Abstract
Let F/Qp 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 Qp and the totally ramified cases to an arbitrary extension F/Qp). Using these results we then construct, for an unramified F/Qp, quotients of the universal supersingular modules which have as quotients all the supersingular representations of GL2(F) with a GL2(OF)-socle that is expected to appear in the mod p local Langlands correspondence. A construction in the case of an extension of Qp with inertia degree 2 and suitable ramification index is also presented.
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