belyi

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

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

Search another word or see belyion Dictionary | Thesaurus |Spanish
Copyright © 2014 Dictionary.com, LLC. All rights reserved.
  • Please Login or Sign Up to use the Recent Searches feature
FAVORITES
RECENT

;