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. 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.

References

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