On some modular representations of the Borel subgroup of GL2(Qp)

Abstract

AbstractColmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a $\overline {\mathbf {F}}_p$-representation W of $\mathrm {Gal}(\overline {\mathbf {Q}}_p / \mathbf {Q}_p)$ by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.