Every compact Riemann surface X defines a complex algebraic curve. Belyi’s Theorem states that this curve is defined over a number field if and only if there is a meromorphic function β : X → P1(C), the Riemann sphere, whose critical values lie in the set {0, 1, ∞}. A Riemann surface that admits a Belyi function is called a Belyi surface.

Applications

Belyi theory has applications to Galois theory, algebraic geometry, Riemann surfaces, combinatorics associated to maps on surfaces, and number theory.

History

The importance of this theory was noted by the great mathematician Grothendieck who in his important manuscript “Esquisse d’un Programme” of 1984 says of Belyi’s Theorem that “to me its essential message is that there is a profound identity between the combinatorics of finite maps on the one hand and the geometry of algebraic curves defined over number fields on the other. This deep result, together with the algebraic-geometric interpretation of maps opens the door onto a new unexplored world within reach of all, who pass by without seeing it.”  Since the Esquisse there has been worldwide activity on Belyi Theory in all its aspects. However, even before the Esquisse, there had been much activity in Southampton in developing the theory. (Except that we did not notice the deep connections with number fields - after all Belyi’s Theorem had not been proved.) Mathematicians in Southampton working on Belyi Theory (or dessins d’enfants as it is more popularly known) include:

Professor Gareth Jones, Dr. Bernhard Koeck & Professor David Singerman

We also have current research students: Zoe Laing & Daniel Pinto

In the past we have supervised at least 15 research students on this topic, many of whom have done important work.

Examples

The first example is Klein’s quartic of genus 3, which has equation X3Y + Y3Z + Z3X = 0: