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Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of 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.

Let p be a prime number and F a complete local field with residue field of characteristic p. In 1993, Barthel and Livné proved the existence of a new kind of $\overline{\bf F}_{p}$-representations of GL2(F) that they called ‘supersingular’ and on which one knows almost nothing. In this article, we determine all the supersingular representations of GL2(Qp) with their intertwinings. This classification shows a natural bijection between the set of isomorphism classes of supersingular representations of GL2(Qp) and the set of isomorphism classes of two-dimensional irreducible $\overline{\bf F}_{p}$-representations of ${\rm Gal}(\overline{\bf Q}_{p}/{\bf Q}_{p})$.

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

Sur quelques représentations modulaires et p-adiques de GL2(Qp): I: (On Some Modular Representations and p-adics of

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

Sur quelques représentations modulaires et p-adiques de GL2(Qp): I: (On Some Modular Representations and p-adics of