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Rebecca Bellovin
Generic smoothness for $G$-valued potentially semi-stable deformation rings
(Lissité générique pour les anneaux de déformations potentiellement semi-stables à valeurs dans $G$)
Annales de l'institut Fourier, 66 no. 6 (2016), p. 2565-2620, doi: 10.5802/aif.3072
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Class. Math.: 11S20, 20G15
Keywords: $p$-adic Hodge theory, deformation rings, algebraic groups

Résumé - Abstract

We extend Kisin’s results on the structure of characteristic $0$ Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups $G$. In particular, we show that such Galois deformation rings are complete intersections. In addition, we study explicitly the structure of the moduli space $X_{\varphi ,N}$ of (framed) $(\varphi ,N)$-modules when $G=\operatorname{GL}_n$. We show that when $G=\operatorname{GL}_3$ and $K_0=\mathbb{Q}_p$, $X_{\varphi ,N}$ has a singular irreducible component, and we construct a moduli-theoretic resolution of singularities.

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