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Complex Multiplication and Lifting Problems
 
Ching-Li Chai University of Pennsylvania, Philadelphia, PA
Brian Conrad Stanford University, Stanford, CA
Frans Oort University of Utrecht, Utrecht, The Netherlands
Front Cover for Complex Multiplication and Lifting Problems
Available Formats:
Hardcover ISBN: 978-1-4704-1014-8
Product Code: SURV/195
387 pp 
List Price: $106.00
MAA Member Price: $95.40
AMS Member Price: $84.80
Electronic ISBN: 978-1-4704-1474-0
Product Code: SURV/195.E
387 pp 
List Price: $100.00
MAA Member Price: $90.00
AMS Member Price: $80.00
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $159.00
MAA Member Price: $143.10
AMS Member Price: $127.20
Front Cover for Complex Multiplication and Lifting Problems
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  • Front Cover for Complex Multiplication and Lifting Problems
  • Back Cover for Complex Multiplication and Lifting Problems
Complex Multiplication and Lifting Problems
Ching-Li Chai University of Pennsylvania, Philadelphia, PA
Brian Conrad Stanford University, Stanford, CA
Frans Oort University of Utrecht, Utrecht, The Netherlands
Available Formats:
Hardcover ISBN:  978-1-4704-1014-8
Product Code:  SURV/195
387 pp 
List Price: $106.00
MAA Member Price: $95.40
AMS Member Price: $84.80
Electronic ISBN:  978-1-4704-1474-0
Product Code:  SURV/195.E
387 pp 
List Price: $100.00
MAA Member Price: $90.00
AMS Member Price: $80.00
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $159.00
MAA Member Price: $143.10
AMS Member Price: $127.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1952014
    MSC: Primary 11; 14;

    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
     
     
    • Chapters
    • Introduction
    • Chapter 1. Algebraic theory of complex multiplication
    • Chapter 2. CM lifting over a discrete valuation ring
    • Chapter 3. CM lifting of $p$-divisible groups
    • Chapter 4. CM lifting of abelian varieties up to isogeny
    • Appendix A. Some arithmetic results for abelian varieties
    • Appendix B. CM lifting via $p$-adic Hodge theory
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Volume: 1952014
MSC: Primary 11; 14;

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.

  • Chapters
  • Introduction
  • Chapter 1. Algebraic theory of complex multiplication
  • Chapter 2. CM lifting over a discrete valuation ring
  • Chapter 3. CM lifting of $p$-divisible groups
  • Chapter 4. CM lifting of abelian varieties up to isogeny
  • Appendix A. Some arithmetic results for abelian varieties
  • Appendix B. CM lifting via $p$-adic Hodge theory
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