On Modular Weights of Galois Representations

Let F be a totally real field and ρ : Gal(F/F )→ GL2(Fp) a Galois representation whose restriction to a decomposition group at some place dividing p is irreducible. Suppose that ρ is modular of some weight σ. We specify a set of weights, not containing σ, such that ρ is modular for at least one weight in this set. Let F be a totally real field and p a prime, and consider a Galois representation ρ : Gal(F/F ) → GL2(Fp). A weight is an irreducible Fp-representation of the finite group GL2(OF /p). There exists a notion of ρ being modular of a certain weight; see, for instance, [1] or [6]. The aim of this note is to prove Theorem 5, which states that if ρ is modular of a weight σ, then it is modular for at least one weight in a certain set of weights distinct from σ. For instance, if p is non-split in F and σ is a quotient of the induction Ind B θ of some character θ : B → Fp, where B ⊂ G = GL2(OK/p) is the subgroup of upper triangular matrices, then ρ is also modular for some other Jordan-Hölder constituent of Ind B θ. The earliest version of this statement, in the case F = Q, was proved by Kevin Buzzard but never published. It was written up by Richard Taylor ([7], Lemma 5.1) in the case of p completely split in F and by the author ([5], Corollary 8.4) in general, using a similar argument. The argument here is somewhat more conceptual than the one given in the unpublished [5]. The motivation for proving Theorem 5 is its application to Serre’s epsilon conjecture for Hilbert modular forms in [6], where it is stated without proof as Proposition 5.9. If p is unramified in F and ρ is assumed to be modular of some weight, then a conjectural list of the modular weights of ρ was given in [1]. The results towards this conjecture in [4] and [6] both rely on some combinatorial computations which give weaker than desired results for some “irregular” weights; see Remark 5.10 of [6]. Theorem 5 allows us to get around this problem at places of F with small residue field. In particular, one can prove the cases of the epsilon conjecture required to simplify the proof of Serre’s conjecture by Khare and Wintenberger by eliminating its reliance on [7]. This is discussed in the remark following Theorem 5.