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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.

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 non-singular algebraic curves over the algebraic numbers using combinatorial data.

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

*H*/Γ

(where *H* is the upper half-plane and Γ is a subgroup 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 **P**^{1}(**C**) ramified only over three points, which after a Möbius transformation may be taken to be . Belyi functions may be described combinatorially by dessins d'enfants.

Belyi functions and dessins d'enfants – but not Belyi's theorem – date at least to the work of Felix Klein; he used them in his article (Klein 1879) to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).^{[1]}

Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.

**^**le Bruyn, Lieven (2008),*Klein's dessins d'enfant and the buckyball*.

- Serre, Jean-Pierre (1997).
*Lectures on the Mordell-Weil theorem*. Aspects of Mathematics. Vol. 15. Translated from the French by Martin Brown from notes by Michel Waldschmidt (Third ed.). Friedr. Vieweg & Sohn, Braunschweig. doi:10.1007/978-3-663-10632-6. ISBN 3-528-28968-6. MR 1757192. - Klein, Felix (1879). "Über die Transformation elfter Ordnung der elliptischen Functionen" [On the eleventh order transformation of elliptic functions].
*Mathematische Annalen*(in German).**15**(3–4): 533–555. doi:10.1007/BF02086276. - Belyĭ, Gennadiĭ Vladimirovich (1980). "Galois extensions of a maximal cyclotomic field".
*Math. USSR Izv*.**14**(2). Translated by Neal Koblitz: 247–256. doi:10.1070/IM1980v014n02ABEH001096. MR 0534593.

- Girondo, Ernesto; González-Diez, Gabino (2012),
*Introduction to compact Riemann surfaces and dessins d'enfants*, London Mathematical Society Student Texts, vol. 79, Cambridge: Cambridge University Press, ISBN 978-0-521-74022-7, Zbl 1253.30001 - Wushi Goldring (2012), "Unifying themes suggested by Belyi's Theorem", in Dorian Goldfeld; Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate (eds.),
*Number Theory, Analysis and Geometry. In Memory of Serge Lang*, Springer, pp. 181–214, ISBN 978-1-4614-1259-5