On the Non-Abelian Global Class Field Theory

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On the non-abelian global class field theory

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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 reciprocity 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

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 topological 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. So, ′ endowed with this topology, ∗i∈I(Gi : Oi) is a topological group.

Proposition 2.1. Let {Gi}i∈I be a collection of locally compact topological 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∞. Assume that, for each i ∈ I, a continuous homomorphism

φi : Gi → H is given. Then, there exists a unique continuous homomorphism

φS : GS → H

1 If {Gi}i∈I is a collection of topological groups and ∗i∈I Gi is the free product of this collection together ι with the canonical embeddings io : Gio ֒→ ∗i∈I Gi, for each io ∈ I, then the universal mapping property φ of free products states that, if for each io ∈ I, io : Gio → H is a continuous homomorphism, then there φ φ ι φ exists a unique continuous homomorphism : ∗i∈I Gi → H, such that ◦ io = io , for every io ∈ I. On the non-abelian global class field theory 5 defined for each finite subset S of I satisfying I∞ ⊆ S, and a unique continuous homomorphism φ = limφ : ′ (G : O ) → H −→ S ∗i∈I i i S satisfying φ φ φ cS ′ S = ◦ cS : GS −→ ∗i∈I(Gi : Oi) −→ H, ′ where cS : GS → ∗i∈I(Gi : Oi) is the canonical homomorphism, for every S.

Proof. The collection of continuous homomorphisms {φi : Gi → H}i∈I determines a unique continuous homomorphism φS : GS → H, for each S, by the universal mapping property of free products. Moreover, for S ⊆ T, the diagram G S A φ AA c S AAS AA A ∃!φ " τT S G / H }> < }} }} }} cT } φT GT ′ is commutative, where G = ∗i∈I(Gi : Oi), by the universality of direct limits. Notation 2.2. As a notation, for a topological group G, the n-fold free product n-copies G ∗···∗ G of G is denoted by G∗n. ϕ z }| { Non-abelian idele` group JK of the global ﬁeld K. Now, the following deﬁnition

introduces the major object that we intend to study in this work.

ν h ϕ ϕ ϕ Definition 2.3. For each ∈ K fix a Lubin-Tate splitting Kν and let = { Kν }ν∈hK . ϕ The topological group JK defined by the “restricted free product” ϕ ϕ ϕ 0 ′ ∇( Kν ) ∇( Kν )

JK := ∗ Z Kν : 1 Kν a ν∈h ∪ K K is called the non-abelian idele` group of the global ﬁeld K. In case K is a number ﬁeld, ϕ ϕ ϕ 0 ′ ∇( Kν ) ∇( Kν ) ∗r1 ∗r2

= Z : W W , C JK ∗ Kν 1 Kν ∗ Ê ∗ ν∈hK ϕ ϕ where the ﬁnite (=henselian) part J of J is deﬁned by K,h K ϕ ϕ ϕ 0 ′ ∇( Kν ) ∇( Kν )

J := ∗ Z : , K,h Kν 1 Kν ν∈hK ϕ ϕ and the inﬁnite (=archimedean) part of by JK,a JK ϕ

:= W ∗r1 W ∗r2 .

Ê C JK,a ∗

Here, as usual r1 and r2 denote the number of real and the number of pairs of

complex-complex conjugate embeddings of the global ﬁeld K in C. 6 K. I. Ikeda

ϕ The non-abelian idele` group JK of K is an “extremely large” topological group, whose definition depends only on the global field K. Remark 2.4. The non-abelian local class field theory in the sense of Koch has evolved in two directions (look at [23] and [37]). The non-abelian idele` groups introduced in Definition 2.3 are defined a` la Fesenko; namely, the construction is based on the work [23]. However, it is also possible to start with Laubie class field theory, and define the non-abelian idele` groups accordingly. Taking into account the work ϕ Laubie [21], the latter construction produces nothing new. More precisely, if JK denotes the non-abelian idele` group of K constructed in terms of Laubie class field ϕ Laubie ∼ ϕ theory, then there exists a topological isomorphism JK −→ JK by [21]. ϕ ab The following theorem describes the abelianization J of the topological ϕ K group JK . ϕ ab ϕ Theorem 2.5. The abelianization JK of the topological group JK is indeed

ÂK. Proof. The proof follows by ﬁrst noting that the direct limit functor is exact and then by abelianizing free products of groups.

3. Non-abelian global reciprocity law : A proposal a For ν ∈ hK ∪ K, choose an embedding sep sep . eν : K ֒→ Kν This embedding determines a continuous homomorphism 2 (look at [51] for details) Weil eν : WKν → WK,

and therefore, for each ν ∈ hK, a continuous homomorphism

ϕ Weil ϕ {·,Kν }ϕ Weil ( Kν ) ∇ Kν Kν eν NR : Z −−−−−−→ WKν −−→ WK. Kν Kν ∼ Theorem 3.1 (Global non-abelian norm-residue symbol “Weak form”). There exists a well-deﬁned continuous homomorphism ϕWeil ϕ NRK : JK → WK, (3.1) which satisﬁes

ϕWeil ϕWeil ϕWeil ϕ ϕ NR cS K (NRK )S = NRK ◦ cS : (JK )S −→ JK −−−−−→ WK, ϕ ϕ

where, following Proposition 2.1, cS : (JK )S → JK is the canonical homomor-

a a phism deﬁned for every ﬁnite subset S of hK ∪ K containing K.

2which is unique if K is a function field and unique up to composition with an inner automorphism of o WK defined by an element of the connected component WK of WK if K is a number field. On the non-abelian global class field theory 7

Proof. Proposition 2.1 applied to the collection of continuous homomorphisms

ϕ Weil ϕ ( Kν ) ∇ Kν Weil NRKν : Z Kν → WK eν : WKν → WK ν∈aK ν∈h K [n o completes the proof.

ϕWeil ϕ We conjecture that the continuous homomorphism NRK : JK →WK should be considered as the global non-abelian norm-residue symbol of K. More precisely,

Conjecture 3.2 (Global non-abelian norm-residue symbol “Strong form”). The homomorphism ϕWeil ϕ NRK : JK → WK is open, continuous and surjective.

Regarding this conjecture, the following remark is in order.

Remark 3.3. The conjecture 3.2 seems to be related with : É

(1) the well-known fact that the absolute Galois group GÉ of the global ﬁeld is topologically generated by the inertia subgroups; (2) the “Riemann’s existence theorem” for global ﬁelds (for details, see [44]).

4. Local-global compatibility of the non-abelian norm residue

symbols a Clearly, for each ν ∈ hK ∪ K, there exists a natural homomorphism

(ϕKν ) ∇ ν h ϕ Z Kν , ∈ K ϕ ν ν

q : (J ) := ν a → J , Ê WÊ

K  , ∈ K,  K

ν a C  WC , ∈ K,  which is deﬁned explicitly via the commutative triangle ϕ (J ) 7 K S (S) nn ιν nnn nnn nnn ϕ n cS (JK )ν PPP PPP PPP qν PPP PP' ϕ

a a ν where S is a ﬁnite subset of hK ∪ K satisfying K ⊆ S and ∈ S. Note that, the ϕ ϕ

deﬁnition of the continuous homomorphism qν : (JK )ν → JK does not depend

a a on the choice of S. In fact, if T is another ﬁnite subset of hK ∪ K satisfying K ⊆ T 8 K. I. Ikeda and ν ∈ T, then by the universal mapping property of free products and by the ϕ ϕ τS deﬁnition of the connecting morphism S∩T : (JK )S∩T → (JK )S, the diagram ϕ (J ) 6 K S∩T ι(S∩T) nn ν nnn nnn nnn ϕ nn τS (J )ν S∩T K QQ QQQ QQQ (S) QQ ιν QQ Q( ϕ (JK )S commutes. Thus, ι(S∩T) τS ι(S∩T) ι(S) cS∩T ◦ ν = cS ◦ S∩T ◦ ν = cS ◦ ν = qν , which also proves that (T) (S) cT ◦ ιν = cS ◦ ιν . ϕ The next Theorem is the “local-global compatibility” of {.,Kν }ϕKν and NRK for

ν ∈ hK.

Theorem 4.1. For each ν ∈ hK, the following square

ϕ qν ϕ ∇( Kν ) / Z Kν JK

ϕ Weil {.,Kν }ϕ NR Kν K WKν / W Weil K eν is commutative.

Proof. Note that, Weil Weil (ϕKν ) eν ◦ {.,Kν }ϕKν = NRKν . Now, to prove that ϕ Weil Weil (ϕKν ) NRK ◦ qν = NRKν ,

(S)

a a first recall that qν = cS ◦ιν , where ν ∈ S a finite subset of hK ∪ K satisfying K ⊂ S. Then, ϕ ϕ Weil Weil ι(S) NRK ◦ qν = NRK ◦ (cS ◦ ν ) ϕ Weil ι(S) = (NRK )S ◦ ν Weil (ϕKν ) = NRKν , which completes the commutativity of the square. On the non-abelian global class field theory 9

ϕ 5. ℓ-adic representations of the non-abelian idele` group JK of K By Theorem 3.1, and also by Conjecture 3.2 as well, it is evident that any representation σ :WK → GL(V) of the absolute Weil group of the global field K on an F- vector ϕ ϕ σ space V defines naturally a representation ∗ : JK → GL(V) of the non-abelian ϕ Weil ϕ ϕ ϕ NR σ K idele` group JK of K on the F-linear space V via the composition ∗ : JK −−−−−→ σ WK −→ GL(V). Here, F denotes any field. Moreover, if F is assumed to be a topological field and the F-representation V of WK is a continuous representation, then ϕ Weil ϕ the continuity of the non-abelian norm-residue symbol NR : J → WK of K K K ϕ yields the continuity of the corresponding F-representation V of J . Thus, the ϕ K 3 (continuous) F-representation theory of the topological group JK is closely related to the (continuous) F-representation theory of the absolute Weil group WK of the global field K. In this work, we shall focus on the continuous ℓ-adic representations of the ϕ non-abelian idele` group JK of K and their basic “analytic invariants”, namely the L-functions and the ε-factors attached to these representations, whose definitions will be made precise below. By the preceding paragraph, such representations are deeply connected with the ℓ-adic representations of the absolute Weil group WK of the global field K, or equivalently4 to the ℓ-adic representations of the absolute Galois group GK of K.

Remark 5.1. In fact, instead of considering the ℓ-adic representations of the non- ϕ abelian idele` group J of the global ﬁeld K, we can, and we should, more generally K ϕ L consider the “ℓ-adic Hecke-Langlands parameters” JK → H(Éℓ) of H, where H is any connected reductive group over K with the dual group H, which is a connected

3In fact, we have not seen any work on the representation theory of the restrictedb free product of topological groups. The closest work in the mathematical literature in this direction are the 1995 paper of Młotkowski [42] and the 2010 paper of Hebisch and Młotkowski [16]. 4 We reproduce the following observation of Brian Conrad : Let K be a global ﬁeld and WK be the absolute

Weil group of K equipped with the natural continuous homomorphism φ : WK → GK with dense image ξ É (look at [51]). Let H be a ﬁnite-type Éℓ-group. Then any continuous homomorphism : WK → H( ℓ) factors continuously through GK as

G nn7 K φ nn nnn nnn nnn n ξ WK OO o OOO OOO ξ OO OO'

H(Éℓ). ξ É In fact, the target H(Éℓ) is totally-disconnected. Therefore, such an arrow : WK → H( ℓ) should kill o φ the identity component WK of WK . In the number ﬁeld case, : WK → GK is a surjective topological φ o φ

quotient map with ker( )= WK . In the function ﬁeld case, : WK → GK is the inclusion and WK is a

ξ É ξ É dense subgroup of GK . Thus, in both cases, : WK → H( ℓ) uniquely deﬁnes o : GK → H( ℓ). 10 K. I. Ikeda

L 5 L

É É reductive group over C, and H( ℓ) is the ℓ-rational points of the L-group H := H ⋊WK of H. These continuous homomorphisms are closely related with the “ℓ- L adic Langlands parameters” WK → H(É ℓ) of H, which are the “ℓ-adic avatars” L ofb the Langlands parameters LK → H of H, where LK denotes the global Langlands group of K, which is a conjectural group in case K is a number ﬁeld. For details, look at [1, 5, 14, 25, 49]. We shall return to this discussion in Section 8. ϕ Continuous ℓ-adic representations of JK . Let ϕ ρ : JK → GL(V) ϕ be a continuous representation of the topological group J on an n-dimensional ϕK ρ vector space V over É . Such a representation ( ,V) of J is called a continuous ℓ ϕ K ϕ ϕ ℓ-adic representation of JK . Then, the natural homomorphism qν : (JK )ν → JK deﬁnes a local continuous representation

(ϕKν ) ∇ ν h

Z , ∈ ϕ Kν K qν ϕ ρ ρν ρ ν ν

= ◦ q : (J ) := ν a −→ J −→ GL(V) Ê WÊ

K  , ∈ K,  K

ν a C  WC , ∈ K, 

ν a h of the local group (JK )ν on the vector space V over Éℓ, for each ∈ K ∪ K. Moreover, we have the following important theorem, which is essentially the “rep-

resentation theoretic” incarnation of Proposition 2.1. h Theorem 5.2. For each ν ∈ aK ∪ K, let ϕ ρ ν : (JK )ν → GL(V) ϕ

be a continuous representation of the local group (JK )ν on the vector space V over

É ρ h ℓ. Then the collection { ν }ν∈aK ∪ K deﬁnes a unique continuous representation ϕ ρ : JK → GL(V)

on the vector space V over Éℓ, such that ϕ ϕ ρ ρ qν

ν : (JK )ν −→ JK −→ GL(V), h for every ν ∈ aK ∪ K.

ρ h Proof. By Proposition 2.1, for the collection { ν }ν∈a ∪ , there exists a unique

ϕ K K ρ É

continuous representation S of (JK )S on the linear space V over ℓ deﬁned for

h a each ﬁnite subset S of aK ∪ K satisfying K ⊆ S, and a unique continuous represen-

ϕ ρ É tation of JK on the linear space V over ℓ satisfying ϕ ϕ ρ ρ ρ cS S = ◦ cS : (JK )S −→ JK −→ GL(V) for every such set S, which further satisﬁes (S) (S)

ρν = ρS ◦ ιν = ρ ◦ cS ◦ ιν = ρ ◦ qν , h for every ν ∈ aK ∪ K.

5 É The complex reductive group H is deﬁned over Z. Thus, we can consider the the group of ℓ-rational points of the L-group LH. b On the non-abelian global class ﬁeld theory 11

This theorem immediately yields the following corollary, whose proof is straightforward. Corollary 5.3. There exists a bijective correspondence ϕ ρ ϕ

ν ∀ ⇄ cont−n ρ É ν

h ν { ν }ν∈aK ∪ K : (J ) −−→ GL(n, ℓ), Rep (J ) K É K cont. ℓ

n ρ o h between the set of all collections { ν }ν∈a ∪ consisting of n-dimensional contin-

ϕ K K

ν a h uous ℓ-adic representations ρν : (J )ν → GL(n, É ) for each ∈ K ∪ K and the ϕ K ℓ ϕ set Repcont−n(J ) of all n-dimensional continuous ℓ-adic representations of J Éℓ K K deﬁned by Theorem 5.2. Hecke-Weil L-functions. To simplify the discussion, from now on, ﬁx an isomor-

∼ C phism Éℓ −→ using the axiom of choice. Our aim now is to define the “Hecke-Weil” L-function LHecke−Weil(s,ρ) at ϕ ρ s ∈ C attached the continuous ℓ-adic representation of JK on the n-dimensional vector space V over Éℓ. Definition 5.4. Let ϕ ρ : JK → GL(V) ϕ be a continuous representation of JK on an n-dimensional vector space V over Éℓ. Hecke−Weil The “Hecke-Weil” L-function L (s,ρ) at s ∈ C attached to the representa- ϕ ρ tion : JK → GL(V) is defined (formally) by the Euler product

Hecke−Weil Hecke−Weil Hecke−Weil

L (s,ρ) := ∏ L (s,ρν ) ∏ L (s,ρν ) ν h ν∈hK ∈ K ℓ∤qν ℓ|qν Hecke−Weil × ∏ L (s,ρν ), s ∈ C,

ν∈aK κ where for each ν ∈ hK, the cardinality of the residue class ﬁeld Kν of Kν is denoted Hecke−Weil by qν . The local “Hecke-Weil” L-factor L (s,ρν ) at s ∈ C is deﬁned

– for ν ∈ hK with ℓ ∤ qν , by ϕ

Hecke−Weil Artin−Weil Φ( Kν ) ρ ρ Φ C L (s, ν ) := L (s, ν ◦ Kν ), s ∈ , where the right-hand side is deﬁned as the usual local Artin-Weil L-factor of ϕ ρ Φ( Kν )

the local representation ν ◦ Kν : WKν → GL(V) of the local absolute Weil C group WKν of Kν on the vector space V over Éℓ at s ∈ ;

– for ν ∈ hK with ℓ | qν , again by ϕ

Hecke−Weil Artin−Weil Φ( Kν ) ρ ρ Φ C L (s, ν ) := L (s, ν ◦ Kν ), s ∈ ,

where this time, the right-hand side is deﬁned in terms of “Fontaine’s Dpst- functor” via the “p-adic Hodge theory”, which we shall not discuss in the text, and refer the reader to [10, 11, 52]; Γ ρ – for ν ∈ aK, to be the “gamma-factor” (s, ν ) deﬁned explicitly for example in [30], which we shall not discuss in the text, and refer the reader to [30]. 12 K. I. Ikeda

Remark 5.5. Deninger introduced a uniﬁed construction of local L-factors of motives (pure as well as mixed motives) as a certain power series on inﬁnite-dimensional cohomologies (for details, look at [8]). It would be very interesting to study Artin- Weil and Hecke-Weil L-functions in the framework of Deninger’s theory.

Hecke−Weil Hecke-Weil ε-factors. Next, we shall define the ε-factor ε (s,ρ) at s ∈ C ϕ ρ attached the continuous ℓ-adic representation of JK on the n-dimensional vector space V over Éℓ. In order to do so, let us briefly review the construction of the global ε-factors of continuous ℓ-adic representations of the absolute Weil group WK of the global field K introduced by Deligne, Dwork, and Langlands (look at [51] for details). Let σ : WK → GL(V)

be a continuous representation of WK on an n-dimensional vector space V over Éℓ.

C A

Let ψ : AK → be a non-trivial additive unitary character of K which is trivial on h K, namely, ψ is a non-trivial global additive character of K. For each ν ∈ aK ∪ k, × the local component ψν : (Kν )+ → C is a non-trivial additive character of the local + +µ ﬁeld Kν . Let d µ be the Haar measure on A normalized by d (x) = 1. Fix K K\AK +µ ∏ +µ +µ ν a decomposition d = ν d ν of d , where for almost allR the local measure d+µν is a “normalized” additive Haar measure on Kν , normalized in the sense that Artin−Weil vol(OKν ) = 1. Now, the global ε-factor ε (s,σ) of the ℓ-adic representation

σ of WK on V at s ∈ C is deﬁned by the product Artin−Weil Artin−Weil +

ε (s,σ) = ∏ ε (s,σν ,ψν ,d µν ) h ν∈aK ∪ K of local factors εArtin−Weil(s,σν ,ψν ,d+µν ). Artin−Weil + 6 Recall that, the local factor ε (σν ,ψν ,d µν ), for ν ∈ hK, is deﬁned explicitly for the case dim (V) = n = 1 by Éℓ −1 + σν (ArtKν (x)) ψν (x/c)d µν (x) Artin−Weil + UK ε σν ψν µν σν Art ( , ,d ) = ( Kν (c)) R −1 + , | σν (ArtKν (x)) ψν (x/c)d µν (x) | UK R ab ∼ × × where the arrow ArtKν : WKν −→ Kν is the Artin reciprocity law of Kν , and c ∈ Kν satisﬁes ν(c) = a(σν ) + n(ψν ). Recall that, the number n(ψν ) is the conductor of × σ the additive character ψν : (Kν )+ → C and a( ν ) is the Artin conductor of the × quasi-character σν : WKν → C . On the other hand, there is no explicit formula for the local ε-factor εArtin−Weil(σν ,ψν ,d+µν ) in case dim (V) = n > 1. The best we Éℓ have is the “existence and uniqueness theorem of Deligne, Dwork and Langlands for local ε-factors”. Remark 5.6. It seems possible to give an explicit formula for the local ε-factor εArtin−Weil(σν ,ψν ,d+µν ) for the general case dim (V) = n ≥ 1 in terms of the Éℓ

6 Therefore, if σν : WKν → GL(V) is a 1-dimensional representations of WKν on the Éℓ-linear space V, then there exists an explicit expression of the local ε-factor εArtin−Weil (σν ,ψν ,d+µν ) in terms of the local Artin reciprocity law of Kν . On the non-abelian global class ﬁeld theory 13

(ϕKν ) ∼ (ϕKν ) Φ ∇ ν non-abelian local reciprocity law Kν : WKν −→ Z Kν of the local ﬁeld K (look at [22])7. Also, for a naive attempt in this direction, look at [19]. Deﬁnition 5.7. Let ϕ ρ : JK → GL(V) ϕ be a continuous representation of JK on an n-dimensional vector space V over

ψ A C ν Éℓ. Let : K → be a non-trivial global additive character of K. For each ∈

× +

h ψ ψ C µ aK ∪ k, the local component of is denoted by ν : (Kν )+ → . Let d be the +µ +µ Haar measure on A normalized by d (x) = 1. Fix a decomposition d = K K\AK ∏ +µ +µ ν +µ ν d ν of d , where for almostR all the local measure d ν is a normalized additive Haar measure on Kν , normalized in the sense that vol(OKν ) = 1. The global ϕ ε εHecke−Weil ρ ρ “Hecke-Weil” -factor (s, ) of the representation : JK → GL(V) at s ∈ C is deﬁned (formally) by the product Hecke−Weil Hecke−Weil +

ε (s,ρ) := ∏ ε (s,ρν ,ψν ,d µν ), s ∈ C. h ν∈aK ∪ K

Hecke−Weil +

h ε ε ρ ψ µ For each ν ∈ aK ∪ K, the local “Hecke-Weil” -factor (s, ν , ν ,d ν )

at s ∈ C is deﬁned by ϕ

Hecke−Weil + Artin−Weil Φ( Kν ) + ε ρ ψ µ ε ρ Φ ψ µ C (s, ν , ν ,d ν ) := (s, ν ◦ Kν , ν ,d ν ), s ∈ . Remark 5.8. Note that, Deﬁnition 5.7 is provisional. In fact, it seems that, to deﬁne the global Hecke-Weil ε-factors from the local factors properly in the light of Re- ϕ mark 5.6, we have to construct the “non-commutative Tamagawa measure” on JK following the lines sketched in [28, 53].

(ϕ) The assignement σ σ∗ . Now, let

σ : WK → GL(V)

be a continuous representation of WK on an n-dimensional vector space V over Éℓ. Then we have the following theorem.

Theorem 5.9. The ℓ-adic continuous representation σ : WK → GL(V) deﬁnes naturally a continuous representation ϕ ϕ σ ∗ : JK → GL(V) ϕ of JK on the n-dimensional space V over Éℓ via the composition ϕ Weil ϕ ϕ NR σ K σ ∗ : JK −−−−−→ WK −→ GL(V),

7As Tate points out in [51], “..., we have no way to define ε(χ,ψ,dx) without using the reciprocity law isomor- ∗ ab phism F ≈WF . In fact it was his [Langlands’] idea about “nonabelian reciprocity laws” relating representations of degree n of WF to irreducible representations π of GL(n,F), and the possibility of defining ε(π,ψ,dx) for the latter, which led Langlands to conjecture and prove ...” 14 K. I. Ikeda which satisfies ϕ ϕ σ Φ( Kν ) σ σ Weil ∗ ν ◦ Kν = ν = ◦ eν ,

for every ν ∈ hK. Moreover, the following identities hold : ϕ Artin−Weil Hecke−Weil L (s,σ) = L (s,σ∗ ). and ϕ Artin−Weil Hecke−Weil ε (s,σ) = ε (s,σ∗ ). Proof. By Theorem 4.1 on the local-global compatibility of non-abelian norm residue symbols, note that ϕ ϕ σ∗ ν = σ∗ ◦ qν ϕ σ Weil = ◦ NRK ◦ qν σ Weil = ◦ eν ◦ {.,Kν }ϕKν . That is, ϕ ϕ σ Φ( Kν ) σ Weil σ ∗ ν ◦ Kν = ◦ eν = ν . Now the proof follows directly from Theorem 3.1 and Deﬁnitions 5.4 and 5.7.

ϕ ϕ Weil Notation 5.10. Let IK denote the image im(NRK ) of the global non-abelian ϕ Weil ϕ norm-residue symbol NRK : JK → WK of K.

Remark 5.11. If σ1 and σ2 are n-dimensional continuous ℓ-adic representations of WK, then clearly by Theorem 5.9, ϕ ϕ σ σ σ ϕ σ ϕ ( 1)∗ = ( 2)∗ ⇔ 1 |I = 2 |I . K K ϕ Moreover, if Conjecture 3.2 holds; namely, if IK = WK, then the assignement ϕ σ σ∗ introduced in Theorem 5.9, where σ is the continuous representation of WK on the n-dimensional space V over Éℓ is an injection. Remark 5.12. If Conjecture 3.2 holds, then the assignement ϕ σ σ∗ introduced in Theorem 5.9, where σ is the continuous representation of WK on the n-dimensional space V over Éℓ clearly preserves irreducibility and semi-simplicity. That is, ϕ – σ is irreducible ⇒ σ is irreducible; ∗ ϕ – σ is semi-simple ⇒ σ∗ is semi-simple.

Note that, 1-dimensional ℓ-adic representations of the absolute Weil group WK of K can be identified with the ℓ-adic Hecke characters of K via abelian global class field theory. Thus, an immediate consequence of Theorem 5.9 is the following corollary, whose proof is straightforward. On the non-abelian global class field theory 15

Corollary 5.13. An ℓ-adic Hecke character

× ×

χ Â É : K \ K → ℓ of K deﬁnes naturally a continuous 1-dimensional ℓ-adic representation ϕ ϕ × X∗ : JK → Éℓ (ϕ) of the non-abelian idele` group JK of K via the composition X

ϕ Weil ϕ ϕ NR Art χ K can. ab K × % ×

/ / / Â / X∗ : JK WK WK K \ K Éℓ , which satisﬁes ϕ ϕ Φ( Kν ) Weil X∗ ν ◦ Kν = Xν = X ◦ eν , for every ν. Moreover, the following identities hold : ϕ Artin−Weil Hecke Hecke−Weil L (s,X) = L (s, χ) = L (s,X∗ ). and ϕ Artin−Weil Hecke Hecke−Weil ε (s,X) = ε (s, χ) = ε (s,X∗ ). Thus, Hecke and Artin-Weil L-functions are special cases of Hecke-Weil L- functions via Theorem 5.9 and its Corollary 5.13 as described in the following diagram :

Hecke-Weil L-functions > Ba | BB || B non-abelian global CFT || BB Corollary 5.13 | BB Theorem 5.9 || BB || BB || BB || BB || B Hecke / Artin-Weil . L-functions abelian global CFT L-functions Also, Hecke and Artin-Weil ε-factors are special cases of Hecke-Weil ε-factors via Theorem 5.9 and its Corollary 5.13 as described in the following diagram :

Hecke-Weil ε-factors ? B` ~ BB ~~ B non-abelian global CFT ~~ BB Corollary 5.13 ~ BB Theorem 5.9 ~~ BB ~~ BB ~~ BB ~~ BB ~~ B Hecke / Artin-Weil . ε-factors abelian global CFT ε-factors 16 K. I. Ikeda

Analytic properties of Hecke-Weil L-functions. Note that, the assignement ϕ σ σ∗ introduced in Theorem 5.9, where σ is the continuous representation of WK on the

n-dimensional space V over É that is under consideration, also sheds light on the ℓ ϕ Hecke−Weil basic analytic properties of the Hecke-Weil L-function L (s,σ∗ ) attached ϕ ϕ σ to the corresponding continuous representation ∗ of JK on the n-dimensional space V over Éℓ . In fact, it is well-known that, LArtin−Weil(s,σ) is a convergent product for

s ∈ C satisfying Re(s) ≫ 0. Moreover, by Brauer’s induction theorem, the product LArtin−Weil(s,σ) defines a meromorphic function in the whole complex plane, and satisfies the functional equation LArtin−Weil(s,σ) = εArtin−Weil(s,σ)LArtin−Weil(1 − s,σ ∨), where σ ∨ is the contragardient8 of σ. Now, we shall single out the continuous ℓ-adic representations of the non- ϕ abelian idele` group JK of K that have nice “analytic invariants”. ϕ ρ Definition 5.14. An n-dimensional continuous ℓ-adic representation of JK is ϕ Weil called Galois type, if it factors through the non-abelian norm-residue symbol NR : ϕ K JK → WK of K as ρ

ϕ Weil NR ργ ϕ K $

/ / É JK WK GL(n, ℓ)

γ ρ É for some n-dimensional continuous ℓ-adic representation : WK → GL(n, ℓ) of WK.

Remark 5.15. Certainly, if σ is a continuous representation of WK on an n-dimensional ϕ ϕ σ space V over Éℓ, then the continuous ℓ-adic representation ∗ of JK on V assigned by Theorem 5.9 is clearly of Galois type, and the following equality holds ϕ γ σ ϕ σ ϕ ( ∗ ) |I = |I K K by Remark 5.11. Moreover, if Conjecture 3.2 holds, then the assignement ϕ σ σ∗ introduced in Theorem 5.9, where σ is the continuous representation of WK on the n-dimensional space V over Éℓ induces a bijective correspondence ϕ cont−n cont−n;Gal

Rep (WK) ⇄ Rep (J ) É É K ℓ ℓ

8 σ É Recall that, the contragradient of a continuous n-dimensional ℓ-adic representation : WK → GL(n, ℓ)

∨ ∨ t −1

σ É σ σ of WK is defined to be the representation : WK → GL(n, ℓ) of WK defined by (w)= (w ) for every w ∈ WK . On the non-abelian global class field theory 17

cont−n between the set Rep (WK) of all n-dimensional continuous ℓ-adic representa- Éℓ ϕ cont−n;Gal tions of WK and the set Rep (J ) of all Galois type n-dimensional contin-

É K ℓ ϕ ϕ uous ℓ-adic representations of J , which is a subset of Repcont−n(J ). Note that, K Éℓ K by Remark 5.12, this bijective correspondence preserves irreducibility and semi- simplicity. We shall return to this discussion within the framework of Tannakian categories in Section 6.

Theorem 5.16. Assume that ρ is an n-dimensional continuous ℓ-adic Galois type ϕ Hecke−Weil ρ representation of JK . Then the attached Hecke-Weil L-function L (s, ) is a convergent product for s ∈ C satisfying Re(s) ≫ 0. Moreover, the product LHecke−Weil(s,ρ) deﬁnes a meromorphic function in the whole complex plane, and satisﬁes the functional equation

LHecke−Weil(s,ρ) = εHecke−Weil(s,ρ)LHecke−Weil(1 − s,ρ∨),

∨ ρ ρ É where is the contragradient of the representation of JK on the ℓ-linear space V.

ϕ ρ Proof. As the n-dimensional continuous ℓ-adic representation of JK is of Galois type, γ ϕ (ρ )∗ = ρ. Now, by Theorem 5.9, the identity

γ LArtin−Weil(s,ρ ) = LHecke−Weil(s,ρ) follows. Thus, the Hecke-Weil L-function LHecke−Weil(s,ρ) is a convergent product Hecke−Weil ρ for s ∈ C satisfying Re(s) ≫ 0. Moreover, the product L (s, ) deﬁnes a meromorphic function in the whole complex plane, and satisﬁes a functional equation γ LHecke−Weil(s,ρ) = εHecke−Weil(s,ρ)LArtin−Weil(1 − s,(ρ )∨),

γ ∨ γ where (ρ ) is the contragradient of the representation ρ of WK on the É -linear ϕℓ ρ space V. Therefore, it sufﬁces to prove that, for the representation of JK on the

Éℓ-linear space V, the identity ϕ ∨ γ ∨ ρ = ((ρ ) )∗ is satisﬁed. Then, by Theorem 5.9, the equality

γ LArtin−Weil(s,(ρ )∨) = LHecke−Weil(s,ρ∨) yields the functional equation

LHecke−Weil(s,ρ) = εHecke−Weil(s,ρ)LHecke−Weil(1 − s,ρ∨). 18 K. I. Ikeda

ϕ Now, for any x ∈ JK , ϕ ϕ ργ ∨ ργ ∨ Weil (( ) )∗ (x) = ( ) (NRK (x)) ϕ t ργ Weil −1 = [( )(NRK (x) )] ϕ t ργ Weil −1 = [( )(NRK (x)) ] = t [ρ(x)−1] = t ρ(x−1) = ρ∨(x), which completes the proof. “Fine-tuning” Theorem 5.9. In the remaining of this section, we shall “ﬁne-tune” Theorem 5.9 by specializing the continuous representation σ : WK → GL(V) of the

global Weil group WK of K on the n-dimensional linear space V over Éℓ to the σ É following cases : The n-dimensional ℓ-adic representation : WK → GL(n, ℓ) of WK – is assumed to be semi-simple;

– has a model over some finite extension E over Éℓ; – is unramified at almost all places of K; ν – in case K is a number field, is “B-admissible at in the sense of Fontaine” (look at [10, 11]) for each henselian place ν of K satisfying ℓ | qν . That is,

for ν | ℓ (so for such a place ν of K, the extension Kν /Éℓ is ﬁnite), the local Weil eν σ

representation σν : WKν −−→ WK −→ GL(V) is “B-admissible in the sense of Fontaine” (look at [10, 11]); and ﬁnally,

– is pure of weight w ∈ Z. σ É Note that, except for the first two assumptions on : WK → GL(n, ℓ), all the remaining ones on the ℓ-adic representation are defined “locally”. Remark 5.17. We make these assumptions, because the global Langlands reciprocity principle for GL(n) over the global field K asserts the existence of a unique natural bijective correspondence between the collection of all equivalence classes of irreducible n-dimensional ℓ-adic representations of the Weil group WK of K satisfying these assumptions, which are called, in case K is assumed to be a number field, “geometric Galois representations” by Fontaine and Mazur (for details, on ℓ-adic representations of WK, look at [13] for the function field case, and [4, 10, 11, 12, 54] for the number field case), and the collection of all equivalence classes of cuspidal

automorphic representations of the adele` group GL(n, AK), which are furthermore “algebraic” in the sense of Clozel in case K is a number ﬁeld9 10 (for details on cus-

pidal automorphic representations of GL(n, AK), look at [13] for the function ﬁeld

9 For example Maass forms on GL(2, É) do not correspond to Galois representations. 10As Ngoˆ points out in [45],

“In the number fields case, only a part of ℓ-adic representations of WK coming from motives should correspond to a part of automorphic representations.” On the non-abelian global class field theory 19 case, and [6] for the number field case). Moreover, this correspondence satisfies certain “naturality” conditions meaning that the bijection should recover abelian global class field theory when n = 1, should respect the corresponding L-functions and ε- factors, should well-behave with respect to certain linear algebraic operations, and should satisfy local-global compatibility. σ Now, let ν ∈ hK such that ℓ ∤ qν . A continuous ℓ-adic representation ν : WKν → GL(V) of WKν on an ℓ-adic space V is said to be Frobenius semi-simple, σ ϕ if the Éℓ-linear operator ν ( Kν |WKν ) acts semi-simply on V, for some fixed Frobe- ϕ nius Kν |WKν in WKν . Recall that, there exits an equivalence of the following abelian categories : Weil-Deligne representations Continuous Frobenius semi-simple rep-

≈ . É of K with coefﬁcients in É resentations of W on -linear spaces ℓ K ℓ ϕ ρ Deﬁnition 5.18. Let ν ∈ hK such that ℓ ∤ qν . A continuous representation : J → ϕ K ν GL(V) of JK on an ℓ-adic space V is called Frobenius semi-simple at , if ϕ ( Kν ) ϕ Φ ϕ ρ ρ Φ( Kν ) Kν ∇( Kν ) ν ν ◦ : WKν −−−−→ Z −→ GL(V) Kν ∼ Kν is a Frobenius semi-simple representation of WKν on V. That is, the linear operator ϕ ρ Φ( Kν ) ϕ ν ◦ Kν ( Kν |WKν ) acts semi-simply on the ℓ-adic linear space V. ϕ ξ ∇( Kν ) ξ

Let ν : Z → GL(V) ∪ { ν : W ν → GL(V)} be a collec- Kν K ν∈aK ν∈hK ϕ n o ξ ∇( Kν ) tion of continuous ℓ-adic representations on V. If ν : Z Kν → GL(V) is a contin- ϕ ( Kν ) ϕ Φ Φ( Kν ) Kν uous Frobenius semi-simple representation in the sense that ξν ◦Φ :WKν −−−−→ Kν ∼ ϕ ξ ∇( Kν ) ν Z Kν −→ GL(V) is a Frobenius semi-simple representation of WKν on V for each

ν ∈ hK satisfying ℓ ∤ qν , then Theorem 5.2 yields the existence of a unique con- ϕ ϕ ρ tinuous representation : JK → GL(V) of JK on the ℓ-adic space V such that

ρν = ξν for ν ∈ hK satisfying ℓ ∤ qν . Thus, we make the following deﬁnition. ϕ ϕ ρ Deﬁnition 5.19. The continuous representation : JK → GL(V) of JK on the n-dimensional ℓ-adic space V is called Frobenius-admissible,if ρ is Frobenius semi-

simple at ν for all ν ∈ hK satisfying ℓ ∤ qν .

Assume that the continuous ℓ-adic representation σ : WK → GL(V) is further- Weil eν σ more semi-simple. Then the local representation σν : WKν −−→ WK −→ GL(V) is

Frobenius semi-simple for each ν ∈ hK satisfying ℓ ∤ qν by the local-global compatibility of the Langlands correspondence for GL(n) (look at [15] for details). Thus, the following corollary follows now from Theorem 5.9, and Deﬁnitions 5.18 and 5.19.

Corollary 5.20. Let σ : WK → GL(V) be a continuous ℓ-adic semi-simple representation of WK on the n-dimensional linear space V over Éℓ. Then, the naturally 20 K. I. Ikeda deﬁned continuous representation ϕ ϕ σ ∗ : JK → GL(V) ϕ

of JK on V is Frobenius-admissible. σ É Recall that, a continuous n-dimensional ℓ-adic representation :WK → GL(n, ℓ)

has a model over some ﬁnite extension E over Éℓ, if the corresponding ℓ-adic Galois σ É

representation o : GK → GL(n, ℓ) has a model over some ﬁnite extension E over σ É Éℓ. That is, if the continuous representation o : GK → GL(n, ℓ) factors through a continuous homomorphism σ• : GK → GL(n,E) as

σ• σ É ,(o : GK −→ GL(n,E) ֒→ GL(n, ℓ where the continuity of the arrow σ• is deﬁned with respect to the Krull topology on GK and the ℓ-adic topology on GL(n,E). Thus, it is natural to make the following deﬁnition.

ϕ ρ É Deﬁnition 5.21. A continuous representation : JK → GL(n, ℓ) is said to have a

ﬁnite model over some ﬁnite extension E over É , if it factors through a continuous ϕ ℓ ρ homomorphism • : JK → GL(n,E) as

ϕ ρ• ρ É ,(JK −→ GL(n,E) ֒→ GL(n, ℓ : where the continuity of the arrow ρ is deﬁned with respect to the restricted free ϕ • product topology on JK and the ℓ-adic topology on GL(n,E).

By Baire category theorem, it follows that any continuous representation GK → É GL(n, Éℓ) has a model over some ﬁnite extension E over ℓ (look at [25] and [29] for details). Therefore, the following corollary immediately follows from Theorem

5.9 and from Deﬁnition 5.21. σ É Corollary 5.22. Let : WK → GL(n, ℓ) be a continuous ℓ-adic representation of WK. The naturally deﬁned continuous ℓ-adic representation

ϕ ϕ σ É ∗ : JK → GL(n, ℓ) ϕ of JK has a model over some finite extension E over Éℓ. More generally, the following proposition follows from Definition 5.14 and from Definition 5.21 directly.

Proposition 5.23. Assume that ρ is an n-dimensional continuous ℓ-adic Galois type

ϕ ρ É

representation of JK . Then has a model over some finite extension E over ℓ. σ É Recall that, a continuous n-dimensional ℓ-adic representation :WK → GL(n, ℓ) σ of WK is said to be unramified at ν ∈ hK, if ν (IKν ) = 1. Thus, we introduce the fol- ϕ lowing type of ℓ-adic representations of the non-abelian idele` group JK of the global field K. On the non-abelian global class field theory 21

ϕ ϕ

ν h ρ Deﬁnition 5.24. Let ∈ K. A continuous representation : JK → GL(V) of JK on an ℓ-adic space V is called unramiﬁed at ν, if

ϕ ϕ 0 ρ Φ( Kν ) (2.1) ρ ∇( Kν ) ν Kν (IKν ) = ν 1 Kν = 1; ϕ ρ ∇( Kν ) that is, if the local representation ν : Z Kν → GL(V) is unramiﬁed. Otherwise,

the representation is called ramified at ν. σ É The continuous n-dimensional ℓ-adic representation : WK → GL(n, ℓ) of WK is said to be unramified at almost all places of K, if σν (IKν ) = 1 for every ν ∈/ S for some finite subset S := S(σ) of hK. Thus, we introduce the following type of ϕ ℓ-adic representations of the non-abelian idele` group JK of the global field K. ϕ ϕ ρ Definition 5.25. The continuous representation : JK → GL(V) of JK on the ℓ-adic space V is said to be unramified at almost all places of K, if if there exists a ν finite subset S(ρ) =: S of hK such that, for each finite prime ∈/ S, the representation ϕ ρ ν of JK is unramified at . Now, by Theorem 5.9, and by Definitions 5.24 and 5.25, the following result follows immediately.

Corollary 5.26. Let σ : WK → GL(V) be a continuous representation of WK on the ℓ-adic space V, which is unramified at almost all places of K. Then, the naturally defined continuous representation ϕ ϕ σ ∗ : JK → GL(V) ϕ of JK on the ℓ-adic space V is unramified at almost all places of K.

Recall that, a continuous ℓ-adic representation σ : WK → GL(V) of WK on the

space V is pure of weight w ∈ Z, if ν – there exists a finite subset S of hK such that, for each finite prime ∈/ S, the local representation σν : WKν → GL(V) is unramified; namely, σν (IKν ) is trivial; and σ ϕ – the eigenvalues of ν ( Kν |WKν ) are algebraic integers whose complex conju- w/2 gates have complex absolute value qν . Thus, we make the following definition.

ϕ ϕ ρ É Deﬁnition 5.27. A continuous representation : JK → GL(V) of JK on a ℓ- linear space V is called pure of weight w ∈ Z, if

– there exists a finite subset S(ρ) =: S of hK such that, for each finite prime (ϕKν ) ν ρν ∇ ρ ∈/ S, the local representation : Z Kν → GL(V) is unramified; that is, is unramified at ν, and ϕ ρ Φ( Kν ) ϕ – the eigenvalues of ν Kν Kν |WKν are algebraic integers whose com- w/2 plex conjugates have complex absolute value qν . 22 K. I. Ikeda

Corollary 5.28. Assume that the continuous ℓ-adic representation σ : WK → GL(V) is pure of weight w. Then, the naturally defined representation ϕ ϕ σ ∗ : JK → GL(V) ϕ of JK on the n-dimensional linear space V over Éℓ is pure of weight w. Proof. Follows from Theorem 5.9, Definitions 5.24, 5.25 and Corollary 5.26, and Definition 5.27.

Finally, in case K is a number ﬁeld, a continuous ℓ-adic semi-simple repre- σ É

sentation : WK → GL(V) of WK on an n-dimensional vector space V over ℓ is ν ν B

called B-admissible at for each henselian place of K satisfying ℓ | qν , where É denotes a topological ℓ-algebra endowed with a continuous linear action of GÉℓ ,

if a corresponding model σ• : GK → GL(V•) on an n-dimensional linear space V•

É B ν over some ﬁnite extension E over Éℓ of degree [E : ℓ] = no is -admissible at for each henselian place ν of K satisfying ℓ | qν . That is, the local Galois representation (σ•)ν : GKν → GL(V•) on the linear space V• over E is B-admissible for each ﬁnite place ν satisfying ℓ | qν . Recall that, following closely [4, 10, 11], for each

ﬁnite place ν satisfying ℓ | qν , (σ•)ν : GKν → GL(V•) is called B-admissible, if the

following equality É dim GKν (DB (V•)) = dim (V•)

B ℓ holds, where

GKν

B É DB (V•) := ( ⊗ ℓ V•)

GKν GKν B is a B -linear space (note that is a field). σ ν Remark 5.29. Note that, the definition of B-admissibility of : WK → GL(V) at for each henselian place ν of K satisfying ℓ | qν does not depend on the choice of a model σ• : GK → GL(V•) on an n-dimensional linear space V• over some finite extension E over Éℓ.

Thus, we make the following deﬁnition. É Deﬁnition 5.30. Let B denote a topological -algebra endowed with a continuous ℓ ϕ ρ linear action of GÉ . A continuous Galois type representation : J → GL(V) of

ϕ ℓ K

B ν ν JK on a Éℓ-linear space V is called -admissible at for each henselian place of

γ ρ É K satisfying ℓ | qν , if there exists a representation : WK → GL(V) on the ℓ-linear space V satisfying ρ

ϕ Weil γ ϕ NR ρ # K / / JK WK GL(V) ν ν and which is B-admissible at for each henselian place of K satisfying ℓ | qν .

Remark 5.31. In fact, following [10, 11], it is possible to deﬁne B-admissible repre- ϕ sentations, at a henselian place ν of K satisfying ℓ | qν , of J on a É -linear space

ϕ K ℓ B V for any regular (Éℓ,(JK )ν )-ring . However, for this work, it suffices to restrict to the special case introduced in Definition 5.30. On the non-abelian global class field theory 23

Certainly, by Theorem 5.9 and by Deﬁnition 5.30, the following corollary follows directly.

Corollary 5.32. Let σ : WK → GL(V) be a continuous representation of WK on the ν ν ℓ-adic space V, which is B-admissible at for each henselian place of K satisfying ℓ | qν . Then, the naturally deﬁned continuous representation ϕ ϕ σ ∗ : JK → GL(V) ϕ ν ν of JK on the ℓ-adic space V is B-admissible at for each henselian place of K

satisfying ℓ | qν . B Remark 5.33. In what follows, we are primarily interested in the case B = dR. That is, we have established the following table : ϕ ϕ σ cont. σ cont. : WK −−→ GL(V) ∗ : JK −−→ GL(V) semi-simple Frobenius-admissible

has a model over some has a model over some É finite extension E/Éℓ finite extension E/ ℓ unramified at almost unramified at almost

all places of K =⇒ all places of K Z pure of weight w ∈ Z pure of weight w ∈

(if K is a number ﬁeld) (if K is a number ﬁeld) B

B-admissible at each -admissible at each ν h ν ∈ hK s.t. ℓ | qν ∈ K s.t. ℓ | qν

6. Tannakian categories ϕ Weil ϕ Note that, the global non-abelian norm-residue symbol NRK : JK → WK of K naturally produces a functor ϕ cont cont;Gal

Rep (WK) → Rep (J ) (6.1) É

É K ℓ ℓ cont from the category Rep (WK) of ℓ-adic continuous representations of WK to the Éϕℓ ϕ category Repcont;Gal(J ) of Galois type continuous representations of J with

É K K ℓ

coefficients É introduced by Definition 5.14, which is defined by pulling back a ℓ ϕ representation of WK on a Éℓ-linear space V to JK . Moreover, Theorem 5.9 states that this functor – preserves the respective representation spaces; – preserves the respective L-functions and ε-factors; and Corollaries 5.20, 5.26, and 5.28 state that it furthermore

– sends semi-simple ℓ-adic continuous representations of WK to the Frobenius- ϕ admissible representations of JK ; – sends ℓ-adic continuous representations of WK which are unramiﬁed at almost ϕ all places of K to ℓ-adic continuous representations of JK which are unram- iﬁed at almost all places of K; 24 K. I. Ikeda

– sends ℓ-adic continuous pure representations of WK of weight w to ℓ-adic con- ϕ tinuous pure representations of JK of weight w. If K is a number ﬁeld, Corollary 5.32 state that the functor (6.1) furthermore

– sends ℓ-adic continuous representations of WK which are B-admissible at each

ϕ ν h

∈ K such that ℓ | qν to ℓ-adic continuous representations of JK which are ν h B-admissible at each ∈ K such that ℓ | qν . ϕ ϕ Weil Notation 6.1. Let NK denote the kernel ker(NRK ) of the global non-abelian ϕ Weil ϕ norm-residue symbol NRK : JK → WK of K. Remark 6.2. Assume that Conjecture 3.2 holds. Let ϕ ρ : JK → GL(V) ϕ be any (that is, not necessarily of Galois type !) continuous representation of JK on a vector space V over Éℓ. Then the following conditions are clearly equivalent : ϕ – we have an inclusion N ⊆ ker(ρ); K ϕ ρ – the representation ( ,V) of JK factors through ϕ ϕ ϕ ρ ρ canonical o : JK −−−−−→ NK \JK −→ GL(V), cK ϕ ϕ ∼ where NK \JK −−−→ϕ WK under the global non-abelian norm-residue symbol NR Ko ϕ NR of K. K ϕ ρ – the representation ( ,V) of JK is of Galois type. ϕ Thus, by Remark 6.2, we get an alternative description of the category Repcont;Gal(J )

É K ℓ as follows. ϕ Let Repcont(J ) ϕ denote the category of continuous ℓ-adic representations Éℓ K N ϕ K ρ ( ,V) of JK satisfying the “congruence relation” ϕ ρ NK ⊆ ker( ). Assuming the validity of Conjecture 3.2, Remark 6.2 yields the equality ϕ ϕ cont;Gal

Rep (J ) = Repcont(J ) ϕ , É

É K K N ℓ ℓ K ϕ which is an alternative description of the category Repcont;Gal(J ). Moreover, com-

É K ℓ bining Remarks 5.15 and 6.2, there exist an equivalence ϕ cont cont

Rep (WK) ≈ Rep (J ) ϕ É É K N ℓ ℓ K ϕ cont cont

of the categories Rep (WK) and Rep (J ) ϕ , “natural” in the sense that É É K N ℓ ℓ K it preserves the respective representation spaces, preserves irreducibility and semi- simplicity, L-functions and ε-factors. More precisely, we have the following theorem. On the non-abelian global class ﬁeld theory 25

Theorem 6.3. Assume that Conjecture 3.2 holds. Then the functor ϕ cont cont

Rep (WK) → Rep (J ) É Éℓ ℓ K deﬁned by Theorem 5.9 induces an equivalence ϕ cont cont

Rep (WK) ≈ Rep (J ) ϕ (6.2) É É K N ℓ ℓ K ϕ cont cont

between the categories Rep (WK) and Rep (J ) ϕ , which preserves the re- É É K N ℓ ℓ K spective representation spaces, preserves irreducibility and semi-simplicity, as well as the respective L-functions and the ε-factors. Proof. Following Remark 6.2, deﬁne the functor ϕ cont cont

Rep (J ) ϕ → Rep (WK) É É K N ℓ K ℓ by ϕ ρ V ρ NR −1 V ( , ) 7→ ( o ◦ ( Ko ) , ) ϕ ϕ ρ for every representation ( ,V) of JK satisfying the “congruence relation” NK ⊆ ker(ρ). Then, by Theorem 5.9, ϕ ϕ ρ NR −1 V ρ NR −1 V ( o ◦ ( Ko ) , ) 7→ (( o ◦ ( Ko ) )∗, ), where ϕ ϕ ϕ ρ NR −1 ρ NR −1 NR o ◦ ( Ko ) )∗ = o ◦ ( Ko ) ◦ K ϕ ϕ ρ NR −1 NR c = o ◦ ( Ko ) ◦ Ko ◦ K = ρo ◦ cK = ρ, which shows that the composition ϕ ϕ cont cont cont

Rep (J ) ϕ → Rep (WK) → Rep (J ) ϕ

É É É K N K N ℓ K ℓ ℓ K ϕ is the identity functor on Repcont(J ) ϕ . É K N ℓ K Now, let σ : WK → GL(V) be any continuous representation of WK on an n- dimensional vector space V over Éℓ. Then, the corresponding continuous representation ϕ ϕ σ ∗ : JK → GL(V) ϕ of the topological group J on the n-dimensional vector space V over É satis- K ϕ ϕ ϕ ϕ ℓ ﬁes the “congruence relation” N ⊆ ker(σ∗ ), as α ∈ N yields σ∗ (α) = σ ◦ ϕ K K α σ NRK( ) = (1WK ) = 1V . Moreover, ϕ ϕ ϕ σ V σ NR −1 V ( ∗ , ) 7→ ( ∗o ◦ ( Ko ) , ), where ϕ ϕ σ NR −1 σ ∗o ◦ ( Ko ) = , 26 K. I. Ikeda

ϕ ϕ because for any w ∈ WK, there exists an αw ∈ J such that NR (αw) = w. There- ϕ ϕ ϕ ϕ ϕ K ϕ K fore, σ NR −1 w σ α σ α σ NR α σ w . Hence, ∗o ◦( Ko ) ( ) = ∗o (NK w) = ∗ ( w) = ◦ K( w) = ( ) the composition ϕ cont cont cont

Rep (WK) → Rep (J ) ϕ → Rep (WK)

É É É K N ℓ ℓ K ℓ cont is the identity functor on Rep (WK). Éℓ Thus, there exists an equivalence ϕ cont cont

Rep (WK) ≈ Rep (J ) ϕ É É K N ℓ ℓ K ϕ cont cont

between the categories Rep (WK) and Rep (J ) ϕ , which clearly preserves É É K N ℓ ℓ K the respective representation spaces, preserves irreducibility and semi-simplicty, as well as the respective L-functions and the ε-factors.

Remark 6.4. Moreover, the equivalence (6.2) induces the equivalences : – ϕ cont ss cont Frob−adm

Rep (WK) ≈ Rep (J ) ϕ É É K N ℓ ℓ K cont ss cont

between the full subcategory Rep (WK) of Rep (WK) whose objects are É Éℓ ℓ ϕ cont Frob−adm the semi-simple objects of Rep (W ) and the full subcategory Rep ( ) ϕ

K É J Éℓ ℓ K N ϕ K of Repcont(J ) ϕ whose objects are the Frobenius-admissible objects of Éℓ K N ϕ K Repcont(J ) ϕ ; Éℓ K N – K ϕ cont nr−ae cont nr−ae

Rep (WK) ≈ Rep (J ) ϕ É É K N ℓ ℓ K cont nr−ae cont

between the full subcategory Rep (WK) of Rep (WK) whose objects É Éℓ ℓ cont are the objects of Rep (WK) which are unramiﬁed at almost all places of K Éℓ ϕ ϕ nr−ae cont and the full subcategory Rep ( ) ϕ of Rep ( ) ϕ whose objects

É J J K É K N ℓ N ℓ K ϕ K are the objects of Repcont(J ) ϕ which are unramiﬁed at almost all places É K N ℓ K of K; – ϕ cont pure,w cont pure,w

Rep (WK) ≈ Rep (J ) ϕ É Éℓ ℓ K N K cont pure,w cont

between the full subcategory Rep (WK) of Rep (WK) whose objects É Éℓ ℓ cont are the objects of Rep (WK) which are pure of weight w and the full subcat- ϕ Éℓ ϕ pure,w cont egory Rep ( ) ϕ of Rep ( ) ϕ whose objects are the objects of

É J J K É K N ℓ N ℓ K ϕ K Repcont(J ) ϕ which are pure of weight w. É K N ℓ K If K is furthermore assumed to be a number ﬁeld, then –

ϕ B cont B−adm cont −adm

Rep (WK) ≈ Rep (J ) ϕ É É K N ℓ ℓ K On the non-abelian global class ﬁeld theory 27

cont B−adm cont

between the full subcategory Rep (WK) of Rep (WK) whose objects É Éℓ ℓ

cont ν h are the objects of Rep (WK) which are B-admissible at each ∈ K satis- Éℓ ϕ ϕ

B−adm cont fying ℓ | qν and the full subcategory Rep ( ) ϕ of Rep ( ) ϕ

É J J K É K N ℓ N ℓ K ϕ K cont ϕ whose objects are the objects of Rep (J ) which are B-admissible at É K N ℓ K

each ν ∈ hK satisfying ℓ | qν .

7. On the global Langlands reciprocity principle On the other hand, the proper generalization to the non-abelian setting of the idele-` class character of abelian global class ﬁeld theory is the notion of automorphic rep-

resentation of GL(n, AK) for 1 ≤ n. So, it is natural to ask the relationship between automorphic representations of GL(n, AK) and the n-dimensional ℓ-adic representa- ϕ tions of the non-abelian idele` group JK of K. Now, suppose that π is an irreducible admissible smooth representation of

GL(n, AK) (cf. [6]). Then, following Flath [9], there exists the restricted tensor prod- ′ uct decomposition π = ⊗ν<∞ πν ⊗ π∞ of π, where – πν is an irreducible admissible representation of the local group GL(n,Kν ),

for each ν ∈ hK; – πν is an unramiﬁed representation of the local group GL(n,Kν ), for almost all

ν ∈ hK.

Fix a rational prime ℓ. We shall deﬁne the set Sℓ as usual by a Sℓ := {ν ∈ hK | ℓ | qν } ∪ K. By the (non-archimedean) local Langlands reciprocity principle for GL(n), which is now a theorem of Laumon-Rapoport-Stuhler [38], Harris-Taylor [15] and Henniart ν [17], for each ν ∈ hK satisfying ℓ ∤ qν , that is ∈/ Sℓ, there is a correspondence π (n) π ν 7−→ LKν ( ν ), where

(n) π É LKν ( ν ) : WKν → GL(n, ℓ) is an (equivalence class of) n-dimensional Frobenius semi-simple ℓ-adic representation of the Weil group WKν of Kν such that Langlands π Artin−Weil (n) π L (s, ν ) = L (s,LKν ( ν )) and εLanglands π ψ +µ εArtin−Weil (n) π ψ +µ (s, ν , ν ,d ν ) = (s,LKν ( ν ), ν ,d ν ). π Moreover, for almost all ν ∈ hK, as the local admissible representation ν of GL(n,Kν ) (n) π is unramified, the corresponding ℓ-adic representation LKν ( ν ) of WKν is unramified as well. On the other hand, if ν ∈ hK satisfies ℓ | qν , then for the time being there is unfortunately no known well-formulated conjectural statement neither for the “p-adic Langlands functoriality” nor for the “p-adic reciprocity principle for

any reductive group G”, where only for the case G = GL(2) and Kν = É p some results are known (look at [33] for the theory of p-adic automorphic forms from a 28 K. I. Ikeda

very general perspective). Thus, for this case and for the case ν ∈ aK as well, we shall “forget” the corresponding irreducible admissible representation πν . To sum up, we have the following

Proposition 7.1. If π is an irreducible admissible smooth representation of GL(n, AK), then there exists a unique collection (n)

L (πν ) : WKν → GL(n, É ) Kν ℓ ν ∈/Sℓ consisting of (equivalencen classes of) n-dimensional Frobo enius semi-simple ℓ-adic representations of WKν for ﬁnite primes ν of K satisfying ℓ ∤ qν , where almost all of them are unramiﬁed. Now, we can state the theorem. Theorem 7.2. For any (equivalence class of) irreducible admissible smooth repre-

sentation π of GL(n, AK), there corresponds an (equivalence class of) ℓ-adic contin-

(n) ϕ λ π É uous Frobenius-admissible representation K ( ) : JK → GL(n, ℓ) unramiﬁed at almost all ν ∈ hK, satisfying ϕ λ (n) π Φ( Kν ) (n) π K ( )ν ◦ Kν = LKν ( ν ) for every ν ∈/ Sℓ, and the equalities LArtin−Hecke(s,λ (n)(π)) = LLanglands(s,π) Sℓ K Sℓ and εArtin−Hecke(s,λ (n)(π)) = εLanglands(s,π). Sℓ K Sℓ Moreover the arrow ϕ (n) cont−n o λ : Π(GL(n, AK)) → Rep (J ) K Éℓ K

is bijective, where Π(GL(n, AK)) denotes the category of irreducible admissible ϕ cont−n o smooth representations of GL(n, AK) and Rep (J ) denotes the category of

É K ℓ ϕ Frobenius-admissible n-dimensional ℓ-adic representations of JK which are un- ramiﬁed at almost all ν ∈ hK. Proof. In fact, the injectivity of the arrow follows from the “strong multiplicity

one” theorem for GL(n, AK) of Piatetski-Shapiro (cf. [47]). For the surjectivity of the arrow, note that, by the local (archimedean and non-archimedean) Langlands ϕ cont−n

ρ o πρ a reciprocity for GL(n), any ∈ Rep (J ) yields a collection { ν }ν∈hK ∪ K Éℓ K consisting of irreducible admissible representations πρν of GL(n,Kν ) as ν runs over finite and infinite places of K, where for almost all finite ν, πν is unramified. Then, ′ by Flath’s theorem, π = ⊗ν<∞ πν ⊗ π∞ is an irreducible admissible smooth repre- λ (n) π ρ sentation of GL(n, AK) satisfying K ( ) = and the equalities LArtin−Hecke(s,ρ) = LLanglands(s,π) Sℓ Sℓ and εArtin−Hecke(s,ρ) = εLanglands(s,π), Sℓ Sℓ which completes the proof. On the non-abelian global class field theory 29

Note that, Theorem 7.2 has a very interesting consequence, which we state

now as the following corollary.

a ξ Corollary 7.3. For a ﬁnite subset S of hK ∪ K satisfying Sℓ ⊆ S, let { ν }ν∈/S be a collection consisting of continuous n-dimensional Frobenius semi-simple ℓ-adic

ϕ ϕ

ξ É ν representations ν : (JK )ν → GL(n, ℓ) of the local group (JK )ν for ∈/ S, where ξ almost all of then are unramiﬁed. Then, the collection { ν }ν∈/S uniquely determines a continuous n-dimensional Frobenius-admissible ℓ-adic representation

ϕ ρ É : JK → GL(n, ℓ)

ϕ ν h of the non-abelian idele` group JK of K, which is unramiﬁed at almost all ∈ K, and which satisﬁes ρν = ξν for every ν ∈/ Sℓ. Clearly, Theorem 7.2 is closely related with the global Langlands reciprocity principle for GL(n) (for details [6, 7, 13, 14, 20, 36]). Let Πaut(GL(n)) denote the category of equivalence classes of automorphic representations of GL(n, AK), and let Πisob(GL(n)) and Πcusp(GL(n)) denote the full subcategories of Πaut(GL(n)) whose objects are the equivalence classes of isobaric and the equivalence classes of

cuspidal automorphic representations of GL(n, AK) respectively. Following closely π [27, 35], an automorphic representation π of GL(n, AK) is called isobaric, if ≃

π1 ⊞···⊞πm with each πi a cuspidal automorphic representation of GL(ni, AK), and n1 + n2 + ···+ nm = n. The automorphic representation π1 ⊞ ··· ⊞ πm of GL(n, AK), called the Langlands sum of π1,··· ,πm, is deﬁned to be the automorphic representation of GL(n, AK), which is unique up to equivalence, satisfying m ⊞m π π LS(s, i=1 i) = ∏LS(s, i). i=1

The existence of the Langlands sum operation ⊞ : Πisob(GL(n)) × Πisob(GL(n)) → Πisob(GL(n)) follows from the theory of Eisenstein series. The global reciprocity principle of Langlands for GL(n) predicts a unique and “natural” bijective correspondence

cont−n o,dR cont−n o

Π (GL(n)) o / Rep (WK) Π (GL(n)) o / Rep (WK) É

isob É isob O ℓ O O ℓ O

? ? ? cont−n o,dR ? cont−n o Π GL n o / Rep (W ) Πcusp(GL(n)) o / Rep (WK) cusp( ( )) K É irr Éℓ irr ℓ K : number ﬁeld K : function ﬁeld between the category Πisob(GL(n)) of equivalence classes of isobaric representa- cont−n o cont−n o,dR

tions of GL(n, AK) and the category Rep (WK) (resp. the category Rep (WK) ) É Éℓ ℓ of equivalence classes of n-dimensional continuous semi-simple ℓ-adic (resp. n- dimensional continuous semi-simple ℓ-adic and de Rham) representations of WK unramified at almost all ν ∈ hK in case K is a function field (resp. in case K is a number field), inducing a bijective correspondence between the full subcategory 30 K. I. Ikeda

Πcusp(GL(n)) of Πisob(GL(n)) consisting of the cuspidal objects of Πisob(GL(n)) cont−n o cont−n o,dR cont−n o

and the full subcategory Rep (WK) (resp. Rep (WK) ) of Rep (WK)

É É Éℓ irr ℓ irr ℓ cont−n o,dR cont−n o

(resp. Rep (WK) ) consisting of the irreducible objects of Rep (WK) É Éℓ ℓ cont−n o,dR (resp. Rep (WK) ) which recovers abelian global class field theory when Éℓ n = 1, preserves the corresponding L-functions and ε-factors, which is well-behaved with respect to certain linear algebraic operations, and which is compatible with the local Langlands reciprocity principle for GL(n). Altough the global reciprocity principle for GL(n) is still conjectural if K is a number field, in case K is a function field the global correspondence for GL(n) is now a theorem of L. Lafforgue [31]. Thus, it would be very interesting to understand the following diagram :

λ (n) K ϕ / cont−n o Π(GL(n, AK)) Rep (J ) o Éℓ K O −1 λ (n) O K

? ? Πaut(GL(n)) o / ? O O

? ? Πisob(GL(n)) o / ? O O

? ? Πcusp(GL(n)) o / ?

Deﬁnition 7.4. An n-dimensional continuous ℓ-adic Frobenius-admissible repre-

ϕ ρ ν h sentation of JK which is unramiﬁed at almost all ∈ K is called automorphic ρ λ (n) π type (resp. isobaric automorphic type, cuspidal automorphic type), if = K ( ) for some automorphic (resp. isobaric automorphic, cuspidal automorphic) represen-

tation π of GL(n, AK).

Now, assume that Conjecture 3.2 holds. Let π be an irreducible admissible

smooth representation of GL(n, AK) such that the corresponding continuous ℓ-adic

(n) ϕ λ π É Frobenius-admissible representation K ( ) : JK → GL(n, ℓ) which is unrami- 11 ﬁed at almost all ν ∈ hK satisﬁes the “congruence relation” ϕ λ (n) π NK ⊆ ker( K ( ));

11As Arthur points out in [2], “However, the condition that π be automorphic is very rigid. It imposes deep and interesting relationships among the components {πν } of π.” Also, look at the Takagi Lectures of M. Harris [14] for a detailed account on the non-abelian generalization of the “congruence conditions” appearing in abelian global class ﬁeld theory. On the non-abelian global class ﬁeld theory 31

ϕ (n) cont−n o namely, λ (π) is an object in the category Rep (J ) ϕ of n-dimensional K Éℓ K N Kϕ ρ continuous ℓ-adic Frobenius-admissible representations of JK unramiﬁed at almost all ν ∈ hK and satisfying the congruence relation ϕ ρ NK ⊆ ker( ). ϕ cont−n o cont−n o

Then, as Rep (J ) ϕ is equivalent to the category Rep (WK) of n- É É K N ℓ K ℓ dimensional continuous ℓ-adic semi-simple representations of WK unramiﬁed at almost all ν ∈ hK by Theorem 6.3 and Remark 6.4, there exists an n-dimensional (n) π continuous semi-simple ℓ-adic representation LK ( ) of WK which is unramiﬁed

(n)

ν h λ π at almost all ∈ K corresponding to K ( ) and satisfying the equalities

(n) (n) LArtin−Hecke(s,λ (π)) = LArtin−Weil(s,L (π)) Sℓ K Sℓ K and (n) (n) εArtin−Hecke(s,λ (π)) = εArtin−Weil(s,L (π)). Sℓ K Sℓ K Thus, Theorem 7.2 has the following important consequence.

Corollary 7.5. Assume that Conjecture 3.2 holds. Then, there exists a bijective arrow

L (n) K

λ (n) K ϕ equiv.(6.2) ' ϕ / cont−n o / cont−n o Π(GL(n, AK))N Rep (J ) ϕ Rep (WK)

É K K o ℓ N o Éℓ −1 K −1 λ (n) equiv.(6.2) K deﬁned by (n) π λ (n) π (n) π LK : K ( ) LK ( ), and satisfying the equalities

Langlands (n) L (s,π) = LArtin−Weil(s,L (π)) Sℓ Sℓ K and Langlands (n) ε (s,π) = εArtin−Weil(s,L (π)), Sℓ Sℓ K

ϕ π A for each from the category Π(GL(n, K))N of all irreducible admissible smooth K representations π of GL(n, AK) satisfying the congruence relation

ϕ λ (n) π NK ⊆ ker( K ( )). 32 K. I. Ikeda

To proceed our discussion, first assume that K is a function field. As the global reciprocity principle for GL(n) over a function field K is now a theorem of L. Laf- forgue (cf. [31]), assuming that Conjecture 3.2 holds, the diagram

ϕ Π(GL(n, AK))N KTi T TT Corollary7.5 TTTT TTTT TTT) cont−n o Rep (WK) j4 Éℓ jjjj jjjj jjjLafforgue jt jjj Πisob(GL(n)) and, taking into account Theorem 6.3 and Corollary 7.5, the diagram

1 Π(GL(n, A )) ϕ K N KTi TTT Corollary7.5 TTTT TTTT TTT) cont−n o Rep (WK) Éℓ irr j4 jjjj jjjj jjjLafforgue jt jjj Πcusp(GL(n))

1 ϕ A where Π(GL(n, A )) ϕ denotes the full subcategory of Π(GL(n, )) consist- K N K N K K n

ϕ ( )

π A λ π ing of the objects of Π(GL(n, K))N for which K ( ) is irreducible, yield the K following equivalences on a given irreducible admissible smooth representation π

of GL(n, AK) :

Proposition 7.6. Let K be a function ﬁeld. Assume that π is an irreducible admis-

sible smooth representation π of GL(n, AK). Then,

ϕ (n)

π A λ π is an isobaric representation of GL(n, K) ⇐⇒ NK ⊆ ker( K ( )), and

ϕ λ (n) π NK ⊆ ker( K ( )), π is a cuspidal representation of GL(n, A ) ⇐⇒ K λ (n) π ( K ( ) is irreducible provided that Conjecture 3.2 holds. On the non-abelian global class ﬁeld theory 33

Next, assume that K is a number ﬁeld. Assuming that Conjecture 3.2 holds, we then expect the following diagrams

ϕ Corollary7.5 cont−n o Π(GL(n, AK))N o / Rep (WK) O K Éℓ O

? ? Global Langlands cont−n o,dR Πisob(GL(n)) o / Rep (WK) Reciprocity for GL(n) Éℓ and

1 Corollary7.5 cont−n o Π(GL(n, AK)) ϕ o / Rep (WK)

N É O K ℓ O irr

? ? Global Langlands cont−n o,dR Πcusp(GL(n)) o / Rep (WK) Reciprocity for GL(n) Éℓ irr

1 where Π(GL(n, A )) ϕ is deﬁned as in the function ﬁeld case. It is then natural to K N K pose the following conjecture.

Conjecture 7.7. Let K be a number ﬁeld. Assume that Conjecture 3.2 holds. Let π

be an admissible smooth representation of GL(n, AK). Then, (1) The following statements are equivalent :

– π is an isobaric representation of GL(n, AK); – the Frobenius-admissible continuous n-dimensional ℓ-adic representa-

(n) ϕ λ π ν h tion K ( ) of JK unramiﬁed at almost all ∈ K satisﬁes ϕ λ (n) π

NK ⊆ ker( K ( )) ν h and is BdR-admissible at each ∈ K satisfying ℓ | qν . (2) The following statements are equivalent :

– π is a cuspidal representation of GL(n, AK); – the Frobenius-admissible continuous n-dimensional ℓ-adic representa-

(n) ϕ λ π ν h tion K ( ) of JK unramiﬁed at almost all ∈ K is irreducible and satisﬁes ϕ λ (n) π

NK ⊆ ker( K ( )) ν h and is BdR-admissible at each ∈ K satisfying ℓ | qν . Certainly, Conjectures 3.2 and 7.7 imply the global Langlands reciprocity for GL(n) over number ﬁelds. 34 K. I. Ikeda

8. Langlands group LK of a number field K In the remaining of the text, we shall closely follow the work of Arthur [1]. It seems possible to apply the ideas developed in this work to the construction of the “hypothetical” Langlands group LK of a given global field K, especially to the construction of the Langlands group LK in case K is a number field, whose existence is one of the major problems in the framework of the global Langlands reciprocity and the global functoriality principles. In case K is a function field, then LK = WK and we have already covered this case in the previous sections of this work.

Therefore, we shall from now on assume that K is a number ﬁeld. Then, for a

each ν ∈ hK ∪ K, the local Langlands group LKν is deﬁned by ν h WAKν = WKν × SL(2, C), if ∈ K;

LKν = ν a (WKν , if ∈ K.

Note that, if ν ∈ hK, instead of the traditional Weil-Deligne group WDKν of the non-archimedean local ﬁeld Kν , we shall use the topological group WKν ×SL(2, C), which is denoted by WAKν and called the Weil-Arthur group of Kν (cf. Langlands [32]). ϕ Now, introduce the “thickened” version W A of the non-abelian idele` group ϕ K

JK of a number ﬁeld K as follows.

ν h ϕ ϕ ϕ Definition 8.1. For each ∈ K fix a Lubin-Tate splitting Kν and let = { Kν }ν∈hK . ϕ The topological group W A K defined by the “restricted free product” ϕ ϕ ϕ 0

′ ( Kν ) ( Kν ) ∗r1 ∗r2

∇ C ∇ C

:= Z × SL(2, ) : × SL(2, ) W W C W A K ∗ Kν 1 Kν ∗ Ê ∗

ν∈h K is called the Weil-Arthur idele` group of the number field K. The finite (=henselian) ϕ ϕ part W A of W A is defined by K,h K ϕ ϕ ϕ 0

′ ( Kν ) ( Kν )

∇ C ∇ C

W A := ∗ Z × SL(2, ) : × SL(2, ) , K,h Kν 1 Kν ν∈hK ϕ ϕ and the inﬁnite (=archimedean) part of is deﬁned by W A K,a W A K ϕ

:= W ∗r1 W ∗r2 .

Ê C W A K,a ∗ Here, as usual r1 and r2 denote the number of real and the number of pairs of

complex-complex conjugate embeddings of the global field K in C. ϕ The topological group W A K is an “extremely big” group, whose definition depends only on K. ϕ ab The following theorem describes the abelianization W A of the topological ϕ K group W A K. ϕ ab ϕ Theorem 8.2. The abelianization W A K of the topological group W A K is indeed ÂK. Proof. The proof follows by first noting that the direct limit functor is exact and then by abelianizing free products of groups. On the non-abelian global class field theory 35

Remark 8.3. From now on, unless otherwise stated, we shall assume the existence

of the global Langlands group LK of a number ﬁeld K. a For ν ∈ hK ∪ K, as before, choose an embedding sep sep . eν : K ֒→ Kν This embedding determines a continuous homomorphism12 Langlands eν : LKν → LK,

and therefore, for each ν ∈ hK, a continuous homomorphism

Langlands {·,Kν }ϕ ×id Langlands

(ϕKν ) ϕKν Kν SL(2,C) eν ∇ C NR : Z × SL(2, ) −−−−−−−−−−−−→ WAKν −−−−−→ LK. Kν Kν ∼ Theorem 8.4 (Ultimate global non-abelian norm-residue symbol “Weak form”). There exists a well-deﬁned continuous homomorphism ϕLanglands ϕ NRK : W A K → LK, (8.1) which satisﬁes

ϕLanglands ϕLanglands ϕLanglands ϕ ϕ NR cS K (NRK )S = NRK ◦ cS : (W A K)S −→ W A K −−−−−−−→ LK, ϕ ϕ

where, following Proposition 2.1, cS : (W A K)S → W A K is the canonical homo-

a a morphism defined for every finite subset S of hK ∪ K containing K. ϕLanglands ϕ We conjecture that the continuous homomorphism NRK : W A K → LK should be considered as the “ultimate form” of the global non-abelian norm-residue symbol of K. More precisely, we pose the following conjecture (to be precise, the following meta-conjecture). Conjecture 8.5 (Ultimate global non-abelian norm-residue symbol “Strong form”). The homomorphism ϕLanglands ϕ NRK : W A K → LK is open, continuous and surjective. Finally, we have the following remark. Remark 8.6. (1) In fact, via the same lines of reasoning of this work, we can study the relationship between the automorphic representations π of a general reductive group G over the number field K with the global Langlands parameters L φ : LK → G of G, which is the content of the global reciprocity principle of Langlands for a general reductive group G. (2) It would be interesting to compare Arthur’s construction of LK, which uses the classification of automorphic representations of Langlands in the sense ϕ of “beyond endoscopy” [34], with the topological group W A K constructed in this paper. This comparison may reveal an unconditional definition of the global Langlands group LK of the number field K.

12which is unique up to conjugacy. 36 K. I. Ikeda

Acknowledgement The author would like to express his gratitude to the organizers of the Adrasan Workshop on “L-functions and Algebraic Numbers” for inviting him to deliver a lecture series on “Langlands correspondence for GL(n)”, and the hospitality that he received during the workshop. The author would also like to thank to Katsuya Miyake for his encouragment and interest in this work, where the ideas in his papers [40, 41] turned out to be central in our work, and to Ali Altugˇ for very carefully reading a preliminary version of this work and pointing out numerous inaccuracies.

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Kazımˆ Ilhan˙ Ikeda˙ Yeditepe University Department of Mathematics In˙ on¨ u¨ Mah., Kayıs¸dagıˇ Cad. 26 Agustosˇ Yerles¸imi 34755 Atas¸ehir, Istanbul Turkey e-mail: [email protected]

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