Finiteness properties of the category of mod $p$ representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$

We establish Bernstein-centre type of results for the category of mod $p$ representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$. We treat all the remaining open cases, which occur when $p$ is $2$ or $3$. Our arguments carry over for all primes $p$. This allows us to remove the restrictions on the residual representation at $p$ in Lue Pan’s recent proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of $$\mathrm{G}\mathrm{a}\mathrm{l}$(\overline{\mathbb Q}/$\mathbb{Q}$)$ with equal Hodge–Tate weights.