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Belyi's theorem

## Summary

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.

## Quotients of the upper half-plane

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

H

(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.

## Belyi functions

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.

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]

## Applications

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

## References

1. ^ 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