Irreducibility of versal deformation rings in the \((p,p)\)-case for 2-dimensional representations
type: Item
instance of: scholarly article
title: Irreducibility of versal deformation rings in the \((p,p)\)-case for 2-dimensional representations
zbMATH Open document ID: 1354.11042
author: Gebhard Böckle
author: Ann-Kristin Juschka
published in: Journal of Algebra
publication date: 2015-10-13T00:00:00Z
review text: Let \(K\) be a finite extension of \({\mathbb Q}_p\) and \(G_K\) be its absolute Galois group. Let further \({\overline\rho}: G_{K} \rightarrow GL_{n}({\mathbb F}) \) be a continuous residual representation for \({\mathbb F}\) a finite field of characteristic \(p\). Denote by \(W ({\mathbb F})\) the ring of Witt vectors of \({\mathbb F}\) and by \(\mathcal O\) the ring of integers of a finite totally ramified extension of \(W ({\mathbb F})[1/p].\) Denote also by \({\mathfrak m}_{\mathcal O}\) its maximal ideal and by \({\overline{\omega}}_{\mathcal O}\) a uniformizer. For \({\overline\rho},\) there exists the deformation functor from the category \({\widehat{Ar_{\mathcal O}}}\) of complete Noetherian local \({\mathcal O}\)-algebras \((R,{\mathfrak m}_R)\) to the category of sets defined by \textit{Ph. Cassou-Noguès} and \textit{M. J. Taylor} (cf. [Prog. Math. 75, 35--64 (1988; Zbl 0714.11078)]) \[ D_{\overline{\rho}}(R):=\{{\rho}: G_{K} \rightarrow GL_{n}(R) \mid \rho \operatorname{mod} {\mathfrak m}_R ={\overline{\rho}}{\text{ and }}\rho {\text{ is a cont. representation}}\}/{\sim} \] where \({\rho}\sim {\rho}'\) if there exists \(A\in {\text{ker}} \,\{ GL_{n}(R) \rightarrow GL_{n}({\mathbb F})\}\) such that \({\rho}' = A{\rho}A^{-1}.\) A deformation of \({\overline{\rho}}\) is an equivalence class \([{\rho}] \) of \({\rho}\) under \(\sim\). The authors study the versal deformation space \({\mathfrak X}({\overline{\rho}})\) of \({\overline{\rho}}.\) In particular the question of its irreducibility. They show that the answer is affirmative for \(n = 2\) and \(p > 2\) and that the crystalline points are Zariski dense in the versal deformation space \({\mathfrak X}({\overline{\rho}}).\)
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11S25
zbMATH DE Number: 6493620
zbMATH Keywords: Galois representations
zbMATH Keywords: local fields
zbMATH Keywords: deformation theory
zbMATH Keywords: crystalline representations
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.07.009
OpenAlex ID: W1155669821
cites work: The generic fiber of the universal deformation space associated to a tame Galois representation
cites work: Q4484967
cites work: Lifting mod \(p\) representations to characteristics \(p^{2}\).
cites work: Irreducible Components of Deformation Spaces: Wild 2-Adic Exercises
cites work: Modularity of certain potentially Barsotti-Tate Galois representations
cites work: About the density of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) crystalline representations
cites work: Q3619554
cites work: Q3579569
cites work: Crystalline Dieudonné module theory via formal and rigid geometry
cites work: SERRE WEIGHTS FOR LOCALLY REDUCIBLE TWO-DIMENSIONAL GALOIS REPRESENTATIONS
cites work: Q4317713
cites work: The symplectic nature of fundamental groups of surfaces
cites work: The deformation theory of representations of fundamental groups of compact Kähler manifolds
cites work: When is a regular sequence super regular?
cites work: Endliche Gruppen I
cites work: Base change, lifting, and Serre's conjecture
cites work: Classification of Demushkin Groups
cites work: Q3199524
cites work: Q4369165
cites work: Q3992979
cites work: Deformations of trianguline B-pairs and Zariski density of two dimensional crystalline representations
cites work: Zariski density of crystalline representations for any p-adic field
cites work: Q3139241
cites work: Form rings and regular sequences
cites work: Q4369161
DOI: 10.1016/J.JALGEBRA.2015.07.009
description: scientific article; zbMATH DE number 6493620