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Galois Representations for General Symplectic Groups 3 Galois representations for general symplectic groups ArnoKret SugWooShin October 16, 2020 Abstract We prove the existence of GSpin-valued Galois representations correspond- ing to cohomological cuspidal automorphic representations of general sym- plectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard–Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G2 in the cohomology of Siegel modular va- rieties of genus three. Under some additional local hypotheses we also show automorphic multiplicity one as well as meromorphic continuation of the spin L-functions. Keywords. Automorphic representations, Galois representations, Langlands correspondence, Shimura varieties Contents 1 Conventions and recollections 14 arXiv:1609.04223v4 [math.NT] 15 Oct 2020 2 Arthur parameters for symplectic groups 18 3 Zariski connectedness of image 23 4 Weak acceptability and connected image 26 A. Kret: Korteweg-de Vries Institute Science Park 105 1090 GE Amsterdam, Netherlands; e- mail: [email protected] S. W. Shin: Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA // Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea; e-mail: [email protected] Mathematics Subject Classification (2010): Primary 11R39; Secondary 11F70, 11F80, 11G18 1 2 Arno Kret, Sug Woo Shin 5 GSpin-valued Galois Representations 30 6 The trace formula with fixed central character 32 7 Cohomology of certain Shimura varieties of abelian type 40 8 Galois representations in the cohomology 45 9 Construction of the GSpin-valued representation 53 10 Compatibility at unramified places 59 11 Galois representations for the exceptional group G2 62 12 Automorphic multiplicity 65 13 Meromorphic continuation of the Spin L-function 68 A Lefschetz functions 70 B Conjugacy in the standard representation 81 Introduction Let G be a connected reductive group over a number field F . The conjectural global Langlands correspondence for G predicts a correspondence between certain auto- morphic representations of G(AF ) and certain ℓ-adic Galois representations valued in the L-group of G. Let us recall from [BG14, 3.2] a rather precise conjecture § on the existence of Galois representations for a connected reductive group G over a number field F . Let π be a cuspidal L-algebraic automorphic representation of G(AF ). (We omit their conjecture in the C-normalization, cf. [BG14, Conj. 5.40], but see Theorem 9.1 below.) Denote by G(Qℓ) the Langlands dual group of G over L Qℓ, and by G(Qℓ) the L-group of G formed by the semi-direct product of G(Qℓ) with Gal(F /F ). According to their conjecture,b for each prime ℓ and each field iso- morphism ι : C Q , there should exist a continuous representation b ≃ ℓ ρ : Gal(F/F ) LG(Q ), π,ι → ℓ which is a section of the projection LG(Q ) Gal(F/F ), such that the following ℓ → holds: at each place v of F where πv is unramified, the restriction ρπ,ι,v : Gal(F v/Fv) L → G(Qℓ) corresponds to πv via the unramified Langlands correspondence. Moreover ρπ,ι should satisfy other desiderata, cf. Conjecture 3.2.2 of loc. cit. For instance at Galois representations for general symplectic groups 3 places v of F above ℓ, the localizations ρπ,ι,v are potentially semistable and have Hodge-Tate cocharacters determined by the infinite components of π. Note that if G is a split group over F , we may as well take ρπ,ι to have values in G(Qℓ). To simplify notation, we often fix ι and write ρπ and ρπ,v for ρπ,ι and ρπ,ι,v, understanding that these representations do depend on the choice of ι in general. b Our main result confirms the conjecture for general symplectic groups over totally real fields in a number of cases (up to Frobenius semisimplification, meaning that we take the semisimple part in (ii) of Theorem A below). We find these groups interesting for two reasons. Firstly they naturally occur in the moduli spaces of polarized abelian varieties and their automorphic/Galois representations have been useful for arithmetic applications (such as the study of L-functions, modularity and the Sato-Tate conjecture). Secondly new phenomena (as the semisimple rank grows) make the above conjecture sufficiently nontrivial, stemming from the nature of the dual group of a general symplectic group: e.g. faithful representations have large dimensions and locally conjugate representations may not be globally conjugate. Let F be a totally real number field. Let n 2. Let GSp denote the split general ≥ 2n symplectic group over F , equipped with similitude character sim : GSp G over 2n → m F . The dual group GSp2n is the general spin group GSpin2n+1, which we view over Qℓ (or over C via ι), admitting the spin representation d spin: GSpin GL n . 2n+1 → 2 Consider the following hypotheses on π. (St) There is a finite F -place vSt such that πvSt is the Steinberg represen- tation of GSp2n(FvSt ) twisted by a character. (L-coh) π sim n(n+1)/4 is ξ-cohomological for an irreducible algebraic repre- ∞| | sentation ξ = ξ of the group (Res GSp ) C GSp y:F֒→C y F/Q 2n ⊗Q ≃ y:F֒→C 2n,C⊗ (Definition 1.12 below). Q A trace formula argument shows that there are plenty of (in particular infinitely many) π satisfying (St) and (L-coh), cf. [Clo86]. Let Sbad denote the finite set of rational primes p such that either p = 2, p ramifies in F , or πv ramifies at a place v of F above p. Theorem A (Theorem 10.3). Suppose that π satisfies hypotheses (St), (L-coh). Let ℓ be a prime number and ι: C ∼ Q a field isomorphism. Then there exists a repre- → ℓ sentation ρ = ρ : Gal(F /F ) GSpin (Q ), π π,ι → 2n+1 ℓ unique up to GSpin2n+1(Qℓ)-conjugation, attached to π and ι such that the following hold. 4 Arno Kret, Sug Woo Shin (i) The composition ρ Gal(F /F ) π GSpin (Q ) SO (Q ) GL (Q ) → 2n+1 ℓ → 2n+1 ℓ ⊂ 2n+1 ℓ is the Galois representation attached to a cuspidal automorphic Sp2n(AF )-sub- representation π♭ contained in π. Further, the composition ρ Gal(F /F ) π GSpin (Q )/Spin (Q ) GL (Q ) → 2n+1 ℓ 2n+1 ℓ ≃ 1 ℓ corresponds to the central character of π via class field theory and ι. (ii) For every finite place v which is not above S ℓ , the semisimple part bad ∪{ } of ρπ(Frobv) is conjugate to ιφπv (Frobv) in GSpin2n+1(Qℓ), where φπv is the unramified Langlands parameter of πv. (iii) For every v ℓ, the representation ρ is de Rham (in the sense that r ρ is | π,v ◦ π,v de Rham for all representations r of GSpin ). Moreover 2n+1,Qℓ (a) The Hodge-Tate cocharacter of ρπ,v is explicitly determined by ξ. More precisely, for all y : F C such that ιy induces v, we have → n(n+1) µHT(ρπ,v, ιy)= ιµHodge(ξy)+ 4 sim (for µHT and µHodge see Definitions 1.10 and 1.14 below). (b) If πv has nonzero invariants under a hyperspecial (resp. Iwahori) subgroup of GSp2n(Fv) then either ρπ,v or a quadratic character twist is crystalline (resp. semistable). (c) If ℓ / S then ρ is crystalline. ∈ bad π,v (iv) For every v , ρ is odd (see Definition 1.8 and Remark 10.4 below). |∞ π,v (v) The Zariski closure of the image of ρπ maps onto one of the following three subgroups of SO2n+1: (a) SO2n+1, (b) the image of a principal SL2 in SO2n+1, or (c) (only possible when n =3) G2 embedded in SO7. ′ (vi) If ρ : Gal(F /F ) GSpin2n+1(Qℓ) is any other continuous morphism such → ′ that, for almost all F -places v where ρ and ρπ are unramified, the semisimple ′ ′ parts ρ (Frobv)ss and ρπ(Frobv)ss are conjugate, then ρ and ρ are conjugate. Our theorem is new when n 3. When n = 2, a better and fairly complete result ≥ without conditions (St) has been known by [Tay93,Lau05,Wei05,Urb05,Sor10,Jor12, GT19]. (Often π is assumed globally generic in the references but this is no longer necessary thanks to [GT19].) Galois representations for general symplectic groups 5 The above theorem in particular associates a weakly compatible system of λ-adic representations with π. See also Proposition 13.1 below for precise statements on the weakly compatible system consisting of spin ρ . It is worth noting that the unique- ◦ π ness in (vi) would be false for general GSpin2n+1(Qℓ)-valued Galois representations in view of Larsen’s example in 5 below.3 Our proof of (vi) relies heavily on the fact § that ρπ contains a regular unipotent element in the image, coming from condition (St). When n = 3 and F = Q, we employ the strategy of Gross and Savin [GS98] to construct a rank 7 motive over Q with Galois group of exceptional type G2 in the cohomology of Siegel modular varieties of genus 3. The point is that ρπ as in the above theorem factors through G (Q ) ֒ GSpin (Q ) if π comes from an 2 ℓ → 7 ℓ automorphic representation on (an inner form of) G2(A) via theta correspondence. In particular we get yet another proof affirmatively answering a question of Serre in the case of G2, cf.
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