Elsevier

Journal of Number Theory

Volume 228, November 2021, Pages 188-207
Journal of Number Theory

General Section
Potentially diagonalizable modular lifts of large weight

https://doi.org/10.1016/j.jnt.2021.03.023Get rights and content

Abstract

We prove that for a Hecke cuspform fSk(Γ0(N),χ) and a prime l>max{k,6} such that lN, there exists an infinite family {kr}r1Z such that for each kr, there is a cusp form fkrSkr(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.

Section snippets

Definitions, notations and preliminary facts

Throughout this paper, K will denote a finite extension of Ql and O its ring of integers. Fix algebraic closures Q of Q, K of K and Ql of Ql respectively, fix an embedding of Q in Ql and denote as usual GK=Gal(K/K), the absolute Galois group of K. Given a local ring R of residual characteristic l, denote by mR its unique maximal ideal and by F its residual field. Denote by GLm(R)1 the kernel of the reduction map GLm(R)GLm(F). For a representation ρ:ΓGLm(R) and for gGLm(R), we will

The ordinary case

Assume f is l-ordinary so that, as is well known, ρf,l is ordinary in the sense of Theorem 1.4 a). A simple strategy to prove our main theorem in this case is by using Hida families. Next, we introduce the precise definitions and relevant facts, enough for our purposes. First of all, we need to recall the concept of ordinary l-stabilisation:

Definition 2.1

Let fSk(Γ1(N),χ) be an ordinary Tl-eigenform and al(f) its eigenvalue. Consider the Hecke characteristic polynomial X2al(f)X+χ(l)lk1 and label its roots α

The non-ordinary case

Let now fSk(Γ0(N),χ) be non-ordinary. Since we are assuming that 0kl1 and since Ql is totally unramified and since lN, ρf,l|Ql is crystalline, hence by Theorem 1.4 b), ρf,l|Ql is potentially diagonalizable.

If we try to repeat the argument of the proof of Theorem 2.6 replacing Hida families by Coleman families in our non-ordinary case, we find that each specialization fkr is neither ordinary nor its weight is in the Fonaine-Lafaille range, as it increases with r. In this section we deal

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1

Partially supported by MTM2016-79400-P.

2

Partially supported by PID2019-107297GB-I00.

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