Let $F/\mathbb{ Q}_p$ be a finite extension. We explore the universal supersingular mod $p$ representations of $\mathrm{ GL}_2(F)$ through computing a basis of their invariant space under the pro-$p$ Iwahori subgroup. This generalizes works of Breuil and Schein from $\mathbb{ Q}_p$ and the totally ramified cases to the arbitrary one. Using these results we then construct for an unramified $F/\mathbb{ Q}_p$ a quotient of the universal supersingular module which has as quotients all the supersingular representations of $\mathrm{ GL}_2(F)$ with a $\mathrm{ GL}_2(\mathcal{ O}_F)$-socle that is expected to appear in the mod $p$ local Langlands correspondence. A construction for the case of an extension of $\mathbb{ Q}_p$ with inertia degree 2 and suitable ramification index is also presented.
Categories: math.NT math.RT