Local constancy for reductions of two-dimensional crystalline representations

We prove the existence of local constancy phenomena for reductions in a general (odd) prime power setting two-dimensional irreducible crystalline representations Gal(? ¯ p /? ). These depend on two parameters: trace and weight k. They appear example context classical modular forms tame level. find an (explicit) result with respect to using Fontaine’s theory (?,?)-modules, its refinement due Berger via Wach modules their continuity properties. The k (for ?0) will follow from study Colmez’s rigid analytic space parametrizing trianguline representations. This work extends some results obtained residually semi-simple case.