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

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.

arXiv commentTo appear in Annals of Math studies. Fixing minor typos found in copyediting, should agree with published version apart from US/UK spelling and other minor issues of style. A pdf of errata and clarifications will be maintained on our websites

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Cited by in corpus (5)

• On local Galois deformation rings
• Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$
• A semisimple mod $p$ Langlands correspondence in families for $G{L}_2({\mathbb{Q}}_p)$
• Crystalline condition for $A_{\inf }$-cohomology and ramification bounds
• Moduli stacks of (phi,Gamma)-modules: a survey