Moduli Stacks of à tale (?, G)-Modules and the Existence of Crystalline Lifts: (AMS-215) (Annals of Mathematics Studies, 408)

Emerton, Matthew,Gee, Toby

ISBN 10: 069124135X / ISBN 13: 9780691241357

Published by Princeton University Press, 2022

About this Item

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Bibliographic Details

• Title: Moduli Stacks of à tale (?, G)-Modules and the Existence of Crystalline Lifts: (AMS-215) (Annals of Mathematics Studies, 408)
• Publisher: Princeton University Press
• Publication Date: 2022
• Language: English
• ISBN 10: 069124135X
• ISBN 13: 9780691241357
• Binding: paperback
• Condition: Good

About this title

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 the Author

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