Serre weights for three-dimensional ordinary Galois representations

Abstract

Let F/Q be a CM field where p splits completely and let r¯:Galfalse(Q¯/Ffalse)→ GL 3false(boldF¯pfalse) be a Galois representation whose restriction to Gal(Q¯p/Fw) is ordinary and strongly generic for all places w above p. In this paper, we specify the set of Serre weights in which r¯ can be modular. To this aim, we develop a technique in integral p‐adic Hodge theory to describe extensions of rank‐one Breuil modules.

Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

“…The inclusion W elim (r) ⊂ W ? (r) (weight elimination) is proven in [EGH13,HLM17,MP]. Recent improvements in weight elimination results show that W elim (r) can be replaced by W (r) and 'automorphic of some reachable Serre weight' with just 'automorphic', see Remark 7.10 for a precise discussion.…”

Crystalline liftings and the weight part of Serre’s conjecture

“…And we are notified that there is the independent work [LMP14] to show that these weights are modular. Finally we want to remark that [BLGG14,EGH13,MP14] can treat (or produce) Serre weights (whose corresponding Hodge-Tate weights are) outside the [0, p] range, whereas our paper can only treat the [0, p] range.…”

“…However, the method can only treat ρ absolutely irreducible, and the weight cycling method seems quite difficult to generalize to higher dimensions. We also note that in the preprint [MP14], when K = Q p , d = 3 and ρ is upper triangular, by the method of weight elimination, they have produced a set of possible modular weights. And we are notified that there is the independent work [LMP14] to show that these weights are modular.…”

General Serre weight conjectures

“…First, [HLM17] verify that in this setting there exists an ordinary crystalline lift of ρ that witnesses the containment F (a − 1, b, c + 1) ∈ W ∃ cris (ρ). Second, when the maximal non-splitness assumption above is dropped, an upper bound on the set of Serre weights of ρ was obtained by Morra-Park [MP17]. 7.4.…”