Potentially crystalline lifts of certain prescribed types

TL;DR

This paper investigates the existence of potentially crystalline lifts of certain Galois representations with specified properties, combining local and global methods to advance automorphy lifting techniques.

Contribution

It provides new results on the existence of potentially crystalline lifts with prescribed weights and types, enhancing the understanding of Galois representations in number theory.

Findings

Proved local existence results for crystalline lifts

Established global automorphy lifting applications

Extended the theory of Galois representations and automorphy

Abstract

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/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.