On the universal mod $p$ supersingular quotients for $\mathrm {GL}_2(F)$ over $\overline{\mathbb F}_p$ for a general $F/\mathbb{Q}_p$

TL;DR

This paper generalizes the understanding of universal supersingular mod p representations of GL2 over finite extensions of Qp, providing explicit bases and constructing quotients that relate to the mod p local Langlands correspondence.

Contribution

It extends previous results from Qp to arbitrary finite extensions, constructing new quotients of the universal supersingular module with anticipated Langlands correspondence properties.

Findings

Computed bases of invariant spaces under pro-p Iwahori subgroup.

Constructed quotients of the universal supersingular module for unramified extensions.

Presented a construction for extensions with inertia degree 2 and specific ramification.

Abstract

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.