By Brian Conrad, Frans Oort
Abelian types with advanced multiplication lie on the origins of sophistication box conception, and so they play a important position within the modern idea of Shimura types. they're specified in attribute zero and ubiquitous over finite fields. This e-book explores the connection among such abelian forms over finite fields and over arithmetically attention-grabbing fields of attribute zero through the examine of a number of ordinary CM lifting difficulties which had formerly been solved simply in distinct situations. as well as giving whole ideas to such questions, the authors offer a variety of examples to demonstrate the final conception and current a close therapy of many primary effects and ideas within the mathematics of abelian forms, equivalent to the most Theorem of advanced Multiplication and its generalizations, the finer features of Tate's paintings on abelian kinds over finite fields, and deformation thought. This booklet presents an incredible representation of ways smooth thoughts in mathematics geometry (such as descent conception, crystalline tools, and team schemes) will be fruitfully mixed with classification box idea to respond to concrete questions on abelian types. it will likely be an invaluable reference for researchers and complicated graduate scholars on the interface of quantity idea and algebraic geometry.
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