On the computation of Belyi functions: A numerical approach

We present a new method for the calculation of Belyi functions for a given Dessin d'enfant. Based on Belyi's theorem, we associate to the Dessin d'enfant a finite index subgroup G of a triangle group. The zeros of the Belyi function we want to calculate are the values of the hauptmodul of the subgroup G at its elliptic points. The singularities of the Belyi function are the values at the cusps. Hence, we need to calculate the hauptmodul of G. The idea of Monien was that the hauptmodul is a holomorphic function on a Riemann surface (without the cusps) and has therefore an harmonic real and imaginary part. These harmonic functions can be calculated with the help of numerical methods.

In this talk, I will explain, how we constructed a complete algorithm from this idea for the calculation of Belyi functions. Furthermore, we present some results we where able to obtain with the help of this method.