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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