$\hat{G}$-local systems on smooth projective curves are potentially automorphic

Böckle, Gebhard; Harris, Michael; Khare, Chandrashekhar; Thorne, Jack A.

doi:10.4310/acta.2019.v223.n1.a1

Cited by 27 publications

(31 citation statements)

References 51 publications

(48 reference statements)

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“…In Subsection 4.3 we explain how to use tools from representation theory to study compatible systems. We recall the notion of a Gcompatible system for G a reductive group from [BHKT16]. Then we show, using [Chi04] and [BHKT16], that any compatible system with split motivic group M can be recovered from a M -compatible system (with Zariski dense image) together with its split motivic representation.…”

Section: Geometric Monodromymentioning

confidence: 99%

“…We recall the notion of a Gcompatible system for G a reductive group from [BHKT16]. Then we show, using [Chi04] and [BHKT16], that any compatible system with split motivic group M can be recovered from a M -compatible system (with Zariski dense image) together with its split motivic representation. We show in Corollary 4.14 that any connected compatible system with semisimple split motivic group has, up to a (uniform) finite kernel, the same monodromy groups as a suitable connected absolutely irreducible compatible system.…”

Section: Geometric Monodromymentioning

confidence: 99%

“…If E is sufficiently large and if the G λ are semisimple, then Chin essentially shows that the compatible system ρ • in fact arises from an M -compatible system r • = (r λ : π 1 (X) −→ M (E λ )) λ∈P ′ E , with Zariski dense image, where M is split semisimple over E, by composing r • with a representation α : M → GL n , defined over E that is independent of λ. It is stated in Corollary 4.13, following [Chi04] and [BHKT16,Prop.6.6]. Choosing an irreducible (almost faithful) representation α ′ instead, and using α ⊕ α ′ as an intermediate step, makes our argument possible.…”

Section: Introductionmentioning

confidence: 99%

“…In Proposition 4.6, we generalize a result of Chin from pure to arbitrary semisimple compatible systems. In Subsection 4.3, building on [Chi04] and [BHKT16], we explain how to use tools from representation theory to study compatible systems. We recall the notion of a G-compatible system for G a reductive group.…”

Section: Introductionmentioning

confidence: 99%

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On the semisimplicity of reductions and adelic openness for 𝐸-rational compatible systems over global function fields

Böckle

Gajda

Petersen

2019

Trans. Amer. Math. Soc.

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Let X be a normal geometrically connected variety over a finite field κ of characteristic p. Let (ρ λ : π1(X) → GLn(E λ )) λ be any semisimple E-rational compatible system where E is a number field and λ ranges over the finite places of E not above p. We derive new properties on the monodromy groups of such systems for almost all λ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense.A key input to our results are automorphic methods and the Langlands correspondence over global function fields proved in [Laf02] by L. Lafforgue.To say more, let (ρ λ : π1(X) → GLn(k λ )) λ be the corresponding mod-λ system, where for every λ by O λ and k λ we denote the valuation ring and the residue field of E λ , and where the reduction is done with respect to some π1(X)-stable O λ -lattice Λ λ of E n λ . Let also G geo λ be the Zariski closure of ρ λ (π1(Xκ)) in GLn,E and let G geo λ be its schematic closure in AutO λ (Λ λ ). Assume in the following that the algebraic groups G geo λ are connected. We prove that for almost all λ the group scheme G geo λ is semisimple over O λ and its special fiber agrees with the Nori envelope of ρ λ (π1(Xκ)). A comparable result under different hypotheses was proved in [CHT17] by Cadoret, Hui and Tamagawa using other methods. As an intermediate result, we show for X a curve that any potentially tame compatible system of mod-λ representations can be lifted to a compatible system over a number field, cf. [Dri15]; this implies for almost all λ the semisimplicity of the restriction ρ λ | π 1 (X κ ) . Finally we establish adelic openness for (ρ λ | π 1 (X κ ) ) λ in the sense of Hui-Larsen [HL15], for E = Q in general, and for E Q under additional hypotheses.

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Section: Geometric Monodromymentioning

confidence: 99%

Section: Geometric Monodromymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the semisimplicity of reductions and adelic openness for 𝐸-rational compatible systems over global function fields

Böckle

Gajda

Petersen

2019

Trans. Amer. Math. Soc.

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“…The proof makes use of the proof of the global Langlands correspondence for GL n by L. Lafforgue [Laf02], together with Chin's application of this work to the analysis of compatible families [Chi04]; see [BHKT,§6].…”

Section: Let T : γmentioning

confidence: 99%

POTENTIAL AUTOMORPHY OF Ĝ-LOCAL SYSTEMS

Thorne¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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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:

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