Complex multiplication and lifting problems
著者
書誌事項
Complex multiplication and lifting problems
（Mathematical surveys and monographs, v. 195）
American Mathematical Society, c2014
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注記
Includes bibliographical references (p. 379-383) and index
内容説明・目次
内容説明
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 generalisations, 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.
目次
Introduction
Algebraic theory of complex multiplication
CM lifting over a discrete valuation ring
CM lifting of $p$-divisible groups
CM lifting of abelian varieties up to isogeny
Some arithmetic results for abelian varieties
CM lifting via $p$-adic Hodge theory
Notes on quotes
Glossary of notations
Bibliography
Index
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