Moduli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts : : (AMS-215) / / Matthew Emerton, Toby Gee.

• Superior document: Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2022 English
• VerfasserIn: Emerton, Matthew,; Gee, Toby,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2022]; ©2023
• Year of Publication: 2022
• Language: English
• Series: Annals of Mathematics Studies ; 408
• Physical Description: 1 online resource (312 p.)

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

Table of Contents

Frontmatter -- Contents -- Chapter One Introduction -- Chapter Two Rings and coefficients -- Chapter Three Moduli stacks of ϕ-modules and (ϕ, Γ)-modules -- Chapter Four Crystalline and semistable moduli stacks -- Chapter Five Families of extensions -- Chapter Six Crystalline lifts and the finer structure of Xd,red -- Chapter Seven The rank 1 case -- Chapter Eight 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