On automorphism groups of a curve as linear groups (Q1087605)

Let X be a complete non-singular curve over \({\mathbb{C}}\) (or any algebraically closed field), G a cyclic group of order n of automorphisms of X, and let \(\chi_ q\) denote the characters of the natural representation of G on the space of q-differentials on X. The relations between these characters are studied and it is shown that the first \(n/\prod p_ i\) characters \((p_ i\) prime, \(p_ i| n)\) determine all others, correcting a result of \textit{I. Guerrero} [Ill. J. Math. 26, 212-225 (1982; Zbl 0504.32018)]. The traces of the \(\chi_ q\) are characterized by a sequence of class functions of G. They classify the actions of cyclic groups on X up to homeomorphism. This is proved by looking at the corresponding Fuchsian groups.