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E-raamat: Dessins d'Enfants on Riemann Surfaces

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This volume provides an introduction to dessins d"enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces.  Dessins d"Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.

Arvustused

The book under review provides a very useful insight into the theory of dessins, and on its deep relationship with a number of mathematical distant fields. It is excellently written, and the Chapters are somewhat independent, in the sense that they have their own abstract, keywords and references. Of course, it is worth noting that the authors have contributed themselves to many of the results included in the book, most of which are really recent. (José Javier Etayo, zbMATH, August, 2017)

This book allows the reader familiar with the analytic aspects of the theory of Riemann surfaces to develop a feeling for the arithmetic aspects of these objects. The reader interested in diving deeper into any of these aspects will most certainly find this well-written book to be of much help. (Ariyan Javanpeykar, Mathematical Reviews, January, 2017)

The book is intended for properly trained mathematics graduate students or researchers. The text is also regularly interrupted by formulations of exercises which ask to prove some intermediate result or apply a definition in a particular situation (some brief hints are added at the end of the book). The book can thus be used as a textbook for a course on this fascinating topic. (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, April, 2016)

Part I Basic Material
1 Historical and Introductory Background
3(38)
1.1 Introduction
3(3)
1.1.1 History: Topics of the Book
3(2)
1.1.2 Prerequisities: Suggestions for the Reader
5(1)
1.2 Compact Riemann Surfaces and Algebraic Curves
6(18)
1.2.1 Examples and Some Basic Facts
6(7)
1.2.2 Algebraic Curves
13(2)
1.2.3 An Alternative Approach to Riemann Surfaces of Genus 1
15(2)
1.2.4 A Moduli Problem for Tori
17(5)
1.2.5 Sketch of a Proof of Theorem 1.1
22(2)
1.3 Appendix: Existence of Enough Meromorphic Functions
24(2)
1.4 Belyi Functions and Their Dessins
26(15)
1.4.1 Belyi's Theorem
26(1)
1.4.2 Existence of Belyi Functions: Simple Examples
27(2)
1.4.3 Dessins
29(4)
1.4.4 Belyi Algorithms
33(4)
References
37(4)
2 Graph Embeddings
41(18)
2.1 Graphs, Maps, and Hypermaps
41(13)
2.1.1 Bipartite Maps
41(4)
2.1.2 Algebraic Bipartite Maps
45(3)
2.1.3 Dessins as Hypermaps
48(3)
2.1.4 Morphisms of Hypermaps
51(1)
2.1.5 Maps and Hypermaps
51(2)
2.1.6 An Instructive Example
53(1)
2.2 Appendix: The Finite Simple Groups
54(5)
2.2.1 The Cyclic Groups of Prime Order
54(1)
2.2.2 The Alternating Groups
55(1)
2.2.3 The Simple Groups of Lie Type
55(1)
2.2.4 The Sporadic Simple Groups
56(1)
References
57(2)
3 Dessins and Triangle Groups
59(30)
3.1 Uniformisation and Fuchsian Groups
59(19)
3.1.1 Uniformisation
60(4)
3.1.2 Triangle Groups
64(4)
3.1.3 More General Facts About Fuchsian Groups
68(3)
3.1.4 Triangle Groups and Belyi Functions
71(1)
3.1.5 Inclusions Between Fuchsian Groups
72(4)
3.1.6 Klein's Quartic Curve
76(2)
3.2 Appendix: Group Presentations
78(2)
3.3 From Dessins to Holomorphic Structures
80(9)
3.3.1 Coverings
80(1)
3.3.2 Triangle Groups and Bipartite Maps
81(2)
3.3.3 Holomorphic Structures
83(2)
3.3.4 Non-cocompact Triangle Groups
85(1)
References
86(3)
4 Galois Actions
89(26)
4.1 Galois Theory
89(7)
4.1.1 Basic Galois Theory
89(2)
4.1.2 The Absolute Galois Group
91(3)
4.1.3 Coverings and Galois Groups
94(2)
4.2 Moduli Fields and Fields of Definition
96(11)
4.2.1 Basic Facts and Definitions
96(5)
4.2.2 Galois Action: An Example and Some Invariants
101(2)
4.2.3 Non-invariants of Galois Action
103(1)
4.2.4 Faithful Galois Action on Families of Dessins
104(3)
4.3 Appendix: Another Proof of Theorem 4.9
107(8)
References
110(5)
Part II Regular Dessins
5 Quasiplatonic Surfaces, and Automorphisms
115(38)
5.1 Quasiplatonic Surfaces: Construction and Counting
115(16)
5.1.1 Definitions and Properties
115(2)
5.1.2 Hurwitz Groups and Surfaces
117(4)
5.1.3 Kernels and Epimorphisms
121(2)
5.1.4 Direct Counting
123(2)
5.1.5 Counting by Character Theory
125(3)
5.1.6 Counting by Mobius Inversion
128(3)
5.2 Low Genera
131(8)
5.2.1 Genus 2
131(3)
5.2.2 Genus 3
134(2)
5.2.3 Genus 4
136(3)
5.3 Infinite Families
139(3)
5.4 Fields of Definition Again
142(3)
5.5 Appendix: Linear and Projective Groups
145(8)
References
149(4)
6 Regular Maps
153(10)
6.1 Regularity
153(3)
6.2 Classification by Genus
156(3)
6.3 Classification by Group
159(4)
References
161(2)
7 Regular Embeddings of Complete Graphs
163(16)
7.1 Examples of Regular Embeddings
163(4)
7.1.1 Examples of Genus 0 and 1
164(2)
7.1.2 Cayley Maps
166(1)
7.2 The Biggs Maps
167(12)
7.2.1 Construction of the Biggs Maps
167(2)
7.2.2 Frobenius Groups
169(1)
7.2.3 Classifying the Regular Embeddings
170(5)
7.2.4 Regular Embeddings and Cyclotomic Polynomials
175(2)
References
177(2)
8 Wilson Operations
179(14)
8.1 A Class of Map Operations
179(3)
8.2 Wilson Operations and Galois Actions
182(3)
8.3 The Group of Operations on Dessins
185(8)
References
191(2)
9 Further Examples
193(20)
9.1 Galois Orbits
193(10)
9.1.1 Generalised Paley Maps
194(3)
9.1.2 Complete Bipartite Maps
197(6)
9.2 Regular Dessins and Equations
203(10)
9.2.1 Kp,q-Dessins, Classification
203(1)
9.2.2 Kp,q-Dessins, Abelian Case
203(2)
9.2.3 Kp,q-Dessins, Semidirect Product Case
205(2)
9.2.4 Equations
207(2)
References
209(4)
Part III Applications
10 Arithmetic Aspects
213(12)
10.1 Abc for Function Fields
213(5)
10.1.1 Digression: Motivation from Number Theory
213(2)
10.1.2 The Polynomial Case
215(1)
10.1.3 Abc on Riemann Surfaces
216(1)
10.1.4 Belyi Functions: The Worst Case
217(1)
10.2 Genus 1 Dessins and Complex Multiplication
218(7)
10.2.1 Unramified Self-covers of Elliptic Curves
219(1)
10.2.2 Complex Multiplication by Roots of Unity
220(1)
10.2.3 Complex Multiplication, General Case
221(3)
References
224(1)
11 Beauville Surfaces
225(26)
11.1 Basic Properties and First Examples
225(8)
11.1.1 Basic Definitions
225(2)
11.1.2 Beauville's Original Example
227(3)
11.1.3 Enumeration of Beauville Surfaces
230(3)
11.2 Beauville Structures for Specific Families of Groups
233(6)
11.2.1 Beauville Surfaces and Simple Groups
233(2)
11.2.2 Beauville Structures for Symmetric Groups
235(2)
11.2.3 Beauville Structures for Alternating Groups
237(1)
11.2.4 Beauville Structures for L2(q)
238(1)
11.3 Further Properties of Beauville Surfaces
239(12)
11.3.1 Invariants of a Beauville Surface
239(4)
11.3.2 Real Beauville Surfaces
243(2)
11.3.3 The Action of the Absolute Galois Group on Beauville Surfaces
245(2)
References
247(4)
Hints for Selected Exercises 251(6)
Index 257
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