Potentially diagonalizable modular lifts of large weight

We prove that for a Hecke cuspform f∈Sk(Γ0(N),χ) and a prime l>max⁡{k,6} such that l∤N, there exists an infinite family {kr}r≥1⊆Z such that for each kr, there is a cusp form fkr∈Skr(Γ0(N),χ) such that the Deligne representation ρfkr,l is a crystalline and potentially diagonalizable lift of ρ‾f,l. When f is l-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Böckle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of GL2n-type representations in the higher level case.