Complex multiplication and lifting problems
By: Ching-Li Chai, Brian Conrad, Frans Oort
Contributors: Conrad, Brian, 1970- author; Oort, Frans, 1935- author
Series: Mathematical surveys and monographs -- volume 195
Publisher: American Mathematical Society
Published: 2014
Place of publication: Providence, Rhode Island
Language: eng
Number of pages: 387
ISBN-13: 9781470410148
Subjects: Complex Multiplication; Abelian varieties; Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Complex multiplication and moduli of abelian varieties; Algebraic geometry -- Abelian varieties and schemes -- Isogeny; Algebraic geometry -- Algebraic groups -- Formal groups, $p$-divisible groups; Algebraic geometry -- Abelian varieties and schemes -- Arithmetic ground fields; Algebraic geometry -- Families, fibrations -- Formal methods; deformations
Description: 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. -- Provided by publisher.
Notes: Includes bibliographical references (pages 379-383) and index.