Deformation rings and base change
type: Item
instance of: scholarly article
title: Deformation rings and base change
zbMATH Open document ID: 0939.11023
published in: Journal of Number Theory
publication date: 2000-07-02T00:00:00Z
review text: Let \(\rho\) be a linear representation of the absolute Galois group of \({\mathbb Q}\) in \(\text{GL}_n({\mathbb F})\), where \({\mathbb F}\) is a finite field of characteristic \(p\). On the Galois group of the maximal extension of \({\mathbb Q}\) which is unramified outside some fixed finite set \(S\) of places there exists a universal deformation \(\rho^{\mathfrak univ} \) of \(\rho\) in \(\text{GL}_n(R_{\mathbb Q})\) where \(R_{\mathbb Q}\) is a local noetherian \(W({\mathbb F})\)-algebra with residue field \({\mathbb F}\). If \(F\) is a finite extension of \({\mathbb Q}\), there is an analog universal deformation of the restriction \(\rho_F\) of \(\rho\) to the absolute Galois group of \(F\), which takes values in some \(\text{GL}_n(R_F)\). Due to universality there is a local ring homomorphism \(\alpha:R_F \rightarrow R_{\mathbb Q}\) which is the main objective of the present paper. The first half of this is devoted to studying \(\alpha\) in the case that \(p\) does not divide \([F:{\mathbb Q}]\). If \(I\) denotes the kernel of \(\alpha\), then one gets a split extension \(R_F/I^2 \rightarrow R_{\mathbb Q}.\) It remains open whether \(\alpha\) itself is a split extension. In the second half of the paper it is shown that for \(n=2\) this splitting would imply the existence of a \(p\)-adic lifting of \(\rho.\) A condition is given under which \(p\) is not a zero divisor in \(R_{\mathbb Q}.\)
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1389288
zbMATH Keywords: mod \(p\) Galois representation
zbMATH Keywords: universal deformation
author: Chandrashekhar Khare
MaRDI profile type: Publication
full work available at URL: jnth.1999.2418
OpenAlex ID: W2017598101
cites work: Families of Galois representations - increasing the ramification
cites work: Q4233881
cites work: Q3882586
cites work: Base change, lifting, and Serre's conjecture
cites work: A finiteness theorem for the symmetric square of an elliptic curve
cites work: Q3199524
cites work: Lifting Galois representations
DOI: 10.1006/JNTH.1999.2418
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description: scientific article; zbMATH DE number 1389288