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 ρf,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.Agencia Estatal de Investigació