Introduction to Belyi functions and Dessins d'enfants

Abstract

[en] The aim of this work is to introduce the reader to the world of dessins d'enfants and the action of the absolute Galois group of the rational numbers, that is the Galois group of the field extension $\overline{\mathbb{Q}} \mid \mathbb{Q}$, where $\overline{\mathbb{Q}}$ stands for the algebraic closure of the rationals, on them. In order to do that, we first define the concept of Riemann surface and check its correspondence with algebraic curves and fields of functions. Then, we prove the key theorem that made possible the action just mentioned, the Belyi Theorem. Finally, in the last chapter we deal with this action, its invariants and give some explicit examples of Belyi functions on the Riemann sphere.