On Arithmetic Dijkgraaf-Witten Theory

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Introduction

In [Ki] Minhyong Kim initiated to study arithmetic Chern-Simons theory for number rings, which is based on the ideas of Dijkgraaf-Witten theory for 3-manifolds ([DW]) and the analogies between 3-manifolds and number rings, knots and primes in arithmetic topology ([Mo2]). We note that Dijkgraaf- Witten theory may be seen as a 3-dimensional Chern-Simons gauge theory with finite gauge group (cf. [FQ], [G], [Wa], [Y] etc). Among other things, Kim constructed an arithmetic analog of the Chern-Simons functional, which is defined on a space of Galois representations over a totally imaginary number field. In the subsequent paper [CKKPY] Kim and his collaborators showed a decomposition formula for arithmetic Chern-Simons invariants and applied it to concrete computations for some examples. Later, Kim’s construction was extended over arbitrary number field which may have real primes ([H], [LP]). Computations of arithmetic Chern-Simons invariants have also been carried out for some examples, by employing number-theoretic considerations in [AC], [BCGKPT], [CKKPPY], [H] and [LP]. In [CKKPPY], the arithmetic Chern- Simons correlation functions for finite cyclic gauge groups were computed in terms of arithmetic linking numbers. It should be noted that Kim also consid- ered arithmetic Chern-Simons functionals for the case where the gauge groups are p-adic Lie groups ([Ki; 3]). By arithmetic Dijkgraaf-Witten theory in the title, we mean arithmetic Chern-Simons theory with finite gauge group in the sense of Kim.

The purpose of this paper is to add some basic constructions and properties to Kim’s theory and lay a foundation for arithmetic Dijkgraaf-Witten theory along the line of topological quantum ﬁeld theory, TQFT for short, in the sense of Atiyah ([A1]). TQFT is a framework to produce topological

2 invariants for manifolds. For example, the Jones polynomials of knots can be obtained in the context of (2+1)-dimensional Chern-Simons TQFT with compact connected gauge group (cf. [A2], [Ko], [Wi]). For the TQFT structure of Dijkgraaf-Witten theory, we consult [DW], [FQ], [G], [Wa], [Y]. In this paper, following Gomi’s treatment [G] and Kim’s original ideas [Ki], we construct an arithmetic analogue of Dijkgraaf-Witten TQFT in a certain special situation, namely, we construct arithmetic analogues, for a finite set S of finite primes of a number field k, of the prequantization bundles, the Chern-Simons 1-cocycle, the Chern-Simons functional, the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function. Arithmetic Dijkgraaf- Witten invariants are new arithmetic invariants for a number field, which may be seen as variants of (non-abelian) Gaussian sums.

We fix a finite group G and a 3-cocycle c Z3(G, R/Z). For an ori- ∈ ented compact manifold X with a fixed triangulation, let X be the space of gauge fields associated to G and let be the gauge groupF Map(X,G) GX acting on X . Note that X and X are finite sets and that the quotient space F:= / is identifiedF withG Hom(π (X),G)/G if X is connected, MX FX GX 1 where Hom(π1(X),G)/G is the quotient of the set of homomorphisms from the fundamental group π1(X) of X to G by the conjugate action of G. As for the classical theory in the sense of physics, we construct, using the 3-cocycle c, the following correspondences (0.1) oriented closed surface Σ 1-cocycle λ Z1( , Map( , R/Z)), Σ ∈ GΣ FΣ oriented compact 3-manifold M 0-chain CS C0( , Map( , R/Z)), M ∈ GM FM which satisfy

(0.2) dCSM = res∗λ∂M , where res : M (resp. M ) ∂M (resp. ∂M ) is the restriction map and d : C0( , Map(F , R/GZ) →C F1( , Map( G , R/Z)) is the coboundary ho- GM FM → GM FM momorphism of group cochains. The key ingredient to construct λΣ and CSM i i d is the transgression homomorphism C (G, R/Z) C − ( , Map( , R/Z)) → GX FX with d = dim X and, in fact, λΣ and CSM are given by the images of c for i = 3, X = Σ and M, respectively ([G]). Then we can construct a -equivariant GΣ principal R/Z-bundle Σ and the associated complex line bundle LΣ over Σ, using λ , and hence theL complex line bundle L over . In fact, isF the Σ Σ MX LΣ product bundle Σ R/Z on which Σ acts by (ρΣ, m).g =(ρΣ.g, m+λΣ(g, ρΣ)) for ρ , m F R×/Z and g G. We call λ the Chern-Simons 1-cocycle. Σ ∈ FΣ ∈ ∈ GΣ Σ The line bundle LΣ (or LΣ) is called the prequantization complex line bundle

3 for a surface Σ. The 0-chain CSM is called the Chern-Simons functional for a 3-manifold M. We see that CSM is a M -equivariant section of res∗ Σ over . G L FM As for the quantum theory, the formalism of (2 + 1)-dimensional TQFT is given by the following correspondences (functor from the cobordism category of surfaces to the category of complex vector spaces)

oriented closed surface Σ quantum Hilbert space , (0.3) Σ oriented 3-manifold M partition function Z H , M ∈ H∂M which satisfy several axioms (cf. [A1]). Here we notice the following two axioms: (0.4) functoriality: An orientation preserving homeomorphism f : Σ ≈ Σ′ → induces an isomorphism ∼ ′ of Hilbert quantum spaces. Moreover, HΣ → HΣ if f extends to an orientation preserving homeomorphism M ≈ M ′, with → ∂M = Σ, ∂M ′ = Σ′, then ZM is sent to ZM ′ under the induced isomorphism ∼ ′ . H∂M → H∂M (0.5) multiplicativity and involutority: For disjoint surfaces Σ1, Σ2 and the surface Σ∗ = Σ with the opposite orientation, we require

Σ1 Σ2 = Σ1 Σ2 , Σ∗ =( Σ)∗, H ⊔ H ⊗ H H H where ( )∗ is the dual space of . Moreover, if ∂M = Σ Σ , ∂M = HΣ HΣ 1 1 ⊔ 2 2 Σ∗ Σ and M is the 3-manifold obtained by gluing M and M along Σ , 2 ⊔ 3 1 2 2 then we require

= ZM ,

∗ where < , >: Σ1 Σ2 Σ Σ3 Σ1 Σ3 is the natural gluing pairing · · H ⊔ × H 2⊔ → H ⊔ of quantum Hilbert spaces. This multiplicative property is indicative of the “quantum” feature of the theory (cf. [A1]). The construction of the Hilbert space is phrased as the geometric quan- HΣ tization. We note that Σ is known to be isomorphic to the space of conformal blocks for the surface ΣH when the gauge group is a compact connected group (cf. [Ko]). Elements of are called (non-abelian) theta functions (cf. [BL]). HΣ For Dijkgraaf-Witten theory, Σ is constructed, in an analogous manner, as the space of -equivariant sectionsH of the prequantization line bundle L over GΣ Σ Σ, in other words, the space of sections of LΣ over Σ: F(0.6) M 2π√ 1λΣ(g)(ϑ) = ϑ : C ϑ(̺ .g)= e − ϑ(̺ ) g , ̺ HΣ { FΣ → | Σ Σ ∀ ∈ GΣ Σ ∈FΣ} = Γ( , L ). MΣ Σ

4 In quantum field theories, partition functions are given as path integrals. In Dijkgraaf-Witten theory, the Dijkgraaf-Witten partition function ZM ∂M is defined by the following finite sum fixing the boundary condition: ∈ H

1 2π√ 1CSM (̺) (0.7) Z (̺ )= e − (̺ ). M ∂M #G ∂M ∈F∂M ̺ M ∈F res(̺X)=̺∂M The value Z (̺ ) is called the Dijkgraaf-Witten invariant of ̺ . M ∂M ∂M ∈ F∂M We note that when [c] is trivial and S is empty, then Σ = and the Dijkgraaf-Witten invariant Z ( ), denoted by Z(M), coincidesF with{∗} the (av- M ∗ eraged) number of homomorphism from π1(M) to G: #Hom(π (M),G) (0.8) Z(M)= 1 , #G which is the classical invariant for the connected 3-manifold M.

Now let us turn to the arithmetic. First, let us recall the basic analogies in arithmetic topology which bridges 3-dimensional topology and number theory ([Mo2]. See also [Mh]). Let k a number field of finite degree over the rationals Q. Let be the ring of integers of k and set X := Spec( ). Let X∞ Ok k Ok k denote the set of infinite primes of k and set X := X X∞. We see X , X∞ k k ⊔ k k k and Xk as analogues of a non-compact 3-manifold M, the set of ends and the end-compactification M, respectively. A maximal ideal p of is identified Ok with the residue field Spec( k/p)= K(Z, 1) (Z being the profinite completion of Z), which is seen as anO analogue of the circle S1 = K(Z, 1). We see the mod p reduction map Spec(Fp) ֒ Xk asb an analogueb of a knot, an embedding 1 → .S ֒ M. Let p be the ring of p-adic integers and let kp be the p-adic field → O We denote Spec( p) and Spec(kp) by Vp and ∂Vp, respectively . We see Vp and O ∂Vp as analogue of a tubular neighborhood of a knot and its boundary torus, respectively. So we see the étale fundamental group Πp of Spec(kp), which is the absolute Galois group Gal(kp/kp) (kp being an algebraic closure of kp), as an analogue of the peripheral group of a knot. (To be precise, the tame quotient of Πp may be seen as a closer analogue of the peripheral group. cf. [Mo2; Chapter 3]) Let S = p1,..., pr be a finite set of maximal ideals of k. Let XS := X S. We see{ S and X} as an analogue of a link in a 3-manifoldO and the link k \ S complement, respectively. We may also see XS as an analogue of a compact 3-manifold with boundary (union of tori), where ∂V := Spec(kp ) S 1 ⊔···⊔ Spec(kpr ) plays an analogous role of the boundary tori, “∂XS = ∂VS ”. The

5 modified étale fundamental group ΠS of XS, introduced in [H; 2.1] taking real primes into account, is the Galois group of the maximal subextension kS of k which is unramified at any (finite and infinite) prime outside S, as an analogue of the link group. We list herewith some analogies which will be used in this paper.

oriented, connected, closed compactified spectrum of 3-manifold M number ring X = Spec( ) k Ok knot prime S1 ֒ M p = Spec( /p) ֒ X : K → { } Ok → k link finite set of maximal ideals = S = p ,..., p L K1 ⊔···⊔Kr { 1 r} tubular n.b.d of a knot p-adic integer ring V Vp = Spec( p) K O boundary torus p-adic field ∂V ∂Vp = Spec(kp) K peripheral group local absolute Galois group π1(∂V ) Πp = Gal(kp/kp) K tubular n.b.d of a link union of pi-adic integer rings

V = V 1 V r VS = Spec( p1 ) Spec( pr ) L K ⊔···⊔ K O ⊔···⊔ O boundary tori union of pi-adic ﬁelds

∂V = ∂V 1 ∂V r ∂VS = Spec(kp1 ) Spec(kpr ) L K ⊔···⊔ K ⊔···⊔ link complement complement of a ﬁnite set of primes X = M Int(V ) XS = Xk S L \ L \ link group maximal Galois group with Π = π1(X ) given ramiﬁcation ΠS = Gal(kS/k) L L

Based on the above analogies, we construct an arithmetic analogue of Dijkgraaf- Witten TQFT in a special situation, which corresponds to the case that M is a link complement and Σ is the boundary tori of a tubular neighborhood of a link and M is a link complement. Notations being as above, let N be an integer > 1 and assume that the number field k contains a primitive N-th root ζ of unity. We fix a finite group G and a 3-cocycle c Z3(G, Z/NZ). Let N ∈ F be a subfield of C such that ζN is contained in F and F = F (F being the complex conjugate). Let S be a finite set of finite primes S = p1,..., pr of k such that any finite prime dividing N is contained in S. Let {X := X } S S k \ and let ∂V := Spec(kp ) Spec(kp ) as before so that ∂V plays a role S 1 ⊔···⊔ r S of the boundary of XS, “∂X S = ∂VS”. For arithmetic analogues of the spaces

6 r of gauge fields and , we consider := Hom (Πp ,G) and FΣ FM FS i=1 cont i XS := Homcont(ΠS,G), respectively, where Homcont( ,G) denotes the set Fof continuous homomorphisms to G. For an arithmeticQ − analog of the gauge groups Σ and M , we simply take the group G acting on S and XS by conjugation.G SetG := /G. F F MS FS As for the classical theory in the arithmetic side, we firstly develop a local theory at a finite prime p, namely, we construct the arithmetic prequantization principal Z/NZ-bundle p and the associated arithmetic pre- L quantization F -line bundle Lp for ∂Vp, which are G-equivariant bundles over p := Homcont(Πp,G). By choosing a section xp Γ( p, p), we construct the F xp 1 ∈ F L arithmetic Chern-Simons 1-cocycle λ Z (G, Map( p, Z/NZ)). The key p ∈ F idea for the constructions is due to M. Kim ([Ki]), who used the conjugate G-action on c and the canonical isomorphism

2 invp : H (Πp, Z/NZ) ∼ Z/NZ −→ in the theory of Brauer groups of local ﬁelds. We note that this isomorphism tells us that ∂Vp is “orientable” and we choose (implicitly) the “orientation” of ∂Vp corresponding to 1 Z/NZ. ∈ Getting together the local theory over S, we construct the arithmetic prequantization principal Z/NZ-bundle and the associated arithmetic pre- LS quantization F -line bundle LS for ∂VS , which are G-equivariant bundles over S. By choosing a section xS of S over S, we construct the arithmetic F xS 1L F Chern-Simons 1-cocycle λS Z (G, Map( S, Z/NZ)) and show that S ∈ F xS L (resp. LS) is isomorphic to the product bundle S = S Z/NZ (resp. xS L F x×S LS = S F ) on which G acts by (ρS, m).g = (ρS.g, m + λS (g, ρS)) (resp. F × xS λS (g,ρS ) (ρS, z).g = (ρS.g, zζN )) for ρS S, m Z/NZ, z F and g G. 3 ∈ F ∈ ∈ ∈ By employing H (ΠS, Z/NZ) = 0, the arithmetic Chern-Simons functional

CSXS for XS is deﬁned as a G-equivariant section of resS∗ ( S) over XS , where res : is the restriction map induced by theL naturalF ho- S FXS → FS momorphisms Πp ΠS for p S. Using the section xS, it can be regarded → ∈ xS as a G-equivariant functional CS : X Z/NZ. Thus we construct the XS F S → following correspondences

1 ∂VS 1-cocycle λS Z (G, Map( S, Z/NZ)), (0.9) xS ∈ 0 F XS 0-chain CS C (G, Map( X , Z/NZ)), XS ∈ F S which satisfy

xS xS (0.10) dCS = res∗ λ . XS S S

7 We may regard (0.9), (0.10) as arithmetic analogues of (0.1), (0.2) in a special situation that corresponds to the case Σ is a boundary tori of a link and M is a link complement. As for the quantum theory in the arithmetic side, following the topological side, we deﬁne the arithmetic quantum space for ∂V to be the space of G- HS S equivariant sections of the arithmetic prequantization F -line bundle LS over xS S. Choosing a section xS Γ( S, S), it is isomorphic to the space S Fgiven by ∈ F L H

xS λS(g)(ρS ) S = θ : S F θ(ρS.g)= ζN θ(ρS) g G, ρS S (0.11) H { F →xS | ∀ ∈ ∈F } = Γ( , L ), MS S xS xS where LS is the quotient of LS by the action of G. The arithmetic Dijkgraaf- xS Witten invariant Z (ρS) of ρS S with respect to xS is then defined by XS ∈ F the following finite sum fixing the boundary condition: x CS S (ρ) xS 1 XS (0.12) Z (ρS)= ζN . XS #G ρ ∈FXXS resS(ρ)=ρS

xS xS Then we can show that ZXS S . Since the spaces S , when xS is var- ∈ H H xS ied, are naturally isomorphic each other, S is identiﬁed with ( S )/ , where the equivalence relation identiﬁesH elements via the isomorphismsH be-∼ xS xS ∼ F tween S ’s. Hence Z determine the element ZX S, which we call the H XS S ∈ H arithmetic Dijkgraaf-Witten partition function for XS. Thus we construct the following correspondences (0.13) ∂V arithmetic quantum space , S HS X arithmetic Dijkgraaf-Witten partition function Z , S XS ∈ HS which satisfy some properties similar to the axioms in (2 + 1)-dimensional TQFT. We note that when [c] is trivial and S is empty, then the arithmetic

Dijkgraaf-Witten invariant ZXS , denoted by Z(Xk), coincides with the (averaged) number of continuous homomorphism from the modified étale fundamental group π1(Xk) of Xk ([H;2.1]), which is the Galois group of maximal extension of k unramified at all finite and infinite primes, to G: #Hom (π(X ),G) (0.14) Z(X )= cont k , k #G which is the classical invariant for the number field k. We may regard (0.11), (0.12), (0.13) and (0.14) as an arithmetic analogues of (0.6), (0.7), (0.3) and

8 (0.8) respectively, in a special situation that corresponds to the case Σ is a boundary tori of a link and M is a link complement. We note that elements of may be seen as arithmetic analogs of (non- HS abelian) theta functions. In this respect it may be interesting to observe that the arithmetic Dijkgraaf-Witten invariants ZXS (ρS) in (0.11) look like (non- abelian) Gaussian sums.

Next we show some basic and functorial properties of arithmetic Chern- Simons 1-cocycles, arithmetic prequantization bundles, arithmetic Chern-Simons invariants, arithmetic quantum spaces and arithmetic Dijkgraaf-Witten partition function (i) when we change the 3-cocycle c in the cohomology class [c], (ii) when we change the pair of k and S to the isomorphic one, (iii) when S is an empty set, and (iv) when S is a disjoint union of finite sets of finite primes and when we re- verse the orientation of ∂VS , and As for (ii) and (iv), we show the following properties: ∼ (0.15) functoriality: If there are isomorphisms ξi : kpi kp′ ′ (1 i r), → i ≤ ≤ then they induce the isomorphism S ∼ S′ for S = p1,..., pr ,S′ = ∼ H → H { } p1′ ,..., pr′ . Moreover, if ξ : k k′ is an isomorphism of number fields { } → ∼ such that ξ(pi) = pi′ and xi induces isomorphisms kpi kp′ ′ , then ξ induces → i the isomorphism S ∼ S′ which sends Z to Z . H → H XS XS′ (0.16) multiplicativity and involutority: For disjoint sets S1,S2 of finite sets of finite primes and ∂VS∗ = ∂VS with the opposite orientation (cf. 4.4 below for the meaning), we show

S1 S2 = S1 S2 , S∗ =( S)∗, H ⊔ H ⊗ H H H where ∗ denotes the arithmetic quantum space for ∂V ∗ and ( )∗ is the HS S HS dual space of S. These propertiesH (0.15) and (0.16) may be regarded as arithmetic analogues of the axioms (0.4) and (0.5) in (2 + 1)-dimensional TQFT.

Finally we show decomposition formulas for arithmetic Chern-Simons invariants, which generalize, in our framework, the “decomposition formula” by Kim and his collaborators ([CKKPY]), and show gluing formulas for arithmetic Dijkgraaf-Witten partition functions. Let S1 and S2 be disjoint sets of ﬁnite primes of k, where S1 may be empty and S2 is non-empty. We assume that any prime dividing N is contained in S2 if S1 is empty and that any prime

9 dividing of N is contained in S if S is non-empty. We set S := S S . When 1 1 1 ⊔ 2 S1 is empty, XS1 = Xk and we mean by CS the arithmetic Chern-Simons XS1 functional CSXk defined in [H] (see also [LP]). We can also define the arithmetic Chern-Simons functional CS for V as a section of ˜res∗ ( ) over VS2 S2 S2 LS2 ˜ ˜ ét VS2 := p S2 Homcont(Πp,G), where Πp := π1 (Vp) and ˜resS : VS2 S2 is F ∈ F →F the restriction map induced by the natural homomorphism Πp Π˜ p. Then Q → we have the following decomposition formula ⊞ (0.17) CSX (ρ) CSVS ((ρ up)p S2 )= CSX (ρ ηS), S1 2 ◦ ∈ S ◦ where ρ Hom (Π ,G), and η : Π Π , up : Π˜ p Π are natural ∈ cont S1 S S → S1 → S1 homomorphisms induced by XS XS1 , Vp XS1 for p S2, respectively, and ⊞ : is the natural→ “sum”→ of arithmetic∈ prequantization LS1 ×LS2 → LS principal Z/NZ-bundles (cf. (4.3.1), (4.3.2)). When S1 is empty, the formula (0.13) is a reformulation of the decomposition formula in [CKKPY]. As for arithmetic Dijkgraaf-Witten partition functions, we have the following gluing formula. Note that XS1 may be obtained by gluing XS and VS∗2 along ∂VS2 , where VS∗2 = VS2 with the opposite orientation. Then we have

(0.18) = Z , XS S2 XS

∗ where < , >: S S2 S1 is the gluing pairing of arithmetic quantum spaces (cf.· · (5.2.3)).H × H We→ may H regard (0.16) as an arithmetic analog of the gluing formula in the axiom (0.5) in (2 + 1)-dimensional TQFT.

The contents of this paper are organized as follows. In Section 1, we collect some basic facts on torsors and group cochains, which will be used in the subsequent sections. In Section 2, we construct arithmetic prequantization bundles, arithmetic Chern-Simons 1-cocycles and the arithmetic Chern-Simons functionals. These constructions correspond to the classical theory of topological Dijkgraaf-Witten TQFT. In Section 3, we construct arithmetic quantum spaces and the arithmetic Dijkgraaf-Witten partition functions. These constructions correspond to the quantum theory of topological Dijkgraaf-Witten TQFT. In Section 4, we show some basic and functorial properties of arithmetic prequantization bundles, arithmetic Chern-Simons 1-cocycles, arithmetic Chern- Simons invariants and arithmetic Dijkgraaf-Witten invariants. In Section 5, we show decomposition formulas for arithmetic Chern-Simons invariants and gluing formulas for arithmetic Dijkgraaf-Witten partition functions.

10 Notation. For a G-equivariant ﬁber bundle ̟ : E B for a group G, we → denote by Γ(E, B) (resp. ΓG(E, B)) the set of sections (resp. the set of G- equivariant sections) of ̟. In this paper, we deal with the case where the base space B is a ﬁnite (discrete) set.

Acknowledgment. We would like to thank Kiyonori Gomi, Tomoki Mihara, Yuji Terashima and Masahito Yamazaki for useful communications. We are grateful to Gomi for answering our questions patiently. The ﬁrst author is supported by Grant-in-Aid for JSPS Fellow (DC1) Grant Number 20J21684. The second author was supported by Grant-in-Aid for JSPS Fellow (DC1) Grant Number 17J02472. The third author is supported by Grant-in-Aid for Scientiﬁc Research (KAKENHI) (B) Grant Number JP17H02837.

1. Preliminaries on torsors and group cochains

In this section, we collect some baisc facts on torsors for an additive group and group cochains, which will be used in the subsequent sections.

1.1. Torsors for an additive group. Let A be an additive group, where the identity element of A is denoted by 0. An A-torsor is deﬁned by a non-empty set T equipped with action of A from the right

T A T ; (t, a) t.a, × −→ 7→ which is simply transitive. So, for any elements s, t T , there exists uniquely a A such that s = t.a. We denote such an a by s ∈ t: ∈ − (1.1.1) a = s t def s = t.a. − ⇐⇒

For A-torsors T and T ′, a morphism f : T T ′ is deﬁned by a map of → sets, which satisﬁes

(1.1.2) f(t.a)= f(t).a for all t T and a A. We easily see that any morphism of A-torsors is an isomorphism.∈ ∈ Deﬁning the action of A on A by (t, a) A A t + a A, A itself ∈ × 7→ ∈ becomes an A-torsor. We call it a trivial A-torsor. A morphism f : A A of trivial A-torsors is given by f(a) = a + λ for any a A with λ = f→(0). ∈

11 Choosing an element t T , any A-torsor T is isomorphic to the trivial A-torsor by the morphism ∈

(1.1.3) ϕ : T ∼ A; s ϕ (s) := s t. t −→ 7→ t − We call ϕt the trivialization at t. Here are some properties concerning A-torsors, which will be used in the subsequent sections.

Lemma 1.1.4. (1) Let T be an A-torsor. For s,t,u T and a A, we have the following equality in A: ∈ ∈ s s =0, s u =(s t)+(t u), s.a t =(s t)+ a. − − − − − − (2) Let T, T ′ be A-torsors and let f : T T ′ be a morphism of A-torsors. → Then, for s, t T , we have the following equality in A: ∈ s t = f(s) f(t). − − (3) Let T, T ′ be A-torsors and let f : T T ′ be a morphism of A-torsors. Fix → t T and t′ T ′, and let λ(f; t, t′) := f(t) t′. Then we have the following ∈ ∈ − commutative diagram: f T T ′ −→ ϕt ϕt′ ↓+λ(f;t,t′) ↓ A A. −→ For other choices s T and s′ T ′, we have ∈ ∈ λ(f; s,s′)= λ(f; t, t′)+(s t) (s′ t′). − − − (4) For an A-torsor T and a subgroup B of A, we note that the quotient set T/B is an A/B-torsor by (t mod B).(a mod B) := (t.a mod B) for t T and a A. ∈ ∈ Proof. (1) These equalities follow from the deﬁnition of group action and (1.1.1). (2) This follows from (1.1.1) and (1.1.2). (3) The former assertion follows from (1.1.3). For the latter assertion, we note the following commutative diagram.

id f id T T T ′ T ′ −→ −→ −→ ϕs ϕt ϕt′ ϕs′ ↓ +(s t) ↓ +λ(f;t,t′) ↓ (s′ t′) ↓ A − A A − − A. −→ −→ −→ 12 Since the composite map in the lower row is +λ(f; s,s′) by the former assertion, the latter assertion follows. (4) This is easily seen. ✷

1.2. Conjugate action on group cochains. Let Π be a profinite group and let M be an additive discrete group on which Π acts continuously from the left. Let Cn(Π, M) (n 0) be the group of continuous n-cochains of Π with coefficients in M and let≥ dn+1 : Cn(Π, M) Cn+1(Π, M) be the coboundary homomorphisms defined by → (1.2.1) n+1 n n (d α )(γ1, . . . , γn+1) := γ1α (γ2, . . . , γn+1) n i n + ( 1) α (γ1, . . . , γi 1, γiγi+1, γi+2, . . . , γn+1) − − i=1 +(X1)n+1αn(γ , . . . , γ ) − 1 n n n n n+1 for α C (Π, M) and γ1,...,γn+1 Π. Let Z (Π, M) := Ker(d ) and Bn(Π, M∈) := Im(dn) be the subgroups∈ of Cn(Π, M) consisting of n-cocycles and n-coboundaries, respectively, and let Hn(Π, M) := Zn(Π, M)/Bn(Π, M), the n-th cohomology group of Π with coefficients in M. By convention, we put Cn(Π, M) = 0 for n < 0. We sometimes write d for dn simply if no misunderstanding is caused. Note that Π acts on Cn(Π, M) from the left by n n 1 1 (1.2.2) (σ.α )(γ1,...,γn) := σα (σ− γ1σ,...,σ− γnσ) for αn Cn(Π, M) and σ, γ ,...,γ Π. By (1.2.1) and (1.2.2), we see that ∈ 1 n ∈ this action commutes with the coboundary homomorphisms: (1.2.3) dn+1(σ.αi)= σ.dn+1(αi) (αi Ci(Π, M)). ∈ Now we shall describe the action of Π on Cn(Π, M) in a concrete manner. For σ, σ1, σ2 Π, 0 i j n (n 1), and 1 k n 1, we define ∈ n ≤ ≤n ≤ n+1 ≥ n ≤ ≤ n− n+2 the maps si = si (σ) : Π Π , si,j = si,j(σ1, σ2) : Π Π and n n n 1 → → t = t : Π Π − by k k → 1 1 si(g1,g2,...,gn) := (g1,...,gi, σ, σ− gi+1σ,...,σ− gnσ), 1 1 si,j(g1,g2,...,gn) := (g1,...,gi, σ1, σ1− gi+1σ1,...,σ1− gjσ1, (1.2.4) 1 1 σ2, (σ1σ2)− gj+1σ1σ2,..., (σ1σ2)− gnσ1σ2), tk(g1,g2,...,gn) := (g1,...,gk 1,gkgk+1,gk+2,...,gn) − for (g ,g ,...,g ) Πn, and define the homomorphisms 1 2 n ∈ hn : Cn+1(Π, M) Cn(Π, M), σ −→ Hn : Cn+2(Π, M) Cn(Π, M) σ1,σ2 −→ 13 by

hn(αn+1) := ( 1)i(αn+1 sn(σ)), σ − ◦ i 0 i n (1.2.5) X≤ ≤ Hn (αn+2) := ( 1)i+j(αn+2 sn (σ , σ )) σ1,σ2 − ◦ i,j 1 2 0 i j n ≤X≤ ≤ for αn+1 Cn+1(Π, M) and αn+2 Cn+2(Π, M). For example, explicit forms n n+1∈ n n+2 ∈ of hσ(α ),Hσ1,σ2 (α ) for n =1, 2 are given as follows:

1 2 2 1 2 hσ(α )(g)= α (σ, σ− gσ) α (g, σ). 2 3 3 1− 1 3 1 3 hσ(α )(g1,g2)= α (σ, σ− g1σ, σ− g2σ) α (g1, σ, σ− g2σ)+ α (g1,g2, σ). 1 3 3 1 − 3 1 3 Hσ1,σ2 (α )(g)= α (σ1, σ2, (σ1σ2)− gσ1σ2) α (σ1, σ1− gσ1, σ2)+ α (g, σ1, σ2) 2 4 4 1 − 1 Hσ1,σ2 (α )(g1,g2)= α (σ1, σ2, (σ1σ2)− g1σ1σ2, (σ1σ2)− g2σ1σ2) 4 1 1 4 1 1 α (σ1, σ1− g1σ1, σ2, (σ1σ2)− g2σ1σ2)+ α (σ1, σ1− g1σ1, σ1− g2σ1, σ2) −4 1 4 1 4 +α (g , σ , σ , (σ σ )− g σ σ ) α (g , σ , σ− g σ , σ )+ α (g ,g , σ , σ ) 1 1 2 1 2 2 1 2 − 1 1 1 2 1 2 1 2 1 2 The following Theorem 1.2.6 and Corollary 1.2.7 were shown in Appendices A and B of [Ki]. Here we give an elementary direct proof. See also Remark 1.2.8 below for the background of the proof.

Theorem 1.2.6. Notations being as above, we have the following equalities.

n n n n+1 n n n 1 n σ.α α = h (d (α )) + d (h − (α )), − σ σ n n+1 n n+1 n n+1 n n+2 n+1 n n 1 n+1 σ .h (α ) h (α )+h (α )= H (d (α )) d (H − (α )). 1 σ2 − σ1σ2 σ1 σ1,σ2 − σ1,σ2 for αn Cn(Π, M) and αn+1 Cn+1(Π, M) (n 1). ∈ ∈ ≥ Proof. By (1.2.4), we can see

tk si+1 (k i) si tk = ◦ ≤ ◦ tk+1 si (i

14 By (1.2.1) and (1.2.5), we have, for any (g ,g ,...,g ) Πn, 1 2 n ∈ n n+1 n n hσ(d (α ))(g1,...,gn) =(σ.α )(g1,...,gn) i n + ( 1) g1.(α si 1)(g2,...,gn) 1 i n − ◦ − ≤ ≤ i+k n + P ( 1) (α tk si)(g1,...,gn) 0 i n,1 k n − ◦ ◦ ≤ ≤ n+≤n+1≤ n +( 1)P α (g1,...,gn) − i+n+1 n + ( 1) (α si)(g1,...,gn 1), 0 i n 1 − ◦ − n n 1 n ≤ ≤ − i n d (hσ− (α ))(g1,...,gn) = P ( 1) g1.(α si)(g2,...,gn) 0 i n 1 − ◦ ≤ ≤ − i+k n + P ( 1) (α si tk)(g1,...,gn) 0 i n 1,1 k n 1 − ◦ ◦ ≤ ≤ − ≤ ≤i+−n n + P( 1) (α si)(g1,...,gn 1), 0 i n 1 − ◦ − ≤ ≤ − P and

n n+2 n+1 Hσ1,σ2 (d (α ))(g1,...,gn) n n+1 i+j n+1 = σ1.hσ2 (α )(g1,...,gn)+ ( 1) g1.(α si 1,j 1)(g2,...,gn) 0

n n+1 n n n 1 n hσ(d (α ))(g1,...,gn)+ d (hσ− (α ))(g1,...,gn) n n =(σ.α )(g1,...,gn) α (g1,...,gn) i+k − n + ( 1) (α tk si)(g1,...,gn) 0 i n,1 k n − ◦ ◦ ≤ ≤ ≤ ≤ i+k n + P ( 1) (α si tk)(g1,...,gn), 0 i n 1,1 k n 1 − ◦ ◦ ≤ ≤ − ≤ ≤ − P

15 and

n n+2 n+1 n n 1 n+1 Hσ1,σ2 (d (α ))(g1,...,gn)+ d (Hσ1−,σ2 (α ))(g1,...,gn) n n+1 n n+1 n n+1 = σ1.hσ2 (α )(g1,...,gn) hσ1σ2 (α )(g1,...,gn)+ hσ1 (α )(g1,...,gn) i+j+k n−+1 + ( 1) (α tk si,j)(g1,...,gn) 0

By (1.2.3), Π acts on Zn(Π, M) from the left. This action is described by Theorem 1.2.6 as follows.

Corollary 1.2.7. Suppose α Zn(Π, M) (n 1). For σ Π, we let ∈ ≥ ∈ n 1 βσ := hσ− (α). Then we have n σ.α = α + d βσ.

For σ, σ′ Π, we have ∈ n 1 βσσ′ = βσ + σ.βσ′ mod B − (Π, M),

n 1 n 1 n 1 namely, the map Π σ β mod B − (Π, M) C − (Π, M)/B − (Π, M) is ∋ 7→ σ ∈ a 1-cocycle.

Proof. The both equalities are obtained immediately from Theorem 1.2.6, since dn+1(α) = 0 by α Zn(Π, M) (n 1). ✷ ∈ ≥ Remark 1.2.8 (Algebro-topological proof of Theorem 1.2.6). For σ Π, ∈ let σ• denote the automorphism of the cochain complex (C•(Π, M),d•) defined by σn(α) := σ.α for α Cn(Π, M). Then Theorem 1.2.6 asserts that ∈ the family of homomorphisms hn : Cn+1(Π, M) Cn(Π, M) gives a ho- { σ → } motopy connecting σ• and idC•(Π,M). Actually our explicit definition (1.2.5), (1.2.6) is obtained by making the following algebro-topological proof concrete: We may assume Π is finite by the limit argument. Let be the one-object category whose morphisms are the elements of Π. We considerE two functors 1 id , σ : defined by id (g) := g, σ(g) := σ− gσ for each morphism E E g Π. LetE →E: Cat Fct(∆op, Set) denote the nerve functor, where Cat is ∈ N → op the categoryb of small categories and Fct(∆b , Set) is the category of simplicial

16 sets. Deﬁne the natural transformation η : σ id by η( ) := σ ( is the → E ∗ ∗ unique object of ). Then η induces a corresponding funcor hη : 1 , where n denotesE the category deﬁned by the set 0, 1,...,n andE × its order.→ E b { } Then h : 1 is a homotopy connecting the two simplicial N η NE ×N → NE maps σ, id : . Let Cn( ) = Z[ (n)] be the group of N N E NE → NE NE NE n-chains of the simplicial set . By [My; Proposition 5.3] and [My; Propo- NE σ sition 6.2],b hη induces a homotopy hn : Cn( ) Cn+1( ) connecting the two chainN maps ( σˆ) , ( id ) :{C ( )NEC→( ). ForNE the} groups of n N • N E • • NE → • NE σ n-cochains C ( , M) = Hom(Cn( ), M), the homotopy hn induces the n NE n+1 NEn { } homotopy hσ : C ( , M) C ( , M) connecting the two cochain { NE → NE } n maps ( σ)•, ( id )• : C•( , M) C•( , M). Since (n) is Π , we N N E NE → NE NE have the isomorphisms for i 0 ≥ b Cn( , M) Map(Πn, M)= Cn(Π, M). NE ≃

Under the above isomorphisms, ( σ)• and ( id )• are identiﬁed with σ• and N n N E idC•(Π,M), respectively, and hence hσ gives a homotopy connecting σ• and ✷ { } idC•(Π,M). b

2. Classical theory

In this section, we construct the arithmetic prequantization bundle and r the arithmetic Chern-Simons 1-cocycle for ∂V := Spec(kp ), where S = S ⊔i=1 i p1,..., pr is a finite set S of primes of a finite algebraic number field k, and the{ arithmetic} Chern-Simons functional over a space of Galois representations unramified outside S. These constructions correspond to the classical theory of topological Dijkgraaf-Witten TQFT. Throughout the rest of this paper, we fix a natural number N > 1and let µN be the group of N-th roots of unity in the field C of complex numbers. We fix a primitive N-th root of unity ζ and the isomorphism Z/NZ µ ; m ζm. N ≃ N 7→ N The base number field k (in C) is supposed to contains µN . Let G be a finite group and let c be a fixed 3-cocycle of G with coefficients in Z/NZ, c Z3(G, Z/NZ), where G acts on Z/NZ trivially. ∈ 2.1. Arithmetic prequantization bundles and arithmetic Chern- Simons 1-cocycles. We firstly develop a local theory at a finite prime. Let p be a finite prime of k and let kp be the p-adic field. We let ∂Vp := Spec(kp), which play a role analogous to the boundary of a tubular neighborhood of a

17 knot (see the dictionary of the analogies in Introduction). Let Πp denote the ´etale fundamental group of ∂Vp with base point Spec(kp)(kp being an algebraic closure of kp), which is the absolute Galois group Gal(kp/kp). Let p be the set of continuous homomorphisms of Πp to G: F

p := Hom (Πp,G). F cont It is a ﬁnite set on which G acts from the right by

1 (2.1.1) p G p; (ρp,g) ρp.g := g− ρpg. F × →F 7→

Let p denote the quotient space by this action: M

p := p/G. M F

Let Map( p, Z/NZ) denote the additive group consisting of maps from p to Z/NZ, onF which G acts from the left by F

(2.1.2) (g.ψp)(ρp) := ψp(ρp.g) for g G, ψp Map( p, Z/NZ) and ρp p. For ρp p and α n ∈ ∈ F ∈ F ∈ F ∈ C (G, Z/NZ), we denote by α ρp the n-cochain of Πp with coeﬃcients in ◦ Z/NZ deﬁned by

(α ρp)(γ ,...,γ ) := α(ρp(γ ),...,ρp(γ )). ◦ 1 n 1 n By (1.2.2) and (2.1.1), we have

(2.1.3) (g.α) ρp = α (ρp.g) ◦ ◦ n for g G,α C (G, Z/NZ) and ρp p. ∈ ∈ ∈F Firstly, we shall construct an arithmetic analog for ∂Vp := Spec(kp) of the prequantization bundle, using the given 3-cocycle c Z3(G, Z/NZ). The key ∈ idea for this is due to Kim ([Ki]), who uses the conjugate G-action on c and the 2nd Galois cohomology group (Brauer group) of the local ﬁeld kp. 3 Let ρp p and so c ρp Z (Πp, Z/NZ). Let d denote the coboundary ∈ F 2 ◦ ∈ 3 homomorphism C (Πp, Z/NZ) C (Πp, Z/NZ). We deﬁne p(ρp) by the quotient set → L

1 2 (2.1.4) p(ρp) := d− (c ρp)/B (Πp, Z/NZ). L ◦ 1 Here we note that d− (c ρp) is non-empty, because the cohomological di- ◦ mension of Πp is 2 ([NSW; Theorem 7.1.8], [S1; Chapitre II, 5.3, Proposition

18 3 1 2 15]) and so H (Πp, Z/NZ) = 0. Thus d− (c ρp) is a Z (Πp, Z/NZ)-torsor in 2◦ the obvious manner and so p(ρp) is an H (Πp, Z/NZ)-torsor by (2.1.4) and L 2 2 Lemma 1.1.4 (4). Since kp contains µN and so H (Πp, Z/NZ) = H (kp,µN ), the theory of Brauer groups (cf. [S2; Chapitre XII]) tells us that there is the canonical isomorphism

2 invp : H (Πp, Z/NZ) ∼ Z/NZ −→ and hence p(ρp) is a Z/NZ-torsor via invp. L Let p be the disjoint union of p(ρp) over all ρp p: L L ∈F

p := p(ρp) L L ρp p G∈F and consider the projection

̟p : p p; αp ρp if αp p(ρp). L −→ F 7→ ∈L 1 Since each ﬁber ̟− (ρp) = p(ρp) is a Z/NZ-torsor, we may regard p as a p L L principal Z/NZ-bundle over p. F Let g G. We deﬁne h C2(G, Z/NZ)/B2(G, Z/NZ) by ∈ g ∈ 2 2 hg := hg(c) mod B (G, Z/NZ),

2 where hg(c) is the 2-cochain deﬁned as in (1.2.5), namely,

2 1 1 1 h (c)(g ,g ) := c(g,g− g g,g− g g) c(g ,g,g− g g)+ c(g ,g ,g), g 1 2 1 2 − 1 2 1 2 where g ,g G. By Corollary 1.2.7, we have 1 2 ∈

(2.1.5) g.c = c + dhg and

(2.1.6) hgg′ = hg + g.hg′ for g,g′ G. By (2.1.3), (2.1.4) and (2.1.5), we have ∈

d(α + h ρp)= c ρp +(g.c c) ρp = g.c ρp = c ρp.g g ◦ ◦ − ◦ ◦ ◦ for α p(ρp) and so we have the isomorphism of Z/NZ-torsors ∈L

(2.1.7) fp(g, ρp): p(ρp) ∼ p(ρp.g); αp αp + h ρp. L −→ L 7→ g ◦ 19 By (2.1.3) and (2.1.6), we have

αp + h ′ ρp = αp +(h + g.h ′ ) ρp gg ◦ g g ◦ = α + h ρp + h ′ (ρp.g) g ◦ g ◦ for g,g′ G. It means that G acts on p from the right by ∈ L

(2.1.8) p G p; αp αp.g := f(g, ρp)(αp). L × →L 7→

By (2.1.7), (2.1.8) and the way of the Z/NZ-action on p, we have the following L commutative diagram

.g p p x Z/NZ L −→ L ̟p ̟p ↓.g ↓ p p, F −→ F namely,

(2.1.9) (αp.m).g =(αp.g).m, ̟p(αp.g)= ̟p(αp).g for αp p, m Z/NZ,g G. So p is a G-equivariant principal Z/NZ- ∈ F ∈ ∈ L bundle over p. Taking the quotient by the action of G, we have the principal F Z/NZ-bundle ̟p : p p. We call ̟p : p p or ̟p : p p the L → M L → F L → M arithmetic prequantization Z/NZ-bundle for ∂Vp := Spec(kp). Let us choose a section xp Γ( p, p), namely, the map ∈ F L

xp : p p such that ̟p xp = id p . F −→ L ◦ F

This means that we ﬁx a “coordinate” on p. In fact, by the trivialization at L xp(ρp) in (1.1.3), we may identify each ﬁber p(ρp) over ρp with Z/NZ: L

ϕ : p(ρp) ∼ Z/NZ; αp αp xp(αp). xp(ρp) L −→ 7→ −

For g G and ρp p, we let ∈ ∈F xp (2.1.10) λ (g, ρp) := fp(g, ρp)(xp(ρp)) xp(ρp.g)= xp(ρp).g xp(ρp.g) p − − so that we have the following commutative diagram by Lemma 1.1.4 (3):

fp(g,ρp) p(ρp) p(ρp.g) L −→ L ϕxp(ρp) ϕxp(ρp.g) ↓xp ↓ +λ (g,ρp) Z/NZ p Z/NZ, −→ 20 namely, for αp p(ρp), we have ∈L xp (2.1.11) αp.g xp(ρp.g)=(αp xp(ρp)) + λ (g, ρp). − − p xp We deﬁne the map λ : G Map( p, Z/NZ) by p → F xp xp (2.1.12) λp (g)(ρp) := λp (g, ρp) for g G and ρp p. ∈ ∈F

Theorem 2.1.13. For g,g′ G, we have ∈ xp xp xp λp (gg′)= λp (g)+(g.λp )(g′).

xp Namely, the map λp is a 1-cocycle:

xp 1 λ Z (G, Map( p, Z/NZ)). p ∈ F

Proof. For g,g′ G and ρp p, we have ∈ ∈F

λp(gg′, ρp) = fp(gg′, ρp)(xp(ρp)) xp(ρp(gg′)) by (2.1.10) − =(xp(ρp)+ h ′ ρp) xp(ρp.(gg′)) by (2.1.7) gg ◦ − =(xp(ρp)+ h ρp + h ′ (ρp.g)) xp(ρp.(gg′)) by (2.1.3), (2.1.6). g ◦ g ◦ − By Lemma 1.1.4 (1), we have

(xp(ρp)+ h ρp + h ′ (ρp.g)) xp(ρp.(gg′)) g ◦ g ◦ − = (xp(ρp)+ h ρp) xp(ρp.g) + (xp(ρp.g)+ h ′ (ρp.g)) xp(ρp.(gg′)) . { g ◦ − } { g ◦ − } Here we see by (2.1.7), (2.1.10) that

xp (xp(ρp)+ hg ρp) xp(ρp.g)= λp (g, ρp), ◦ − xp (xp(ρp.g)+ h ′ (ρp.g)) xp(ρp.(gg′)) = λ (g′, ρp.g). g ◦ − p Combining these, we have

xp xp xp λp (gg′, ρp)= λp (g, ρp)+ λp (g′, ρp.g) for any ρp p. By (2.1.2) and (2.1.12), we obtain the assertion. ✷ ∈F

xp We call λp the Chern-Simons 1-cocycle for ∂Vp with respect to the section xp.

21 xp For a section xp Γ( p, p), we deﬁne by the product (trivial) princi- ∈ F L Lp pal Z/NZ-bundle over p: F xp := p Z/NZ, Lp F × on which G acts from the right by

xp xp xp (2.1.14) G ; ((ρp, m),g) (ρp.g, m + λ (g, ρp)), Lp × →Lp 7→ p and so the projection xp xp ̟ : p p Lp −→ F is G-equivariant.

Proposition 2.1.15. We have the following isomorphism of G-equivariant principal Z/NZ-bundles

xp xp Φ : p ∼ ; αp (̟p(αp),αp xp(̟p(αp))). p L −→ Lp 7→ − In particular, the isomorphism class of xp is independent of the choice of Lp a section xp. In other words, for another section xp′ Γ( p, p), we have x′ ∈ F L p xp as G-equivariant principal Z/NZ-bundles. Lp ≃Lp

xp xp Proof. (i) It is easy to see that ̟ Φ = ̟p. p ◦ p (ii) For αp p and m Z/NZ, we have ∈F ∈ xp Φ (αp.m) =(̟p(αp.m),αp.m xp(̟p(αp.m))) p − =(̟p(αp),αp.m xp(̟p(αp))) − =(̟p(αp), (αp xp(̟p(αp))) + m) by Lemma 1.1.4 (1) xp − = Φp (αp).m.

xp xp 1 (iii) Φ has the inverse deﬁned by (Φ )− ((ρp, m)) := xp(ρp).m for (ρp, m) p p ∈ p Z/NZ. F × xp By (i), (ii), (iii), Φp is an isomorphism of principal Z/NZ-bundles. So it suﬃces to show that Φxp is G-equivarinat. It follows from that

xp Φp (αp.g) =(̟p(αp.g),αp.g xp(̟p(αp.g))) − xp =(̟p(αp).g, (αp xp(ρp)) + λp (g, ρp)) by (2.1.9), (2.1.11) xp − = Φp (α).g by (2.1.14) ✷

Taking the quotient of ̟p : p p by the action of G, we have the principal L →F 22 xp xp xp xp Z/NZ-bundle ̟ : p. We call ̟p : p p or ̟ : p p Lp → M L → F p Lp → M the arithmetic prequantization principal Z/NZ-bundle for ∂Vp with respect to the section xp.

′ xp,xp For xp, x′ Γ( p, p), we deﬁne the map δ : p Z/NZ by p ∈ F L p F → ′ xp,xp (2.1.16) δ (ρp) := xp(ρp) x′ (ρp) p − p for ρp p. ∈F

Lemma 2.1.17 For xp, x′ , x′′ Γ( p, p), we have p p ∈ F L x′ ,x x ,x′ x ,x′ x′ ,x′′ x ,x′′ δxp,xp =0, δ p p = δ p p =0, δ p p + δ p p = δ p p . p p − p p p p

Proof. These equalities follow from Lemma 1.1.4 (1). ✷

xp The following proposition tells us how λp is changed when we change the section xp.

Proposition 2.1.18. For xp, x′ Γ( p, p), we have p ∈ F L x′ x ,x′ x ,x′ λ p (g) λxp (g)= g.δ p p δ p p p − p p − p xp 1 for any g G. So the cohomology class [λ ] H (G, Map( p, Z/NZ)) is ∈ p ∈ F independent of the choice of a section xp.

Proof. By (2.1.10) and Lemma 1.1.4 (1), (2), we have

′ xp xp λ (g, ρp) λ (g, ρp) =(fp(g, ρp)(x′ (ρp)) x′ (ρp.g)) (fp(g, ρp)(xp(ρp)) xp(ρp.g)) p − p p − p − − =(xp(ρp.g) x′ (ρp.g))+(fp(g, ρp)(x′ (ρp)) fp(g, ρp)(xp(ρp))) − p p − =(xp(ρp.g) xp′ (ρp.g))+(xp′ (ρp) xp(ρp)) ′ ′ xp,xp − xp,xp − =(g.δ )(ρp) δ (ρp) by (2.1.2) p − p for any g G and ρp p, hence the assertion. ✷ ∈ ∈F

xp By Proposition 2.1.18, we denote the cohomology class [λp ] by [λp], which we call the arithmetic Chern-Simons 1st cohomology class for ∂Vp. As a corollary of Proposition 2.1.18, we can make the latter statement of Proposition 2.1.15 more precise as follows.

23 Corollary 2.1.19. (1) For xp, x′ Γ( p, p), we have the following iso- p ∈ F L morphism of G-equivariant principal Z/NZ-bundles over p: F ′ ′ ′ xp,xp xp xp xp,xp Φ : ∼ ; (ρp, m) (ρp, m + δ (ρp)), p Lp −→ Lp 7→ p ′ xp,xp where δ : p Z/NZ is the map deﬁned in (2.1.16). p F → (2) For xp, x′ , x′′ Γ( p, p), we have p p ∈ F L ′ ′ xp,xp xp xp Φp Φp = Φp , ◦ ′ ′ ′ ′′ ′ ′′ xp,xp xp,xp xp,xp 1 xp,xp xp,xp xp,xp ( Φp = id xp , Φp = (Φp )− , Φp Φp = Φp Lp ◦ ′ xp,xp Proof. (1) We easily see that Φp is isomorphism of principal Z/NZ-bundles ′ xp,xp and so it suﬃces to show that Φp is G-equivariant. This follows from

′ ′ xp,xp xp,xp xp Φp ((ρp, m).g) = Φp ((ρp.g, m + λp (g, ρp))) by (2.1.14) ′ xp xp,xp =(ρp.g, m + λp (g, ρp)+ ηp (ρp.g)) ′ ′ xp,xp xp =(ρp.g, m + ηp (ρp)+ λp (g, ρp)) by Proposition 2.1.18 ′ xp,xp = Φp (ρp, m).g.

′ xp xp,xp (2) The ﬁrst equality follows from the deﬁnitions of Φp , Φp . The latter equalities follow from Lemma 2.1.17. ✷

Let F be a ﬁeld containing µ . Let Lp be the F -line bundle over p N F ֒ associated to the principal Z/NZ-bundle p and the homomorphism Z/NZ m L → F ×; m ζN , namely, (2.1.20)7→ Lp := p Z/NZ F L × m := ( p F )/(αp, z) (αp.m, ζ− z) (αp p, m Z/NZ, z F ), L × ∼ N ∈L ∈ ∈ on which G acts from the right by

(2.1.21) Lp G Lp; ([(αp, z)],g) [(α.g, z)]. × → 7→ The projection ̟p : Lp p; [(αp, z)] ̟p(α) ,F −→ F 7→ 1 is a G-equivariant F -line bundle. We denote the ﬁber ̟p−,F (ρp) over ρp by Lp(ρp):

(2.1.22) Lp(ρp) := [(αp, z)] Lp ̟p(αp)= ρp, z F { ∈ | ∈ } 24 We have a non-canonical bijection by ﬁxing an αp p(ρp): ∈L

Lp(ρp) ∼ F ; [(αp, z)] z. −→ 7→

Taking the quotient by the action of G, we obtain the F -line bundles ̟p,F : Lp p. We call ̟p : Lp p or ̟p : Lp p the arithmetic → M ,F → F ,F → M prequantization F -line bundle for ∂Vp. xp Let L be the product F -line bundle over p: p F xp L := p F, p F × on which G acts from the right by

xp xp xp λp (g,ρp) (2.1.23) L G L ; ((ρp, z),g) (ρp.g, zζ ), p × → p 7→ N and the projection xp xp ̟ : L p p,F p −→ F is G-equivariant. Then we have the following Proposition similar to Proposi- tion 2.1.13 and Corollary 2.1.17.

Proposition 2.1.24. We have the following isomorphism of G-equivariant F -line bundles over p F xp xp αp xp(̟p(αp)) Φ : Lp ∼ L ; [(αp, z)] (̟p(αp),zζ − ). p,F −→ p 7→ N

For another section xp′ , we have the following isomorphism of G-equivariant F -line bundles over p F ′ ′ ′ xp,xp xp,xp xp xp ηp (ρp) Φ : L ∼ L :(ρp, z) (ρp,zζ ), p,F p −→ p 7→ N ′ xp,xp where δ : p Z/NZ is the map in (2.1.16), and we have the equalities p F → ′ ′ xp,xp xp Φp,F Φp,F xp = Φp,F ′ ′ ′ ′′ ′ ′′ xp,xp ◦ xp,xp xp,xp xp,xp xp,xp xp,xp x 1 ( Φp,F = idL p , Φp,F = (Φp,F )− , Φp,F Φp,F = Φp,F p,F ◦ for xp, x′ , x′′ Γ( p, p). p p ∈ F L

xp xp Proof. (i) It is easy to see that ̟ Φ = ̟p . p,F ◦ p,F ,F (ii) For ρp p, we let ∈F xp xp 1 L (ρp) := (̟ )− (ρp)= (ρp, z) z F F. p p,F { | ∈ } ≃ 25 xp So Φp,F restricted to a ﬁber over ρp

xp xp αp xp(ρp) Φ : Lp(ρp) L (ρp); [(αp, z)] (ρp,zζ − ) p,F |Lp(ρp) −→ p 7→ N is F -linear. (iv) For g G, we have ∈ xp xp Φp ([(αp, z)].g) = Φp ([(αp.g, z)]) by (2.1.21) αp.g xp(̟p(αp.g)) =(̟p(αp.g),zζN − ) xp (αp xp(ρp))+λ (g,ρp) =(̟p(αp).g, zζ − p ) by (2.1.11) xp = Φp,F ([αp, z)]).g by (2.1.22).

xp Hence Φp,F is the isomorphism of G-equivariant F -line bundles over p. The proofs of the latter parts are similar to those of Corollary 2.1.19F (1), (2). ✷

xp xp Taking the quotient of ̟p,F : Lp p by the action of G, we have the F -line xp xp →Fxp xp xp xp bundle ̟ : L p. We call ̟ : L p or ̟ : L p the p,F p → M p,F p → F p,F p → M arithmetic prequantization F -line bundle for ∂Vp with respect to the section xp.

Let S = p ,..., p be a ﬁnite set of ﬁnite primes of k and let ∂V := { 1 r} S ∂Vp ∂Vp . Let be the direct product of p ’s: 1 ⊔···⊔ r FS F i

:= p p . FS F 1 ×···×F r It is a ﬁnite set on which G acts diagonally from the right, namely,

(2.1.25) G ; (ρ ,g) ρ .g := (ρp .g, . . . , ρp .g) FS × →FS S 7→ S 1 r for ρS = (ρp1 ,...,ρpr ) S and let S denote the quotient space by this action ∈ F M := /G. MS FS Let Map( S, Z/NZ) be the additive group of maps from S to Z/NZ, on which G actsF from the left by F

(2.1.26) (g.ψS)(ρS) := ψS(ρS.g) for ψ Map( , Z/NZ),g G and ρ . S ∈ FS ∈ S ∈FS For ρ = (ρp ,...,ρp ) , let (ρ ) be the quotient space of the S 1 r ∈ FS LS S product p (ρp ) p (ρp ): L 1 1 ×···×L r r

(2.1.27) (ρ ) := ( p (ρp ) p (ρp ))/ , LS S L 1 1 ×···×L r r ∼ 26 where the equivalence relation is deﬁned by ∼ r

(2.1.28) (αp ,...,αp ) (α′ ,...,α′ ) (αp α′ )=0. 1 r ∼ p1 pr ⇐⇒ i − pi i=1 X

We see easily that S(ρS) is equipped with the simply transitive action of Z/NZ deﬁned by L

(ρ ) Z/NZ (ρ ); LS S × −→ LS S ([α ], m) [α ].m := [(αp .m, . . . , αp )] = = [(αp ,...,αp .m)] S 7→ S 1 r ··· 1 r for α =(αp ,...,αp ) and hence (ρ ) is a Z/NZ-torsor. S 1 r LS S Let the disjoint union of p(ρ ) for ρ : LS L S S ∈FS (2.1.29) := (ρ ), LS LS S ρS S G∈F on which G acts diagonally from the right by

(2.1.30) G ; ([(αp ,...,αp )],g) [(αp .g, . . . , αp .g)]. LS × −→ LS 1 r 7→ 1 r Consider the projection

̟ : ; α =(αp ) (̟p (αp )), S LS −→ FS S i 7→ i i 1 which is G-equivariant. Since each ﬁber ̟p− (ρS)= S(ρS) is a Z/NZ-torsor, we may regard ̟ : as a G-equivariantL principal Z/NZ-bundle. S LS −→ FS Taking the quotient by the action of G, we have the principal Z/NZ-bundle ̟ : . We call ̟ : or ̟ : the arithmetic S LS → MS S LS → FS S LS → MS prequantization Z/NZ-bundle for ∂V = Spec(kp ) Spec(kp ). S 1 ⊔···⊔ r Let x be a section of ̟ , x Γ( , ). By (2.1.27) and (2.1.29), it S S S ∈ FS LS is written as x = [(xp ,...,xp )], where xp Γ( p , p ) for 1 i r. For S 1 r i ∈ F i L i ≤ ≤ g G and ρ =(ρp ) , we set ∈ S i ∈FS xS xp1 xpr (2.1.31) λ (g, ρ ) := λ (g, ρp )+ + λ (g, ρp ) S S p1 1 ··· pr r and deﬁne the map λxS : G Map( , Z/NZ) by S → FS xS xS (2.1.32) λS (g)(ρS) := λS (g, ρS) for g G and ρ . ∈ S ∈FS

27 Lemma 2.1.33. (1) Let x′ Γ( p , p ) be another section for 1 i r pi ∈ F i L i ≤ ≤ such that [(xp′ 1 ,...,xp′ r )] = xS . Then we have

r r x′ xpi pi λpi (g, ρpi )= λpi (g, ρpi ) i=1 i=1 X X xS for g G and ρp p . So λ (g, ρ ) is independent of the choic of xp ’s ∈ i ∈ F i S S i such that xS = [(xp1 ,...,xpr )]. xS (2) The map λS is a 1-cocycle:

λxS Z1(G, Map( , Z/NZ)). S ∈ FS

Proof. (1) Since (xp (ρp ),...,xp (ρp )) (x′ (ρp ),...,x′ (ρp )), by (2.1.28), 1 1 r r ∼ p1 1 pr r we have r

(xp (ρp ) x′ (ρp ))=0 i i − pi i i=1 X for any ρp p . Therefore we have i ∈F i r r xpi λ (g, ρp ) = (fp (g, ρp )(xp (ρp )) xp (ρp .g)) by (2.1.10) pi i i i i i − i i i=1 i=1 X Xr = ((fp (g, ρp )(xp (ρp )) fp (g, ρp )(x′ (ρp ))) i i i i − i i pi i i=1 Xr + (fp (g, ρp )(x′ (ρp )) x′ (ρp .g)) i i pi i − pi i i=1 Xr + (x′ (ρp .g) xp (ρp .g)) by Lemma 1.1.4 (1) pi i − i i i=1 Xr = (fp (g, ρp )(x′ (ρp )) x′ (ρp .g)) by Lemma 1.1.4 (2) i i pi i − pi i i=1 Xr x′ pi = λpi (g, ρpi ) i=1 X for g G and ρp p . ∈ i ∈F i

28 (2) By Theorem 2.1.13 and (2.1.31), we have

r xS xpi λS (gg′, ρS) = λpi (gg′, ρpi ) i=1 Xr r xpi xpi = λpi (g, ρpi )+ λpi (g′, ρpi .g) i=1 i=1 XxS xS X = λS (g, ρS)+ λS (g′, ρS.g) xS xS =(λS (g)+(g.λS )(g′))(ρS) for g G and ρ =(ρp ) . By (2.1.26) and (2.1.32), we obtaine ∈ S i ∈FS xS xS xS ✷ λS (gg′)= λS (g)+(g.λS )(g′)

We call λS the arithmetic Chern-Simons 1-cocycle for ∂VS with respect to xS.

Proposition 2.1.34. Let xS′ = [(xp′ 1 ,...,xp′ r )] Γ( S, S) be another section ′ ∈ F L of ̟ . We deﬁne the map δxS,xS : Z/NZ by S S FS → r ′ x ,x′ xS,xS pi pi δS (ρS) := δpi (ρpi ) i=1 X x ,x′ pi pi for ρ =(ρp ) , where δ is the map deﬁned (2.1.16). Then we have S i ∈FS pi ′ ′ ′ λxS (g) λxS (g)= g.δxS,xS δxS ,xS S − S S − S for g G. So the cohomology class [λxS ] H1(G, Map( , Z/NZ)) is inde- ∈ S ∈ FS pendent of the choice of xS.

′ xS,xS Proof. First, note that δS is proved to be independent of the choices of xpi ’s in the similar manner to the proof of Lemma 2.1.33 (1). By the deﬁnition ′ xS,xS δS , the formula follows from Proposition 2.1.18 by taking the sum over p S. ✷ i ∈

xS We denote the cohomology class [λS ] by [λS], which we call the arithmetic Chern-Simons 1st cohomology class for ∂VS . Let xS be the product principal Z/NZ-bundle over : LS FS xS := Z/NZ, LS FS × 29 on which G acts from the right by xS G xS ; ((ρ , m),g) (ρ .g, m + λxS (g, ρ )). LS × →LS S 7→ S S S

Proposition 2.1.35. We have the following isomorphism of G-equivariant principal Z/NZ-bundles over : FS r xS xS Φ : ∼ ;[α ] = [(αp ,...,αp )] (̟ ([α ]), (αp xp (̟p (αp ))). S LS −→ LS S 1 r 7→ S S i − i i i i=1 X For another section xS′ , we have the following isomorphism of G-equivariant F -line bundles over FS ′ ′ ′ xS ,x x x xS,x Φ S : S ∼ S :(ρ , m) (ρ , m + δ S (ρ )), S LS −→ LS S 7→ S S S ′ xS,xS where δS : S Z/NZ is the map in Proposition 2.1.34. For xS, xS′ , xS′′ Γ( , ) weF have→ the equalities ∈ FS LS ′ ′ xS,xS xS xS ΦS ΦS = ΦS , ′ ′ ′ ′′ ′ ′′ x ,x ◦ x ,xS xS ,x x ,x xS ,x xS ,x S S x S S 1 S S S S ( ΦS = id S , ΦS = (ΦS )− , ΦS ΦS = ΦS . LS ◦

Proof. First, suppose [(αp1 ,...,αpr )] = [(αp′ 1 ,...,αp′ r )]. Then ̟pi (αpi ) = r ̟p (α′ ) and (α′ αp ) = 0 by (2.1.28). So we have i pi i=1 pi − i r P r (α′ xp (̟p (α′ ))) = (α′ αp )+(αp xp (̟p (α′ ))) pi − i i pi pi − i i − i i pi i=1 i=1 X Xr = (αp xp (̟p (αp ))). i − i i i i=1 X The proofs of the assertions go well in the similar manner to those of Propo- sition 2.1.15 and Corollary 2.1.19, by taking the sum over p S. ✷ i ∈ Taking the quotient by the action of G, we obtain the principal Z/NZ-bundle x x ̟xS : S . We call ̟xS : xS or ̟xS : S the arith- S LS → MS S LS → FS S LS → MS metic prequantization principal Z/NZ-bundle for ∂VS with respect to xS.

Let LS be the F -line bundle associated to the principal Z/NZ-bundle S m L over S and the homomorphism Z/NZ F ×; m ζN : (2.1.36)F → 7→ LS := S Z/NZ F L × m := ( F )/([α ], z) ([α ].m, ζ− z) ([α ] , m Z/NZ, z F ), LS × S ∼ S N S ∈LS ∈ ∈ 30 on which G acts from the right by

(2.1.37) L G L ; ([([α ], z)],g) [([α ].g, z)]. S × −→ S S 7→ S The projection

̟ : L ; [([α ], z)] ̟ ([α ]) S,F S −→ FS S 7→ S S 1 is a G-equivariant F -line bundle. We denote the ﬁber ̟p−,F (ρS) over ρS by L (ρ ), which is non-canonically bijective to F by ﬁxing [α ] (ρ ): S S S ∈LS S (2.1.38) L (ρ ) := [([α ], z)] L ̟ ([α ]) = ρ ∼ F ; [([α ], z)] z. S S { S ∈ S | S S S } → S 7→

Taking the quotient by the action of G, we obtain the F -line bundle ̟S,F : L . We call ̟ : L or ̟ : L the arithmetic S → MS S,F S → FS S,F S → MS prequntization F -line bundle for ∂VS . Let LxS be the trivial F -line bundle over : S FS LxS := F, S FS × on which G acts from the right by

xS LxS G LxS ; ((ρ , z),g) (ρ .g, zζλS (g,ρS )). S × → S S 7→ S N

Proposition 2.1.39. We have the following isomorphism of G-equivariant F -line bundles over : FS r x x P =1(αp xp (̟p (αp ))) Φ S : L ∼ L S ; [(α , z)] (̟ (α ),zζ i i − i i i ) S,F S −→ S S 7→ S S N

For another section xS′ , we the following isomorphism of G-equivariant F -line bundles over FS ′ ′ ′ xS,xS xS ,x x x δ (ρS ) Φ S : L S ∼ L S : [(ρ , z)] [(ρ ,zζ S )], S,F S −→ S S 7→ S N ′ xS,xS where δS : S Z/NZ is the map in Proposition 2.1.25. For xS, xS′ , xS′′ Γ( , ), weF have→ the equalities ∈ FS LS ′ ′ xS,xS xS xS ΦS,F ΦS,F = ΦS,F , ′ ′ ′ ′′ ′ ′′ x ,x ◦ x ,xS xS ,x x ,x xS ,x xS ,x S S x S S 1 S S S S ( ΦS,F = id S , ΦS,F = (ΦS,F )− , ΦS,F ΦS,F = ΦS,F . LS ◦

31 Proof. The assertions can be proved in the similar manner to those of the assertions in Proposition 2.1.20, by taking the sum over p S. ✷ i ∈

xS Taking the quotient by the action of G, we obtain the line F -bundle ̟S,F : x x L S . We call ̟xS : LxS or ̟xS : L S the arithmetic S → MS S,F S → FS S,F S → MS prequantization line F -bundle for ∂VS with respect to xS.

We also give the description of LS in terms of the tensor product of F -line bundles. Let p : p be the i-th projection. Let p∗(Lp ) be the F -line i FS → F i i i bundle over induced from Lp by p : FS i i

p∗(Lp ) := (ρ , [(αp , z )]) Lp p (ρ )= ̟p (αp ) , i i { S i i ∈FS × i | i S i i } and let p∗(̟p ): p∗(Lp ) ; (ρ , [(αp , z )]) ρ i i i i −→ FS S i i 7→ S be the induced projection. The ﬁber over ρS =(ρpi ) is given by

1 p∗(̟p )− (ρ ) = ρ [(αp , z )] Lp ρp = ̟p (αp ), z F i i S { S}×{ i i ∈ i | i i i i ∈ } = Lpi (ρpi ) F, ≃ where Lp (ρp ) is as in (2.1.22). Let Lp ⊠ ⊠ Lp be the tensor product of i i 1 ··· r pi∗(Lpi )’s: Lp ⊠ ⊠ Lp := p∗(Lp ) p∗(Lp ), 1 ··· r 1 1 ⊗···⊗ r r which is an F -line bundle over . An element of Lp ⊠ ⊠ Lp is written by FS 1 ··· r

(ρ , [(αp , z )] [(αp , z )]), S 1 1 ⊗···⊗ r r ⊠ where ρ = (ρp ) , [(αp , z )] Lp (ρp ). Let ̟ : Lp ⊠ ⊠ Lp S i ∈ FS i i ∈ i i S 1 ··· r → FS be the projection. For ﬁber over ρS, we have

r ⊠ 1 (2.1.40) (̟ )− (ρ ) ∼ F ;(ρ , [(αp , z )] [(αp , z )]) z . S S → S 1 1 ⊗···⊗ r r 7→ i i=1 Y ⊠ ⊠ The right action of G on Lp1 Lpr is given by (2.1.41) ··· Lp ⊠ ⊠ Lp G Lp ⊠ ⊠ Lp ; 1 ··· r × → 1 ··· r ((ρ , [(αp , z )] [(αp , z )]),g) (ρ .g, [(αp .g, z )] [(αp .g, z )]). S 1 1 ⊗···⊗ r r 7→ S 1 1 ⊗···⊗ r r 32 ⊠ The projection ̟S is G-equivariant.

Proposition 2.1.42. We have the following isomorphism of G-equivariant F -line bundles over FS ⊠ ⊠ ⊠ ∼ ΦS,F : Lp1 Lpr LS; ··· −→ r (ρ , [(αp , z )] [(αp , z )]) [(α , z )], S 1 1 ⊗···⊗ r r 7→ S i=1 i where ρ =(ρp ) , [(αp , z )] Lp (ρp ), and α = [(Qαp ,...,αp )]. S i ∈FS i i ∈ i i S 1 r

mi r Proof. If (αpi , zi) is changed to (αpi .mi,ζN− zi) for mi Z/NZ,(αS, i=1 zi) is mi r r ∈ changed to (αS.mi,ζN− i=1 zi) (αS, i=1 zi). So, by (2.1.20) and (2.1.36), ⊠ ∼ Q ΦS,F is well-deﬁned. QxS ⊠ Q⊠ (i) It is easy to see that ̟S ΦS,F = ̟S . ⊠ ◦ (ii) By (2.1.40), ΦS,F restricted to a ﬁber over ρS is F -linear. ⊠ (iii) By (2.1.30), (2.1.37) and (2.1.41), we see that ΦS,F is G-equivariant. ⊠ Therefore ΦS,F is a morphism of G-equivariant F -line bundles over S. The inverse is given by F

⊠ 1 (Φ )− : L ∼ Lp ⊠ ⊠ Lp ; S,F S −→ 1 ··· r (α , z) (̟ (α ), [(αp , z)] [(αp , 1)] [(αp , 1)]), S 7→ S S 1 ⊗ 2 ⊗···⊗ r ⊠ ✷ Hence ΦS,F is a G-equivariant isomorphism.

2.2. Arithmetic Chern-Simons functionals. Let be the ring of inte- Ok gers of k. Let X := Spec( ) and let X∞ denote the set of infinite primes of k Ok k k. We set X := X X∞. Let S = p ,..., p be a finite set of finite primes k k ⊔ k { 1 r} of k. Let X := X S. We denote by Π the modified étale fundamental S k \ S group of XS with geometric base point Spec(k)(k being a fixed algebraic closure of k), which is the Galois group of the maximal subextension kS of k over k, unramified outside S (cf. [H; 2.1]). We assume that all maximal ideals of dividing N are contained in S (in particular, S is non-empty). Ok Let denote the set of continuous representations of Π to G: FXS S := Hom (Π ,G), FXS cont S on which G acts from the right by

1 (2.2.1) G ; (ρ, g) ρ.g := g− ρg, FXS × →FXS 7→ and let denote the quotient set by this action: MXS := /G. MXS FXS 33 Let Map( XS , Z/NZ) be the additive group of maps from XS to Z/NZ, on which G actsF from the left by F

(2.2.2) (g.ψ)(ρ) := ψ(ρ.g) for g G, ψ Map( , Z/NZ) and ρ . ∈ ∈ FXS ∈FXS -We ﬁx an embedding k ֒ kpi , which induces the continuous homomor phism for each 1 i r → ≤ ≤ ιp : Πp Π . i i −→ S

Let respi and resS denote the restriction maps (the pull-backs by ιpi ) deﬁned by

resp : p ; ρ ρ ιp , (2.2.3) i FXS −→ F i 7→ ◦ i res := (resp ): ; ρ (ρ ιp ), S i FXS −→ FS 7→ ◦ i which are G-equivariant by (2.1.1), (2.1.25) and (2.2.1). We denote by Respi and ResS the homomorphisms on cochains deﬁned by (2.2.4) n n Respi : C (ΠS, Z/NZ) C (Πpi , Z/NZ); α α ιpi , n −→ r n 7→ ◦ Res := (Resp ): C (Π , Z/NZ) C (Πp , Z/NZ); α (α ιp ). S i S −→ i=1 i 7→ ◦ i Firstly, we note the following Q

Lemma 2.2.5. We have

3 H (ΠS, Z/NZ)=0.

3 Proof. It suffices to show that the p-primary part H (ΠS, Z/NZ)(p) = 0 for 3 any prime number p. Since H (ΠS, Z/NZ)(p)= 0 for p ∤ N, we may assume that p N. | Case that n> 2. Then K is totally imaginary and so ΠS = ΠS X∞ (ΠS X∞ := ∪ k ∪ k πét(Spec( S) being the Galois group of the maximal extension of k unram- 1 Ok \ ified outside S X∞). By our assumption on S, all primes over p are contained ∪ k in S. So the cohomological p-dimension cdp(ΠS) 2 by [NSW; Proposition 3 ≤ 8.3.18]. Hence H (ΠS, Z/NZ)(p) = 0. Case that n = 2 and so p = 2. Since S does not contain any real primes of k, the cohomological 2-dimension cd2(ΠS) 2 by [NSW; Theorem 10.6.7]. 3 ≤ Hence H (ΠS, Z/2Z)(2) = 0. ✷

34 3 Let ρ XS and so c ρ Z (ΠS, Z/NZ). By Lemma 2.2.5, there is β C2(Π∈, Z F/NZ)/B2(Π ◦, Z/N∈Z) such that ρ ∈ S S (2.2.6) c ρ = dβ , ◦ ρ 2 3 where d : C (ΠS, Z/NZ) C (ΠS, Z/NZ) is the coboundary homomorphism. By (2.2.3), (2.2.4) and (2.2.6),→ we see that

(2.2.7) c resp (ρ)= d Resp (β ) ◦ i i ρ for 1 i r. By (2.1.4) , (2.1.27) and (2.2.7), we have ≤ ≤ (2.2.8) [Res (β )] (res (ρ)). S ρ ∈LS S

Let resS∗ ( S) be the G-equivariant principal Z/NZ-bundle over XS induced from L by res : F LS S (2.2.9) res∗ ( ) := (ρ, α ) res (ρ)= ̟ (α ) . S LS { S ∈FXS ×LS | S S S } and let res∗ (̟ ) be the projection res∗ ( ) . The quotient by the S S S LS → FXS action of G is the principal Z/NZ-bundle res∗( ) over induced from LS MXS S by resS. By (2.2.9), a section of resS∗ (̟S) is naturally identiﬁed with a map yL : satisfying ̟ y = res : S FXS →LS S ◦ S S (2.2.10) Γ( , res∗ ( )) = y : ̟ y = res , FXS S LS { S FXS →LS | S ◦ S S} on which G acts by (yS.g)(ρ) := yS(ρ.g) for ρ XS ,g G. We deﬁne the arithmetic Chern-Simons functional CS : ∈ F by∈ XS FXS →LS

(2.2.11) CSXS (ρ) := [ResS(βρ)]

for ρ XS . The value CSXS (ρ) S is called the arithmetic Chern-Simons invariant∈F of ρ. ∈L

Lemma 2.2.12. (1) CSXS (ρ) is independent of the choice of βρ.

(2) CSXS is a G-equivariant section of resS∗ (̟S):

CS Γ ( , res∗ ( )) = Γ( , res∗ ( )). XS ∈ G FXS S LS MXS S LS

2 2 Proof. (1) Let βρ′ C (ΠS, Z/NZ)/B (ΠS, Z/NZ) be another choice satisfying ∈ 2 c ρ = dβ′ . Then we have β′ = β + z for some z H (Π , Z/NZ) and so ◦ ρ ρ ρ ∈ S

Resp (β′ ) Resp (β ) = invp (z) (1 i r). i ρ − i ρ i ≤ ≤ 35 Noting that any primes dividing N is contained in S, Tate-Poitou exact sequence ([NSW; 8.6.10]) implies the following exact sequence

r r r 2 Qi=1 respi 2 Pi=1 invpi H (Π , Z/NZ) H (Πp , Z/NZ) Z/NZ 0, S −→ i −→ −→ i=1 Y which yields r

invpi (z)=0. i=1 X By (2.1.28), we obtain

[ResS(βρ′ )] = [ResS(βρ)].

(2) By (2.2.8), (2.2.10) and (2.2.11), we have

CS Γ( , res∗ ( )). XS ∈ FXS S LS

So it suﬃces to show that CSXS is G-equivariant. By (2.1.5) and (2.2.6), we have β = c (ρ.g)=(g.c) ρ =(c + dh ) ρ = d(β + h ρ). ρ.g ◦ ◦ g ◦ ρ g ◦ 2 for g G and ρ XS . Therefore there is z H (ΠS, Z/Z) such that β =∈β + h ρ +∈z Fand so ∈ ρ.g ρ g ◦ Res (β ) = Res (β )+ h res (ρ) + Res (z) S ρ.g S ρ g ◦ S S = ResS(βρ).g + ResS(z).

By the same argument as in (1) above, we obtain ✷ CSXS (ρ.g) = [ResS(βρ.g)] = [ResS(βρ)].g = CSXS (ρ).g.

xS Let x = [(xp ,...,xp )] Γ( , ) be a section and let be the S 1 r ∈ FS LS LS arithmetic prequantization principal Z/NZ-bundle over S with respect to xS. xS F Let resS∗ ( S ) be the G-equivariant principal Z/NZ-bundle over XS induced x L F from S by res : LS S xS xS res∗ ( ) = (ρ, (ρ , m)) res (ρ)= ρ S LS { S ∈FXS ×LS | S S} = Z/NZ FXS ×

36 xS by identifying (ρ, (ρS, m)) with (ρ, m). So a section of resS∗ ( S ) over XS is identiﬁed with a map Z/NZ: L F FXS → xS Γ( , res∗ ( )) = Map( , Z/NZ), FXS S LS FXS on which G acts by (2.2.2). Therefore, letting MapG( XS , Z/NZ) denote the set of G-equivariant maps Z/NZ, we have theF identiﬁcation FXS → xS ΓG( XS , resS∗ ( S )) = MapG( XS , Z/NZ) F L F x = ψ : Z/NZ ψ(ρ.g)= ψ(ρ)+ λ S (g, res (ρ)) { FXS → | S S for ρ ,g G . ∈FXS ∈ } x x The isomorphism Φ S : ∼ S in Proposition 2.1.35 induces the isomor- S LS → LS phism

x xS ∼ S Ψ :ΓG( XS , resS∗ ( S)) ΓG( XS , resS∗ ( S )) = MapG( XS , Z/NZ) F L −→ x F L F y Φ S y . S 7→ S ◦ S We then deﬁne the arithmetic Chern-Simons functional CSxS : Z/NZ XS XS xS F → with respect to xS by the image of CSXS under Ψ : (2.2.13) CSxS := ΨxS (CS ). XS XS

Theorem 2.2.14. (1) For ρ , we have ∈FXS r xS CS (ρ)= (Respi (βρ) xpi (respi (ρ))), XS − i=1 X which is independent of the choice of βρ. (2) We have the following equality in C1(G, Map( , Z/NZ)) FXS xS xS dCS = res∗(λ ). XS S

xS Proof. (1) This follows from the deﬁnition of ΦS in Proposition 2.1.36 and (2.2.13). (2) Since CS Map ( , Z/NZ), we have XS ∈ G FXS xS xS xS CS (ρ.g)= CS (ρ)+ λ (g, resS(ρ)) XS XS S for g G and ρ , which means the assertion. ✷ ∈ ∈FXS

37 Proposition 2.2.15. Let xS′ Γ( S, S) be another section which yields ′ ′ xS xS ,xS ∈ F L CS , and let δS : S Z/NZ be the map in Proposition 2.1.34. Then XS F → we have ′ ′ xS xS xS,xS CS (ρ) CS (ρ)= δS (resS(ρ)). XS − XS

Proof. By Proposition 2.2.14 (1) and Lemma 1.1.4 (1), we have

r r ′ xS xS CS (ρ) CS (ρ) = (Respi (βρ) xp′ (respi (ρ))) (Respi (βρ) xpi (respi (ρ))) XS − XS − i − − i=1 i=1 Xr X = (xp (resp (ρ)) x′ (resp (ρ))) i i − pi i i=1 ′ XxS,x = δ S (resS(ρ)). ✷

′ ′ xS ,x x x S S ∼ S For xS, xS′ Γ( S, S), the G-equivariant isomorphism ΦS : S S induces the isomorphism∈ F L L →L

′ ′ ′ xS,x xS xS xS xS ,xS x Ψ S :Γ ( , res∗ ( )) ∼ Γ ( , res∗ ( )); ψ Φ ψ S . G FXS S LS −→ G FXS S LS 7→ S ◦ By Proposition 2.1.35, we have

′ ′ ΨxS,xS ΨxS =ΨxS ′ ′ ′ ′′ ′ ′′ xS,xS ◦ x ,xS xS,x 1 x ,x xS,x xS,x ∗ xS S S S S S S Ψ = idΓG( ,res ( )), Ψ = (Ψ )− , Ψ Ψ =Ψ . ( FXS S LS ◦

xS So we can deﬁne the equivalence relation ion the disjoint union of Γ ( , res∗ ( )) ∼ G FXS S LS over x Γ( , ) by S ∈ FS LS ′ ′ ′ ψxS ψxS ΨxS,xS (ψxS )= ψxS ∼ ⇐⇒ ′ xS xS x xS for ψ Γ ( , res∗ ( )) and ψ S Γ ( , res∗ ( )). Then we have ∈ G FXS S LS ∈ G FXS S LS the following identiﬁcation:

Γ ( , res ( )) = Γ ( , res ( xS ))/ ; G XS S∗ S xS Γ( S , S) G XS S∗ S (2.2.16) F L ∈ F L F L ∼ ψ ΨxS (ψ) 7→ F where CS and [CSxS ] are identiﬁed. XS XS

3. Quantum theory

38 In this section, we construct the arithmetic quantum space and the arithmetic Dijkgraaf-Witten invariant over the moduli space of Galois representations. These constructions correspond to the quantum theory of topological Dijkgraaf-Witten TQFT. We keep the same notations and assumptions as in Section 2. We assume that F is a subﬁeld of C such that ζN is contained in F and F = F (F being the complex conjugate).

3.1. Arithmetic quantum spaces. Following the construction of the quantum Hilbert space, we deﬁne the arithmetic quantum space S for ∂VS by the space of G-equivariant sections of the arithmetic prequantizationH F -line bundle ̟ : L : S,F S →FS := Γ ( , L )=Γ( , L ). HS G FS S MS S It is a ﬁnite dimensional F -vector space. xS Let xS = [(xp1 ,...,xpr )] Γ( S, S) be a section and let LS be the arithmetic prequantization F -line∈ bundleF L over with respect to x and let FS S (3.1.1) xS xS xS := ΓG( S, L )=Γ( S, LS ) S S x H F M λ S (g,ρ ) = θ : F θ(ρ .g)= ζ S S θ(ρ ) for ρ ,g G , { FS → | S N S S ∈FS ∈ } which we call the arithmetic quantum space S for ∂VS with respect to xS. The x x H isomorphism Φ S : L ∼ L S in Proposition 2.1.39 induces the isomorphism S,F S → S x x (3.1.2) ΘxS : ∼ S ; θ Φ S θ. HS −→ HS 7→ S,F ◦ xS We call an element of S or S an arithmetic theta function (cf. Remark 3.2.4 below). H H ′ ′ xS ,xS xS xS For x , x′ Γ( , ), the isomorphism Φ : L ∼ L induces the S S ∈ FS LS S,F S → S isomorphism of F -vector spaces:

′ ′ ′ x x xS ,x ΘxS,xS : S ∼ S ; θxS Φ S θxS HS −→ HS 7→ S ◦ and, by Proposition 2.1.39, we have

′ ΘxS,xS ΘxS =ΘxS ′ ′ ′ ′′ ′ ′′ xS,xS ◦ x ,xS xS,x 1 x ,x xS,x xS ,x Θ = id xs , Θ S = (Θ S )− , Θ S S Θ S =Θ S . HS ◦ xS So the equivalence relation is deﬁned on the disjoint union of all S running over x Γ( , ) by ∼ H S ∈ FS LS ′ ′ ′ θxS θxS ΘxS,xS (θxS )= θxS ∼ ⇐⇒ 39 x ′ x′ for θxS S and θxS s . Then we have the following identiﬁcation: ∈ HS ∈ HS (3.1.3) = xS / . HS HS ∼ xS Γ( S , S) ∈ GF L

Remark 3.1.4. The arithmetic quantum space is an arithmetic ana- HS log of the quantum Hilbert space Σ for a surface Σ in (2+1)-dimensional Chern-Simons TQFT. We recall thatH is known to coincides with the space HΣ of conformal blocks ([BL]) and its dimension formula was shown by Verlinde ([V]). It would also be an interesting question in number theory to describe the dimension and a canonical basis of in comparison of Verlinde’s formulas. HS 3.2. Arithmetic Dijkgraaf-Witten partition functions. For ρ , S ∈ FS we deﬁne the subset (ρ ) of by FXS S FXS (ρ ) := ρ res (ρ)= ρ . FXS S { ∈FXS | S S } xS We then deﬁne the arithmetic Dijkgraaf-Witten invariant Z (ρS) of ρS with XS respect to xS by

x CS S (ρ) xS 1 XS (3.2.1) Z (ρS) := ζN . XS #G ρ (ρS ) ∈FXXS

xS Theorem 3.2.2. (1) Z (ρS) is independent of the choice of βρ. XS (2) We have xS xS Z S . XS ∈ H

Proof. (1) This follows from Theorem Theorem 2.2.12 (1). (2) This follows from Theorem 2.2.14 (2) and (3.1.1).

xS xS We call Z S the arithmetic Dijkgraaf-Witten partition function for XS ∈ H XS with respect to xS.

The following proposition tells us how they are changed when we change xS.

Proposition 3.2.3. For sections x , x′ Γ( , ), we have S S ∈ FS LS ′ x x′ ΘxS,xS (Z S )= Z S . XS XS

40 Proof. We have

′ x ,x′ xS,x xS S S Θ S ( )(ρS) =(Φ Z )(ρS) XS S,F XS Z ◦ ′ xS,xS ηS (ρS ) = ZXS (ρS )ζN by Proposition 2.1.39 ′ xS xS ,xS CS (ρ)+ηS (ρS ) XS = ζN by (3.2.1) ρ (ρS ) ∈FXXS x′ CS S (ρ) XS = ζN by Proposition 2.2.15 ρ (ρS ) ∈FXS ′X xS = Z (ρS ) XS for ρ . So we obtain the assertion. ✷ S ∈FS

xS By the identiﬁcation (3.1.3), Z deﬁnes the element ZX of S which is XS S H independent of the choice of xS. We call it the arithmetic Dijkgraaf-Witten partition function for XS.

Remark 3.2.4. In (2+1)-dimensional Chern-Simons TQFT, an element of for a surface Σ may be regarded as a (non-abelian) generalization of the HΣ classical theta function on the Jacobian manifold of Σ (cf. [BL]. It goes back to Weli’s paper [We]. See [Mo1] for an arithmetic analog.) In this respect, it may be interesting to observe that the Dijkgraaf-Witten partition function in (3.2.1) may look like a variant of (non-abelian) Gaussian sums.

4. Some basic and functorial properties In this section, we study some basic and functorial properties of the objects constructed in Sections 2 and 3. We keep the same notations as in Sections 2 and 3.

4.1. Change of the 3-cocycle c. The theory given in Sections 2 and 3 depends on a chosen 3-cocycle c. We shall see in the following that when c is changed in the cohomology class [c], objects are changed to isomorphic ones, and hence the theory depends essentially on the cohomology class [c]. 3 Let c′ Z (G, Z/NZ) be another 3-cocycle representing [c]. The objects con- ∈ structed by using c′ will be denoted by using ′, for example, by p′ , Lp′ ,... etc. L

41 2 There is b C (G, Z/NZ) such that c′ c = db. Then we have the ∈ − isomorphism of Z/NZ-torsors for ρp p: ∈F

p(ρp) ∼ ′ (ρp); αp αp + b ρp, L −→ Lp 7→ ◦ which induces the following isomorphisms of arithmetic quantization bundles:

ξp : p ∼ ′ , ξp : Lp ∼ L′ , (4.1.1) L −→ Lp ,F −→ p ξ : ∼ ′ , ξ : L ∼ L′ . S LS −→ LS S,F S −→ S

Let xp Γ( p, p) and x = [(xp ,...,xp )] Γ( , ), and let x′ ∈ F L S 1 r ∈ FS LS p ∈ Γ( ′, ′ ) and x′ Γ( ′ , ′ ). Denote by λ′ and λ′ the arithmetic Chern- Fp Lp S ∈ FF LS p S Simons 1-cocycles for ∂Vp and ∂VS with respect to xp′ and xS′ , respectively. We deﬁne κp : p Z/NZ and κ : Z/NZ by F → S FS → r

κp(ρp) := (ξp xp)(ρp) x′ (ρp), κ (ρ ) := κp (ρp ) ◦ − p S S i i i=1 X for ρp p and ρ =(ρp ,...,ρp ) , respectively. Then we have ∈F S 1 r ∈FS

λ′ (g) λp(g)= g.κp κp, λ′ (g) λ (g)= g.κ κ . p − − S − S S − S

We note that if we take x′ := ξp xp and x′ := ξ x , κp =0 and so κ = 0. p ◦ S S ◦ S S As in Propositions 2.1.19, 2.1.24, 2.1.36 and 2.1.39, using κp and κS, we have the isomorphisms

′ ′ xp xp xp xp p ∼ p′ , Lp ∼ Lp′ , ′ ′ LxS −→ L x xS −→ x ∼ ′ S , L ∼ L′ S . LS −→ LS S −→ S which are compatible with the isomorphisms in (4.1.1) via the isomorphisms xp xp xS xS p p , Lp Lp , S S and LS LS in Propositions 2.1.15, 2.1.24, L2.1.36≃ L and 2.1.39.≃ L ≃ L ≃ The isomorphism ξ : ∼ ′ induces the isomorphism S LS →LS

Γ ( , res∗ ( )) ∼ Γ ( , res∗ ( ′ )) G FXS S LS −→ G FXS S LS

∼ which sends CSX to CS′ , and the isomorphism ξS,F : S S′ induces the S XS L →L isomorphism ∼ ′ HS −→ HS which sends Z to Z′ . XS XS

42 4.2. Change of number fields. Let k′ be an another number field contains a primitive N-th root of unity and let S′ = p′ ,..., p′ ′ be a finite set { 1 r } of finite primes of k′ such that any finite prime dividing N is contained in S′. The objects constructed by using k′ and S′ will be denoted by, for example, p′ , Lp′ , S′ , LS′ ,... etc, for simplicity of notations . Assume that r = r′ and L L ∼ there are isomorphisms ξi : kpi kp′ ′ for 1 i r. Then ξi’s induces the → i ≤ ≤ following isomorphisms of arithmetic quantization bundles:

ξp : p ∼ p′ , ξp : Lp ∼ Lp′ i L i −→ L i i,F i −→ i

ξ : ∼ ′ , ξ : L ∼ L ′ . S LS −→ LS S,F S −→ S Let xp Γ( p , p ) and x = [(xp ,...,xp )] Γ( , ), and let xp′ i ∈ F i L i S 1 r ∈ FS LS i ∈ Γ( p′ , p′ ) and xS′ = [(xp′ ,...,xp′ )] Γ( S′ , S′ ). Then we have the isomor- F i L i 1 r ∈ F L phisms of arithmetic prequantization bundles with respect to sections

x ′ x ′ xp p xp p i ∼ i i ∼ i pi p′ , Lpi Lp′ L −→ L i −→ i

xS xS′ xS xS′ ∼ ′ , L ∼ L ′ . LS −→ LS S −→ S Suppose further that there is an isomorphism τ : k ∼ k′ of number fields → which sends p to p′ for 1 i r. so that we have the isomorphism i i ≤ ≤ ξ : X := X S ∼ X ′ S′ =: X ′ . S k \ −→ k \ S √3 √3 2π√ 1 For example, let k := Q( 2),k′ := Q( 2ω), ω := exp( 3− ) and so N = 2. 3 3 Let ξ be the isomorphism k ∼ k′ defined by ξ(√2) := √2ω. Noting 2 k = 3 2 3 → 3 O (√2) ,X 2=(X 4)(X 7)(X 20) mod 31, let S := p1 := (√2), p2 := 3 − − 3 − − 3 { (31, √2 4), p2 := (31, √2 7), p4 := (31, √2 20) ,S′ := ξ(S) = p1′ := 3 − 3 − 3 − } 3 { (√2ω), p′ := (31, √2ω 4), p′ := (31, √2ω 7), p′ := (31, √2ω 20) , so that 2 − 3 − 4 − } we have kp1 = kp′ ′ = Q2 and kpi = kp′ ′ = Q31 (2 i 4). So this example 1 i ≤ ≤ satisfies the above conditions. The isomorphism ξ : X ∼ X ′ induces the bijection ξ∗ : ∼ . S S XS′ XS By the constructions in the subsection→ 2.2 and the section 3, weF hav−→e the F following

Proposition 4.2.1. The isomorphism ξ : ∼ ′ induces the bijection S LS →LS ΓG( , res∗ ( S)) ∼ ΓG( , res∗ ′ ( S′ )) FXS S L −→ FXS′ S L which sends CS to CS . The isomorphism ξ : L ∼ L ′ induces the XS XS′ S,F S S isomorphism → ∼ ′ , HS −→ HS 43 sends Z to Z . XS XS′

Remark 4.2.2. Proposition 4.4.1 may be regarded as an arithmetic analogue of the axiom in (2 + 1)-diemnsional TQFT, which asserts that an orientation homeomorphism f : Σ ≈ Σ′ between closed surfaces induces an isomorphism → ∼ ′ of quantum Hilbert spaces and if f extends to an orientation pre- HΣ → HΣ serving homeomorphism M ≈ M ′, with ∂M = Σ, ∂M ′ = Σ′, Z is sent to → M Z ′ under the induced isomorphism ∼ ′ . M H∂M → H∂M 4.3. The case that S is empty. In the theory in Sections 2 and 3, we can include the case that S is the empty set as follows. ∅ We define to be the space of a single point, := . We define ∅ ∅ the arithmeticF prequantization principal Z/NZ-bundleF to{∗} be Z/NZ, on which G acts trivially, so that the map ̟ : is GL-equivariant.∅ So the ∅ L∅ →F∅ arithmetic prequantization F -line bundle L is defined by Z/NZ Z/NZ F = F . ∅ The arithmetic Chern-Simons 1-cocycle λ is defined to be 0. × ∅ Let Π˜ k be the modified étale fundamental group of Xk defined by consider- ing the Artin-Verdier topology on Xk, which takes the real primes into account (cf. [H; 2.1], [AC], [Bi], [Z]). It is the Galois group of the maximal extension of k unramified at all finite and infinite primes. We set := Hom (Π˜ ,G). FXk cont k

Following [H], we define the arithmetic Chern-Simons invariant CSXk (ρ) of ρ by the image of c under the composition ∈FXk ρ∗ H3(G, Z/NZ) H3(Π˜ , Z/NZ) H3(X , Z/Z) Z/NZ, → k → k ≃ where the cohomology group of Xk is the modified étale cohomology defined in the Artin-Verdier topology. Thus we have the arithmetic Chern-Simons functional CS : Z/NZ and so we see that Xk FXk → CSX ΓG( X , res∗( )) = Map( X , Z/NZ), k ∈ F k ∅ L∅ M k where res is the (unique) restriction map X . Then we have ∅ F k →F∅ dCSX = 0 = resS∗ (λ ). k ∅

The arithmetic quantum space is deﬁned by ΓG( , L )= F . Following H∅ F∅ ∅ [H], we deﬁne the arithmetic Dijkgraaf-Witten invariant Z(Xk) of Xk by 1 CS (ρ) CS (ρ) Z(X ) := ζ Xk = ζ Xk k #G N N ρ ρ ∈MXXk ∈MXXk 44 and the arithmetic Dijkgraaf-Witten partition function by ZX : F by k F∅ → ZX ( ) := Z(Xk) for . So we have k ∗ ∗∈F∅

ZX . k ∈ H∅

We note that when [c] is trivial, Z(Xk) coincides with the (averaged) number of continuous homomorphism from Π˜ k to G:

#Hom (Π˜ ,G) Z(X )= cont k , k #G which is the classical invariant for the number ﬁeld k.

4.4. Disjoint union of ﬁnite sets of primes and reversing the orientation of ∂V . Let S = p ,..., p and S = p ,..., p be disjoint S 1 { 1 r1 } 2 { r1+1 r} sets of ﬁnite primes of k and let S = S S . We include the case where S is 1 ⊔ 2 1 empty, but S2 is non-empty. (For the case where S1 and S2 are both empty, the following arguments are trivial.) Then we have

= . FS FS1 ×FS2 For the arithmetic quantization principal Z/NZ-bundles, we deﬁne the map

⊞ : , LS1 ×LS2 −→ LS as follows. For the case that S = (and so S = S), we set 1 ∅ 2 ⊞ (4.4.1) m [αS2 ] := [αS2 ].m

for (m, [αS]) S2 . For the case that S1 = , we set ∈L∅ ×L 6 ∅ ⊞ (4.4.2) [αS1 ] [αS2 ] := [(αS1 ,αS2 )] for ([α ], [α ]) . S1 S2 ∈LS1 ×LS2 For the arithmetic quantization F -line bundles, we let pi∗(LSi ) be the G- equivariant F -line bundle over S induced from LSi by the projection S for i =1, 2: F F → FSi

p∗(L ) := (ρ , [α ]) ρ = ̟ ([α ]) (i =1, 2) i Si { S S1 | Si Si Si } for ρS = (ρS1 , ρS2 ). When S1 = , we think of pi∗(L ) = F simply over ∅ = . Let ∅ F∅ {∗} p∗(̟ ): p∗(L ) i i i Si −→ FS 45 be the projection. The ﬁber over ρS =(ρS1 , ρS2 ) is given by

1 p∗(̟ )− (ρ ) = ρ [([α ], z ) L ρ = ̟ ([α ]), z F i i S { S}×{ Si i ∈ Si | Si Si Si i ∈ } = LSi (ρSi ) F, ≃ where LSi (ρSi ) is as in (2.1.38). We set

L ⊠ L := p∗(L ) p∗(L ), S1 S2 1 S1 ⊗ 2 S2 which is the F -line bundle over and whose element is written by FS (ρ , [([α ], z )] [([α ], z )]), S S1 1 ⊗ S2 2 where ρ =(ρ , ρ ) , ([α ], z ) L (ρ ). The right action on L ⊠L S S1 S2 ∈FS Si i ∈ Si Si S1 S2 is deﬁned by

(ρ , [([α ], z )] [([α ], z )]).g := (ρ .g, [([α ].g, z )] [([α ].g, z )]) S S1 1 ⊗ S2 2 S S1 1 ⊗ S2 2 so that the projection L ⊠ L is G-equivariant. Then, as in Propo- S1 S2 → FS sition 2.1.?, we have the isomorphism of G-equivariant F -line bundles over : FS L ⊠ L ∼ L ; (ρ , [([α ], z )] [([α ], z )]) [([α ], z z )], S1 S2 −→ S S S1 1 ⊗ S2 2 7→ S 1 2 where αS = (αS1 ,αS2 ). Choose xSi Γ( Si , Si ) and let xS := [(xS1 , xS2 )] Γ( , ). Then we see that ∈ F L ∈ FS LS xS1 xS2 xS λS1 (g, ρS1 )+ λS2 (g, ρS2 )= λS (g, ρS) for g G and ρ =(ρ , ρ ). ∈ S S1 S2 x Proposition 4.4.3. For θ Si (i =1, 2), we deﬁne θ θ xS by i ∈ HSi 1 · 2 ∈ HS (θ θ )(ρ ) := θ (ρ )θ (ρ ) 1 · 2 S 1 S1 2 S2 for ρS = (ρS1 , ρS2 ). Then we have the following isomorphism of F -vector spaces xS xS x 1 2 ∼ S ; (θ , θ ) θ θ . HS1 ⊗ HS2 −→ HS 1 2 7→ 1 · 2 For θ (i =1, 2), we deﬁne θ ⊠ θ by i ∈ HSi 1 2 ∈ HS

(θ ⊠ θ )(ρ ) := p∗(θ (ρ )) p∗(θ (ρ )) 1 2 S 1 1 S1 ⊗ 2 2 S2 46 for ρS = (ρS1 , ρS2 ). Then we have the following isomorphism of F -vector spaces ⊠ S1 S2 ∼ S; (θ1, θ2) θ1 θ2. H ⊗ H −→ H 7→ xS xSi i the above isomorphisms are compatible via the isomorphisms Θ : Si Si x H ≃ H (i =1, 2) and ΘxS : S in (3.1.2). HS ≃ HS

xS Proof. For θ S , set θ1(ρS1 ) := θ(ρS1 , 1) and θ2(ρS2 ) := θ2(1, ρS2 ). Then xS ∈ H x xS1 xS2 θ i and the map S ; θ θ θ , gives the inverse i ∈ HSi HS → HS1 ⊗ HS2 7→ 1 ⊗ 2 of the former map. By the deﬁnitions, the second map is compatible with the xS xS i i xS xS ﬁrst one via ΘSi : Si Si (i =1, 2) and Θ : S S and so we have the following commutativeH ≃ H diagram H ≃ H

HS1 ⊗ HS2 −→ HS ΘxS1 ΘxS2 ΘxS ⊗ xS ≀↓xS ↓≀x 1 2 ∼ S , HS1 ⊗ HS2 −→ HS from which the second isomorphism follows. ✷

Remark 4.4.4. Proposition 4.4.3 may be regarded as an arithmetic analog of the multiplicative property that Σ1 Σ2 = Σ1 Σ2 for disjoint surfaces Σ1 H ⊔ H ⊗ H and Σ2 which is one of the axioms required in (2+1)-dimensional TQFT ([A1]).

For a finite prime p of k, the canonical isomorphism 2 invp : H (∂Vp, Z/NZ) ∼ Z/NZ ét −→ indicates that ∂Vp is “orientable” and we choose (implicitly) the “orientation” of ∂Vp corresponding 1 Z/NZ. We let ∂V ∗ = ∂Vp with the “opposite ∈ p orientation”, namely, invp([∂V ∗]) = 1. p − The arithmetic prequantization principal Z/NZ-bundle for ∂Vp∗, denoted by p∗ , is defined (formally) by p with the opposite action of the structure L L group Z/NZ, (αp, m) αp.( m) for αp p∗ and m Z/NZ. So the 7→ − ∈ L ∈ arithmetic prequantization F -line bundle Lp∗ for ∂Vp∗ is the dual bundle of Lp, Lp∗ = Lp∗. Noting Γ( p, p∗ ) = Γ( p, p), the arithmetic Chern-Simons xp F L xp F L 1-cocycle λp∗ for ∂Vp∗ is given by λp for xp Γ( p, p∗ ). The actions of G xp xp − ∈ F L xp on ∗ = p Z/NZ and L ∗ = p F are changed to those via λ ∗ . Lp F × p F × p For a finite set of finite primes S = p1,..., pr , we set ∂VS∗ := ∂Vp∗1 { } xS ⊔ ∗ ∗ ∂Vp∗r . Then the arithmetic prequantization bundles S , LS , S∗ and ···⊔xS L L LS∗ (xS Γ( S, S∗ )=Γ( p, S)) are defined in the similar manner. For the arithmetic∈ Chern-SimonsF L 1-cocycle,F L we have

xS xS λ ∗ = λ . S − S 47 xS Let S∗ be the arithmetic quantum space for ∂VS∗ with respect to xS. Then we see thatH

xS xS λS∗ (g,ρS) S∗ = θ∗ : S F θ∗(ρS.g)= ζN θ∗(ρS) for ρS S,g G H { F → | λS(g,ρS) ∈F ∈ } = θ∗ : S F θ∗(ρS.g)= ζN− θ∗(ρS ) for ρS S,g G { xS F → | ∈F ∈ } = , HS x where S is the complex conjugate of xS . Since the pairing HS HS

xS xS ∗ F ; (θ∗, θ) θ∗(ρ )θ(ρ ) HS × HS −→ 7→ S S ρS S X∈F is a (Hermitian) perfect pairing, together with (3.1.2), we have the following

xS xS Proposition 4.4.5. ∗ and ∗ are the dual spaces of and , re- HS HS HS HS spectively: xS xS ∗ =( )∗, ∗ =( )∗. HS HS HS HS

Remark 4.4.6. Proposition 4.3.5 may be regarded as an arithmetic analog of the involutory property that ∗ = ∗ , where Σ∗ = Σ with the opposite HΣ HΣ orientation, which is one of the axioms required in (2 + 1)-dimensional TQFT ([A1]).

In the subsection 2.2 and the section 3, we have chosen implicitly the orientation of XS so that the boundary ∂XS with induced orientation may be ∗ identiﬁed with ∂VS. Let XS denote XS with the opposite orientation. Then, the arithmetic Chern-Simons functional and the Dijkgraaf-Witten partition ∗ function for XS are given as follows:

x CS S (ρ) xS xS xS − XS (4.4.7) CS ∗ = CS , Z ∗ (ρS)= ζN . XS − XS XS ρ ∈FXXS

5. Decomposition and gluing formulas

In this section, we show a decomposition formula for arithmetic Chern- Simons invariants and a gluing formula for arithmetic Dijkgraaf-Witten partition functions, which generalize the decomposition formula in [CKKPY] in

48 our framework. We keep the same notations and assumptions as in Sections 2, 3 and 4.

5.1. Arithmetic Chern-Simons functionals and arithmetic Dijkgraaf- Witten partition functions for V . For a finite prime p of k, let p denote S O the ring of p-adic integers and we let Vp := Spec( p). For a non-empty finite O set of finite primes S = p , , p of k, let V := Vp Vp , which { 1 ··· r} S 1 ⊔···⊔ r plays a role analogous to a tubular neighborhood of a link, and so ∂VS plays a role of the boundary of VS. In this subsection, we introduce the arithmetic Chern-Simons functional and arithmetic Dijkgraaf-Witten partition function for VS, which will be used for our gluing formula in the next section. Let Π˜ p be the étale fundamental group of Vp, namely, the Galois group of the maximal unramified extension of kp and we set

:= Hom (Π˜ p,G), := . FVp cont FVS FVp1 ×···×FVpr

Since Π˜ p Zˆ (proﬁnite inﬁnite cyclic group), G. G acts on from ≃ FVp ≃ FVS the right by

1 G ; ((˜ρp ) ,g) ρ.g := (g− ρ˜p g) , FVS × →FVS i i 7→ i i and let denote the quotient set by this action: MVS := /G. MVS FVS

Let ˜resp : V p and ˜resS := ( ˜resp ): V S denote the restriction i F pi → F i F S → F maps induced by the natural continuous homomorphisms vp : Πp Π˜ p i i → i (1 i r), which are G-equivariant. We denote by Res˜ p and Res˜ the ≤ ≤ i S homomorphisms on cochains given as the pull-back by vpi :

˜ n ˜ n Respi : C (Πpi , Z/NZ) C (Πpi , Z/NZ); αi αi vpi , r −→n r 7→ n ◦ Res˜ := (Res˜ p ): C (Π˜ p , Z/NZ) C (Πp , Z/NZ);(α ) (α vp ). S i i=1 i −→ i=1 i i 7→ i ◦ i Q 3 ˜ Q 3 ˜ Forρ ˜ = (˜ρpi )i VS , c ρ˜pi Z (Πpi , Z/NZ). Since H (Πpi , Z/NZ) = 0, ∈2 F ◦ ∈ there is β˜p C (Π˜ p , Z/NZ) such that i ∈ i

c ρ˜p = dβ˜p . ◦ i i We see that c ˜resp (˜ρp )= dRes˜ p (β˜p ) ◦ i i i i

49 for 1 i r and we have ≤ ≤

[Res˜ ((β˜p ) )] ( ˜res (˜ρ)). S i i ∈LS S

Let ˜resS∗ ( S) be the G-equivariant principal Z/NZ-bundle over VS induced from byL ˜res : F LS S

˜res∗ ( ) := (˜ρ, α ) ˜res (˜ρ)= ̟ (α ) S LS { S ∈FVS ×LS | S S S } and let ˜resS∗ (̟S) be the projection ˜resS∗ ( S) VS . We deﬁne the arithmetic Chern-Simons functional CS : L by→F VS FVS →LS ˜ ˜ CSVS (˜ρ) := [ResS((βpi )i)] forρ ˜ . The value CS (˜ρ) is called the arithmetic Chern-Simons invari- ∈FVS VS ant ofρ ˜.

˜ Lemma 5.1.1. (1) CSVS (˜ρ) is independent of the choice of βpi .

(2) CSVS is a G-equivariant section of ˜resS∗ (̟S):

CS Γ ( , ˜res∗ ( )) = Γ( , ˜res∗ ( )). VS ∈ G FVS S LS MVS S LS

˜ Proof. (1) This follows from the fact that the cohomological dimension of Πpi is one. (2) The proof of this lemma is almost same as Lemma 2.2.8. (2). ✷

x S ∼ For a section xS = [(xp1 ,...,xpr )] Γ( S, S), the isomorphism ΦS : S x ∈ F L L → S induces the isomorphism LS x ˜ xS ∼ S Ψ :ΓG( VS , ˜resS∗ ( S)) ΓG( VS , ˜resS∗ ( S )) = MapG( VS , Z/NZ); F L −→ x F L F y Φ S y . S 7→ S ◦ S xS We deﬁne the arithmetic Chern-Simons functional CSVS : VS Z/NZ with xS F → respect to xS by the image of CSVS under Ψ .

Proposition 5.1.2. (1) For ρ , we have ∈FVS r xS CS (˜ρ)= (Res˜ ((β˜p ) ) x ( ˜res (˜ρ))). VS S i i − S S i=1 X

50 (2) We have the following equality in C1(G, Map( , Z/NZ)) FVS

xS dCSVS = ˜res∗(λS ).

Proof. (1) This follows from the deﬁnition of Ψ˜ xS . (2) Since CS Map ( , Z/NZ), we have VS ∈ G FVS xS xS xS CSVS (˜ρ.g)= CSVS (˜ρ)+ λS (g, ˜resS(˜ρ)) for g G andρ ˜ , which means the assertion. ✷ ∈ ∈FVS

Proposition 5.1.3. Let xS′ Γ( S, S) be another section, which yields ′ ′ xS xS ,xS ∈ F L CSVS and let δS : S Z/NZ be the map in Proposition 2.1.34. Then we have F → ′ ′ CSxS (˜ρ) CSxS (˜ρ)= δxS,xS ( ˜res (˜ρ)). VS − VS S S

Proof. This follows from Proposition 5.1.2. (1) and Lemma 1.1.4. ✷

For ρ , we deﬁne the subset (ρ ) of by S ∈FS FVS S FVS (ρ ) := ρ˜ ˜res (˜ρ)= ρ . FVS S { ∈FVS | S S}

We then deﬁne the arithmetic Dijkgraaf-Witten invariant ZVS (ρS) of ρS with respect to xS by x x CS S (˜ρ) S VS ZVS (ρS) := ζN . ρ˜ FV (ρS ) ∈ XS xS ˜ Theorem 5.1.4. (1) ZVS (ρS) is independent of the choice of βρpi . (2) We have ZxS xS . VS ∈ HS

Proof. (1) This follows from Proposition 5.1.2. (1). (2) This follows from Proposition 5.1.2. (2). ✷

xS We call ZVS the arithmetic Dijkgraaf-Witten partition function) for VS with respect to xS.

51 Proposition 5.1.5. For sections x , x′ Γ( , ) we see that S S ∈ FS LS ′ x′ xS,xS xS S Θ (ZVS )= ZVS . Proof. This follows from Proposition 5.1.3. ✷

By the identiﬁcation (3.1.3), ZxS deﬁnes the element Z of which is VS VS HS independent of the choice of xS. We call it the arithmetic Dijkgraaf-Witten partition function for VS.

In the above, the orientation of VS is chosen so that it is compatible with that of ∂VS as explained in the subsection 4.4. Let VS∗ denote VS with opposite orientation. Then, following (4.4.7), the arithmetic Chern-Simons functional and the arithmetic Dijkgraaf-Witten partition function are given by

x CS S (ρ) xS xS xS − XS (5.1.6) CS ∗ = CSV , Z ∗ (ρS)= ζN . VS − S VS ρ V ∈FXS 5.2. Gluing formulas for arithmetic Chern-Simons invariants and gluing formulas for arithmetic Dijkgraaf-Witten partition functions. Let S1 and S2 be disjoint sets of ﬁnite primes of k, where S1 may be empty and S2 is non-empty. We assume that any prime dividing N is contained in S2 if S1 is empty and that any prime dividing N is contained in S1 if S1 is non-empty. We let S := S S . We may think of X as the space obtained by gluing 1 ⊔ 2 S1 X and V ∗ along ∂V . Let η : Π Π , ιp : Πp Π , vp : Πp Π˜ p, and S S2 S2 S S → S1 → S → up : Π˜ p Π be the natural homomorphisms, where p S , so that we have → S1 ∈ 2 η ιp = up vp for p S . S ◦ ◦ ∈ 2 ❍ ✟✯ ΠS ❍ηS ιp ✟ ❍ ✟✟ ❍❥ Πp ❍ ✟✯ ΠS1 ❍ ✟✟ vp ❍❍❥ ✟ up Π˜ p

⊞ Let : S1 S2 be the map deﬁned as in (4.3.1) and (4.3.2). Now we L ×L →LS have the following decomposition formula.

Theorem 5.2.1 (Decomposition formula). For ρ Hom (Π ,G), we have ∈ cont S1 ⊞ CSX (ρ) CSVS ((ρ up)p S2 )= CSX (ρ ηS). S1 2 ◦ ∈ S ◦ 52 Proof. Case that S1 = . Although this may be well known, we give a proof for the sake of readers. By∅ the Artin–Verdier Duality for compact support étale cohomologies ([Mil; Chapter II. Theorem 3.1]) and modified étale cohomologies ([Bi; Theorem 5.1]), we have the following isomorphisms for a fixed ζ µ , N ∈ N 3 H (X , Z/NZ) Hom (Z/NZ, G )∼ µ (k)∼ Z/NZ, comp S ∼= XS m,XS ∼= N ∼=

3 H (X , Z/NZ) = Hom (Z/NZ, G )∼ = µ (k)∼ = Z/NZ, k ∼ Xk m,Xk ∼ N ∼ where Gm,XS (resp. Gm,Xk ) is the sheaf of units on XS (resp. Xk) and ( )∼ is given by Hom( , Q/Z). We denote the isomorphisms above by inv′ : −3 − 3 Hcomp(XS, Z/NZ) Z/NZ and inv : H (Xk, Z/NZ) Z/NZ. Now we re- → 3 → call the deﬁnition of Hcomp(XS, Z/NZ) ([Mil; p.165]). We deﬁne the complex Ccomp(ΠS, Z/NZ) by

n n n 1 C (Π , Z/NZ) := C (Π , Z/NZ) C − (Πp, Z/NZ), comp S S × p S Y∈

d(a, (bp)) := (da, (Resp(a) dbp)), − n n 1 n where a C (ΠS, Z/NZ)and(bp) p S C − (Πp, Z/NZ).Hcomp(XS, Z/NZ) is deﬁned∈ by ∈ ∈ Q n n Hcomp(XS, Z/NZ) := H (Ccomp∗ (ΠS, Z/NZ)).

3 Then we can describe inv′ : Hcomp(XS, Z/NZ) Z/NZ as follows. Let 3 →3 [(a, (bp))] Hcomp(XS, Z/NZ). Since da = 0 and H (ΠS, Z/NZ) = 0, there is a cochain∈b C2(Π , Z/NZ) such that db = a. Then we have ∈ S

inv′([(a, (bp)]) = invp([Resp(b) bp]), − p S X∈ 2 where invp : H (Πp, Z/NZ) Z/NZ be the canonical isomorphism given by the theory of Brauer groups.→ We note that the right side of the equation above doesn’t depend on the choice of b. Recall that Π˜ k denotes the modified 3 3 étale fundamental group of Xk. Let j3 : H (Π˜ k, Z/NZ) H (Xk, Z/NZ) be the natural homomorphism induced by the modified Hochschild-Serr→ e spectral sequence ([H; Corollary 2.2.8]). We describe the image of the cohomology class [c ρ] H3(Π , Z/NZ) by the composed map ◦ ∈ k 1 3 3 inv′− inv j : H (Π˜ , Z/NZ) H (X , Z/NZ). ◦ ◦ 3 k → comp S 53 3 3 Since c (ρ ηS) Z (ΠS, Z/NZ) and H (ΠS, Z/NZ) = 0, there exists a ◦ ◦ 2∈ cochain βρ ηS C (ΠS, Z/NZ) such that dβρ ηS = c (ρ ηS). We note that ◦ ∈ ◦ ◦ ◦ 3 ˜ dResp(βρ ηS )= d(βρ ηS ιp)= c ρ up vp. Since c (ρ up) Z (Πp, Z/NZ) 3 ◦˜ ◦ ◦ 2 ˜ ◦ ◦ ◦ ◦ ◦ ∈ ˜ and H (Πp, Z/NZ) = H (Πp, Z/NZ) = 0, there exists a cochain βρ up ◦ 2 ˜ ˜ ˜ ∈ C (Πp, Z/NZ) such that dβρ up = c (ρ up). We set βρ up vp := βρ up vp 2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ∈ C (Πp, Z/NZ). So we have dβρ up vp = c (ρ up vp). Then we obtain ◦ ◦ ◦ ◦ ◦ 1 (inv′− inv j3)([c ρ]) = [(c (ρ ηS), (βρ up ))]. ◦ ◦ ◦ ◦ ◦ ◦

We see that [Resp(βρ ηS )], [βρ up vp ] p(ρ up vp). Thus we obtain ◦ ◦ ◦ ∈L ◦ ◦ CS (ρ) = (inv j )([c ρ]) Xk ◦ 3 ◦ 1 = (inv′ inv′− inv j )([c ρ]) ◦ ◦ ◦ 3 ◦ = inv′([(c (ρ ηS), (βρ up vp ))]) ◦ ◦ ◦ ◦

= invp([Resp(βρ ηS ) βρ up vp )]) ◦ − ◦ ◦ p S X∈ = CSXS (ρ ηS) CSVS ((ρ up)p S). ◦ − ◦ ∈ Case that S = . Let β C2(Π , Z/NZ) be a cochain such that dβ = c ρ. 1 6 ∅ ρ ∈ S1 ρ ◦ We have d(βρ ηS)= c (ρ ηS) and d(βρ ιp)= c (ρ ιp) for p S2. So we obtain ◦ ◦ ◦ ◦ ◦ ◦ ∈

CSXS (ρ)+ CSVS2 ((ρ up)p S2 ) = [(βρ ηS ιp)p S1 ] + [(βρ up vp)p S2 ] 1 ◦ ∈ ◦ ◦ ∈ ◦ ◦ ∈ = [(βρ up vp)p S] ◦ ◦ ∈ = [(βρ ηS ιp)p S] ◦ ◦ ∈ = CS (ρ η ). ✷ XS ◦ S

Let x Γ( , ) (i = 1, 2)) be any sections. We deﬁne the section Si ∈ FSi LSi xS Γ( S, ) by ∈ F LS

xS (ρS1 , ρS2 ) := xS1 (ρS1 )+ xS1 (ρS2 ).

By Theorem 5.1. and Theorem 2.2.9. (1), we obtain the following

Corollary 5.2.2. Notations being as above, we have the following equality in Z/NZ. xS1 xS2 xS CS (ρ)+ CSVS ((ρ up)p S2 )= CS (ρ ηS). XS1 2 ◦ ∈ XS ◦ 54 We consider the situation that we obtain the space XS1 by gluing XS and VS∗2 xS xS2 xS1 along ∂V . We deﬁne the pairing <, >: ∗ by S2 HS × HS2 → HS1

∗ ∗ (5.2.3) < θS, θS2 > (ρS1 ) := θS(ρS1 , ρS2 )θS2 (ρS2 )

ρS S X2 ∈F 2

xS xS2 ∗ for θS , θS ∗ and ρS1 S1 . This induces the pairing <, >: ∈ HS 2 ∈ HS2 ∈ F ∗ H by (3.1.2). Now we prove the following gluing formula. HS × HS2 → S1 Theorem 5.2.4 (Gluing formula). Notations being as above, We have the following equality = Z . XS S2 XS1

Proof. We show the equality

xS xS1 xS1 = Z XS S2 XS1 for any sections x Γ( , ) (i =1, 2). Noting (5.1.6), we have Si ∈ FSi LSi x xS ′ S2 ′ x xS1 CS (ρ ) CSV (˜ρ ) S XS − S2 (ρS1 ) = ζN ζN XS S2 ′ ρS FS ρ F (ρS ,ρS ) ρ˜ FV (ρS ) X2 ∈ 2 ∈ XSX1 2 ∈ XS2 2 x xS ′ S2 CS (ρ ) CSV (˜ρ) XS − S2 = ζN ′ ρS FS (ρ ,ρ˜) F (ρS ,ρS ) FV (ρS ) X2 ∈ 2 ∈ XS X1 2 × S2 2 for ρ . We deﬁne the map S1 ∈FS1

χ(ρS1 ): (ρS1 ) (ρS1 , ρS2 ) V (ρS2 ) FXS1 → FXS ×F S2 ρS S 2G∈F 2 by

χ(ρS1 )(ρ1)=(ρ1 ηS, (ρ1 up)p S2 ) ◦ ◦ ∈ for ρ1 X (ρS1 ). In order to obtain the required statement by Corollary ∈ F S1 5.2.2, it suﬃces to show that χ(ρS1 ) is bijective. (Though this may be seen by noticing that ΠS1 is the push-out of the maps ιp and vp (ΠS1 is the amalgamated product of Π and Π˜ along Πp) for S = p , we give here a straightforward S k 2 { } proof.)

χ(ρS1 ) is injective: Suppose χ(ρS1 )(ρ1) = χ(ρS1 )(ρ1′ ) for ρ1, ρ1′ F (ρS1 ). ∈ XS1 55 Then ρ η = ρ′ η . Since η is surjective, ρ = ρ′ . 1 ◦ S 1 ◦ S S 1 1 χ(ρS1 ) is surjective: Let (ρ, (˜ρp)p S2 ) FX (ρS1 , ρS2 ) FVS (ρS2 ). Then we ∈ S 2 have ∈ ×

resS1 (ρ)= ρS1 , resS2 (ρ)= ρS2 , ˜resS2 ((˜ρp)p S2 )= ρS2 . ∈ Since ˜resp(˜ρp) is unramiﬁed representation of Πp for p S , ρ is unramiﬁed ∈ 2 over S2. Therefore there is ρ1 such that ρ = ρ1 ηS. Since we see that ∈FXS1 ◦

ρ up vp = ρ η ιp = ρ ιp =ρ ˜p vp 1 ◦ ◦ 1 ◦ S ◦ ◦ ◦ for p S2 and vp is surjective, we have ρ1 up =ρ ˜p for p S2. Hence ∈ ◦ ✷ ∈ χ(ρS1 )(ρ1)=(ρ, (˜ρp)p S2 ) and so χ(ρS1 ) is surjective. ∈

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H. Hirano: Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan. e-mail: [email protected]

J. Kim: Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan. Current address: 3-18-3, Megurohoncho, Meguro-ku, Tokyo 152-0002, Japan e-mail: [email protected]

M. Morishita: Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan. e-mail: [email protected]