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 $\overline{\rho }:G_K\to {\mathrm{GL}}_n({\overline{\mathbb{F}}}_p)$ 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 $n\le 5$, 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.