Numerical calculation of automorphic functions for finite index subgroups of triangle groups

Abstract

We present a new method to calculate automorphic functions for finite index subgroups of triangle groups. Since automorphic functions are holomorphic, it is well known that the real and the imaginary part are both harmonic. The central idea of my advisor Monien was to look at the two parts separately. We solve the Laplace equation to find the real and imaginary part of an automorphic function. This solution can be calculated using numerical methods.<br /> To each finite index subgroup of a triangle group we can associate a Belyi function and a dessin d'enfant. The zeros of this Belyi function are the values of the automorphic function we calculated at elliptic points. Hence, we can find an approximation for the coefficients of the Belyi function. The precision of this approximation is increased by the use of Newton's method. Once we have an approximation with high accuracy, we find the correct algebraic number using the LLL algorithm. From the exact Belyi function we can reconstruct the exact automorphic function. <br /> In order to handle finite index subgroups of triangle groups, we introduce the notion of generalized Farey symbols. These symbols are a generalization of the classical Farey symbols for the modular group. They are used to do efficient calculations with subgroups of Hecke groups.