Berger, Laurent On some modular representations of the Borel subgroup of \(\text{GL}_{2}(\mathbb Q_p)\). (English) Zbl 1219.11079 Compos. Math. 146, No. 1, 58-80 (2010). The article is a contribution to the \(p\)-adic Langlands correspondence, and more specifically the ‘mod \(p\)’ correspondence first introduced by C. Breuil in [Compos. Math 138, No. 2, 165–188 (2003; Zbl 1044.11041)]. Breuil exhibited a natural correspondence between irreducible two-dimensional representations \(\rho: \text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\to \text{GL}(W)\), where \(W\) is a two-dimensional vector space over \(\overline{\mathbb F}_p\), and irreducible supersingular \(\overline{\mathbb F}_p\)-representations of \(\text{GL}_2(\mathbb Q_p\)). P. Colmez [Astérisque 330, 281–509 (2010; Zbl 1218.11107)] has given a recipe to associate a smooth modular representation \(\Omega(W)\) of the Borel subgroup of \(\text{GL}_2(\mathbb Q_p)\) to a \(\overline{\mathbb F}_p\)-representation \(W\) of \(\text{Gal}(\overline{\mathbb Q}_p/(\mathbb Q_p)\) (of arbitrary finite dimension \(n\)) by using Fontaine’s theory of \((\varphi,\Gamma)\)-modules. The author computes \(\Omega(W)\) explicitly (Proposition 2.3.3). In the case \(n= 2\), he shows that the Colmez functor is compatible with Breuil’s correspondence, i.e. \(\Omega(W)\) is the restriction to the Borel subgroup of \(\text{GL}_2(\mathbb Q_p)\) of the supersingular representation associated to \(W\) by Breuil’s construction. This recovers the author’s result from his earlier paper [Astérisque 330, 263–279 (2010; Zbl 1233.11060)]. Reviewer: Andrzej Dąbrowski (Szczecin) Cited in 4 Documents MSC: 11F80 Galois representations 11F33 Congruences for modular and \(p\)-adic modular forms 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F85 \(p\)-adic theory, local fields 20C20 Modular representations and characters 20G25 Linear algebraic groups over local fields and their integers 22E35 Analysis on \(p\)-adic Lie groups 22E50 Representations of Lie and linear algebraic groups over local fields Keywords:\(p\)-adic Langlands correspondence; supersingular representations; \((\varphi,\Gamma)\)-modules; Galois representations; Breuil’s correspondence; Colmez functor PDF BibTeX XML Cite \textit{L. Berger}, Compos. Math. 146, No. 1, 58--80 (2010; Zbl 1219.11079) Full Text: DOI arXiv References: [2] doi:10.1017/S1474748003000021 · Zbl 1165.11319 · doi:10.1017/S1474748003000021 [3] doi:10.1023/A:1026191928449 · Zbl 1044.11041 · doi:10.1023/A:1026191928449 [6] doi:10.1006/jnth.1995.1124 · Zbl 0841.11026 · doi:10.1006/jnth.1995.1124 [7] doi:10.1007/BF01405086 · Zbl 0235.14012 · doi:10.1007/BF01405086 [8] doi:10.1215/S0012-7094-94-07508-X · Zbl 0826.22019 · doi:10.1215/S0012-7094-94-07508-X [10] doi:10.1007/s00039-007-0646-3 · Zbl 1346.22010 · doi:10.1007/s00039-007-0646-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.