On a generalization of the Lewittes theorem on Weierstrass points

Abstract

Suppose that X is a compact Riemann surface of genus g ≥ 2, while σ is an automorphism of X of order n, and g* is the genus of the quotient surface X* = X/〈σ〉. In 1951 Schoneberg obtained a sufficient condition for a fixed point P ∈ X of σ to be a Weierstrass point of X. Namely, he showed that P is a Weierstrass point of X if g* ≠ [g/n], where [x] is the integral part of x. Somewhat later Lewittes proved the following theorem, equivalent to Schoneberg’s theorem: If a nontrivial automorphism σ fixes more than four points of X then all of them are Weierstrass points.