Arithmetic Compactifications of Pel Type Shimura Varieties (London Mathematical Society Monographs) - Hardcover

Lan, Kai-wen

 
9780691156545: Arithmetic Compactifications of Pel Type Shimura Varieties (London Mathematical Society Monographs)
Hardcover
ISBN 10: 0691156549 ISBN 13: 9780691156545
Verlag: Princeton University Press, 2013

Inhaltsangabe

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. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.


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. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications:



  • A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures

  • An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings

  • A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary


Through these topics, 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).

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Über die Autorin bzw. den Autor

Kai-Wen Lan is assistant professor of mathematics at the University of Minnesota.

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Arithmetic Compactifications of PEL-Type Shimura Varieties

By Kai-Wen Lan

PRINCETON UNIVERSITY PRESS

Copyright © 2013 Princeton University Press
All rights reserved.
ISBN: 978-0-691-15654-5

Contents

Acknowledgments............................................................xi
Introduction...............................................................xiii
1 Definition of Moduli Problems............................................1
2 Representability of Moduli Problems......................................91
3 Structures of Semi-Abelian Schemes.......................................143
4 Theory of Degeneration for Polarized Abelian Schemes.....................175
5 Degeneration Data for Additional Structures..............................285
6 Algebraic Constructions of Toroidal Compactifications....................373
7 Algebraic Constructions of Minimal Compactifications.....................447
A Algebraic Spaces and Algebraic Stacks....................................487
B Deformations and Artin's Criterion.......................................519
Bibliography...............................................................535
Index......................................................................545

Excerpt

<h2>CHAPTER 1</h2><p><b>Definition of Moduli Problems</b></p><br><p>In this chapter, we give the definition of the moduli problems providing integralmodels of PEL-type Shimura varieties that we will compactify.</p><p>Just to make sure that potential logical problems do not arise in our use ofcategories, we assume that a pertinent choice of a <i>universe</i> has been made (seeSection A.1.1 for more details). This is harmless for our study, and we shall notmention it again in our work.</p><p>The main objective in this chapter is to state Definition 1.4.1.4 with justifications.In order to explain the relation between our definition and those in theliterature, we include also Definition 1.4.2.1 (which, in particular, agrees with thedefinition in [83, §5] when specialized to the same bases), and compare our twodefinitions. All sections preceding them are preparatory in nature. Technical resultsworth noting are Propositions 1.1.2.20, 1.1.5.17, 1.2.2.3, 1.2.3.7, 1.2.3.11,1.2.5.15, 1.2.5.16, and 1.4.3.4. Theorem 1.4.1.11 (on the representability of ourmoduli problems in the category of algebraic stacks) is stated in Section 1.4, but itsproof will be carried out in Chapter 2. The representability of our moduli problemas schemes (when the level is neat) will be deferred until Corollary 7.2.3.10, afterwe have accomplished the construction of the minimal compactifications.</p><br><p><b>1.1 PRELIMINARIES IN ALGEBRA</p><p>1.1.1 Lattices and Orders</b></p><p>For the convenience of readers, we shall summarize certain basic definitions andimportant properties of lattices over an order in a (possibly noncommutative) finite-dimensionalalgebra over a Dedekind domain. Our main reference for this purposewill be [114].</p><p>Let us begin with the most general setting. Let <i>R</i> be a (commutative) noetherianintegral domain with fractional field Frac<i>(R)</i>.</p><p>Definition 1.1.1.1. <i>An R-lattice M is a finitely generated R-module M withno nonzero R-torsion. Namely, for every nonzero m [member of] M, there is no nonzeroelement r [member of] R such that rm = 0.</i></p><p>Note that in this case we have an embedding from <i>M</i> to <i>M</i> [??] Frac<i>(R)</i>.</p><p>Definition 1.1.1.2. <i>Let V be any finite-dimensional Frac(R)-vector space. Afull R-lattice M in V is a finitely generated submodule M of V such that Frac(R)·M = V. In other words, M contains a Frac(R)-basis of V.</i></p><p>Let <i>A</i> be a (possibly noncommutative) finite-dimensional algebra overFrac<i>(R)</i>.</p><p>Definition 1.1.1.3. <i>An R-order O in the Frac(R)-algebra A is a subring ofA having the same identity element as A, such that O is also a full R-lattice in A.</i></p><p>Here are two familiar examples of orders:</p><p>1. If <i>R</i> is a Dedekind domain, and if <i>A = L</i> is a finite separable field extensionof Frac<i>(R)</i>, then the integral closure <i>O</i> of <i>R</i> in <i>L</i> is an <i>R</i>-order in <i>A</i>. In particular, if <i>R = Z</i>, then the rings of algebraic integers <i>O = O<sub>L</sub></i> in <i>L</i> is aZ-order in <i>L</i>.</p><p>2. If <i>A</i> = M<sub><i>n</i></sub>(Frac<i>(R)</i>), then <i>O</i> = M<i><sub>n</sub>(R)</i> is an <i>R</i>-order in <i>A</i>.</p><p>Definition 1.1.1.4. <i>A maximal R-order in A is an R-order not properlycontained in another R-order in A.</i></p><p>Proposition 1.1.1.5 ([114, Thm. 8.7]). <i>1. If the integral closure of R inA is an R-order, then it is automatically maximal.</p><p>2. If O is a maximal R-order in A, then M<sub>n</sub>(O) is a maximal R-order in M<sub>n</sub>(A)for each integer n ≥ 1. In particular, if R is normal (namely, integrallyclosed in Frac(R)), then M<sub>n</sub>(R) is a maximal R-order in M<sub>n</sub>(Frac(R)).</i></p><br><p>Suppose moreover that <i>A</i> is a separable Frac<i>(R)</i>-algebra. By definition, <i>A</i> isArtinian and semisimple. For simplicity, we shall suppress the modifier <i>reduced</i>from traces and norms when talking about such algebras. By [114, Thm. 9.26], theassumption that <i>A</i> is a (finite-dimensional) separable Frac<i>(R)</i>-algebra implies thatthe (redduced) trace pairing...

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Verlag
Princeton University Press
Erscheinungsdatum
2013
Sprache
Englisch
ISBN 10 
0691156549
ISBN 13 
9780691156545
Einband
Gebundene Ausgabe
Anzahl der Seiten
561
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