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.

Ernesto Girondo (Author), Gabino González-Diez (Author)

9780521519632, Cambridge University Press

Hardback, published 22 December 2011

312 pages, 90 b/w illus.
22.9 x 15.2 x 2.1 cm, 0.57 kg

"Overall the text is very well written and easy to follow, partly due to the abundance of good concrete examples in every single section illustrating concepts from the very basic to the very technical."
Aaron D. Wootton, Mathematical Reviews

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.

1. Riemann surfaces and algebraic curves
2. Riemann surfaces and Fuchsian groups
3. Belyi's theorem
4. Dessins d'enfants
References
Index.

Subject Areas: Algebraic topology [PBPD], Complex analysis, complex variables [PBKD]