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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.## References

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 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.

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

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Last updated on Thursday November 22, 2007 at 11:10:50 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday November 22, 2007 at 11:10:50 PST (GMT -0800)

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