Arithmetic compactifications of PEL-type Shimura varieties

1117 Pages·2008·4.58 MB·English

Arithmetic compactifications of PEL-type
Shimura varieties
A dissertation presented
by
Kai-Wen Lan
to
The Department of Mathematics
in partial fulfilment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Mathematics
Harvard University
Cambridge, Massachusetts
May 2008
(cid:13)c 2008 - Kai-Wen Lan
All rights reserved.
Thesis Advisor: Author:
Richard L. Taylor Kai-Wen Lan
Arithmetic compactifications of PEL-type
Shimura varieties
Abstract
In this thesis, we constructed minimal (Satake-Baily-Borel) compactifi-
cations and smooth toroidal compactifications of integral models of general
PEL-type Shimura varieties (defined as in Kottwitz [79]), with descriptions
of stratifications and local structures on them extending the well-known ones
in the complex analytic theory. This carries out a program initiated by Chai,
Faltings, and some other people more than twenty years ago. The approach
we have taken is to redo the Faltings-Chai theory [37] in full generality, with
as many details as possible, but without any substantial case-by-case study.
The essential new ingredient in our approach is the emphasis on level struc-
tures, leading to a crucial Weil pairing calculation that enables us to avoid
unwanted boundary components in naive constructions.
iii
iv
Contents
Abstract iii
Contents v
Acknowledgements xv
Introduction xxi
Notations and Conventions xxxv
1 Definition of Moduli Problems 1
1.1 Preliminaries in Algebra . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Lattices and Orders . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Determinantal Conditions . . . . . . . . . . . . . . . . 8
1.1.3 Projective Modules . . . . . . . . . . . . . . . . . . . . 20
1.1.4 Generalities of Pairings . . . . . . . . . . . . . . . . . . 24
1.1.5 Classification of Pairings By Involutions . . . . . . . . 35
1.2 Linear Algebraic Data . . . . . . . . . . . . . . . . . . . . . . 45
v
1.2.1 PEL-Type O-Lattices . . . . . . . . . . . . . . . . . . . 45
1.2.2 Torsion of Universal Domains . . . . . . . . . . . . . . 57
1.2.3 Self-Dual Symplectic Modules . . . . . . . . . . . . . . 64
1.2.4 Gram-Schmidt Procedures . . . . . . . . . . . . . . . . 82
1.2.5 Reflex Fields . . . . . . . . . . . . . . . . . . . . . . . 88
1.2.6 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . 100
1.3 Geometric Structures . . . . . . . . . . . . . . . . . . . . . . . 103
1.3.1 Abelian Schemes and Quasi-Isogenies . . . . . . . . . . 103
1.3.2 Polarizations . . . . . . . . . . . . . . . . . . . . . . . 112
1.3.3 Endomorphisms Structures . . . . . . . . . . . . . . . . 122
1.3.4 Conditions on Lie Algebras . . . . . . . . . . . . . . . 123
1.3.5 Tate Modules . . . . . . . . . . . . . . . . . . . . . . . 126
1.3.6 Principal Level Structures . . . . . . . . . . . . . . . . 130
1.3.7 General Level Structures . . . . . . . . . . . . . . . . . 140
1.4 Definitions of the Moduli Problems . . . . . . . . . . . . . . . 147
1.4.1 Definition by Isomorphism Classes . . . . . . . . . . . 148
1.4.2 Definition by Z× -Isogeny Classes . . . . . . . . . . . . 154
(2)
1.4.3 Relation Between Two Definitions . . . . . . . . . . . . 158
1.4.4 Definition By Different Set of Primes . . . . . . . . . . 166
2 Representability of Moduli Problems 169
2.1 Theory of Obstructions for Smooth Schemes . . . . . . . . . . 172
2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 173
vi
2.1.2 Deformation of Smooth Schemes . . . . . . . . . . . . . 179
2.1.3 Deformation of Morphisms . . . . . . . . . . . . . . . . 186
2.1.4 Change of Bases . . . . . . . . . . . . . . . . . . . . . 189
2.1.5 Deformation of Invertible Sheaves . . . . . . . . . . . . 195
2.1.6 De Rham Cohomology . . . . . . . . . . . . . . . . . . 206
2.1.7 Kodaira-Spencer Maps . . . . . . . . . . . . . . . . . . 220
2.2 Formal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 225
2.2.1 Local Moduli Functors and Schlessinger’s Criterion . . 225
2.2.2 Rigidity of Structures . . . . . . . . . . . . . . . . . . . 230
2.2.3 Prorepresentability . . . . . . . . . . . . . . . . . . . . 240
2.2.4 Formal Smoothness . . . . . . . . . . . . . . . . . . . . 245
2.3 Algebraic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 260
2.3.1 Grothendieck’s Formal Existence Theory . . . . . . . . 260
2.3.2 Effectiveness of Local Moduli . . . . . . . . . . . . . . 263
2.3.3 Proof of Representability . . . . . . . . . . . . . . . . . 265
2.3.4 Properties of Kodaira-Spencer Maps . . . . . . . . . . 268
3 Structures of Semi-Abelian Schemes 273
3.1 Groups of Multiplicative Type, Tori, and Their Torsors . . . . 274
3.1.1 Groups of Multiplicative Type . . . . . . . . . . . . . . 274
3.1.2 Torsors and Invertible Sheaves . . . . . . . . . . . . . . 276
3.1.3 Construction Using Sheaves of Algebras . . . . . . . . 286
3.1.4 Group Structures on Torsors . . . . . . . . . . . . . . . 295
vii
3.1.5 Group Extensions . . . . . . . . . . . . . . . . . . . . . 305
3.2 Biextensions and Cubical Structures . . . . . . . . . . . . . . 308
3.2.1 Biextensions . . . . . . . . . . . . . . . . . . . . . . . . 308
3.2.2 Cubical Structures . . . . . . . . . . . . . . . . . . . . 310
3.2.3 A Fundamental Example . . . . . . . . . . . . . . . . . 314
3.2.4 The Group G(L) for Abelian Schemes . . . . . . . . . . 315
3.2.5 Descending Structures . . . . . . . . . . . . . . . . . . 317
3.3 Semi-Abelian Schemes . . . . . . . . . . . . . . . . . . . . . . 320
3.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 320
3.3.2 Extending Structures . . . . . . . . . . . . . . . . . . . 325
3.3.3 Raynaud Extensions . . . . . . . . . . . . . . . . . . . 327
3.4 The Group K(L) and Applications . . . . . . . . . . . . . . . 333
3.4.1 Quasi-FiniteSubgroupsofaSemi-AbelianSchemeover
a Henselian Base . . . . . . . . . . . . . . . . . . . . . 333
3.4.2 Statement of the Theorem on the Group K(L) . . . . . 336
3.4.3 Dual Semi-Abelian Schemes . . . . . . . . . . . . . . . 340
3.4.4 Dual Raynaud Extensions . . . . . . . . . . . . . . . . 343
4 Theory of Degeneration for Polarized Abelian Schemes 349
4.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
4.2 Ample Degeneration Data . . . . . . . . . . . . . . . . . . . . 351
4.2.1 Main Definitions and Main Theorem . . . . . . . . . . 352
4.2.2 Equivalence Between ι and τ . . . . . . . . . . . . . . . 361
viii
4.2.3 Equivalence Between Action on L\ and ψ . . . . . . . 366
η
4.2.4 Equivalence Between The Positivity Condition for ψ
and The Positivity Condition for τ . . . . . . . . . . . 376
4.3 Fourier Expansions of Theta Functions . . . . . . . . . . . . . 380
4.3.1 Definition of ψ and τ . . . . . . . . . . . . . . . . . . . 380
4.3.2 Relations Between Theta Representations . . . . . . . 395
4.3.3 Addition Formula . . . . . . . . . . . . . . . . . . . . . 408
4.3.4 Dependence of τ on the Choice of L . . . . . . . . . . . 419
4.4 Equivalences of Categories . . . . . . . . . . . . . . . . . . . . 424
4.5 Mumford’s Construction . . . . . . . . . . . . . . . . . . . . . 433
4.5.1 Relatively Complete Models . . . . . . . . . . . . . . . 434
4.5.2 Construction of The Quotient . . . . . . . . . . . . . . 455
4.5.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . 474
4.5.4 Proof of the Equivalences . . . . . . . . . . . . . . . . 490
4.6 Kodaira-Spencer Maps . . . . . . . . . . . . . . . . . . . . . . 511
4.6.1 Definition for Semi-Abelian Schemes . . . . . . . . . . 512
4.6.2 Definition for Period Maps . . . . . . . . . . . . . . . . 517
4.6.3 Compatibility with Mumford’s Construction . . . . . . 528
5 Degeneration Data for Additional Structures 551
5.1 Data for Endomorphism Structures . . . . . . . . . . . . . . . 552
5.1.1 Analysis of Endomorphism Structures . . . . . . . . . . 552
5.1.2 Analysis of Lie Algebra Conditions . . . . . . . . . . . 557
ix
5.2 Data for Principal Level Structures . . . . . . . . . . . . . . . 562
5.2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . 562
5.2.2 Analysis of Principal Level Structures . . . . . . . . . . 564
5.2.3 Analysis of Splittings for G[n] . . . . . . . . . . . . . 586
η
5.2.4 Weil Pairings in General . . . . . . . . . . . . . . . . . 601
5.2.5 Sheaf-Theoretic Realization of Splittings of G[n] . . . 613
η
5.2.6 Weil Pairings for G[n] via Splittings . . . . . . . . . . 623
η
5.2.7 Construction of Principal Level Structures . . . . . . . 641
5.3 Data for General PEL-Structures . . . . . . . . . . . . . . . . 657
´
5.3.1 Formation of Etale Orbits and Main Result . . . . . . 657
5.3.2 Degenerating Families . . . . . . . . . . . . . . . . . . 670
5.3.3 Criterion for Properness . . . . . . . . . . . . . . . . . 672
5.4 Notion of Cusp Labels . . . . . . . . . . . . . . . . . . . . . . 675
5.4.1 Principal Cusp Labels . . . . . . . . . . . . . . . . . . 675
5.4.2 General Cusp Labels . . . . . . . . . . . . . . . . . . . 686
5.4.3 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . 693
6 Algebraic Constructions of Toroidal Compactifications 711
6.1 Review of Toroidal Embeddings . . . . . . . . . . . . . . . . . 712
6.1.1 Rational Polyhedral Cone Decompositions . . . . . . . 712
6.1.2 Toroidal Embeddings of Torsors . . . . . . . . . . . . . 716
6.2 Construction of Boundary Charts . . . . . . . . . . . . . . . . 719
6.2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . 719
x