International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 609425, 18 pages
doi:10.1155/2008/609425
Research Article

One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions

Antonio F. Costa,1 Milagros Izquierdo,2 and Gonzalo Riera3

1Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Educación a Distancia (UNED), Senda del rey, 9, 28040 Madrid, Spain
2Matematiska Institutionen, Linköpings Universitet, 581 83 Linköping, Sweden
3Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

Received 9 June 2008; Accepted 13 October 2008

Academic Editor: Heinrich Begehr

Copyright © 2008 Antonio F. Costa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Hurwitz spaces are spaces of pairs (S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of (p−1)/2 transpositions and the monodromy group is the dihedral group Dp. We prove that the completion ℋDp¯ of the Hurwitz space ℋDp is uniformized by a non-nomal index p+1 subgroup of a triangular group with signature (0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show that ℋDp is a quotient of the upper half plane by the modular group Γ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projection ℋDp¯→ℂ^ and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.