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# 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.

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

H

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

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

the complex projective line, ramified only over three points - customarily taken to be $\left\{0, 1, infty\right\}$. 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.

## References

• J.-P. Serre, Lectures on the Mordell-Weil Theorem (1989), p.71

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