Potentially diagonalizable modular lifts of large weight

TL;DR

This paper proves the existence of infinite families of modular lifts of large weight that are potentially diagonalizable, using different methods for ordinary and non-ordinary forms, with implications for automorphy in higher rank cases.

Contribution

It establishes the existence of potentially diagonalizable crystalline lifts for large weight modular forms, extending previous results to a broader setting.

Findings

Constructs infinite families of modular lifts with prescribed properties.

Uses Hida theory for ordinary forms and local-to-global methods for non-ordinary forms.

Provides groundwork for automorphy lifting in higher rank cases.

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 $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ 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 B\"ockle 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.