A note on presentations of supersingular representations of $\text{GL}_2(F)$

TL;DR

This paper proves that all smooth irreducible supersingular representations with central character of GL_2(F) over certain fields are not of finite presentation, extending previous results to more general field extensions.

Contribution

It extends Schraen's result by showing that supersingular representations are never of finite presentation for non-Q_p finite field extensions.

Findings

Supersingular representations are not of finite presentation when F ≠ Q_p.

Extends previous quadratic extension results to more general finite field extensions.

Provides new insights into the structure of supersingular representations of GL_2(F).

Abstract

We prove that any smooth irreducible supersingular representation with central character of $\text{GL}_2(F)$ is never of finite presentation when $F$ is a finite field extension of $\mathbb{Q}_p$ such that $F\neq \mathbb{Q}_p$, extending a result of Schraen for quadratic extensions.