Moduli Stacks of Etale ( , )-Modules and the Existence of Crystalline

Moduli Stacks of Etale (φ, Γ)-Modules and the Existence of Crystalline, Volume 1, provides a foundational account of a new construction within the p-adic Langlands correspondence. Authors Matthew Emerton and Toby Gee construct Noetherian formal algebraic stacks over Spf Zp that algebraize Mazur’s formal deformation rings of local Galois representations. These stacks parameterize étale (φ, Γ)-modules, allowing the formal completions at points in their special fibres to recover universal deformation rings. This advanced mathematics text serves as a critical reference for researchers studying the intersection of number theory and geometry.

The work uses these stacks to demonstrate that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero and admit crystalline lifts. The authors explicitly describe the irreducible components of the underlying reduced substacks and explore the connection between stack geometry and the Breuil-Mézard conjecture. This algebraic geometry volume also establishes several foundational results in p-adic Hodge theory.