Moduli Stacks of Étale (?, ?)-Modules and the Existence of Crystalline Lifts: (AMS-215)

by Matthew Emerton, Toby Gee

Overview

A foundational account of a new construction in the p-adic Langlands correspondence

Motivated by the p-adic Langlands program, this book constructs stacks that algebraize Mazur’s formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf Zp that parameterize étale (ϕ, Γ)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. These stacks are then used to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. The book explicitly describes the irreducible components of the underlying reduced substacks and discusses the relationship between the geometry of these stacks and the Breuil–Mézard conjecture. Along the way, it proves a number of foundational results in p-adic Hodge theory that may be of independent interest.

Product Details

ISBN-13:	9780691241340
Publisher:	Princeton University Press
Publication date:	12/13/2022
Series:	Annals of Mathematics Studies , #215
Pages:	312
Product dimensions:	6.12(w) x 9.25(h) x (d)

About the Author

Matthew Emerton is professor of mathematics at the University of Chicago. Toby Gee is professor of mathematics at Imperial College London.

Table of Contents

1 Introduction 1

1.1 Motivation 1

1.2 Our main theorems 2

1.3 (φ, Λ)-modules with coefficients 4

1.4 Families of extensions 5

1.5 Crystalline lifts 6

1.6 Crystalline and semistable moduli stacks 9

1.7 The geometric Breuil-Mézard conjecture and the weight part of Serre's conjecture 10

1.8 Further questions 12

1.9 Previous work 13

1.10 An outline of the book 14

1.11 Acknowledgments 15

1.12 Notation and conventions 16

2 Rings and coefficients 19

2.1 Rings 19

2.2 Coefficients 25

2.3 Almost Galois descent for profinite group actions 35

2.4 An application of almost Galois descent 45

2.5 Étale φ-modules 47

2.6 Frobenius descent 48

2.7 (φ, Λ)-modules 52

3 Moduli stacks of φ-modules and (φ, Λ)-modules 56

3.1 Moduli stacks of φ-modules 56

3.2 Moduli stacks of (φ, Λ)-modules 58

3.3 Weak Wach modules 66

3.4 Χ_d is an Ind-algebraic stack 69

3.5 Canonical actions and weak Wach modules 74

3.6 The connection with Galois representations 76

3.7 (φ, G_Κ)-modules and restriction 80

3.8 Tensor products and duality 84

4 Crystalline and semistable moduli stacks 85

4.1 Notation 85

4.2 Breuil-Kisin modules and Breuih-Kisin-Fargues modules 85

4.3 Canonical extensions of G_K∞-actions 88

4.4 Breuil-Kisin-Fargues G_K-modules and canonical actions 92

4.5 Stacks of semistable and crystalline Breuil-Kisin-Fargues modules 93

4.6 Inertial types 104

4.7 Hodge-Tate weights 106

4.8 Moduli stacks of potentially semistable representations 115

5 Families of extensions 125

5.1 The Herr complex 125

5.2 Residual gerbes and isotrivial families 141

5.3 Twisting families 142

5.4 Dimensions of families of extensions 149

5.5 Χ_d is a formal algebraic stack 157

6 Crystalline lifts and the finer structure of Χ_{d, red} 166

6.1 The fiber dimension of H² on crystalline deformation rings 166

6.2 Two geometric lemmas 168

6.3 Crystalline lifts 172

6.4 Potentially diagonalizable crystalline lifts 175

6.5 The irreducible components of Χ_{d, red} 178

6.6 Closed points 180

6.7 The substack of G_K-representations 182

7 The rank 1 case 185

7.1 Preliminaries 185

7.2 Moduli stacks in the rank 1 case 191

7.3 A ramification bound 197

8 A geometric Breuil-Mézard conjecture 211

8.1 The qualitative geometric Breuil-Mézard conjecture 212

8.2 Semistable and crystalline inertial types 214

8.3 The relationship between the numerical, refined and geometric Breuil-Mézard conjectures 215

8.4 The weight part of Serre's conjecture 217

8.5 The case of GL₂(Q_p) 218

8.6 GL₂(K): potentially Barsotti-Tate types 218

8.7 Brief remarks on GL_d, d>2 223

A Formal algebraic stacks 225

B Graded modules and rigid analysis 242

C Topological groups and modules 257

D Tate modules and continuity 261

E Points, residual gerbes, and isotrivial families 275

F Breuil-Kisin-Fargues modules and potentially semistable representations (by Toby Gee and Tong Liu) 279

Bibliography 289

Index 297