Bigness in Compatible Systems

Abstract

Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if ρ is an ℓ-adic representation of the absolute Galois group of a number field for which the residual representation ρ comes from a modular form then so does ρ. This theorem has numerous hypotheses; a crucial one is that the image of ρ must be "big," a technical condition on subgroups of GLn. In this paper we investigate this condition in compatible systems. Our main result is that in a sufficiently irreducible compatible system the residual images are big at a density one set of primes. This result should make some of the work of Clozel, Harris and Taylor easier to apply in the setting of compatible systems. Elementary properties of bignessIn this section we establish some elementary properties of bigness. Throughout this section, k denotes a finite field of characteristic ℓ, V a finite dimensional vector space over k and G a subgroup of GL(V ).

Residual Irreducibility of Compatible Systems

“…Proof. By [SW,Lemma 6.2], f carries G(O L ) into itself for any tamely ramified finite extension L/K. Therefore, by the previous lemma, f uniquely extends to a map G → G of O-schemes, which is necessarily a group automorphism by uniqueness.…”

“…We claim that this cohomology group vanishes. First, the representation V k of G • k has small norm (see, for instance, [SW,Prop. 3.5]).…”

“…An element of H 1 (G • k , V k ) is represented by an extension of V k by the trivial representation. However, such an extension still has small norm, and is therefore semi-simple (see, for instance, [SW,Prop. 3.3…”

“…But note then that G(k) has no small index subgroups, and so Γ = G(k). The result then follows from the fact that V G = V G(k) , which relies on dim(G) and V der being small compared to char(k) (see, for instance, [SW,Prop. 3.6]).…”

“…Descend V to a representation V ′ over a finite extension K ′ /K, and let V be a G-stable lattice in V ′ (see [La,§1.12] for a proof of existence). Let E = End(V), which is also a representation of small norm (again by [SW,Prop. 3.5]).…”