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 vii

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