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/Qp be a finite extension and suppose (Formula Presented) admits a crystalline lift with Hodge–Tate weights contained in the range [0, p]. Does (ρ̄) admit a potentially diagonalisable crystalline lift of the same Hodge–Tate weights? We answer this question in the affirmative when K = Qp and n ≤ 5, and ρ̄ satisfies a mild ‘cyclotomic-free’ condition. We also prove partial results when K/Qp is unramified and n is arbitrary.