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)
DOI: https://doi.org/http://dx.doi.org/10.1090/surv/195
MathSciNet review: 3137398
MSC: Primary 14K02; Secondary 11G15, 14D15, 14K15, 14L05
Read more about this volume
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
Front/Back Matter
Chapters
View full volume PDF