Automorphy for some l-adic lifts of automorphic mod l Galois representations

Abstract

We extend the methods of Wiles and of Taylor and Wiles from GL 2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL 2 . Following Wiles' method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.

Deformations of Galois representations and exceptional monodromy

“…For simplicity we have not sought optimal generality in this discussion, but what we do is more than enough for our purposes. The reader should note that [CHT08,§2] provides a template for this discussion; those authors work with the L-group of an outer form of GL n × GL 1 .…”

“…When the Weyl group of G does not contain −1, G contains no order 2 element inducing a split Cartan involution, and so there are no 'odd' representations Γ Q → G(k). We instead deform odd representations valued in the L-group of a suitable outer form of G. We develop, only in the degree of generality needed for our application, the basics of deformation theory for L-groups in §9; our task is made easy by the template provided in [CHT08,§2], which treats the case of type A n . With these foundations in place, there are no new difficulties in extending the arguments of earlier sections; we explain the very minor modifications needed in §10.…”

Galois representations, automorphic forms, and the Sato-Tate Conjecture

“…A proof of this theorem is given in [H] but it is conditional on work in progress, including notably the results of [L], [CHL1], [CHL2], [CH], and [Shin]. It is a consequence of the following result, also proved conditionally in [H]:…”

Irreducibility of automorphic Galois representations of GL(n), n at most 5

“…Since r λ (π) is de Rham, so is s λ , so by e.g. Lemma 4.1.3 of [10] there is an algebraic character χ of A × F /F × such that s λ = r λ (χ). Then the weakly compatible system {r λ (χ)} weakly divides {r λ (π)}, so by Lemma 4.8, we see that r λ (π) is irreducible for all λ, a contradiction.…”