By Brian Conrad, Frans Oort

Abelian types with advanced multiplication lie on the origins of sophistication box idea, they usually play a valuable function within the modern conception of Shimura kinds. they're distinctive in attribute zero and ubiquitous over finite fields. This ebook explores the connection among such abelian types over finite fields and over arithmetically fascinating fields of attribute zero through the examine of numerous typical CM lifting difficulties which had formerly been solved simply in specified situations. as well as giving entire suggestions to such questions, the authors offer quite a few examples to demonstrate the overall concept and current an in depth therapy of many basic effects and ideas within the mathematics of abelian forms, resembling the most Theorem of complicated Multiplication and its generalizations, the finer elements of Tate's paintings on abelian kinds over finite fields, and deformation concept. This ebook presents a great representation of ways sleek thoughts in mathematics geometry (such as descent concept, crystalline equipment, and crew schemes) may be fruitfully mixed with category box concept to respond to concrete questions on abelian kinds. it is going to be an invaluable reference for researchers and complicated graduate scholars on the interface of quantity conception and algebraic geometry.

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