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On the Non-Abelian Global Class Field Theory See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/268165110 On the non-abelian global class field theory Article · September 2013 DOI: 10.1007/s40316-013-0004-9 CITATION READS 1 19 1 author: Kazım Ilhan Ikeda Bogazici University 15 PUBLICATIONS 28 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: The Langlands Reciprocity and Functoriality Principles View project All content following this page was uploaded by Kazım Ilhan Ikeda on 10 January 2018. The user has requested enhancement of the downloaded file. On the non-abelian global class field theory Kazımˆ Ilhan˙ Ikeda˙ –Dedicated to Robert Langlands for his 75th birthday – Abstract. Let K be a global field. The aim of this speculative paper is to discuss the possibility of constructing the non-abelian version of global class field theory of K by “glueing” the non-abelian local class field theories of Kν in the sense of Koch, for each ν ∈ hK, following Chevalley’s philosophy of ideles,` and further discuss the relationship of this theory with the global reciprocity principle of Langlands. Mathematics Subject Classification (2010). Primary 11R39; Secondary 11S37, 11F70. Keywords. global fields, idele` groups, global class field theory, restricted free products, non-abelian idele` groups, non-abelian global class field theory, ℓ-adic representations, automorphic representations, L-functions, Langlands reciprocity principle, global Langlands groups. 1. Non-abelian local class field theory in the sense of Koch For details on non-abelian local class field theory (in the sense of Koch), we refer the reader to the papers [21, 23, 24, 50] as well as Laubie’s work [37]. For the basic theory of local fields and for the standard notation that we shall follow, we refer the reader to [3, 39]. Let K be a local field. That is, K is a complete discrete valuation field with fi- f nite residue class field κK of q = p elements. We shall furthermore fix an extension nr sep ϕK of the Frobenius automorphism FrK of K to K . Namely, we fix a Lubin-Tate splitting ϕK = ϕ over K. The non-abelian local class field theory for K establishes an algebraic and topological isomorphism Φ(ϕ) ∼ ∇(ϕ) K : GK −→ K between the absolute Galois group GK of the local field K and a certain topological ∇(ϕ) ϕ group K which depends on K and on the choice of the Lubin-Tate splitting over The author and this work was partially supported by TUB¨ ITAK˙ Project No. 107T728. 2 K. I. Ikeda 1 Φ(ϕ)− Φ(ϕ) K. In this paper, we shall denote the inverse K of the isomorphism K by ∇(ϕ) ∼ {·,K}ϕ : K −→ GK. ∇(ϕ) The construction of the topological group K involves the theory of APF-extensions of K and the fields of norms construction of Fontaine and Wintenberger. Moreover, Φ(ϕ) the isomorphism K , which is called the non-abelian local reciprocity law of K, is “natural” in the sense that properties such as “existence”, “functoriality” and a certain “ramification theoretic” property are all satisfied. The isomorphism {·,K}ϕ is called the non-abelian local norm-residue symbol of K. Remark 1.1. We should point out that the non-abelian local class field theory for K works under a technical assumption on the local field K. That is, the inclusion sep µ p(K ) ⊂ K should be satisfied. Under this assumption, the image of the non- Φ(ϕ) abelian local reciprocity map K can be described explicitly. But, this restriction on the local field K can be dropped without any effect on the general theory. Namely, for any local field K, we glue the non-abelian local class field theory for the local field K(µ) and the abelian local class field theory for the local cyclotomic extension K(µ)/K, where µ is any primitive pth root of unity (for example, look at Section 8 of [23]). So, “non-abelianization” of local class field theory in the sense proposed first by Koch, and developed further by de Shalit, Fesenko, Gurevich, Laubie and others, is now a complete and solid theory. Thus, it is then a natural attempt to construct the non-abelian version of global class field theory of a global field by “glueing” the non-abelian local class field theories of respective completions of this global field following Chevalley’s philosophy of ideles.` ϕ 2. Non-abelian idele` group JK of a global field K From now on K denotes a global field; that is, K is a finite extension of É or a finite extension of Fq(T) (that is, the field of rational functions of a curve defined over a finite field Fq). For details about global fields and the abelian global class field theory, we refer the reader to [39, 46]. Let aK denote the set of all archimedean ν h primes of K (so in case K is a function field, then aK = ∅). For each ∈ K, where hK denotes the set of all henselian (=non-archimedean) primes of K, let Kν denote the completion of K with respect to the ν-adic absolute value. Fixing a Lubin-Tate splitting ϕKν over Kν , the non-abelian local reciprocity law ϕ ϕ Φ( Kν ) ∼ ∇( Kν ) Kν : GKν −→ Kν or equivalently the “Weil form” of the non-abelian local reciprocity law ϕ ϕ Φ( Kν ) ∼ ∇( Kν ) Kν : WKν −→ Z Kν ϕ ∇( Kν ) of the local field Kν is defined. Here, Z Kν is a certain dense subgroup of the ϕ ∇( Kν ) topological group Kν (for details, see [23]). Moreover, following [50], or Section On the non-abelian global class field theory 3 8 of [21] together with [24] for detailed account, for each ν ∈ hK, there exists the ϕ 0 ϕ ∇( Kν ) ∇( Kν ) subgroup 1 Kν of Z Kν satisfying the equality ϕ ϕ 0 Φ( Kν ) 0 ∇( Kν ) Kν GKν = 1 Kν . (2.1) The following two tables summarize the abelian and the non-abelian local class field theories of Kν , where ν ∈ hK : ϕ Abelian local class field theory Non-abelian local C.F.T. ( K fixed) (ϕKν ) ab × G ∇ G Kν Kν Kν Kν ϕ ab × ∇( Kν ) W Z WKν Kν and Kν Kν 0 ϕ 0 ab c 0 ( Kν ) W U ν ∇ Kν K WKν 1 Kν δ i ab δ i δ (ϕKν ) WKν , ∈ (i − 1,i] UKν δ ∇ WKν , ∈ (i − 1,i] 1 Kν Recall that, the passage from abelian local class field theory to abelian global class field theory follows via the idele` group ÂK of the global field K, where the topological group ÂK is defined by the “restricted direct product” ′ × ÂK := ∏ Kν : UKν a ν∈hK ∪ K × a ν h of the collection {Kν }ν∈hK ∪ K with respect to the collection {UKν } ∈ K , equipped with the restricted direct product topology. Namely, abelian global class field theory for K establishes an algebraic and topological isomorphism −1 \× ∼ ab (·,K) = ArtK : K \ÂK −→ Gal(K /K) or equivalently the “Weil form” of abelian global class field theory for K establishes an algebraic and topological isomorphism −1 × ∼ ab (·,K) = ArtK : K \ÂK −→ WK × satisfying certain “naturality” conditions. The noncompact group K \ÂK is called \× × Â the idele` class group of K and K \ÂK denotes the profinite completion of K \ K. The construction of the global norm residue symbol (·,K) or the Artin reci- procity law ArtK of the global field K can roughly be sketched as follows : First, for each finite abelian extension L/K, consider the well-defined homomorphism RL/K : ÂK → Gal(L/K) defined by RL/K : (xν )ν 7→ ∏∏(xν ,Lµ /Kν ) ν ν|µ ν for every (xν )ν ∈ ÂK. Almost all are unramified in the extension L/K which yields the well-definedness of the map RL/K. Next, pass to the projective limit ab limR : Â → limGal(L/K) = Gal(K /K) ←− L/K K ←− L/K L/K 4 K. I. Ikeda over all possible such L/K. Finally, check that the morphism limR induces a ←− L/K L/K homomorphism × ab K \ÂK → GK which factors as × ∼ ab ab K \ÂK −→ WK → GK . Thus, it is natural to push this idea to the extreme, and introduce the non- abelian idele` group JK of K, following the analogy between the tables above and taking into account the philosophy of Miyake [40, 41] (also look at Iwasawa [26]), as follows. Restricted free products. Let {Gi}i∈I be a collection of locally compact topolog- ical groups and for all but finitely many i ∈ I let Oi be a compact open subgroup of Gi. Denote the finite subset of I consisting of all i ∈ I for which Oi is not defined by I∞. For every finite subset S of I satisfying I∞ ⊆ S, define the topological group GS := ∗ Oi ∗ ∗ Gi i∈/S i∈S as the free product of the topological groups Oi, for i ∈ I −S, and Gi, for i ∈ S, which exists in the category of topological groups (cf. Morris [43]). Then, the restricted free product of the collection {Gi}i∈I with respect to the collection {Oi}i∈I−I∞ , ′ which is denoted by ∗i∈I(Gi : Oi), is defined by the injective limit ′ ∗ (Gi : Oi) := limGS i∈I −→ S defined over all possible such S, where the connecting morphism τT S : GS → GT for S ⊆ T is defined naturally by the “universal mapping property of free products” 1 ′ (cf. Hilton-Wu [18] and Morris [43]). The topology on ∗i∈I(Gi : Oi) is defined ′ by declaring X ⊆ ∗i∈I(Gi : Oi) to be open if X ∩ GS is open in GS for every S.
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