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Arithmetic Compactifications of PEL-Type Shimura Varieties$
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Kai-Wen Lan

Print publication date: 2013

Print ISBN-13: 9780691156545

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691156545.001.0001

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Definition of Moduli Problems

Definition of Moduli Problems

(p.1) Chapter One Definition of Moduli Problems
Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan

Princeton University Press

This chapter lays down the foundations and the definition of the moduli problems to be considered in the rest of this volume. For the purposes of proving representability and constructing compactifications, the chapter uses the definition by isomorphism classes of abelian schemes with additional structures, at the same time revealing that there is also the definition by isogeny classes of abelian schemes with additional structures. This chapter explains that there is a canonical isomorphism from each of the moduli problems defined by isomorphism classes to a canonically associated moduli problem defined by isogeny classes. Consequently, the complex fibers of these moduli problems contain (complex) Shimura varieties associated with some reductive groups as open and closed subalgebraic stacks.

Keywords:   moduli problems, isomorphism classes, abelian schemes, isogeny classes, isomorphism, reductive groups, algebraic stacks

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