Représentations cristallines irréductibles de ${\mathrm{GL}}_2({\mathbf{Q}}_p)$

Abstract

In \cite[\S1.3]{Br2}, some unitary representations of ${\mathrm{GL}}_2({\mathbf{Q}}_p)$ on $p$-adic Banach spaces are associated to 2-dimensional irreducible crystalline representations of $\mathrm{Gal}({\bar{\mathbf{Q}}}_p)/{\mathbf{Q}}_p)$. 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 $(\varphi ,\Gamma )$-modules associated to the corresponding crystalline representations.

44 pages, in french