# Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016  pp. 659–674.

Automorphisms of Non-Cyclic p-Gonal Riemann Surfaces

Authors:  Antonio F. Costa (1) and Ruben A. Hidalgo (2)
Author institution:(1) Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, 28040 Madrid, Spain
(2) Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, 4780000 Temuco, Chile

Summary:

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. Moreover, 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 𝒜4, 𝒜5 and Σ4. Examples in each case are also 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,π), where S is a non-cyclic p-gonal Riemann surface and π 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.

2010 Math. Subj. Class. 30F10; 14H37.

Keywords: Riemann surface, Fuchsian group, automorphisms.