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 V_A∘ 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 V_A∘ 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 V_A∘ is the universal deformation of V_A∘⊗ _A∘ 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.