viii CONTENTS

4.3. Tensor construction for p-divisible groups 224

4.4. Self-duality and CM lifting 228

4.5. Striped and supersingular Lie types 233

4.6. Complex conjugation and CM lifting 240

Appendix A. Some arithmetic results for abelian varieties 249

A.1. The p-part of Tate’s work 249

A.2. The Main Theorem of Complex Multiplication 257

A.3. A converse to the Main Theorem of Complex Multiplication 292

A.4. Existence of algebraic Hecke characters 296

Appendix B. CM lifting via p-adic Hodge theory 321

B.1. A generalization of the toy model 321

B.2. Construct CM lifting by p-adic Hodge theory 333

B.3. Dieudonn´ e theories over a perfect field of characteristic p 343

B.4. p-adic Hodge theory and a formula for the closed fiber 359

Notes on Quotes 371

Glossary of Notations 373

Bibliography 379

Index 385