Contents

Preface vii

Introduction 1

Notation and conventions 11

Part A. Heights 13

Chapter I. The concept of a height 15

1. The naive height on the projective space over 15

2. Generalization to number fields 17

3. Geometric interpretation 21

4. The adelic Picard group 25

Chapter II. Conjectures on the asymptotics of points of bounded height 35

1. A heuristic 35

2. The conjecture of Lang 38

3. The conjecture of Batyrev and Manin 40

4. The conjecture of Manin 44

5. Peyre’s constant I—the factor α 47

6. Peyre’s constant II—other factors 50

7. Peyre’s constant III—the actual definition 59

8. The conjecture of Manin and Peyre—proven cases 62

Part B. The Brauer group 81

Chapter III. On the Brauer group of a scheme 83

1. Central simple algebras and the Brauer group of a field 84

2. Azumaya algebras 89

3. The Brauer group 93

4. The cohomological Brauer group 94

5. The relation to the Brauer group of the function field 98

6. The Brauer group and the cohomological Brauer group 101

7. The theorem of Auslander and Goldman 103

8. Examples 107

Chapter IV. An application: the Brauer–Manin obstruction 119

1. Adelic points 119

2. The Brauer–Manin obstruction 122

3. Technical lemmata 126

4. Computing the Brauer–Manin obstruction—the general strategy 129

5. The examples of Mordell 132

v