Belyi's theorem

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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.

It follows that the Riemann surface in question can be taken to be

H/Γ

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

$\mathbf P^1(\mathbb{C}),$

the complex projective line, ramified only over three points - customarily taken to be $\{0, 1, \infty\}$ . Belyi's theorem is an existence theorem for such functions. It has subsequently been much used in the inverse Galois problem.