Contents
Preface vii
Introduction xi
0.1. Motivation and strategy xi
0.2. Rough outline of the theory xiv
0.3. Comparison with other theories xix
Part 1. Main concepts and results 1
Chapter 1. Preliminaries from algebraic geometry 3
1.1. Algebro-geometric terminology 3
1.2. Categorical quotients in algebraic geometry 11
1.3. Analytic uniformization and critical finiteness 15
Chapter 2. Outline of S—geometry 31
2.1. Main concepts of 5—geometry 31
2.2. Main conjectures 55
2.3. Main results: a sample 56
2.4. Appendix: Axiomatic characterization of J—geometries 61
Part 2. General theory 69
Chapter 3. Global theory 71
3.1. p—jet spaces of schemes 71
3.2. Behavior with respect to etale maps 75
3.3. Link between p—jet spaces and 5—functions 78
3.4. Galois covers 82
3.5. S—base loci versus postcritical loci 85
3.6. S—tangent maps and S—differentials of S—functions 88
Chapter 4. Local theory 107
4.1. Local analogue of the main conjectures 107
4.2. Bounding the rank of the module of 5—invariants 119
4.3. 5—invariants for the formal multiplicative group 121
4.4. p—jets of formal group laws 123
4.5. Local versus global picture: fixed points and cycles 126
4.6. Some general global converse results 132
Chapter 5. Birational theory 141
5.1. The basic graded rings 141
5.2. £—Galois groups 145
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