Belyi's theorem

In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.

This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.

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.

A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P¹(C) ramified only over three points, which after a Möbius transformation may be taken to be ${\displaystyle \{0,1,\infty \}}$. Belyi functions may be described combinatorially by dessins d'enfants.

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