The generic fiber of the universal deformation space associated to a tame Galois representation
type: Item
instance of: scholarly article
title: The generic fiber of the universal deformation space associated to a tame Galois representation
zbMATH Open document ID: 0958.11041
author: Gebhard Böckle
published in: Manuscripta Mathematica
publication date: 1999-05-09T00:00:00Z
review text: Let \(E\) be a finite extension of \(\mathbb{Q}\), \(Q\) a finite set of primes in \(E\), \(\overline\rho: G_E\rightarrow \text{ GL}_2(k)\) a representation of the absolute Galois group of \(E\) over the finite field \(k\) of characteristic \(l\), unramified outside \(Q\). The equivalence classes of lifts of \(\overline\rho\) to complete Noetherian local rings with residue field \(k\) are parametrized by a universal deformation ring \(R_Q\). The author shows: if \(\overline\rho\) is tame and irreducible, and if the Leopoldt conjecture holds for the splitting field of \(\text{ ad}_{\overline\rho}\), then the spectrum of \(R_Q\otimes {\mathbb Q}_l\) (viewed \(l\)-adically analytically) is smooth near the trivial lift. This implies that \(R_Q\) can be written as a quotient of \(W(k)T_1,\dots ,T_n\) (power series over the ring of Witt vectors) by \(m\) relations, where \(n\) and \(m\) are prescribed by cohomological data.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 14B12
zbMATH DE Number: 1182243
zbMATH Keywords: Galois representations
zbMATH Keywords: deformation
zbMATH Keywords: Witt vectors
zbMATH Keywords: Leopoldt conjecture
MaRDI profile type: Publication
full work available at URL: s002290050064
OpenAlex ID: W2033832865
DOI: 10.1007/S002290050064
description: scientific article; zbMATH DE number 1182243