**Mathematical Surveys and Monographs**

Volume: 195;
2014;
387 pp;
Hardcover

MSC: Primary 11; 14;

Print ISBN: 978-1-4704-1014-8

Product Code: SURV/195

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

**Electronic ISBN: 978-1-4704-1474-0
Product Code: SURV/195.E**

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

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#### Supplemental Materials

# Complex Multiplication and Lifting Problems

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*Ching-Li Chai; Brian Conrad; Frans Oort*

Abelian varieties with complex multiplication lie at the origins of
class field theory, and they play a central role in the contemporary
theory of Shimura varieties. They are special in characteristic 0 and
ubiquitous over finite fields. This book explores the relationship
between such abelian varieties over finite fields and over
arithmetically interesting fields of characteristic 0 via the study of
several natural CM lifting problems which had previously been solved
only in special cases. In addition to giving complete solutions to
such questions, the authors provide numerous examples to illustrate
the general theory and present a detailed treatment of many
fundamental results and concepts in the arithmetic of abelian
varieties, such as the Main Theorem of Complex Multiplication and its
generalizations, the finer aspects of Tate's work on abelian varieties
over finite fields, and deformation theory.

This book provides an ideal illustration of how modern techniques
in arithmetic geometry (such as descent theory, crystalline methods,
and group schemes) can be fruitfully combined with class field theory
to answer concrete questions about abelian varieties. It will be a
useful reference for researchers and advanced graduate students at the
interface of number theory and algebraic geometry.

#### Readership

Graduate students and research mathematicians interested in algebraic number theory and algebraic geometry.

#### Table of Contents

# Table of Contents

## Complex Multiplication and Lifting Problems

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Introduction 112 free
- Algebraic theory of complex multiplication 1324 free
- CM lifting over a discrete valuation ring 91102
- CM lifting of 𝑝-divisible groups 137148
- CM lifting of abelian varieties up to isogeny 195206
- Some arithmetic results for abelian varieties 249260
- CM lifting via 𝑝-adic Hodge theory 321332
- Notes on quotes 371382
- Glossary of notations 373384
- Bibliography 379390
- Index 385396 free
- Back Cover Back Cover1402