Contents
Preface vii
Foreword
John Tate ix
Non-archimedean analytic geometry: first steps
Vladimir Berkovich 1
Chapter 1. Several approaches to non-archimedean geometry
Brian Conrad 9
Introduction 9
1. Affinoid algebras 10
2. Global rigid-analytic spaces 15
3. Coherent sheaves and Raynaud’s theory 30
4. Berkovich spaces I 43
5. Berkovich spaces II 52
Bibliography 63
Chapter 2. The p-adic upper half plane
Samit Dasgupta and Jeremy Teitelbaum 65
Introduction 65
1. Geometry of the p-adic upper half plane 66
2. Boundary distributions and integrals 75
3. L-invariants and modular symbols 89
4. Breuil duality and p-adic Langlands theory 106
Bibliography 119
Chapter 3. An introduction to Berkovich analytic spaces and
non-archimedean potential theory on curves
Matthew Baker 123
Introduction 123
1. The Berkovich projective line 125
2. Further examples of Berkovich analytic spaces 134
3. Harmonic functions 141
4. Laplacians 150
5. Introduction to potential theory on Berkovich curves 164
Bibliography 173
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