Ball and Surface Arithmetics

Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 6
1.1 Cyclic Points......Page 14
1.2 Graphs of Abelian Points......Page 18
1.3 Geometric Interpretation......Page 20
1.4 Derived Representations......Page 23
1.5 The Differential Relation......Page 28
1.6 Stepwise Resolutions of Cyclic Points......Page 32
1.7 Continued Fractions and Selfintersection Numbers......Page 34
1.8 Reciprocity Law for Geometric Sums......Page 41
1.9 Explicit Dedekind Sums......Page 44
1.10 Eisenstein Sums......Page 47
1.11 Hirzebruch\'s Sum......Page 50
1.12 Geometric Interpretation......Page 53
1.13 Quotients and Coverings of Modifications......Page 59
1.14 Selfintersections of Quotient Curves......Page 62
1.15 The Bridge Algorithm......Page 66
1.16 First Orbital Properties......Page 69
1.17 Local Orbital Euler Numbers......Page 76
1.18 Absorptive Numbers......Page 83
2.1 Point Arrangements on Curves......Page 89
2.2 Euler Heights of Orbital Curves......Page 91
2.3 The Geometric Local-Global Principle......Page 95
2.4 Signature Heights of Orbital Curves......Page 98
3.1 Regular Arrangements on Surfaces......Page 107
3.2 Basic Invariants and Fixed Point Theorem......Page 111
3.3 EULER Heights......Page 120
3.4 Signature Heights......Page 128
3.5 Quasi-homogeneous Points, Quotient Points and Cusp Points......Page 137
3.6 Quasi-smooth Orbital Surfaces......Page 149
3.7 Open Orbital Surfaces......Page 158
3.8 Orbital Decompositions......Page 166
4.1 Ball Lattices......Page 179
4.2 Neat Ball Cusp Lattices......Page 183
4.3 Invariants of Neat Ball Quotient Surfaces......Page 192
4.4 r -Rational Discs......Page 199
4.5 Cusp Singularities, Reflections and Elliptic Points......Page 208
4.6 Orbital Ball Quotient Surfaces and Molecular......Page 213
4.7 Invariants of Disc Quotient Curves......Page 226
4.8 Invariants of Ball Quotient Surfaces......Page 232
4,9 Global Proportionality......Page 244
4.10 Orbital Decompositions and the Finiteness Theorem......Page 246
4.11 Leading Examples......Page 252
4.12 Towards the Count of Ball Metrics on Non-Compact Surfaces......Page 266
5.1 Classification Diagram......Page 272
5.2 Picard Modular Surface of the Field of Eisenstein Numbers......Page 274
5.3 Picard Modular Surface of the Field of Gauss-Numbers......Page 278
5.4 Kodaira Classification of Picard Modular Surfaces......Page 287
5.5 Special Results and Examples......Page 305
5A Volumes of Fundamental Domains of Picard Modular Groups......Page 313
5A.1 The Order of Finite Unitary Groups......Page 314
5A.2 Index of Congruence Subgroups......Page 321
SA.3 Local Volutnina......Page 329
5A.4 The Global Volume......Page 334
6.1 Introduction......Page 343
6.2 Arrangements with Rational Coefficients......Page 346
6.3 Finite Morphisms of Q`Orbital Surfaces......Page 350
6.4 Functorial Properties for Rational Invariants......Page 355
6.5 Euler and Signature Heights......Page 360
6.6 Reduction of Galois-Finite Morphisms......Page 365
6.7 Local Base Changes......Page 372
6.8 Global Base Changes......Page 376
6.9 Explicit Hurwitz Formulas for Finite Surface Coverings......Page 387
6.10 Finite Coverings of Ruled Surfaces and the Inequality ci < 2c2......Page 403
Index......Page 414
Bibliography......Page 419