AUTOMORPHISMS OF NON-CYCLIC p-GONAL RIEMANN SURFACES

In this paper we prove that the order of a holomorphic automorphism of a non-cyclic p-gonal compact Riemann surface S of genus g > (p - 1)(2) is bounded above by 2(g + p - 1). We also show that this maximal order is attained for infinitely many genera. This generalises the similar result for the particular case p = 3 recently obtained by Costa-Izquierdo. More over, we also observe that the full group of holomorphic automorphisms of S is either the trivial group or is a finite cyclic group or a dihedral group or one of the Platonic groups A(4), A(5) and Sigma(4). Examples in each case a real so provided. If S admits a holomorphic automorphism of order 2 ( g + p - 1), then its full group of automorphisms is the cyclic group generated by it and every p-gonal map of S is necessarily simply branched. Finally, we note that each pair ( S, pi), where S is a non-cyclic p-gonal Riemann surface and pi is a p-gonal map, can be defined over its field of moduli. Also, if the group of automorphisms of S is different from a non-trivial cyclic group and g > ( p - 1)(2), then S can be also be defined over its field of moduli.