1. Historical and Introductory Background

This chapter begins with a brief historical introduction to the theory of dessins d’enfants, from the early discovery of the platonic solids, through nineteenth-century work on Riemann surfaces, algebraic curves and holomorphic functions, and twentieth-century research on regular maps, to the fundamental and far-reaching ideas circulated by Grothendieck in the 1980s, and subsequent efforts to implement his programme. After this we summarise the background knowledge we will assume, together with suggestions for further reading. The second section gives a brief introduction to compact Riemann surfaces, including the Riemann-Hurwitz formula for the genus of a surface, and the equivalence of the categories of compact Riemann surfaces and of smooth complex projective algebraic curves. Elliptic curves (Riemann surfaces of genus 1) are treated in detail, as simple examples of subtler phenomena encountered later. The third section contains technical results on the existence of meromorphic functions with specific properties. In the final section we define Belyĭ functions and prove one direction of Belyĭ’s theorem, that such functions characterise algebraic curves defined over number fields, by using an algorithm which constructs a Belyĭ function on such a curve. We give a first definition of dessins d’enfants as the pre-images of the unit real interval [0, 1] under Belyĭ functions, and we discuss several simple examples of dessins.