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

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Matthew Emerton Toby Gee

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

AMS-215

Matthew Emerton, Toby Gee

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English

Princeton University Press
01 March 2023

Algebraic geometry

Series: Annals of Mathematics Studies

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.

By: Matthew Emerton, Toby Gee
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 235mm, Width: 156mm,
ISBN: 9780691241357
ISBN 10: 069124135X
Series: Annals of Mathematics Studies
Pages: 312
Publication Date: 01 March 2023
Audience: General/trade , ELT Advanced
Format: Paperback
Publisher's Status: Active

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