Potentially semi-stable deformation rings

Source abstract

Let K / Q p K/\mathbb {Q}_p be a finite extension and G K G_K the absolute Galois group of K K . For ( A ∘ , m ) (A^{\circ }, \mathfrak {m}) a complete local ring with finite residue and V A ∘ V_{A^{\circ }} a finite free A ∘ A^{\circ } -module equipped with an action of G K G_K , we show that A ∘ [ 1 / p ] A^{\circ }[1/p] has a maximal quotient over which the representation V A ∘ 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 p -adic Hodge type. If V A ∘ V_{A^{\circ }} is the universal deformation of V A ∘ ⊗ A ∘ A ∘ / m V_{A^{\circ }}\otimes _{A^{\circ }} A^{\circ }/\mathfrak {m} , then we compute the dimension of A ∘ [ 1 / p ] 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 p -adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at p p .