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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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Algebraic Constructions of Minimal Compactifications

Algebraic Constructions of Minimal Compactifications

Chapter:
(p.447) Chapter Seven Algebraic Constructions of Minimal Compactifications
Source:
Arithmetic Compactifications of PEL-Type Shimura Varieties
Author(s):

Kai-Wen Lan

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691156545.003.0007

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.

Keywords:   toroidal compactifications, minimal compactifications, automorphic forms, Fourier–Jacobi expansions, arithmetic minimal compactifications, moduli problems, Koecher's principle, arithmetic toroidal compactifications, codimension counting

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