Repr\'esentations cristallines irr\'eductibles de ${\rm GL}_2(\mathbf{Q}_p)$

Abstract

In \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 $(\phi,\Gamma)$-modules associated to the corresponding crystalline representations.