Automorphisms with fixed points and Weierstrass points of compact Riemann surfaces

Abstract

Let M be a compact Riemann surface of genus g'^2 and T be a conformal automorphism of order N with t fixed points. We denote the cyclic group generated by T and M/(T} the surface by identifying the equivalent points on M under the elements of . M is considered as a covering surface of M/ and the behavior of ramifications depends on the gap sequences of the fixed points. Lewittes [7] proved that if t^5, then every fixed point of T is 1-Weierstrass point, and Guerrero [4] proved that if t=l and the fixed point is not a 1-Weierstrass point, then T has order 6, g = l (mod 6) and the fixed point is a 3, then every fixed point of T is a g-Weierstrass point for q^>2 (q^2 (mod 3)). Farkas and Kra [3] proved that if T is of prime order TV and t^3, then every fixed point is a ^-Weierstrass point for q^>2 (q=l (mod AT)). Accola [1] proved that if T is of prime order N and t^S, then every fixed point is a iV-Weierstrass point. Recently Horiuchi and Tanimoto [5] gave a sufficientcondition for fixed points to be <7-Weierstrasspoint (<?^2) and showed that the results mentioned above are obtained by using the condition and studied the case where ^3 and T is of order 5. Almost all of the results mentioned above, however, are obtained under the condition that T is of prime order. In thispaper we investigate the properties of automorphisms without the condition that T is of prime order. In the first