Introduction to Compact Riemann Surfaces and Dessins d’Enfants

An elementary account of the theory of compact Riemann surfaces and an introduction to the Belyi–Grothendieck theory of dessins d'enfants.

Starting with a friendly account of the theory of compact Riemann surfaces, this 2011 book then introduces the Belyi–Grothendieck theory of dessins d'enfants, taking the reader with no previous knowledge of the subject to the forefront of current research. Many worked examples and illustrations are provided.Few books on the subject of Riemann surfaces cover the relatively modern theory of dessins d'enfants (children's drawings), which was launched by Grothendieck in the 1980s and is now an active field of research. In this 2011 book, the authors begin with an elementary account of the theory of compact Riemann surfaces viewed as algebraic curves and as quotients of the hyperbolic plane by the action of Fuchsian groups of finite type. They then use this knowledge to introduce the reader to the theory of dessins d'enfants and its connection with algebraic curves defined over number fields. A large number of worked examples are provided to aid understanding, so no experience beyond the undergraduate level is required. Readers without any previous knowledge of the field of dessins d'enfants are taken rapidly to the forefront of current research.