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On some modular representations of the Borel subgroup of \(\text{GL}_{2}(\mathbb Q_p)\). (English) Zbl 1219.11079
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)].

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
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References:
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