Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.001.0001
- Subject:
- Mathematics, Geometry / Topology
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical ...
More
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.Less
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.
Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.003.0005
- Subject:
- Mathematics, Geometry / Topology
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 ...
More
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.Less
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.
