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)

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ISBN 10: 0691156549 ISBN 13: 9780691156545

Publisher: Princeton University Press, 2013

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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).

"synopsis" may belong to another edition of this title.

About the Author

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

Excerpt. � Reprinted by permission. All rights reserved.

Arithmetic Compactifications of PEL-Type Shimura Varieties

By Kai-Wen Lan

PRINCETON UNIVERSITY PRESS

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>Definition of Moduli Problems In this chapter, we give the definition of the moduli problems providing integralmodels of PEL-type Shimura varieties that we will compactify.Just to make sure that potential logical problems do not arise in our use ofcategories, we assume that a pertinent choice of a universe 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.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. 1.1 PRELIMINARIES IN ALGEBRA1.1.1 Lattices and OrdersFor 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].Let us begin with the most general setting. Let R be a (commutative) noetherianintegral domain with fractional field Frac(R).Definition 1.1.1.1. 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.Note that in this case we have an embedding from M to M [??] Frac(R).Definition 1.1.1.2. 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.Let A be a (possibly noncommutative) finite-dimensional algebra overFrac(R).Definition 1.1.1.3. 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.Here are two familiar examples of orders:1. If R is a Dedekind domain, and if A = L is a finite separable field extensionof Frac(R), then the integral closure O of R in L is an R-order in A. In particular, if R = Z, then the rings of algebraic integers O = OL in L is aZ-order in L.2. If A = Mn(Frac(R)), then O = Mn(R) is an R-order in A.Definition 1.1.1.4. A maximal R-order in A is an R-order not properlycontained in another R-order in A.Proposition 1.1.1.5 ([114, Thm. 8.7]). 1. If the integral closure of R inA is an R-order, then it is automatically maximal.2. If O is a maximal R-order in A, then Mn(O) is a maximal R-order in Mn(A)for each integer n ≥ 1. In particular, if R is normal (namely, integrallyclosed in Frac(R)), then Mn(R) is a maximal R-order in Mn(Frac(R)). Suppose moreover that A is a separable Frac(R)-algebra. By definition, A isArtinian and semisimple. For simplicity, we shall suppress the modifier reducedfrom traces and norms when talking about such algebras. By [114, Thm. 9.26], theassumption that A is a (finite-dimensional) separable Frac(R)-algebra implies thatthe (redduced) trace pairing TrA/Frac(R) : A x A -> Frac(R) is nondegenerate (as pairings on Frac(R)-vector spaces).An important invariant of an order defined by the trace pairing is the discriminant.Definition 1.1.1.6. Let t = [A : Frac(R)]. The discriminantDDisc = DiscO/Ris the ideal of R generated by the set of elements{DetFrac(R)(TrA/Frac(R)(xixj))1≤i≤t,1≤j≤t : x1, ..., xt [member of] O}.Remark 1.1.1.7. If O has a free R-basis e1, ..., et, then each of the t elementsx1, ..., xt can be expresseeeed as an R-linear combination of e1, ..., et. Hencein this case Disc is generated by a single elementDetFrac(R)(TrA/Frac(R)(eiej))1≤i≤t,1≤j≤t.Another important invariant is the following definition:Definition 1.1.1.8. The inverse differentDiff-1 = Diff-1O/Rof O over R is defined byDiff-1O/R := {x [member of] A : TrA/Frac(R)(xy) [subset] R [for all]y [member of] O}.It is clear from the definition that Diff-1O/R is a two-sided ideal in A, and thatthe formation of inverse differents is compatible with localizations.Lemma 1.1.1.9. Suppose O is locally free as an R-module. Then Diff-1 islocally free as an R-module, and TrA/Frac(R) induces a perfect pairingTrA/Frac(R) : O x Diff-1 -> R.Moreover, if {ei}1≤i≤t is any R1-basis of O [??] R1for some localization R1 of R, then there exists a unique dual R1-basis{fi}1≤i≤t of Diff-1 [??] R1 such thatTrA/Frac(R)(eifj) = δij for all 1 ≤i ≤ t and 1 ≤ j ≤ t.Proof. We may localize R and assume that O is free over R. Let {ei}1≤i≤tbe any basis of O over R. Then {ei}1≤i≤t is also a basis of A over Frac(R). Bynondegeneracy of the trace pairing TrA/Frac(R) : A x A -> Frac(R), there existsa unique basis {fi}1≤i≤t of A over Frac(R), which is dual to{ei}1≤i≤t in thesense that TrA/FracR(eifj) = δijfor all 1 ≤ i ≤ t and 1 ≤ j ≤ t. If[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]A satisfies TrA/Frac(R)(xy) [member of] R for all x [member of] O, then in particular,cj = TrA/Frac(R)(ejy) [member of] R for all1 ≤ j ≤ t. Thus {fj}1≤j≤t is also a basis ofDiff-1. This shows the perfectness of the pairing TrA/Frac(R): O x Diff-1 -> Rand the existence of the dual bases, as desired.Suppose R is a noetherian normal domain, and A is a finite-dimensional separableFrac(R)-algebra.Proposition 1.1.1.10 ([114, Cor. 10.4]). Let R be a noetherian normal domain,and let A be a finite-dimensional separable Frac(R)-algebra. Then everyR-order in A is contained in a maximal R-order in A. There exists at least onemaximal R-order in A.For each ideal p of R, we denote by Rp the localization of R at p, and by [??]pthe completion of Rp with respect to its maximal ideal pRp. (The slight deviationof this convention is that we shall denote by Z(p) the localization of Z at (p), andby Zp the completion of Z(p).) If R is local, then we denote simply by [??] itscompletion at its maximal ideal.Definition 1.1.1.11. Let R be a noetherian normal domain, and let p be aprime ideal of R. We say that an R-order O in A is maximal at p if O [??]Rp is maximal in A.Proposition 1.1.1.12 ([114, Thm. 11.1, Cor. 11.2]). Let R be a noetheriannormal domain. An R-order O in A is maximal if and only if O is maximal at everyprime ideal of R, or equivalently at every maximal ideal of R.Proposition 1.1.1.13 ([114, Thm. 11.5]). Let R be a local noetherian normaldomain. Suppose [??] is a noetherian domain. Then an R-order O in A ismaximal if and only if O [??] [??] is an [??]-maximal order in A [??] Frac(R).Remark 1.1.1.14. The statement and proof of [114, Thm. 11.5] make senseonly when [??] is a noetherian integral domain.Corollary 1.1.1.15. Let R be a noetherian normal domain, and let p bea prime ideal of R. Suppose [??]p is a noetherian domain. Then an R-order O ismaximal at p if and only if O [??] [??]p is maximal in A [??] Frac([??]p).Now suppose R is a Dedekind domain. In particular, R is noetherian and normal,and the completions of R at localizations of its prime ideals are noetheriandomains.Proposition 1.1.1.16. LetDiff = DiffO/R := {z [member of] A : z Diff-1O/R [subset] O}be the inverse ideal of Diff-1O/R. Then this is a two-sided ideal of O, and the discriminantDiscO/R is related to Diff-1O/R byDiscO/R = NormA/Frac(R)(DiffO/R) = [Diff-1O/R : O]R. (1.1.1.17)If O is a maximal order, then this is just [114, Thm. 25.2]. The same proof vialocalizations works in the case where O is not maximal as well:Proof of Proposition 1.1.1.16. By replacing R with its localizations, wemay assume that every R-lattice is free over R. Let t = [A : Frac(R)]. Let{ei}1≤i≤t be any R-basis of O, and let{fi}1≤i≤t be the dual R-basis of Diff-1given by Lemma 1.1.1.9. Since O [subset] Diff-1, we may express each ei as[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some aij [member of] R. By definition,NormA/Frac(R)(Diff) = [Diff-1 : O]R = (DetFrac(R)(aij)).On the other hand,[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].Hence Disc = (DetFrac(R)(TrA/Frac(R)(eixj)))= (DetFrac(R)(aij)), verifyingequation (1.1.1.17) as desired.Definition 1.1.1.18. We say that the prime ideal p of R is unramified in Oif p [??] DiscO/R.Proposition 1.1.1.19 ([114, Thm. 25.3]). Every two maximal R-orders O1and O2 in A have the same discriminant over R.Therefore it makes sense to say,Definition 1.1.1.20. An ideal p of R is unramified in A if it is unramified inone (and hence every) maximal R-order of A.For each global field K, we shall denote the rings of integers in K by OK. Thisis in conflict with the notation O with no subscripts, but the correct interpretationshould be clear from the context.Proposition 1.1.1.21. Let R be a Dedekind domain such that Frac(R) isa global field, let A be a finite-dimensional Frac(R)-algebra with center E, andlet O be an R-order in A. Suppose p is a nonzero prime ideal of R such thatp [??] DiscO/R. Then,1. O is maximal at p;2. O [??] [??]p is isomorphic to a product of matrix algebras containing OE [??][??]p as its center;3. p is unramified in A and in E. Proof. If O [subset] O' are two orders, then necessarilyO [subset] O' [subset] Diff-1O'/R [subset] Diff-1O/R.In particular, if p is a prime ideal of R such that p [??] DiscO/R, then the relationO [??] Rp = Diff-1O/R [??] Rp forces O [??] Rpto be maximal. This proves the first statement.According to [114, Thm. 10.5], O [??] Rp is a maximal Rp-order if and only ifit is a maximal OE [??] Rp-order. Then [114, Thm. 25.7] implies thatO [??] [??]p is a product of matrix algebras containing OE [??] [??]p as its center.This is the second statement.Finally, by taking the matrix with only one element on the diagonal, we see thatO [??} Rp = Diff-1O/R [??] Rpforces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is, p [??] DiscO/Rforces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the third statement follows from Proposition 1.1.1.19and Definition 1.1.1.20.Definition 1.1.1.22. A (left) O-module M is called an O-lattice if it is anR-lattice. Namely, it is finitely generated and torsion-free as an R-module.Proposition 1.1.1.23 (see [114, Thm. 21.4 and Cor. 21.5]). Every maximalorder O over a Dedekind domain is hereditary in the sense that all O-lattices areprojective O-modules.Proposition 1.1.1.24 (see [114, Cor. 21.5 and Thm. 2.44]). Every projectivemodule over a maximal order is a direct sum of left ideals.Remark 1.1.1.25. Propositions 1.1.1.23 and 1.1.1.24 imply that, althoughO-lattices might not be projective, their localizations or completions become projectiveas soon as O itself becomes maximal after localization or completion. 1.1.2 Determinantal ConditionsLet C be a finite-dimensional separable algebra over a field k. By definition (suchas [114, p. 99]), the center E of C is a commutative finite-dimensional separablealgebra over k. Let K be a (possibly infinite) field extension of k. Unless otherwisespecified, all the homomorphisms below will be k-linear.Fix a separable closure Ksep of K, and consider the possible k-algebra homomorphismsτ from E to Ksep. Note that Homk(E,Ksep) has cardinality [E : k],because E is separable over k. The Gal(Ksep/K)-orbits [t ] of such homomorphismsτ : E -> Ksep can be classified in the following way: Consider the equivalenceclasses of pairs of the form (Kτ, τ), where Kτ is isomorphic over K to thecomposite of K and the image of τ in Ksep, and where τ is the induced homomorphismfrom E to Kτ. Here Kτ is necessarily separable over K with degree atmost [E : k]. Two such pairs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are considered equivalent ifthere is an isomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]over K such that τ2 = σ [??] τ1. We shalldenote such an equivalence class by [τ] : E -> K[τ]. By abuse of notation, thiswill also mean an actual representative τ : E -> Kτ, which can be considered as ahomomorphism τ : E -> Ksep as well. Note that[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]where each Ksepτ' means a copy of Ksep withτ' : E -> Ksep in the equivalenceclass [τ].Lemma 1.1.2.1. We have a decomposition[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]into a product of separable extensions E [τ] of K.Corollary 1.1.2.2. We have a decomposition[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]into simple K-algebras.Let us quote the following weaker form of the Noether–Skolem theorem:LEMMA 1.1.2.4 (see, for example, [68, Lem. 4.3.2]). Let C' be a simple Artinianalgebra. Then all simple C'-modules are isomorphic to each other.A useful reformulation of Lemma 1.1.2.4 is as follows:Corollary 1.1.2.5. Let C' be a simple Artinian algebra with center E'. Thenan irreducible representation of C' with coefficients in some field K' is determinedup to isomorphism by its restriction to E'.(Continues...)

(Continues...)

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

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Hardback. Condition: New. 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). Seller Inventory # LU-9780691156545

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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

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

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Arithmetic Compactifications of PEL-Type Shimura Varieties

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