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arXiv · 1908.07185

Moduli stacks of \'etale (phi,Gamma)-modules and the existence of crystalline lifts

Matthew Emerton·Toby Gee
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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 \'etale (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-M\'ezard conjecture.

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Matthew Emerton, Toby Gee. 2019-08-20. Moduli stacks of \'etale (phi,Gamma)-modules and the existence of crystalline lifts. https://arxiv.org/abs/1908.07185

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