Potentially semi-stable deformation rings

Let K/ Qₚ be a finite extension and GK the absolute Galois group of K. For (A^, m) a complete local ring with finite residue and V₀^ a finite free A^ -module equipped with an action of GK, we show that A^ 1/p has a maximal quotient over which the representation V₀^ is semi-stable with Hodge-Tate weights in a given range. We show an analogous result for representations which are potentially semi-stable of fixed Galois type and p-adic Hodge type. If V₀^ is the universal deformation of V₀^ ₀^ A^ / m, then we compute the dimension of A^ 1/p and we show that these rings are sometimes smooth. Finally we apply this theory to show, in some new cases, the compatibility of the p-adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at p.