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.
ANDREI BELYI, RESEARCHER ON EURO-RUSSIAN ENERGY ISSUES : SECURITY OF ENERGY DEMAND A CRUCIAL ISSUE FOR RUSSIA.(Interview)
May 22, 2006; Russia's Andrei V. belyi specialises in Russian-European energy issues. His thesis presented at the Universite Libre de...
INTERVIEW WITH ANDREI BELYI, ASSOCIATE PROFESSOR AT THE HIGHER SCHOOL OF ECONOMICS, MOSCOW : EU-RUSSIA: "COLD PEACE" RATHER THAN A NEW "COLD WAR".(Interview)
May 31, 2007; Russian-born Andrei V. belyi is from the Centre for Energy Studies at the Institute of World Economy and International...