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

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 $$F\ne \mathbb {Q}_p$$ , extending a result of Schraen in (J Reine Angew Math (Crelle’s J) 2015(704):187–208, 2015) for quadratic extensions.