Infinitesimal actions in the \(p\)-adic local Langlands correspondence
type: Item
instance of: scholarly article
title: Infinitesimal actions in the \(p\)-adic local Langlands correspondence
zbMATH Open document ID: 1259.11110
author: Gabriel Dospinescu
published in: Mathematische Annalen
publication date: 2012-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.4788
review text: Let \(V\) be a two-dimensional absolutely irreducible \(\overline{\mathbb Q}_p\)-representation of the absolute Galois group \(G_{\mathbb Q_p}= \text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). Let \(\Pi(V)\) be the \(\text{GL}_2(\mathbb Q_p)\) Banach representation associated by \textit{P. Colmez}'s \(p\)-adic Langlands correspondence [Astérisque 330, 281--509 (2010; Zbl 1218.11107)]. The author establishes a link between the action of the Lie algebra of \(\text{GL}_2(\mathbb Q_p)\) on the locally analytic vectors \(\Pi(V)^{\text{an}}\) of \(\Pi(V)\), the connection \(\nabla\) on the \((\phi,\Gamma)\)-module associated to \(V\) and the Sen polynomial of \(V\) (Theorem 1). This answers a question of Harris (see Remark 3.3.8 in [\textit{M. Emerton}, Pure Appl. Math. Q. 2, No. 2, 279--393 (2006; Zbl 1254.11106)]). Using this result the author gives a new (and more direct) proof of a theorem of Colmez (loc. cit., thms. VI 6.13 and VI 6.18), stating that \(\Pi(V)\) has non-zero locally algebraic vectors if and only if \(V\) is potentially semi-stable with distinct Hodge-Tate weights (Theorem 4). Let us stress that Theorem 4 is crucial in the proofs of (a part of) Fontaine-Mazur conjecture by \textit{M. Emerton} [Local-global compatibility in the \(p\)-adic Langlands programme for \(\text{GL}_2/\mathbb Q\), preprint] and by \textit{M. Kisin} [J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)].
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 22E57
zbMATH DE Number: 6087548
zbMATH Keywords: \(p\)-adic Langlands correspondence
zbMATH Keywords: \((\phi,\Gamma)\)-modules
zbMATH Keywords: \(p\)-adic Hodge theory
zbMATH Keywords: Banach representation
zbMATH Keywords: Fontaine-Mazur conjecture
zbMATH Keywords: Sen polynomial
MaRDI profile type: Publication
OpenAlex ID: W1977701254
arXiv ID: 1102.4788
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DOI: 10.1007/S00208-011-0736-2
description: scientific article; zbMATH DE number 6087548