It follows that the Riemann surface in question can be taken to be
with H the upper half-plane and Γ of finite index in the modular group, compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.
This is a result of G. V. Belyi from 1979; it was at that time considered surprising. A Belyi function is a holomorphic map from a compact Riemann surface to
the complex projective line, ramified only over three points - customarily taken to be . Belyi functions may be described combinatorially by dessins d'enfants. Belyi's theorem is an existence theorem for such functions. It has subsequently been much used in the inverse Galois problem.