Correspondance de Langlands locale $p$-adique et changement de poids

Abstract

If $f$ is a modular form of weight $\ge 1$, the representation of ${\mathbf{GL}}_2({\mathbf{Q}}_p)$ attached to it by the $p$-adic local Langlands correspondence encodes the representation attached by the classical correspondence. We introduce weight shifting techniques that allow us to extend the result to weight $1$. We also give a full description of the Jordan–Hölder components of the locally analytic representations appearing in the correspondence.