Irreducible modular representations of the Borel subgroup of GL_2(Q_p)
  Let E be a finite extension of Fp. Using Fontaine's theory of (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. 
Laurent Berger, Mathieu Vienney
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