On the universal mod p supersingular quotients for GL₂(F) over F‾_p for a general F/Q_p

Let F/Q_p be a finite extension. We explore the universal supersingular mod p representations of GL₂(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 GL₂(F) with a GL₂(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.