Moduli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts (Annals of Mathematics Studies, 215) - Hardcover

Emerton, Matthew; Gee, Toby

Hardcover

ISBN 10: 0691241341 ISBN 13: 9780691241340

Publisher: Princeton University Press, 2022

Synopsis

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.

About this title

About the Author

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

Edition details

• Publisher: Princeton University Press
• Publication date: 2022
• Language: English
• ISBN 10: 0691241341
• ISBN 13: 9780691241340
• Binding: Hardcover
• Number of pages: 312