Potentially Crystalline Lifts of Certain Prescribed Types

Abstract

We prove several results concerning the existence of potentially crystalline lifts of prescribed Hodge-Tate weights and inertial types of a given representation $\overline{r}:G_K\to {\mathrm{GL}}_n({\overline{\mathbb{F}}}_p)$, where $K/{\mathbb{Q}}_p$ is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.