Complex Multiplication and Lifting Problems
Ching-Li Chai · Brian Conrad · Frans Oort
dec. 2013 · Mathematical Surveys and Monographs Cartea 195 · American Mathematical Soc.
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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.;)
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.
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