Finiteness properties of the category of mod p representations of GL2(Qp)

We establish the Bernstein-centre type of results for the category of mod p representations of. We treat all the remaining open cases, which occur when p is or. 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 with equal Hodge-Tate weights.