complete study of the cases where the prime orders are small. 1. Let M be a compact Riemann surface of genus g > 2. We denote Aut M the group of conformal automorphisms of M, v(T) the number of fixed points of an automorphism T E Aut M and Hq(M) the space of holomorphic q-differentials on M. Lewittes proved that if v(T) > 5, then every fixed point is a 1-Weierstrass point [5], and in this relation, some cases have been studied by Accola [1], Duma [2], Farkas and Kra [3] for higher-order Weierstrass points (see Corollaries 1, 2, 3, 4 below). Guerrero [4] proved that if v(T) = 1 and the fixed point is not a 1 -Weierstrass point, then T has order 6, g 1 mod 6 and the fixed point is a q-Weierstrass point for all q > 2. It is known that if the order of T is prime, then v(T) > 2 [3]. Guerrero also gave examples of Riemann surfaces with automorphisms of prime order whose two fixed points are not q-Weierstrass points for q > 2. The purpose of this paper is to give a sufficient condition for fixed points to be q-Weierstrass points (q > 2) and to supplement the results mentioned above. We will show that if v(T)(2s + 1 n) $4 2(n5 r), then the fixed points of T are q-Weierstrass points, and study the case where v (T) > 3 and the order of T is 5. 2. For T E Aut M, let e be the rotation constant of T at a fixed point of T, i.e. locally T : z -* ez. There is a basis for the space of holomorphic q-differentials such that the linear map induced by T on this space is given by the matrix diag(eYl +q 6 Y2-2+q ... ,6Yd l+q) for each q > 2, where d = (2q 1)(g 1), and 1 = Y1 < Y2 < ..< yd < 2q(g 1) + 2 is the q-gap Received by the editors June 27, 1988 and, in revised form, August 29, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14F07; Secondary 30F10. ( I1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page