Potentially semi-stable deformation rings

Abstract:

Let $K/\mathbb {Q}_p$ 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 $p$.