Sur quelques repr\'esentations potentiellement cristallines de GL_2(Q_p)

TL;DR

This paper constructs Banach space representations of GL_2(Q_p) from certain p-adic Galois representations, demonstrating their irreducibility and admissibility using (phi,Gamma)-modules.

Contribution

It introduces a new method to associate Banach space representations to crystalline Galois representations and proves their irreducibility and admissibility.

Findings

B(V) is nonzero for the considered representations

B(V) is topologically irreducible

B(V) is admissible

Abstract

To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is moreover phi-semi-simple, we use the (phi,Gamma)-module and the Wach module associated to V to show that the representation B(V) is nonzero, topologically irreducible and admissible.