Riemann surfaces and dessins d'enfants

Abstract

In this thesis, we will discuss the basic theory of Riemann surfaces and prove the equivalence between the categories of compact Riemann surfaces, irreducible, non-singular algebraic curves over the complex plane and function fields in one variable over the complex plane as field of constants. We will state and prove Belyi's theorem and define dessin d'enfants as a consequence of this theorem. We will end with a short discussion on the action of the absolute Galois group on these dessins.