2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72729In \cite[\S1.3]{Br2}, some unitary representations of ${\rm GL}_2(\mathbf{Q}_p)$ on $p$-adic Banach spaces are associated to 2-dimensional irreducible crystalline representations of ${\rm Gal}(\bar{\mathbf{Q}}_p)/\mathbf{Q}_p)$. Some conjectures are formulated concerning those Banach spaces (non triviality, topological irreducibility, admissibility). We prove those conjectures by reinterpreting those Banach as spaces of functions of a certain type on $\mathbf{Q}_p$, and then by using the theory of $(ϕ,Γ)$-modules associated to the corresponding crystalline representations.44 pages, in frenchNumber TheoryRepresentation Theory11Ftext