Moduli stacks of Étale (ϕ, Γ)-modules and the existence of crystalline lifts : annals and mathematics studies-215 / Matthew Emerton and Toby Gee.

Format:

Book

Author/Creator:

Emerton, Matthew, author.

Gee, Toby, 1980- author.

Series:

Annals of mathematics studies ; Volume 215.

Annals of Mathematics Studies ; Volume 215

Language:

English

Subjects (All):

Algebraic stacks.

Moduli theory.

Geometry, Algebraic.

Physical Description:

1 online resource (313 pages)

Place of Publication:

Princeton, N. J. : Princeton University Press, [2022]

System Details:

Mode of access: World Wide Web.

Summary:

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.

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

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9780691241364

0691241368

OCLC:

1350413710