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

Abstract

Hurwitz spaces are spaces of pairs(S,f)whereSis a Riemann surface andf:S→ℂ^a meromorphic function. In this work, we study1-dimensional Hurwitz spacesℋDpof meromorphicp-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of(p−1)/2transpositions and the monodromy group is the dihedral groupDp. We prove that the completionℋDp¯of the Hurwitz spaceℋDpis uniformized by a non-nomal indexp+1subgroup of a triangular group with signature(0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show thatℋDpis a quotient of the upper half plane by the modular groupΓ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projectionℋDp¯→ℂ^and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.