Introduction to Compact Riemann Surfaces and Dessins d’Enfants

Cover......Page 1
LONDON MATHEMATICAL SOCIETY STUDENT TEXTS......Page 2
Frontispiece......Page 3
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 10
1.1.1 Riemann surfaces – examples......Page 14
1.1.2 Morphisms of Riemann surfaces......Page 23
1.2.1 The topological surface underlying a compact Riemann surface......Page 39
1.2.2 The fundamental group......Page 49
1.2.3 The Euler–Poincar´ e characteristic......Page 52
1.2.4 The Riemann–Hurwitz formula for morphisms to the sphere......Page 54
1.2.5 Coverings......Page 60
1.2.6 Ramified coverings......Page 71
1.2.7 Auxiliary results about the compactification of Riemann surfaces and extension of maps......Page 76
1.3 Curves, function fields and Riemann surfaces......Page 80
1.3.1 The function field of a Riemann surface......Page 84
2.1 Uniformization......Page 94
2.1.1 PSL(2,R) as the group of isometries of hyperbolic
space spaces......Page 96
2.1.2 Groups uniformizing Riemann surfaces of genus g ≥ 2......Page 102
2.2 The existence of meromorphic functions......Page 113
2.2.1 Existence of functions in genus g = 1......Page 114
2.2.2 Existence of functions in genus g ≥ 2......Page 117
2.3 Fuchsian groups......Page 120
2.4.1 Triangles in hyperbolic space......Page 125
2.4.3 Construction of triangle groups......Page 128
2.4.4 The modular group PSL(2,Z)......Page 134
2.5 Automorphisms of Riemann surfaces......Page 139
2.5.1 The action of the automorphism group on the function field......Page 146
2.5.2 Uniformization of Klein’s curve of genus three......Page 151
2.6.1 The moduli space M1......Page 157
2.6.2 The moduli space Mg for g > 1......Page 160
2.7 Monodromy......Page 161
2.7.1 Monodromy and Fuchsian groups......Page 163
2.7.2 Characterization of a morphism by its monodromy......Page 165
2.8 Galois coverings......Page 168
2.9 Normalization of a covering of P1......Page 170
2.9.1 The covering group of the normalization......Page 173
3
 Belyi’s Theorem......Page 182
3.1 Proof of part (a) ⇒ (b) of Belyi’s Theorem......Page 184
3.1.1 Belyi’s second proof of part (a) ⇒ (b)......Page 188
3.2 Algebraic characterization of morphisms......Page 189
3.3 Galois action......Page 193
3.4 Points and valuations......Page 196
3.4.1 Galois action on points......Page 204
3.5 Elementary invariants of the action of Gal(C)......Page 206
3.6 A criterion for definability over Q......Page 208
3.6.1 Proof of part (b) ⇒ (a) of Belyi’s Theorem......Page 209
3.7 Proof of the criterion for definibility over Q......Page 210
3.7.1 Specialization of transcencendental coeffcients......Page 211
3.7.2 Infinitesimal specializations......Page 212
3.7.3 End of the proof......Page 215
3.8 The field of definition of Belyi functions......Page 217
4.1 Definition and first examples......Page 220
4.1.1 The permutation representation pair of a dessin......Page 222
4.2 From dessins d’enfants to Belyi pairs......Page 225
4.2.1 The triangle decomposition associated to a dessin......Page 226
4.2.2 The Belyi function associated to a dessin......Page 228
4.3 From Belyi pairs to dessins......Page 236
4.3.1 The monodromy of a Belyi pair......Page 241
4.4 Fuchsian group description of Belyi pairs......Page 242
4.4.1 Uniform dessins......Page 251
4.4.2 Automorphisms of a dessin......Page 254
4.4.3 Regular dessins......Page 257
4.5 The action of Gal(Q) on dessins d’enfants......Page 263
4.5.1 Faithfulness on dessins of genus 0......Page 265
4.5.3 Faithfulness on dessins of genus g > 1......Page 268
4.6.1 Some dessins of genus 0......Page 270
4.6.2 Examples in genus g = 1......Page 287
4.6.3 Some examples in genus g ≥ 2......Page 295
References......Page 304
Index......Page 310