p-ADIC REPRESENTATIONS OF GL2(L) AND DERIVED CATEGORIES

We construct some locally a"e (p) -analytic representations of GL(2)(L), L a finite extension of a"e (p) , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel'd's upper half space has a structure of rank 2 admissible filtered (phi, N)-module. Finally, we prove that this filtered module is associated, via Fontaine's theory, to the initial Galois representation.