General SectionPotentially diagonalizable modular lifts of large weight
Section snippets
Definitions, notations and preliminary facts
Throughout this paper, K will denote a finite extension of and its ring of integers. Fix algebraic closures of , of K and of respectively, fix an embedding of in and denote as usual , the absolute Galois group of K. Given a local ring R of residual characteristic l, denote by its unique maximal ideal and by its residual field. Denote by the kernel of the reduction map . For a representation and for , we will
The ordinary case
Assume f is l-ordinary so that, as is well known, 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 be an ordinary -eigenform and its eigenvalue. Consider the Hecke characteristic polynomial and label its roots α
The non-ordinary case
Let now be non-ordinary. Since we are assuming that and since is totally unramified and since , is crystalline, hence by Theorem 1.4 b), 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 is neither ordinary nor its weight is in the Fonaine-Lafaille range, as it increases with r. In this section we deal
References (18)
On automorphic points in polarized deformation rings
Am. J. Math.
(2019)- et al.
Automorphy of in the self-dual case
- et al.
Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula
(1989) - et al.
Potential automorphy and change of weight
Ann. Math. (2)
(2014) - et al.
Serre weights for rank two unitary groups
Math. Ann.
(2013) On the isomorphism . Appendix to: Khare, Chandrashekhar: on isomorphisms between deformation rings and Hecke rings
Invent. Math.
(2003)- et al.
Diagonal cycles and Euler systems I: a p-adic Gross-Zagier formula
Ann. Sci. Éc. Norm. Supér.
(2014) The weight in Serre's conjectures on modular forms
Invent. Math.
(1992)- et al.
A geometric perspective on the Breuil–Mezard conjecture
J. Inst. Math. Jussieu
(2014)
Cited by (1)
Modular supercuspidal lifts of weight 2
2023, arXiv