Appears in collection : Conference on Arithmetic Geometry in honor of Luc Illusie

In 2005, Colmez defined an exact functor from the category of finite length admissible smooth representations of GL_2(Q_p) over a field of characteristic p to the category of finite length continuous representations of the absolute Galois group of Q_p. This functor has played a crucial role in the p-adic Langlands program for GL_2(Q_p). In this talk, I will review the construction of Colmez’s functor and a generalization due to Breuil. I will discuss the exactness and finiteness of this (generalized) functor. This is a joint work with Breuil, Herzig, Morra and Schraen.