# Potentially Crystalline Lifts of Certain Prescribed Types

### Toby Gee

Department of Mathematics, Imperial College London, London SW7 2AZ, UK### Florian Herzig

Department of Mathematics, University of Toronto, Canada### Tong Liu

Department of Mathematics, Purdue University, USA### David Savitt

Department of Mathematics, Johns Hopkins University

## Abstract

We prove several results concerning the existence of potentially crystalline lifts of prescribed Hodge-Tate weights and inertial types of a given representation $\overline{r}:G_{K}\to\mathrm{GL}_{n}(\overline{\Bbb{F}}_p)$, where $K/\Bbb Q_p$ 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.

## Cite this article

Toby Gee, Florian Herzig, Tong Liu, David Savitt, Potentially Crystalline Lifts of Certain Prescribed Types. Doc. Math. 22 (2017), pp. 397–422

DOI 10.4171/DM/569