Irreducible modular representations of the Borel subgroup of GL_2(Q_p)

Laurent Berger (1) , Mathieu Vienney (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

Let E be a finite extension of Fp. Using Fontaine's (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every such representation of B_2(Qp) arises in this way. Our proof extends to algebraically closed fields E of characteristic p. In this case, infinite dimensional smooth irreducible E-linear representations of B_2(Qp) having a central character arise in a similar way from irreducible E-linear representations of the Weil group of Qp.

Domaines: Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

Laurent Berger, Mathieu Vienney. Irreducible modular representations of the Borel subgroup of GL_2(Q_p). Cambridge Univ. Press. Automorphic Forms and Galois Representations, 414, 2014, London Math. Soc. Lecture Note Ser. ⟨hal-00671261⟩