Potentially semi-stable deformation rings

Let $K/\Q_p$ be a finite extension and $G_K$ the absolute Galois group of $K$. For $(A^{\circ}, \mathfrak{m})$ a complete local ring with finite residue and $V_{A^{\circ}}$ a finite free $A^{\circ}$-module equipped with an action of $G_K$ , we show that $A^{\circ}[1/p]$ has a maximal quotient over which the representation $V_{A^{\circ}}$ 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 $V_{A^{\circ}}$ is the universal deformation of $V_{A^{\circ}}\otimes_{A^{\circ}} A^{\circ}/\mathfrak{m}$, then we compute the dimension of $A^{\circ}[1/p]$ 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 $p$.