Laumon Sheaf and the mod p Langlands Program for GL_2 of a Finite Degree Extension of Qp

Let E be a finite degree extension of Qp. Given a mod p representation of the absolute Galois group of E we construct a sheaf on a punctured absolute Banach-Colmez space that should give the first step in the construction of the mod p local Langlands correspondence as a representation of the mirabolic subgroup of GL2(E). We will explain the construction of this sheaf, its conjectural properties and the link with the recent work of Breuil, Herzig, Hu Morra, and Schraen about local-global compatibility.