Contents

Preface ix

Introduction 1

References 8

Notation and terminology 9

Chapter 1. Algebraic theory of complex multiplication 13

1.1. Introduction 13

1.2. Simplicity, isotypicity, and endomorphism algebras 15

1.3. Complex multiplication 23

1.4. Dieudonn´ e theory, p-divisible groups, and deformations 33

1.5. CM types 65

1.6. Abelian varieties over finite fields 70

1.7. A theorem of Grothendieck and a construction of Serre 76

1.8. CM lifting questions 86

Chapter 2. CM lifting over a discrete valuation ring 91

2.1. Introduction 91

2.2. Existence of CM lifting up to isogeny 102

2.3. CM lifting to a normal domain up to isogeny: counterexamples 109

2.4. Algebraic Hecke characters 117

2.5. Theory of complex multiplication 127

2.6. Local methods 130

Chapter 3. CM lifting of p-divisible groups 137

3.1. Motivation and background 137

3.2. Properties of a-numbers 143

3.3. Isogenies and duality 146

3.4. Some p-divisible groups with small a-number 156

3.5. Earlier non-liftability results and a new proof 161

3.6. A lower bound on the field of definition 164

3.7. Complex multiplication for p-divisible groups 166

3.8. An upper bound for a field of definition 182

3.9. Appendix: algebraic abelian p-adic representations of local fields 185

3.10. Appendix: questions and examples on extending isogenies 191

Chapter 4. CM lifting of abelian varieties up to isogeny 195

4.1. Introduction 195

4.2. Classification and Galois descent by Lie types 211

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