Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci

Contents
Preface
0. Introduction
0.1 Background and Aim
0.2 Overview
0.3 Outline of the Constructions
0.4 What is Known, What is New, and What Can Be Studied Next
0.5 What to Note and to Skip in Special Cases
0.6 Notation and Conventions
0.7 Acknowledgements
1. Theory in Characteristic Zero
1.1 PEL-type Moduli Problems and Shimura Varieties
1.1.1 Linear Algebraic Data for PEL Structures
1.1.2 PEL-type Moduli Problems
1.1.3 PEL-type Shimura Varieties
1.2 Linear Algebraic Data for Cusps
1.2.1 Cusp Labels
1.2.2 Cone Decompositions
1.2.3 Rational Boundary Components
1.2.4 Parameters for Kuga Families
1.3 Algebraic Compactifications in Characteristic Zero
1.3.1 Toroidal and Minimal Compactifications of PEL-Type
Moduli Problems
1.3.2 Boundary of PEL-Type Moduli Problems
1.3.3 Toroidal Compactifications of PEL-Type Kuga Families and
Their Generalizations
1.3.4 Justification for the Parameters for Kuga Families
1.4 Automorphic Bundles and Canonical Extensions in
Characteristic Zero
1.4.1 Automorphic Bundles
1.4.2 Canonical Extensions
1.4.3 Hecke Actions
1.5 Comparison with the Analytic Construction
2. Flat Integral Models
2.1 Auxiliary Choices
2.1.1 Auxiliary Choices of Smooth Moduli Problems
2.1.2 Auxiliary Choices of Toroidal and Minimal Compactifications
2.2 Flat Integral Models as Normalizations and Blow-Ups
2.2.1 Flat Integral Models for Minimal Compactifications
2.2.2 Flat Integral Models for Projective Toroidal Compactifications
2.2.3 Hecke Actions
2.2.4 The Case When p is a Good Prime
3. Ordinary Loci
3.1 Ordinary Semi-Abelian Schemes and Serre's Construction
3.1.1 Ordinary Abelian Schemes and Semi-Abelian Schemes
3.1.2 Serre's Construction for Ordinary Abelian Schemes
3.1.3 Extensibility of Isogenies
3.2 Linear Algebraic Data for Ordinary Loci
3.2.1 Necessary Data for Ordinary Reductions
3.2.2 Maximal Totally Isotropic Submodules at p
3.2.3 Compatibility with Cusp Labels
3.3 Level Structures
3.3.1 Level Structures Away from p
3.3.2 Hecke Twists Defined by Level Structures Away from p
3.3.3 Ordinary Level Structures at p
3.3.4 Hecke Twists Defined by Ordinary Level Structures at p
3.3.5 Comparison with Level Structures in Characteristic Zero
3.3.6 Valuative Criteria
3.4 Ordinary Loci
3.4.1 Naive Moduli Problems with Ordinary Level Structures
3.4.2 Ordinary Loci as Normalizations
3.4.3 Properties of Kodaira{Spencer Morphisms
3.4.4 Hecke Actions
3.4.5 The Case When p is a Good Prime
3.4.6 Quasi-Projectivity of Coarse Moduli
4. Degeneration Data and Boundary Charts
4.1 Theory of Degeneration Data
4.1.1 Degenerating Families of Type (PE, )
4.1.2 Common Setting for the Theory of Degeneration
4.1.3 Degeneration Data for Polarized Abelian Schemes with
Endomorphism Structures
4.1.4 Degeneration Data for Principal Ordinary Level Structures
4.1.5 Degeneration Data for General Ordinary Level Structures
4.1.6 Comparison with Degeneration Data for Level Structures in
Characteristic Zero
4.2 Boundary Charts of Ordinary Loci
4.2.1 Constructions with Level Structures but without Positivity
Conditions
4.2.2 Toroidal Embeddings, Positivity Conditions, and Mumford
Families
4.2.3 Extended Kodaira{Spencer Morphisms and Induced
Isomorphisms
5. Partial Toroidal Compactifications
5.1 Approximation and Gluing Along the Ordinary Loci
5.1.1 Ordinary Good Formal Models
5.1.2 Ordinary Good Algebraic Models
5.1.3 Gluing in the Etale Topology
5.2 Partial Toroidal Compactifications of Ordinary Loci
5.2.1 Main Statements
5.2.2 Hecke Actions
5.2.3 The Case When p is a Good Prime
5.2.4 Boundary of Ordinary Loci
6. Partial Minimal Compactifications
6.1 Homogeneous Spectra and Their Properties
6.1.1 Construction of Quasi-Projective Models
6.1.2 Local Structures and Stratifications
6.2 Partial Minimal Compacti cations of Ordinary Loci
6.2.1 Main Statements
6.2.2 Hecke Actions
6.2.3 Quasi-Projectivity of Partial Toroidal Compactifications
6.3 Full Ordinary Loci in p-Adic Completions
6.3.1 Hasse Invariants
6.3.2 Nonordinary and Full Ordinary Loci
6.3.3 Nonemptiness of Ordinary Loci
7. Ordinary Kuga Families
7.1 Partial Toroidal Compactifications
7.1.1 Parameters for Ordinary Kuga Families
7.1.2 Boundary of Ordinary Loci, Continued
7.1.3 Ordinary Kuga Families and Their Generalizations
7.1.4 Main Statements
7.2 Main Constructions of Compactifications and Morphisms
7.2.1 Partial Toroidal Boundary Strata
7.2.2 Justification for the Parameters
7.2.3 Extensibility of f
7.2.4 Properness of ftor
7.2.5 Log Smoothness of ftor
7.2.6 Equidimensionality of ftor
7.2.7 Hecke Actions
7.3 Calculation of Formal Cohomology
7.3.1 Setting
7.3.2 Formal Fibers of ftor
7.3.3 Relative Cohomology and Local Freeness
7.3.4 Degeneracy of the (Relative) Hodge Spectral Sequence
7.3.5 Extended Gauss–Manin Connections
7.3.6 Identification of Rbf*tor ( Nord,tor)
8. Automorphic Bundles and Canonical Extensions
8.1 Constructions over the Ordinary Loci
8.1.1 Technical Assumptions
8.1.2 Automorphic Bundles
8.1.3 Canonical Extensions
8.1.4 Hecke Actions
8.2 Higher Direct Images to the Minimal Compactifications
8.2.1 Some Vanishing Theorems
8.2.2 Formal Fibers of ord
8.2.3 Relative Cohomology of Formal Fibers of ord
8.2.4 Formal Fibers of Canonical Extensions
8.2.5 End of the Proof
8.3 Constructions over the Total Models
8.3.1 Principal Bundles
8.3.2 Automorphic Bundles
8.3.3 Canonical Extensions
8.3.4 Compatibility with the Constructions over the Ordinary Loci
8.3.5 Pushforwards to the Total Minimal Compactifications
8.3.6 Hecke Actions
Bibliography
Index