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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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