##
**Introduction to compact Riemann surfaces and dessins d’enfants.**
*(English)*
Zbl 1253.30001

London Mathematical Society Student Texts 79. Cambridge: Cambridge University Press (ISBN 978-0-521-74022-7/pbk; 978-0-521-51963-2/hbk). xii, 298 p. (2012).

As the authors point out in the preface, the present text is an expanded version of the lecture notes for a course on Riemann surfaces and A. Grothendieck’s “dessins d’enfants” (French for “children’s drawings”), which has been taught for several years to students of the masters degree in mathematics at Universidad Autónoma de Madrid, Spain. Accordingly, the text is geared toward seasoned students who have taken courses in linear algebra, general topology, field theory, and basic complex analysis. Otherwise, the prerequisites are kept to a minimum. As the title of the book indicates, the material is divided into two major parts, consisting of two chapters each. Whilst the first part provides an elementary introduction to the classical theory of compact Riemann surfaces, the second part is devoted to the much more recent topic of Bely theory and the allied framework of “dessins d’enfants”. In this regard, the book under review is among the very few primers on Riemann surfaces that also cover the comparatively modern aspects of the latter topic. Actually, the excellent textbook “Algèbre et théories galoisiennes” by R. Douady and A. Douady [Algèbre et théories galoisiennes. 2ème éd., revue et augmentée. Paris: Cassini (2005; Zbl 1076.12004)] covers much related material, also in an introductory, however by far more abstract fashion, and may therefore be seen as a useful companion to the present representation.

As for the precise contents, the first chapter of the book gives the basic definitions concerning Riemann surfaces, including holomorphic maps, meromorphic functions, differentials, automorphisms, algebraic curves, and numerous concrete examples. Furthermore, the topology of Riemann surfaces, coverings, the Riemann-Hurwitz formula for ramified coverings, the function field of a compact Riemann surface, and the functorial equivalences between compact Riemann surfaces, function fields in one variable, and irreducible algebraic curves, respectively, are discussed in great detail.

Chapter 2 extends the study of compact Riemann surfaces by using the uniformization theorem as a basic tool, thereby explaining the existence of sufficiently many meromorphic functions, Fuchsian groups and their fundamental domains, hyperbolic geometry and Fuchsian triangle groups, automorphism groups of compact Riemann surfaces, monodromy groups, Galois coverings, and the normalization of a covering of \(\mathbb{P}^1_{\mathbb{C}}\).

In the remaining two chapters of the book, this approach to compact Riemann surfaces (or algebraic curves, respectively) is applied to give an introduction to the Grothendieck-Belyi theory of “dessins d’enfants”, and its connection to Riemann surfaces definable over Chapter 3 is devoted to a proof of G. V. Belyi’s celebrated theorem from [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267–276 (1979; Zbl 0409.12012)], which is stated here in the following form: Let \(C\) be a compact Riemann surface. Then the algebraic curve associated to \(C\) is given by a polynomial \(F(X,Y)\) with coefficients in \(\overline{\mathbb{Q}}\subset\mathbb{C}\) if and only if there is a morphism \(f:C\to\mathbb{P}^1_{\mathbb{C}}\) with at most three branch points.

The proof given here is a novel, tailor-made and more elementary variant of the original proof, which entirely remains within the scope of the present text. The authors’ criterion for definability over \(\overline{\mathbb{Q}}\) was first published by the second author in [Q. J. Math. 57, No. 3, 339–354 (2006; Zbl 1123.14016)], and this is the crucial ingredient of the proof of Belyi’s theorem as presented in this chapter of the book.

Chapter 4 introduces Grothendieck’s “dessins d’enfants”, that is, a type of graph drawing suitable for the study of Riemann surfaces and their combinatorial invariants with respect to the action of the Galois group \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on certain coverings (Belyi pairs).

The main goal, in this context, is to give a proof of the so-called Grothendieck correspondence. This fascinating result from the 1980s establishes a one-to-one correspondence between graphs dividing an orientable surface into a disjoint union of cells on the one hand, and algebraic curves \(C\) endowed with a function \(f:C\to P^1_{\mathbb{C}}\) ramified over three points, both with coefficients in \(\overline{\mathbb{Q}}\), on the other. These particular graphs are called “dessins d’enfants”, whereas the above data \((C,f)\) are commonly known as “Belyi pairs”. The authors give a very detailed description of Grothendieck’s correspondence, including the study of Belyi pairs via Fuchsian groups and some properties of the action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on “dessins d’enfants”, thereby instructively-illustrating this beautiful relationship between topology, graph theory, complex analysis, algebra, and arithmetic geometry. In particular, this chapter is of very topical character and leads the reader to the forefront of current research in Grothendieck-Belyi theory.

Apart from the overall utmost lucid, detailed, and careful presentation of the material, another outstanding feature of the book under review is the great wealth of concrete, instructive examples illustrating the respective theoretical topics. In fact, about one third of the entire text is devoted to completely worked out examples and clarifying remarks, which certainly helps the reader understand the various abstract concepts, methods, and interrelations discussed in the course of the text. On the other hand, there are no exercises accompanying the main text. Nevertheless, the ambitious reader might try to work out many of the given examples independently, which would provide a large number of exercises, too.

All in all, this textbook gives a very profound first introduction to compact Riemann surfaces and the related Grothendieck-Belyi theory via “dessins d’enfants”, and it should be seen as a solid, highly valuable basis for the study of the current research literature in the field simultaneously.

As for the precise contents, the first chapter of the book gives the basic definitions concerning Riemann surfaces, including holomorphic maps, meromorphic functions, differentials, automorphisms, algebraic curves, and numerous concrete examples. Furthermore, the topology of Riemann surfaces, coverings, the Riemann-Hurwitz formula for ramified coverings, the function field of a compact Riemann surface, and the functorial equivalences between compact Riemann surfaces, function fields in one variable, and irreducible algebraic curves, respectively, are discussed in great detail.

Chapter 2 extends the study of compact Riemann surfaces by using the uniformization theorem as a basic tool, thereby explaining the existence of sufficiently many meromorphic functions, Fuchsian groups and their fundamental domains, hyperbolic geometry and Fuchsian triangle groups, automorphism groups of compact Riemann surfaces, monodromy groups, Galois coverings, and the normalization of a covering of \(\mathbb{P}^1_{\mathbb{C}}\).

In the remaining two chapters of the book, this approach to compact Riemann surfaces (or algebraic curves, respectively) is applied to give an introduction to the Grothendieck-Belyi theory of “dessins d’enfants”, and its connection to Riemann surfaces definable over Chapter 3 is devoted to a proof of G. V. Belyi’s celebrated theorem from [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267–276 (1979; Zbl 0409.12012)], which is stated here in the following form: Let \(C\) be a compact Riemann surface. Then the algebraic curve associated to \(C\) is given by a polynomial \(F(X,Y)\) with coefficients in \(\overline{\mathbb{Q}}\subset\mathbb{C}\) if and only if there is a morphism \(f:C\to\mathbb{P}^1_{\mathbb{C}}\) with at most three branch points.

The proof given here is a novel, tailor-made and more elementary variant of the original proof, which entirely remains within the scope of the present text. The authors’ criterion for definability over \(\overline{\mathbb{Q}}\) was first published by the second author in [Q. J. Math. 57, No. 3, 339–354 (2006; Zbl 1123.14016)], and this is the crucial ingredient of the proof of Belyi’s theorem as presented in this chapter of the book.

Chapter 4 introduces Grothendieck’s “dessins d’enfants”, that is, a type of graph drawing suitable for the study of Riemann surfaces and their combinatorial invariants with respect to the action of the Galois group \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on certain coverings (Belyi pairs).

The main goal, in this context, is to give a proof of the so-called Grothendieck correspondence. This fascinating result from the 1980s establishes a one-to-one correspondence between graphs dividing an orientable surface into a disjoint union of cells on the one hand, and algebraic curves \(C\) endowed with a function \(f:C\to P^1_{\mathbb{C}}\) ramified over three points, both with coefficients in \(\overline{\mathbb{Q}}\), on the other. These particular graphs are called “dessins d’enfants”, whereas the above data \((C,f)\) are commonly known as “Belyi pairs”. The authors give a very detailed description of Grothendieck’s correspondence, including the study of Belyi pairs via Fuchsian groups and some properties of the action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on “dessins d’enfants”, thereby instructively-illustrating this beautiful relationship between topology, graph theory, complex analysis, algebra, and arithmetic geometry. In particular, this chapter is of very topical character and leads the reader to the forefront of current research in Grothendieck-Belyi theory.

Apart from the overall utmost lucid, detailed, and careful presentation of the material, another outstanding feature of the book under review is the great wealth of concrete, instructive examples illustrating the respective theoretical topics. In fact, about one third of the entire text is devoted to completely worked out examples and clarifying remarks, which certainly helps the reader understand the various abstract concepts, methods, and interrelations discussed in the course of the text. On the other hand, there are no exercises accompanying the main text. Nevertheless, the ambitious reader might try to work out many of the given examples independently, which would provide a large number of exercises, too.

All in all, this textbook gives a very profound first introduction to compact Riemann surfaces and the related Grothendieck-Belyi theory via “dessins d’enfants”, and it should be seen as a solid, highly valuable basis for the study of the current research literature in the field simultaneously.

Reviewer: Werner Kleinert (Berlin)

### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30F10 | Compact Riemann surfaces and uniformization |

11G32 | Arithmetic aspects of dessins d’enfants, Belyĭ theory |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

14H15 | Families, moduli of curves (analytic) |

14H30 | Coverings of curves, fundamental group |