Appears in collection : Conférence de mi-parcours du programme ANRThéorie de Hodge p-adique et Développements (ThéHopaD)

This is joint work with P. Colmez and V. Paškūs, whose goal is the complete classification of absolutely irreducible unitary admissible Bach space representations of $GL_2(Q_p)$ (and their reduction modulo p) for all primes p (completing previous work of Paskus). In this talk we will explain how the injectivity of the Montreal functor is related to this problem (and several other results) and sketch the proof of this injectivity.