On the universal mod p supersingular quotients for GL2(F) over (F)over-barp for a general F/Qp

Let F/Q(p) be a finite extension. We explore the universal supersingular mod p representations of GL(2)(F) by computing a basis for their spaces of invariants under the pro-p Iwahori subgroup. This generalizes works of Breuil and Schein (from Q(p) and the totally ramified cases to an arbitrary extension F/Q(p)). Using these results we then construct, for an unramified F/Q(p), quotients of the universal supersingular modules which have as quotients all the supersingular representations of GL(2)(F) with a GL(2)(O-F)-socle that is expected to appear in the mod p local Langlands correspondence. A construction in the case of an extension of Q(p) with inertia degree 2 and suitable ramification index is also presented. (C) 2018 Elsevier Inc. All rights reserved.