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/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.

Belyi functions

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

Applications

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

References

• Serre, J.-P. (1989), Lectures on the Mordell-Weil Theorem, p.71
• Klein, F. (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh order transformation of elliptic functions)", Mathematische Annalen 15 (3-4): 533–555, doi:10.1007/BF02086276, collected as pp. 140–165 in Oeuvres, Tome 3 
• Belyĭ, G. V. (1980). "On Galois Extensions of a Maximal Cyclotomic Field". Mathematics of the USSR-Izvestiya 14 (2): 247. doi:10.1070/IM1980v014n02ABEH001096. 

Further reading

• Wushi Goldring (2012), "Unifying themes suggested by Belyi's Theorem", in Dorian Goldfeld; Jay Jorgenson; Peter Jones et al., Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 181-214, ISBN 978-1-4614-1259-5