Coefficient systems and supersingular representations of $GL_2(F)$

TL;DR

This paper constructs numerous supersingular smooth irreducible mod p representations of GL_2(F) over non-Archimedean local fields, extending known results and conjecturing completeness for all such fields.

Contribution

It introduces a new construction of supersingular representations for GL_2(F) and conjectures this captures all such representations for any non-Archimedean local field.

Findings

Constructed many supersingular representations for GL_2(F)

Matches known results for F=Q_p with Breuil's work

Conjectures completeness of the construction for all F

Abstract

Let $F$ be a non-Archimedean local field with the residual characteristic $p$. We construct a "good" number of smooth irreducible $\bar{\mathbf{F}}_p$-representations of $GL_2(F)$, which are supersingular in the sense of Barthel and Livn\'e. If $F=\mathbf{Q}_p$ then results of Breuil imply that our construction gives all the supersingular representations up to the twist by an unramified quasi-character. We conjecture this is true for arbitrary $F$.