C. Breuil [J. Inst. Math. Jussieu 2, No. 1, 23–58 (2003; Zbl 1165.11319); Ann. Sci. Éc. Norm. Supér. (4) 37, No. 4, 559–610 (2004; Zbl 1166.11331)] has associated to a \(p\)-adic 2-dimensional representation \(V\) of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\), crystalline or semi-stable, a certain representation \(\Pi(V)\) of \(\text{GL}_2(\mathbb Q_p)\). It is known (Colmez, Breuil) that \(\Pi(V)\) is non-trivial, irreducible and admissible. Such a correspondence has been extended to all 2-dimensional representations by P. Colmez [Representations of \(\text{GL}_2(\mathbb Q_p)\) and \((\varphi,\Gamma)\)-modules. Astérisque 330, 281–509 (2010; Zbl 1218.11107)]; it encodes the classical local Langlands correspondence for \(\text{GL}_2(\mathbb Q_p)\). C. Breuil [Compos. Math. 138, No. 2, 165–188 (2003; Zbl 1044.11041); J. Inst. Math. Jussieu 2, No. 1, 23–58 (2003; Zbl 1165.11319)] has also defined a variant of the above correspondence in characteristic \(p\), and conjectures that it is compatible with the characteristic zero case.
The author proves Breuil’s conjecture for trianguline representations. The main ingredient of the proof is the study of some smooth irreducible representations of the Borel subgroup \(B(\mathbb Q_p)\) of \(\text{GL}_2(\mathbb Q_p)\) through models built using the theory of \((\varphi,\Gamma)\)-modules.
For the entire collection see [Zbl 1192.11001].