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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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Degeneration Data for Additional Structures

Degeneration Data for Additional Structures

Chapter:
(p.285) Chapter Five Degeneration Data for Additional Structures
Source:
Arithmetic Compactifications of PEL-Type Shimura Varieties
Author(s):

Kai-Wen Lan

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691156545.003.0005

This chapter supplies a theory of degeneration data for endomorphism structures, Lie algebra conditions, and level structures, based on the theory of degeneration in Chapter 4. People often claim that the degeneration theory for general PEL-type structures is just a straightforward consequence of the functoriality of the merely polarized case. However, the Weil-pairing calculation carried out in this chapter may suggest that this is not true. As this chapter shows, functoriality does not seem to imply properties about pairings in an explicit way. There are conceptual details to be understood beyond simple implications of functoriality. However, this chapter is able to present a theory of degeneration data for abelian varieties with PEL structures, together with the notion of cusp labels.

Keywords:   degeneration theory, degeneration data, endomorphism structures, Lie algebra conditions, level structures, PEL-type structures, functoriality, Weil-pairing calculation, cusp labels

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