By Brian Conrad, Frans Oort
Abelian types with advanced multiplication lie on the origins of sophistication box idea, they usually play a vital position within the modern thought of Shimura forms. they're specific in attribute zero and ubiquitous over finite fields. This ebook explores the connection among such abelian forms over finite fields and over arithmetically attention-grabbing fields of attribute zero through the learn of numerous normal CM lifting difficulties which had formerly been solved purely in designated circumstances. as well as giving whole options to such questions, the authors supply quite a few examples to demonstrate the final conception and current a close remedy of many basic effects and ideas within the mathematics of abelian types, reminiscent of the most Theorem of complicated Multiplication and its generalizations, the finer features of Tate's paintings on abelian types over finite fields, and deformation idea. This ebook presents a terrific representation of ways glossy innovations in mathematics geometry (such as descent idea, crystalline tools, and staff schemes) may be fruitfully mixed with type box thought to respond to concrete questions on abelian forms. will probably be an invaluable reference for researchers and complicated graduate scholars on the interface of quantity conception and algebraic geometry.
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