\(p\)-adic-representations of \(\text{GL}_2(L)\) and derived categories
type: Item
instance of: scholarly article
title: \(p\)-adic-representations of \(\text{GL}_2(L)\) and derived categories
zbMATH Open document ID: 1210.11066
author: Benjamin Schraen
published in: Israel Journal of Mathematics
publication date: 2010-06-29T00:00:00Z
review text: The article concerns the interesting correspondence between \(p\)-adic two-dimensional representations \(\rho\) of the Galois group \(G_p\) of \(\mathbb{Q}_p\), and unitary representations \(\pi\) of \(\text{GL}(2,\mathbb Q_p)\) on \(p\)-adic Banach spaces. Breuil associated to a semistable non-cristalline two-dimensional \(\rho\) a locally analytic \(\pi\). An invariant \(\mathfrak{L}\) appears. It does not appear in the ``classical'' (complex) Langlands correspondence. The unitary completion of the \(\pi\) is admissible and nonzero, and its mod \(p\) reduction is compatible with the mod \(p\) correspondence defined by Breuil. The main result of this paper is that on considering the space of morphisms, in a suitable derived category, between the dual of \(\pi\) and the de Rham complex of Drinfeld's half-plane -- a rank two admissible filtered \((\varphi,N)\)-module -- one finds the Fontaine module associated with the original \(\rho\). The work extends from \(\mathbb Q_p\) to a finite extension \(L\). Then the locally \(\mathbb Q_p\)-analytic representations \(\pi\) of \(\text{GL}(2,L)\) become parametrized not by a single \(\mathfrak{L}\), but by \((\mathfrak{L}_\sigma)\), where \(\sigma\) ranges over the embeddings of \(L\) in the algebraic closure of \(\mathbb Q_p\). The author explains that the \(\pi\) obtained might not be the best objects when \(L\not=\mathbb{Q}_p\). For example, it is not clear that their unitary completion is admissible. But he hopes they are subobjects of the correct representations.
reviewed by: Yuval Z. Flicker
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 5728352
zbMATH Keywords: \(p\)-adic representations of GL(2) over a finite extension of \(\mathbb Q_p\)
zbMATH Keywords: Fontaine's theory
zbMATH Keywords: Galois representations
MaRDI profile type: Publication
full work available at URL: s11856-010-0031-z
OpenAlex ID: W1994402871
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DOI: 10.1007/S11856-010-0031-Z
description: scientific article; zbMATH DE number 5728352