POTENTIAL AUTOMORPHY OF Ĝ-LOCAL SYSTEMS

Abstract

Vincent Lafforgue has recently made a spectacular breakthrough in the setting of the global Langlands correspondence for global fields of positive characteristic, by constructing the 'automorphic-to-Galois' direction of the correspondence for an arbitrary reductive group G. We discuss a result that starts with Lafforgue's work and proceeds in the opposite ('Galoisto-automorphic') direction.G(Q ℓ ) of Zariski dense image, and show that any such representation is potentially automorphic, in the sense that there exists a Galois cover Y → X such that the pullback of ρ to πé t 1 (Y ) is contained in the image of Lafforgue's construction. We will guide the reader through the context surrounding this result, and discuss some interesting open questions that are suggested by our methods.2 Review of the case G = GL n We begin by describing what is known about the Langlands conjectures in the setting of the general linear group G = GL n . In this case very complete results were obtained by Laurent Lafforgue [Laf02]. As in the introduction, we write F q for the finite field with q elements, and let X be a smooth, projective, connected curve over F q . The Langlands correspondence predicts a relation between representations of the absolute Galois group of K and automorphic representations of the group GL n (A K ). We now describe each of these in turn.Let K s be a fixed separable closure of K. We write Γ K = Gal(K s /K) for the absolute Galois group of K, relative to K s . It is a profinite group. If S ⊂ X is a finite set of closed points, then we write K S ⊂ K s for the maximal extension of K which is unramified outside S, and Γ K,S = Gal(K S /K) for its Galois group. This group has a geometric interpretation: if we set U = X − S, and write η for the geometric generic point of U corresponding to K s , then there is a canonical identification Γ K,S ∼ = πé t 1 (U, η) of the Galois group with theétale fundamental group of the open curve U .Fix a prime ℓ ∤ q and a continuous character ω : Γ K → Q × ℓ of finite order. If n ≥ 1 is an integer, then we write Gal n,ω for the set of conjugacy classes of continuous representations ρ : Γ K → GL n (Q ℓ ) with the following properties: