Moduli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts

Matthew Emerton, Toby Gee

• Publisher: Princeton University Press
• Year: 2022
• Print ISBN: 9780691241357, 9780691241340
• eBook ISBN: 9780691241364
• Topic: Mathematics
• Subtopic: Algebraic Geometry
• Index: Mathematics

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 (?, ?)-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.

Table of contents

• Cover
• Contents
• 1. Introduction
• 2. Rings and coefficients
• 3. Moduli stacks of φ-modules and (φ, Γ)-modules
• 4. Crystalline and semistable moduli stacks
• 5. Families of extensions
• 6. Crystalline lifts and the finer structure of Xd,red
• 7. The rank 1 case
• 8. A geometric Breuil–Mézard conjecture
• Appendix A. Formal algebraic stacks
• Appendix B. Graded modules and rigid analysis
• Appendix C. Topological groups and modules
• Appendix D. Tate modules and continuity
• Appendix E. Points, residual gerbes, and isotrivial families
• Appendix F. Breuil–Kisin–Fargues modules and potentially semistable representations (by Toby Gee and Tong Liu)
• Bibliography
• Index