Potentially diagonalizable modular lifts of large weight

Abstract

We prove that for a Hecke cuspform $f\in S_k({\Gamma }_0(N),\chi )$ and a prime $l>\max \{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r{\}}_{r\ge 1}\subseteq \mathbb{Z}$ such that for each $k_r$, there is a cusp form $f_{k_r}\in S_{k_r}({\Gamma }_0(N),\chi )$ such that the Deligne representation ${\rho }_{f_{k_r,l}}$ is a crystaline and potentially diagonalizable lift of ${\overline{\rho }}_{f,l}$. 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 ${\mathrm{GL}}_{2n}$-representations in the higher level case.