Serre Weights and Wild Ramification in Two-Dimensional Galois Representations

Abstract

A generalization of Serre's Conjecture asserts that if F is a totally real field, then certain characteristic p representations of Galois groups over F arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over p. This characterization of the weights, which is formulated using p-adic Hodge theory, is known under mild technical hypotheses if p > 2. In this paper we give, under the assumption that p is unramified in F, a conjectural alternative description for the set of weights. Our approach is to use the Artin-Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using p-adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

Explicit Serre weights for two-dimensional Galois representations

“…In this paper, we prove [DDR16,Conj. 7.2], which says that the recipes of [BDJ10] and [DDR16] agree. This is a purely local conjecture, and our proof is purely local.…”

“…This description is not useful for practical computations, and the recent paper [DDR16] proposed an alternative recipe in terms of local class field theory, along with the Artin-Hasse exponential, which can be made completely explicit in concrete examples. (Indeed, [DDR16,[9][10] gives substantial numerical evidence for their conjecture. )…”

“…Suppose that ρ ∼ = χ 1 * 0 χ 2 , and set χ = χ 1 χ −1 2 . For a given Serre weight, the recipes of [BDJ10] and [DDR16] determine subspaces L BDJ and L DDR of H 1 (G K , χ), and we have to prove that L BDJ = L DDR .…”

“…The fourth author's PhD thesis [Mav16] proved [DDR16,Conj. 7.2] in generic cases using similar techniques to those of this paper in the setting of (ϕ, Γ)-modules (using the results of [CD11] where we appeal to [GLS14]), while the first three authors arrived separately at the strategy presented here for resolving the general case.…”

Minimal weights of Hilbert modular forms in characteristic

“…Consider the Galois representation defined in Example IIIb 1 of [16, §9], so F = Q( √ 5), p = 3 and ρ : G F → GL 2 (F 9 ) is absolutely irreducible and has the property that ρ| GK ≃ χ 1 * 0 χ 0 where χ i | IK = ǫ τi for appropriately chosen τ i : O F /p ≃ F 9 . Setting χ = χ 1 χ −1 0 , we have χ| IK = ǫ −2 τ0 , and the discussion in [16] shows that the associated extension class lies in the line V χ of Lemma 11.2.1. It follows that ρ has a crystalline lift of weight ((3, 1), (0, −1)).…”