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 xi Introduction xv Notations and Conventions xxv 1 Definition of Moduli Problems 1 1.1 Preliminaries in Algebra . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Lattices and Orders . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Determinantal Conditions . . . . . . . . . . . . . . . . 6 1.1.3 Projective Modules . . . . . . . . . . . . . . . . . . . . 14 1.1.4 Generalities of Pairings . . . . . . . . . . . . . . . . . . 17 1.1.5 Classification of Pairings By Involutions . . . . . . . . 24 1.2 Linear Algebraic Data . . . . . . . . . . . . . . . . . . . . . . 31 1.2.1 PEL-Type O-Lattices . . . . . . . . . . . . . . . . . . . 31 1.2.2 Torsion of Universal Domains . . . . . . . . . . . . . . 38 1.2.3 Self-Dual Symplectic Modules . . . . . . . . . . . . . . 43 1.2.4 Gram-Schmidt Procedures . . . . . . . . . . . . . . . . 54 1.2.5 Reflex Fields . . . . . . . . . . . . . . . . . . . . . . . 58 1.2.6 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.3 Geometric Structures . . . . . . . . . . . . . . . . . . . . . . . 68 1.3.1 Abelian Schemes and Quasi-Isogenies . . . . . . . . . . 68 1.3.2 Polarizations . . . . . . . . . . . . . . . . . . . . . . . 75 1.3.3 Endomorphisms Structures . . . . . . . . . . . . . . . . 81 v 1.3.4 Conditions on Lie Algebras . . . . . . . . . . . . . . . 82 1.3.5 Tate Modules . . . . . . . . . . . . . . . . . . . . . . . 84 1.3.6 Principal Level Structures . . . . . . . . . . . . . . . . 87 1.3.7 General Level Structures . . . . . . . . . . . . . . . . . 94 1.4 Definitions of the Moduli Problems . . . . . . . . . . . . . . . 99 1.4.1 Definition by Isomorphism Classes . . . . . . . . . . . 99 1.4.2 Definition by Z× -Isogeny Classes . . . . . . . . . . . . 103 (2) 1.4.3 Relation Between Two Definitions . . . . . . . . . . . . 106 1.4.4 Definition By Different Set of Primes . . . . . . . . . . 111 2 Representability of Moduli Problems 115 2.1 Theory of Obstructions for Smooth Schemes . . . . . . . . . . 116 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 117 2.1.2 Deformation of Smooth Schemes . . . . . . . . . . . . . 121 2.1.3 Deformation of Morphisms . . . . . . . . . . . . . . . . 126 2.1.4 Change of Bases . . . . . . . . . . . . . . . . . . . . . 128 2.1.5 Deformation of Invertible Sheaves . . . . . . . . . . . . 132 2.1.6 De Rham Cohomology . . . . . . . . . . . . . . . . . . 139 2.1.7 Kodaira-Spencer Maps . . . . . . . . . . . . . . . . . . 149 2.2 Formal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.2.1 Local Moduli Functors and Schlessinger’s Criterion . . 152 2.2.2 Rigidity of Structures . . . . . . . . . . . . . . . . . . . 156 2.2.3 Prorepresentability . . . . . . . . . . . . . . . . . . . . 162 2.2.4 Formal Smoothness . . . . . . . . . . . . . . . . . . . . 166 2.3 Algebraic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 175 2.3.1 Grothendieck’s Formal Existence Theory . . . . . . . . 175 2.3.2 Effectiveness of Local Moduli . . . . . . . . . . . . . . 177 2.3.3 Proof of Representability . . . . . . . . . . . . . . . . . 178 2.3.4 Properties of Kodaira-Spencer Maps . . . . . . . . . . 180 3 Structures of Semi-Abelian Schemes 183 3.1 Groups of Multiplicative Type, Tori, and Their Torsors . . . . 183 3.1.1 Groups of Multiplicative Type . . . . . . . . . . . . . . 183 3.1.2 Torsors and Invertible Sheaves . . . . . . . . . . . . . . 185 3.1.3 Construction Using Sheaves of Algebras . . . . . . . . 191 3.1.4 Group Structures on Torsors . . . . . . . . . . . . . . . 198 3.1.5 Group Extensions . . . . . . . . . . . . . . . . . . . . . 205 3.2 Biextensions and Cubical Structures . . . . . . . . . . . . . . 207 vi 3.2.1 Biextensions . . . . . . . . . . . . . . . . . . . . . . . . 207 3.2.2 Cubical Structures . . . . . . . . . . . . . . . . . . . . 208 3.2.3 A Fundamental Example . . . . . . . . . . . . . . . . . 211 3.2.4 The Group G(L) for Abelian Schemes . . . . . . . . . . 212 3.2.5 Descending Structures . . . . . . . . . . . . . . . . . . 213 3.3 Semi-Abelian Schemes . . . . . . . . . . . . . . . . . . . . . . 215 3.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 215 3.3.2 Extending Structures . . . . . . . . . . . . . . . . . . . 218 3.3.3 Raynaud Extensions . . . . . . . . . . . . . . . . . . . 219 3.4 The Group K(L) and Applications . . . . . . . . . . . . . . . 223 3.4.1 Quasi-FiniteSubgroupsofaSemi-AbelianSchemeover a Henselian Base . . . . . . . . . . . . . . . . . . . . . 223 3.4.2 Statement of the Theorem on the Group K(L) . . . . . 225 3.4.3 Dual Semi-Abelian Schemes . . . . . . . . . . . . . . . 228 3.4.4 Dual Raynaud Extensions . . . . . . . . . . . . . . . . 230 4 Theory of Degeneration for Polarized Abelian Schemes 233 4.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.2 Ample Degeneration Data . . . . . . . . . . . . . . . . . . . . 234 4.2.1 Main Definitions and Main Theorem . . . . . . . . . . 235 4.2.2 Equivalence Between ι and τ . . . . . . . . . . . . . . . 241 4.2.3 Equivalence Between Action on L\ and ψ . . . . . . . 244 η 4.2.4 Equivalence Between The Positivity Condition for ψ and The Positivity Condition for τ . . . . . . . . . . . 251 4.3 Fourier Expansions of Theta Functions . . . . . . . . . . . . . 253 4.3.1 Definition of ψ and τ . . . . . . . . . . . . . . . . . . . 253 4.3.2 Relations Between Theta Representations . . . . . . . 262 4.3.3 Addition Formula . . . . . . . . . . . . . . . . . . . . . 271 4.3.4 Dependence of τ on the Choice of L . . . . . . . . . . . 278 4.4 Equivalences of Categories . . . . . . . . . . . . . . . . . . . . 282 4.5 Mumford’s Construction . . . . . . . . . . . . . . . . . . . . . 287 4.5.1 Relatively Complete Models . . . . . . . . . . . . . . . 287 4.5.2 Construction of The Quotient . . . . . . . . . . . . . . 301 4.5.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . 313 4.5.4 Proof of the Equivalences . . . . . . . . . . . . . . . . 324 4.6 Kodaira-Spencer Maps . . . . . . . . . . . . . . . . . . . . . . 338 4.6.1 Definition for Semi-Abelian Schemes . . . . . . . . . . 338 4.6.2 Definition for Period Maps . . . . . . . . . . . . . . . . 342 vii 4.6.3 Compatibility with Mumford’s Construction . . . . . . 349 5 Degeneration Data for Additional Structures 365 5.1 Data for Endomorphism Structures . . . . . . . . . . . . . . . 365 5.1.1 Analysis of Endomorphism Structures . . . . . . . . . . 365 5.1.2 Analysis of Lie Algebra Conditions . . . . . . . . . . . 369 5.2 Data for Principal Level Structures . . . . . . . . . . . . . . . 373 5.2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . 373 5.2.2 Analysis of Principal Level Structures . . . . . . . . . . 373 5.2.3 Analysis of Splittings for G[n] . . . . . . . . . . . . . 388 η 5.2.4 Weil Pairings in General . . . . . . . . . . . . . . . . . 398 5.2.5 Sheaf-Theoretic Realization of Splittings of G[n] . . . 406 η 5.2.6 Weil Pairings for G[n] via Splittings . . . . . . . . . . 412 η 5.2.7 Construction of Principal Level Structures . . . . . . . 425 5.3 Data for General PEL-Structures . . . . . . . . . . . . . . . . 436 ´ 5.3.1 Formation of Etale Orbits and Main Result . . . . . . 436 5.3.2 Degenerating Families . . . . . . . . . . . . . . . . . . 445 5.3.3 Criterion for Properness . . . . . . . . . . . . . . . . . 446 5.4 Notion of Cusp Labels . . . . . . . . . . . . . . . . . . . . . . 448 5.4.1 Principal Cusp Labels . . . . . . . . . . . . . . . . . . 448 5.4.2 General Cusp Labels . . . . . . . . . . . . . . . . . . . 455 5.4.3 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . 460 6 Algebraic Constructions of Toroidal Compactifications 473 6.1 Review of Toroidal Embeddings . . . . . . . . . . . . . . . . . 473 6.1.1 Rational Polyhedral Cone Decompositions . . . . . . . 474 6.1.2 Toroidal Embeddings of Torsors . . . . . . . . . . . . . 476 6.2 Construction of Boundary Charts . . . . . . . . . . . . . . . . 478 6.2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . 478 6.2.2 Construction without Positivity Condition and Level Structures . . . . . . . . . . . . . . . . . . . . . . . . . 480 6.2.3 Construction with Principal Level Structures . . . . . . 487 6.2.4 Construction with General Level Structures . . . . . . 507 6.2.5 Construction with Positivity Condition . . . . . . . . . 511 6.2.6 Identifications Between Parameter Spaces . . . . . . . 522 6.3 Approximation and Gluing . . . . . . . . . . . . . . . . . . . . 524 6.3.1 Good Formal Models . . . . . . . . . . . . . . . . . . . 524 6.3.2 Good Algebraic Models . . . . . . . . . . . . . . . . . . 531 viii ´ 6.3.3 Etale Presentation and Gluing . . . . . . . . . . . . . . 541 6.4 Arithmetic Toroidal Compactifications . . . . . . . . . . . . . 552 6.4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 552 6.4.2 Towers of Toroidal Compactifications . . . . . . . . . . 556 6.4.3 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . 560 7 Algebraic Constructions of Minimal Compactifications 565 7.1 Automorphic Forms and Fourier-Jacobi Expansions . . . . . . 566 7.1.1 Automorphic Forms . . . . . . . . . . . . . . . . . . . 566 7.1.2 Fourier-Jacobi Expansions . . . . . . . . . . . . . . . . 567 7.2 Arithmetic Minimal Compactifications . . . . . . . . . . . . . 574 7.2.1 Positivity of Hodge Invertible Sheaves . . . . . . . . . . 574 7.2.2 Stein Factorizations and Finite Generation . . . . . . . 575 7.2.3 Main Construction . . . . . . . . . . . . . . . . . . . . 578 7.2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . 586 7.2.5 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . 592 7.3 Projectivity of Toroidal Compactifications . . . . . . . . . . . 594 7.3.1 Convexity Conditions on Cone Decompositions . . . . . 594 7.3.2 Generalities on Normalizations of Blow-Ups . . . . . . 598 7.3.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . 599 A Algebraic Spaces and Algebraic Stacks 609 A.1 Some Category Theory . . . . . . . . . . . . . . . . . . . . . . 609 A.1.1 A Set-Theoretical Remark . . . . . . . . . . . . . . . . 609 A.1.2 2-Categories and 2-Functors . . . . . . . . . . . . . . . 610 A.2 Grothendieck Topology . . . . . . . . . . . . . . . . . . . . . . 615 A.3 Algebraic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 618 A.3.1 Properties of an Algebraic Space . . . . . . . . . . . . 620 A.3.2 Quasi-Coherent Sheaves on an Algebraic Space . . . . . 625 A.3.3 Points and the Zariski Topology of an Algebraic Space 625 A.4 Category Fibred in Groupoids . . . . . . . . . . . . . . . . . . 627 A.5 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 A.6 Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . . . . . 639 A.6.1 Properties of Algebraic Stacks . . . . . . . . . . . . . . 641 A.6.2 Quasi-Coherent Sheaves on Algebraic Stacks . . . . . . 644 A.6.3 Points and the Zariski Topology of an Algebraic Stack 646 A.6.4 Coarse Moduli Spaces . . . . . . . . . . . . . . . . . . 647 ix B Deformations and Artin’s Criterion 649 B.1 Infinitesimal Deformations . . . . . . . . . . . . . . . . . . . . 649 B.1.1 Structure of Complete Local Rings . . . . . . . . . . . 653 B.2 Existence of Algebraization . . . . . . . . . . . . . . . . . . . 657 B.2.1 Generalization from Sets to Groupoids . . . . . . . . . 661 B.3 Artin’s Criterion for Algebraic Stacks . . . . . . . . . . . . . . 668 Bibliography 678 Index 689 x