Pseudodeformations
type: Item
instance of: scholarly article
title: Pseudodeformations
zbMATH Open document ID: 1295.11055
author: Joël Bellaïche
published in: Mathematische Zeitschrift
publication date: 2012-05-08T00:00:00Z
review text: From the text: ``Mazur's deformation theory of representations of (Galois) groups played a very important role in the striking progresses that have been made in algebraic number theory in the last fifteen years (to name a few: the proof of Shimura-Taniyama-Weil conjecture, of Serre's conjecture on the modularity of odd \(\mod p\) Galois representations, of Fontaine-Mazur's conjecture, and of Sato-Tate conjecture). The theory works as its best only when the representation to be deformed, \(\overline\rho: G\to \mathrm{GL}_d(\kappa)\) (where \(\kappa\) is a field), satisfies the following condition (1): \(\dim_\kappa \mathrm{End}_G(\overline\rho, \overline\rho)= 1\). By Schur's lemma, this happens for example if \(\overline\rho\) is absolutely irreducible. In this case, under some finiteness hypotheses on the group \(G\) that are satisfied in applications to Galois theory, Mazur's deformation functor \(A\to D_{\overline\rho}(A)\) which attaches to an Artinian local ring \(A\) with residue field \(\kappa\) the set of strict equivalence classes of deformations \(\rho: G\to\mathrm{GL}_d(A)\) of \(\overline\rho\) is pro-representable by a complete Noetherian local ring \(R\) (see [\textit{B. Mazur}, Math. Sci. Res. Inst. 16, 385--437 (1989; Zbl 0714.11076)]. The tangent space of this local ring is identified with the space \(\mathrm{Ext}^1_G(\overline\rho, \overline\rho)= H^1(G, \mathrm{ad\,}\overline\rho)\). Lifting questions are also solved by the construction of obstruction classes in \(\mathrm{Ext}^2_G(\overline\rho, \overline\rho)= H^2(G, \mathrm{ad\,}\overline\rho\)).'' ``We consider here the problem of finding analogues of those results in the absence of condition (1).'' ``We study the deformation functor of a reducible pseudo character. We show that there is a natural filtration (the complexity filtration) on such a functor, and that this filtration induces a filtration on the tangent space, whose graded pieces can be described in terms of the extension spaces betw…
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 6031785
zbMATH Keywords: Galois representation
zbMATH Keywords: Mazur's deformation functor
zbMATH Keywords: lifting of Galois representation
zbMATH Keywords: obstruction theory
zbMATH Keywords: pseudodeformations
MaRDI profile type: Publication
full work available at URL: s00209-011-0846-2
OpenAlex ID: W3191772289
cites work: Q3558905
cites work: The \(p\)-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings
cites work: Base change, lifting, and Serre's conjecture
cites work: Pseudo representations
cites work: Characterization of characters and pseudo-characters
cites work: Residually reducible representations and modular forms
cites work: Galois representations associated to Siegel modular forms of low weight
DOI: 10.1007/S00209-011-0846-2
description: scientific article; zbMATH DE number 6031785