Complex Multiplication and Lifting Problems
About this Title
Ching-Li Chai, University of Pennsylvania, Philadelphia, PA, Brian Conrad, Stanford University, Stanford, CA and Frans Oort, University of Utrecht, Utrecht, The Netherlands
Publication: Mathematical Surveys and Monographs
Publication Year 2014: Volume 195
ISBNs: 978-1-4704-1014-8 (print); 978-1-4704-1474-0 (online)
MathSciNet review: 3137398
MSC: Primary 14K02; Secondary 11G15, 14D15, 14K15, 14L05
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.
Graduate students and research mathematicians interested in algebraic number theory and algebraic geometry.
Table of Contents
- 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