Finiteness properties of the category of mod p representations of

Abstract

Abstract We establish the Bernstein-centre type of results for the category of mod p representations of $\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$ . 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 $\operatorname {\mathrm {Gal}}(\overline {\mathbb Q}/\mathbb {Q})$ with equal Hodge–Tate weights.