Repr\'esentations modulaires de ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$ et repr\'esentations galoisiennes de dimension 2

We prove Breuil’s conjecture concerning the reduction modulo $p$ of trianguline representations $V$ and of the representations $\mathrm{\Pi }(V)$ of ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$ associated to them by the $p$-adic Langlands correspondence. The main ingredient of the proof is the study of some smooth irreducible representations of $\mathrm{B}({\mathbf{Q}}_p)$ through models built using the theory of $(\varphi ,\mathrm{\Gamma })$-modules.