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

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.

30 pages, comments welcome. Revised version: bad alignment of equations, typos and some layout issues fixed. Conclusion 1.3 about endomorhpisms of the universal supersingular quotients added. Proofs in Sections 3.2 and 3.3 made clearer. Main results unchanged

References in corpus (2)

• The weight part of Serre's conjecture for GL(2)
• Serre's Modularity Conjecture

Cited by in corpus (2)

• A note on presentations of supersingular representations of ${\text{GL}}_2(F)$
• Iwahori-Hecke model for mod p representations of GL(2,F)