Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline lifts

2019/08/20 by Emerton, Matthew, Gee, Toby
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1908.07185

Abstract

We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize étale (phi,Gamma)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. We use these stacks 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. We also discuss the relationship between the geometry of our stacks and the Breuil-Mézard conjecture.

Related

• Emerton--Gee stacks, Serre weights, and Breuil--Mézard conjectures for GSp₄
• Local Shtukas, Hodge-Pink Structures and Galois Representations
• Local geometry of moduli stacks of two-dimensional Galois representations
• Moduli stacks of two-dimensional Galois representations
• Loose crystalline lifts and overconvergence of étale (φ, τ)-modules