Potentially diagonalisable lifts with controlled Hodge--Tate weights

Abstract

Motivated by the weight part of Serre's conjecture we consider the following question. Let $K/\mathbb{Q}_p$ be a finite extension and suppose $\overline{\rho} \colon G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p)$ admits a crystalline lift with Hodge--Tate weights contained in the range $[0,p]$. Does $\overline{\rho}$ admits a potentially diagonalisable crystalline lift of the same Hodge--Tate weights? We answer this question in the affirmative when $K = \mathbb{Q}_p$ and $n \leq 5$, and $\overline{\rho}$ satisfies a mild `cyclotomic-free' condition. We also prove partial results when $K/\mathbb{Q}_p$ is unramified and $n$ is arbitrary.