$$ \displaystyle{  Z(\overline{X}_k) := \frac{1}{\# G} \sum_{\rho \in {\cal M}_{\overline{X}_k}} \zeta_N^{CS_{\overline{X}_k}(\rho)} = \sum_{\rho \in {\cal M}_{\overline{X}_k}} \zeta_N^{CS_{\overline{X}_k}(\rho)}    }$$

and the arithmetic Dijkgraaf-Witten partition function by $Z_{\overline{X}_k} : {\cal F}_{\emptyset} \rightarrow F$ by $Z_{\overline{X}_k}(*) := Z(\overline{X}_k)$ for $* \in {\cal F}_{\emptyset}$. So we have

$$ Z_{\overline{X}_k} \in {\cal H}_{\emptyset}.$$

We note that when $[c]$ is trivial,  $Z({\overline{X}_k})$ coincides with the (averaged) number of continuous homomorphism from $\tilde{\Pi}_k$  to $G$:

$$ Z(\overline{X}_k) = \frac{ \# {\rm Hom}_{\rm cont}(\tilde{\Pi}_k,G) }{\# G}, $$

which is the classical invariant for the number field $k$.

4.4. Disjoint union of finite sets of primes and reversing the orientation of $\partial V_S$. Let $S_1 = \{ \frak{p}_1, \dots, \frak{p}_{r_1} \}$ and $S_2 = \{ \frak{p}_{r_1+1}, \dots , \frak{p}_r \}$ be disjoint sets of finite primes of $k$ and let $S = S_1 \sqcup S_2$. We include the case where $S_1$ is empty, but $S_2$ is non-empty.  (For the case where $S_1$ and $S_2$ are both empty, the following arguments are trivial.) Then we have

$$ {\cal F}_S = {\cal F}_{S_1} \times {\cal F}_{S_2}.$$

For the arithmetic quantization principal $\mathbb{Z}/N\mathbb{Z}$-bundles, we define the map

$$ \boxplus  :  {\cal L}_{S_1} \times {\cal L}_{S_2} \longrightarrow {\cal L}_S,$$

as follows. For the case that $S_1 = \emptyset$ (and so $S_2 = S$), we set

$$ m \boxplus [\alpha_{S_2}] := [\alpha_{S_2}].m   \leqno{(4.4.1)}$$

for $(m, [\alpha_S]) \in {\cal L}_{\emptyset} \times {\cal L}_{S_2}$. For the case that $S_1 \neq \emptyset$, we set

$$ [\alpha_{S_1}] \boxplus [\alpha_{S_2}] := [(\alpha_{S_1}, \alpha_{S_2})]   \leqno{(4.4.2)}$$

for $([\alpha_{S_1}],[\alpha_{S_2}]) \in {\cal L}_{S_1} \times {\cal L}_{S_2}$.

For the arithmetic quantization $F$-line bundles, we let $p_i^{*}(L_{S_i})$ be the $G$-equivariant $F$-line bundle over ${\cal F}_S$ induced from $L_{S_i}$ by the projection ${\cal F}_S \rightarrow {\cal F}_{S_i}$ for $i = 1, 2$:

$$ p_i^{*}(L_{S_i}) := \{ ( \rho_S, [\alpha_{S_1}]) \, | \, \rho_{S_i} = \varpi_{S_i}([\alpha_{S_i}]) \; (i=1, 2) \}$$

for $\rho_S = (\rho_{S_1}, \rho_{S_2})$.  When $S_1 = \emptyset$, we think of $p_i^{*}(L_{\emptyset}) = F$ simply over ${\cal  F}_{\emptyset} = \{*\}$. Let

$$ p_i^{*}(\varpi_i) : p_i^{*}(L_{S_i}) \longrightarrow {\cal F}_S $$

be the projection. The fiber over $\rho_S = (\rho_{S_1}, \rho_{S_2})$ is given by

$$ \begin{array}{ll} p_i^{*}(\varpi_i)^{-1}(\rho_S) & = \{ \rho_S \} \times \{ [ ([\alpha_{S_i}],z_i) \in L_{S_i} \; |\; \rho_{S_i} = \varpi_{S_i}([\alpha_{S_i}]), z_i \in F \}\\
& = L_{S_i}(\rho_{S_i})\\
& \simeq F,
\end{array}$$

where $L_{S_i}(\rho_{S_i})$ is as in (2.1.38). We set

$$ L_{S_1} \boxtimes L_{S_2} := p_1^{*}(L_{S_1}) \otimes p_2^{*}(L_{S_2}),$$

which is the $F$-line bundle over ${\cal F}_S$ and whose element is written by

$$ (\rho_S, [ ([\alpha_{S_1}],z_1)] \otimes [ ([\alpha_{S_2}],z_2)]),$$

where $\rho_S = (\rho_{S_1}, \rho_{S_2}) \in {\cal F}_S$, $([\alpha_{S_i}],z_i) \in L_{S_i}(\rho_{S_i})$. The right action  on $L_{S_1} \boxtimes L_{S_2}$ is defined by

$$ (\rho_S, [ ([\alpha_{S_1}],z_1)] \otimes [ ([\alpha_{S_2}],z_2)]).g := (\rho_S.g, [ ([\alpha_{S_1}].g, z_1)] \otimes [ ([\alpha_{S_2}].g, z_2)])$$

so that the projection $L_{S_1} \boxtimes L_{S_2} \rightarrow {\cal F}_S$ is $G$-equivariant.
Then, as in Proposition 2.1.?, we have the isomorphism of $G$-equivariant $F$-line bundles over ${\cal F}_S$:

$$ L_{S_1} \boxtimes L_{S_2} \stackrel{\sim}{\longrightarrow} L_S; \; (\rho_S, [ ([\alpha_{S_1}],z_1)] \otimes [ ([\alpha_{S_2}],z_2)]) \mapsto [([\alpha_S],z_1z_2)],$$

where $\alpha_S = (\alpha_{S_1},\alpha_{S_2})$. Choose $x_{S_i} \in \Gamma({\cal F}_{S_i}, {\cal L}_{S_i})$ and let $x_S := [(x_{S_1},x_{S_2})] \in \Gamma({\cal F}_S,{\cal L}_S)$. Then we see that

$$ \lambda_{S_1}^{x_{S_1}}(g, \rho_{S_1}) + \lambda_{S_2}^{x_{S_2}}(g, \rho_{S_2}) = \lambda_S^{x_S}(g,\rho_S)$$

for $g \in G$ and $\rho_S = (\rho_{S_1},\rho_{S_2})$.

Proposition 4.4.3. For $\theta_i \in {\cal H}_{S_i}^{x_{S_i}}$ ($i=1,2$), we define $\theta_1\cdot \theta_2 \in {\cal H}_S^{x_S}$ by

$$ (\theta_1 \cdot\theta_2)(\rho_S) := \theta_1(\rho_{S_1}) \theta_2(\rho_{S_2})$$

for $\rho_S = (\rho_{S_1}, \rho_{S_2})$. Then we have the following isomorphism of $F$-vector spaces

$$ {\cal H}_{S_1}^{x_{S_1}} \otimes {\cal H}_{S_2}^{x_{S_2}} \stackrel{\sim}{\longrightarrow} {\cal H}_{S}^{x_S};\; (\theta_1, \theta_2) \mapsto \theta_1 \cdot \theta_2.$$

For $\theta_i \in {\cal H}_{S_i}$ ($i=1,2$), we define $\theta_1 \boxtimes \theta_2 \in {\cal H}_S$ by

$$ (\theta_1 \boxtimes \theta_2)(\rho_S) := p_1^{*}(\theta_1(\rho_{S_1}))\otimes p_2^{*}(\theta_2(\rho_{S_2}))$$

for $\rho_S = (\rho_{S_1}, \rho_{S_2})$. Then we have the following isomorphism of $F$-vector spaces

$$ {\cal H}_{S_1} \otimes {\cal H}_{S_2} \stackrel{\sim}{\longrightarrow} {\cal H}_S; \; (\theta_1, \theta_2) \mapsto \theta_1 \boxtimes \theta_2.$$

the above isomorphisms are compatible via the isomorphisms $\Theta^{x_{S_i}} : {\cal H}_{S_i} \simeq {\cal H}_{S_i}^{x_{S_i}}$ $(i=1,2)$ and $\Theta^{x_S} : {\cal H}_S \simeq {\cal H}_S^{x_S}$ in (3.1.2).

Proof. For $\theta \in {\cal H}_S^{x_S}$, set $\theta_1(\rho_{S_1}) := \theta(\rho_{S_1},1)$ and $\theta_2(\rho_{S_2}) := \theta_2(1,\rho_{S_2})$. Then $\theta_i \in {\cal H}_{S_i}^{x_{S_i}}$ and the map ${\cal H}_S^{x_S} \rightarrow {\cal H}_{S_1}^{x_{S_1}}\otimes {\cal H}_{S_2}^{x_{S_2}}$; $\theta \mapsto \theta_1 \otimes \theta_2$, gives the inverse of the former map. By the definitions, the second map is compatible with the first one via $\Theta_{S_i}^{x_{S_i}} : {\cal H}_{S_i} \simeq {\cal H}_{S_i}^{x_{S_i}}$ $(i=1,2)$ and $\Theta^{x_S} : {\cal H}_S \simeq {\cal H}_S^{x_S}$  and so we have the following commutative diagram

$$ \begin{array}{rcc}
{\cal H}_{S_1} \otimes {\cal H}_{S_2} & \longrightarrow & {\cal H}_S \\
\Theta^{x_{S_1}}\otimes \Theta^{x_{S_2}}\; \wr \downarrow \;\;\;\; \;\;\;\; & & \;\;\;\;\;\;\;\; \downarrow \wr \; \Theta^{x_S}\\
{\cal H}_{S_1}^{x_{S_1}} \otimes {\cal H}_{S_2}^{x_{S_2}} & \stackrel{\sim}{\longrightarrow} & {\cal H}_S^{x_S}, \\
\end{array}

from which the second isomorphism follows. $\;\;\Box$

Remark 4.4.4. Proposition 4.4.3 may be regarded as an arithmetic analog of the multiplicative property that ${\cal H}_{\Sigma_1 \sqcup \Sigma_2} = {\cal H}_{\Sigma_1} \otimes {\cal H}_{\Sigma_2}$ for disjoint surfaces $\Sigma_1$ and $\Sigma_2$ which is one of the axioms required in $(2+1)$-dimensional TQFT ([A1]).

For a finite prime $\frak{p}$ of $k$, the canonical isomorphism

$$ {\rm inv}_{\frak{p}} :  H^2_{\mbox{\'{e}t}}(\partial V_{\frak{p}}, \mathbb{Z}/N\mathbb{Z}) \stackrel{\sim}{\longrightarrow} \mathbb{Z}/N\mathbb{Z} $$

indicates  that  $\partial V_{\frak{p}}$ is “orientable" and we choose (implicitly) the “orientation" of $\partial V_{\frak{p}}$ corresponding $1 \in \mathbb{Z}/N\mathbb{Z}$. We let $\partial V^{*}_{\frak{p}} = \partial V_{\frak{p}}$ with the “opposite orientation", namely, ${\rm inv}_{\frak{p}}([\partial V^*_{\frak{p}}]) = -1$.

The arithmetic prequantization principal $\mathbb{Z}/N\mathbb{Z}$-bundle  for $\partial V_{\frak{p}}^*$, denoted by  ${\cal L}_{\frak{p}^*}$, is defined (formally) by ${\cal L}_{\frak{p}}$ with the opposite action of the structure group $\mathbb{Z}/N\mathbb{Z}$, $(\alpha_{\frak{p}},m) \mapsto \alpha_{\frak{p}}.(-m)$ for $\alpha_{\frak{p}} \in {\cal L}_{\frak{p}^*}$ and $m \in \mathbb{Z}/N\mathbb{Z}$. So the arithmetic prequantization $F$-line bundle $L_{\frak{p}^*}$ for $\partial V_{\frak{p}}^*$ is the dual bundle of $L_{\frak{p}}$,  $L_{\frak{p}^*} = L_{\frak{p}}^*$. Noting $\Gamma({\cal F}_{\frak{p}}, {\cal L}_{\frak{p}^*}) = \Gamma({\cal F}_{\frak{p}}, {\cal L}_{\frak{p}})$, the arithmetic Chern-Simons $1$-cocycle $\lambda_{\frak{p}^*}^{x_{\frak{p}}}$ for $\partial V_{\frak{p}}^*$ is given by  $- \lambda_{\frak{p}}^{x_{\frak{p}}}$ for $x_{\frak{p}} \in \Gamma({\cal F}_{\frak{p}}, {\cal L}_{\frak{p}^*})$. The actions of $G$ on ${\cal L}_{\frak{p}^*}^{x_{\frak{p}}} = {\cal F}_{\frak{p}} \times \mathbb{Z}/N\mathbb{Z}$ and $L_{\frak{p}^*}^{x_{\frak{p}}} = {\cal F}_{\frak{p}} \times F$ are changed to those via $\lambda_{\frak{p}^*}^{x_{\frak{p}}}$.

For a finite set of finite primes $S = \{ \frak{p}_1, \dots , \frak{p}_r \}$, we set $\partial V_S^* := \partial V_{\frak{p}_1}^* \sqcup \cdots \sqcup \partial V_{\frak{p}_r}^*$. Then the arithmetic prequantization bundles ${\cal L}_{S^*}, L_{S^*}, {\cal L}_{S^*}^{x_S}$ and $L_{S^*}^{x_S}$ ($x_S \in \Gamma({\cal F}_{S}, {\cal L}_{S^*}) = \Gamma({\cal F}_{\frak{p}}, {\cal L}_{S})$) are defined in the similar manner. For the arithmetic Chern-Simons 1-cocycle, we have

$$ \lambda_{S^*}^{x_S} = - \lambda_S^{x_S}.$$

Let  ${\cal H}_{S^*}^{x_S}$ be the arithmetic quantum space for $\partial V_S^{*}$ with respect to $x_S$. Then we see that

$$ \begin{array}{ll} {\cal H}_{S^*}^{x_S} & = \{ \theta^* : {\cal F}_S \rightarrow F \; | \; \theta^*(\rho_S.g)  = \zeta_N^{\lambda_{S^*}^{x_S}(g,\rho_S)}\theta^*(\rho_S) \; \mbox{for}\; \rho_S \in {\cal F}_S, g \in G \}\\
& = \{ \theta^* : {\cal F}_S \rightarrow F \; | \; \theta^*(\rho_S.g)  = \zeta_N^{-\lambda_S(g,\rho_S)}\theta^*(\rho_S) \; \mbox{for}\; \rho_S \in {\cal F}_S, g \in G \}\\
&  = \overline{{\cal H}}_S^{x_S},
\end{array}

where $\overline{{\cal H}}_S^{x_S}$ is the complex conjugate of ${\cal H}_S^{x_S}$. Since the pairing

$$ {\cal H}_{S^*}^{x_S} \times {\cal H}_S^{x_S} \longrightarrow F; \; (\theta^*, \theta) \mapsto \displaystyle{  \sum_{\rho_S \in {\cal F}_S} \theta^*(\rho_S) \theta(\rho_S) }$$

is a (Hermitian) perfect pairing, together with (3.1.2), we have the following

Proposition 4.4.5. ${\cal H}_{S^*}^{x_S}$ and ${\cal H}_{S^*}$ are the dual spaces of ${\cal H}_S^{x_S}$ and ${\cal H}_S$, respectively:

$$ {\cal H}_{S^*}^{x_S} = ({\cal H}_S^{x_S})^*,\;\; {\cal H}_{S^*} = ({\cal H}_S)^*.$$

Remark 4.4.6. Proposition 4.3.5 may be regarded as an arithmetic analog of the involutory property that ${\cal H}_{\Sigma^*} = {\cal H}_{\Sigma}^*$, where  $\Sigma^* = \Sigma$ with the opposite 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 $\overline{X}_S$ so that the boundary $\partial \overline{X}_S$ with induced orientation may be identified with $\partial V_S$. Let $\overline{X}_S^{*}$ denote $\overline{X}_S$ with the opposite orientation. Then,  the arithmetic Chern-Simons functional and the Dijkgraaf-Witten partition function for $\overline{X}_S^{*}$ are given as follows:

$$  CS_{\overline{X}_S^{*}}^{x_S} = - CS_{\overline{X}_S}^{x_S}, \;\; Z_{\overline{X}_S^{*}}^{x_S}(\rho_S) = \sum_{\rho \in {\cal F}_{\overline{X}_S}} \zeta_N^{- CS_{\overline{X}_S}^{x_S}(\rho)}.	\leqno{(4.4.7)}$$

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 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 $V_S$.
For a  finite prime $\frak{p}$ of $k$, let ${\cal O}_{\frak{p}}$ denote the ring of $\frak{p}$-adic integers and we let $V_{\frak{p}} := {\rm Spec}({\cal O}_{\frak{p}})$. For a non-empty finite set of finite primes $S = \{ \frak{p}_1, \cdots , \frak{p}_r \}$ of $k$, let $V_S := V_{\frak{p}_1} \sqcup \cdots \sqcup V_{\frak{p}_r}$, which plays a role analogous to a tubular neighborhood of a link, and so $\partial V_S$ plays a role of the boundary of $V_S$.   In this subsection, we introduce the arithmetic Chern-Simons functional and arithmetic Dijkgraaf-Witten partition function for $V_S$, which will be used for our gluing formula in the next section.

Let $\tilde{\Pi}_{\frak{p}}$ be the étale fundamental group of $V_{\frak{p}}$, namely, the Galois group of the maximal unramified extension of $k_{\frak{p}}$ and we set

$${\cal F}_{V_{\frak{p}}} := {\rm Hom}_{\rm cont}(\tilde{\Pi}_{\frak{p}}, G),\;\; {\cal F}_{V_{S}} := {\cal F}_{V_{\frak{p}_1}} \times \cdots \times  {\cal F}_{V_{\frak{p}_r}}.$$

Since $\tilde{\Pi}_{\frak{p}} \simeq \hat{\mathbb{Z}}$ (profinite infinite cyclic group), ${\cal F}_{V_{\frak{p}}} \simeq G$. $G$ acts on ${\cal F}_{V_{S}}$ from the right by

$$ {\cal F}_{V_S} \times G \rightarrow {\cal F}_{V_S}; \;\; ((\tilde{\rho}_{\frak{p}_i})_i, g) \mapsto \rho.g := (g^{-1} \tilde{\rho}_{\frak{p}_i} g)_i,  $$

and let ${\cal M}_{V_S}$ denote the quotient set by this action:

$$ {\cal M}_{V_S} := {\cal F}_{V_S}/G.$$

Let $\tilde{{\rm res}}_{\frak{p}_i} : {\cal F}_{V_{\frak{p}_i}} \rightarrow {\cal F}_\frak{p} $ and $\tilde{{\rm res}}_S := (\tilde{\rm res}_{\frak{p}_i}) : {\cal F}_{V_S} \rightarrow {\cal F}_{S} $ denote the restriction maps induced by the natural continuous homomorphisms $v_{\frak{p}_i} : \Pi_{\frak{p}_i} \rightarrow \tilde{\Pi}_{\frak{p}_i}$ $(1 \leq i \leq r)$,   which are $G$-equivariant. We denote by $\tilde{{\rm Res}}_{\frak{p}_i}$ and $\tilde{{\rm Res}}_S$  the homomorphisms on cochains given as the pull-back by $v_{\frak{p}_i}$:

$$ \begin{array}{l}
\tilde{{\rm Res}}_{\frak{p}_i} : C^n(\tilde{\Pi}_{\frak{p}_i},\mathbb{Z}/N\mathbb{Z}) \longrightarrow C^n(\Pi_{\frak{p}_i}, \mathbb{Z}/N\mathbb{Z});  \alpha_i \mapsto \alpha_i \circ v_{\frak{p}_i},\\
\tilde{{\rm Res}}_S := (\tilde{{\rm Res}}_{\frak{p}_i}) : \prod_{i=1}^r  C^n(\tilde{\Pi}_{\frak{p}_i},\mathbb{Z}/N\mathbb{Z}) \longrightarrow \prod_{i=1}^r C^n(\Pi_{\frak{p}_i}, \mathbb{Z}/N\mathbb{Z}); (\alpha_i) \mapsto (\alpha_i \circ v_{\frak{p}_i}).\\
\end{array}$$

For $ \tilde{\rho}=(\tilde{\rho}_{{\frak{p}_i}})_i \in {\cal F}_{V_S}$, $c \circ \tilde{\rho}_{{\frak{p}_i}} \in Z^3(\tilde{\Pi}_{{\frak{p}_i}}, \mathbb{Z}/N\mathbb{Z})$. Since $H^3(\tilde{\Pi}_{\frak{p}_i},\mathbb{Z}/N \mathbb{Z})=0$, there is $\tilde{\beta}_{{\frak{p}_i}} \in C^2(\tilde{\Pi}_{{\frak{p}_i}}, \mathbb{Z}/N\mathbb{Z})$ such that

$$ c \circ \tilde{\rho}_{{\frak{p}_i}} = d \tilde{\beta}_{{\frak{p}_i}}.$$

We see that

$$c \circ  \tilde{{\rm res}}_{{\frak{p}_i}} (\tilde{\rho}_{{\frak{p}_i}}) = d \tilde{{\rm Res}}_{{\frak{p}_i}} (\tilde{\beta}_{{\frak{p}_i}}) $$

for  $1\leq i \leq r$ and we have

$$ [\tilde{{\rm Res}}_{S}((\tilde{\beta}_{{\frak{p}_i}})_i)]  \in {\cal L}_{S}(\tilde{{\rm res}}_S(\tilde{\rho})).$$

Let $\tilde{{\rm res}}_S^{*}({\cal L}_S)$ be the $G$-equivariant principal $\mathbb{Z}/N\mathbb{Z}$-bundle over ${\cal F}_{V_S}$ induced from ${\cal L}_S$ by $\tilde{{\rm res}}_S$:

$$  \tilde{{\rm res}}_S^{*}({\cal L}_S) := \{ (\tilde{\rho}, \alpha_S) \in {\cal F}_{V_S} \times {\cal L}_S \, | \, \tilde{{\rm res}}_S(\tilde{\rho}) = \varpi_S(\alpha_S) \}$$

and let $\tilde{{\rm res}}_S^{*}(\varpi_S)$ be the projection $ \tilde{{\rm res}}_S^{*}({\cal L}_S) \rightarrow {\cal F}_{V_S}$.
We define the arithmetic Chern-Simons functional $CS_{V_S} : {\cal F}_{V_S} \rightarrow {\cal L}_S$ by

$$ CS_{V_S}(\tilde{\rho}) := [ \tilde{{\rm Res}}_S((\tilde{\beta}_{{\frak{p}_i}})_i)]$$

for $\tilde{\rho} \in {\cal F}_{V_S}$. The value $CS_{V_S}(\tilde{\rho})$ is called the arithmetic Chern-Simons invariant of $\tilde{\rho}$.

Lemma 5.1.1.  (1) $CS_{V_S}(\tilde{\rho})$ is independent of the choice of $\tilde{\beta}_{{\frak{p}_i}}$.

(2) $CS_{V_S}$ is a $G$-equivariant section of $\tilde{{\rm res}}_S^{*}(\varpi_S)$:

$$ CS_{V_S} \in \Gamma_G({\cal F}_{V_S}, \tilde{{\rm res}}_S^{*}({\cal L}_S)) = \Gamma({\cal M}_{V_S}, \tilde{{\rm res}}_S^{*}(\overline{{\cal L}}_S)).$$

Proof. (1) This follows from the fact that the cohomological dimension of $\tilde{\Pi}_{\frak{p}_i}$ is one.

(2) The proof of this lemma is almost same as Lemma 2.2.8. (2). $\;\; \Box$

For a section $x_S = [ (x_{\frak{p}_1}, \dots , x_{\frak{p}_r})] \in \Gamma({\cal F}_S, {\cal L}_S)$, the isomorphism $\Phi_S^{x_S} : {\cal L}_S \stackrel{\sim}{\rightarrow} {\cal L}_S^{x_S}$ induces the isomorphism

$$ \begin{array}{rcl} \tilde{\Psi}^{x_S} : \Gamma_G({\cal F}_{V_S}, \tilde{{\rm res}}_S^{*}({\cal L}_S)) & \stackrel{\sim}{\longrightarrow} & \Gamma_G({\cal F}_{V_S}, \tilde{{\rm res}}_S^{*}({\cal L}_S^{x_S})) = {\rm Map}_G({\cal F}_{V_S},\mathbb{Z}/N\mathbb{Z}); \\
y_S  & \mapsto &  \Phi_S^{x_S} \circ y_S.\\
\end{array}$$

We  define the arithmetic Chern-Simons functional $CS_{V_S}^{x_S} : {\cal F}_{V_S} \rightarrow \mathbb{Z}/N\mathbb{Z}$ with respect to $x_S$ by the image of $ CS_{V_S}$ under $\Psi^{x_S}$.

Proposition 5.1.2. (1) For $\rho \in {\cal F}_{V_S}$, we have

$$ CS_{V_S}^{x_S}(\tilde{\rho}) = \sum_{i=1}^r (\tilde{{\rm Res}}_{S}((\tilde{\beta}_{{\frak{p}_i}})_i)  - x_S(\tilde{{\rm res}}_{S}(\tilde{\rho}))). $$

(2) We have the following equality in  $ C^1(G, {\rm Map}({\cal F}_{V_S}, \mathbb{Z}/N\mathbb{Z}))$

$$ d CS_{V_S} = \tilde{{\rm res}}^*(\lambda_S^{x_S}). $$

Proof. (1) This follows from the definition of $\tilde{\Psi}^{x_S}$.

(2) Since $CS_{V_S} \in {\rm Map}_G({\cal F}_{V_S}, \mathbb{Z}/N\mathbb{Z})$, we have

$$ CS_{V_S}^{x_S}(\tilde{\rho}.g) = CS_{V_S}^{x_S}(\tilde{\rho}) + \lambda_S^{x_S}(g, \tilde{{\rm res}}_S(\tilde{\rho}))$$

for $g \in G$ and $\tilde{\rho} \in {\cal F}_{V_S}$, which means the assertion.  $\;\; \Box$

Proposition 5.1.3. Let $x'_S \in \Gamma({\cal F}_S, {\cal L}_S)$ be another section, which yields $CS_{V_S}^{x'_S}$ and let $\delta_S^{x_S, x'_S} : {\cal F}_S \rightarrow \mathbb{Z}/N\mathbb{Z}$ be the map in Proposition 2.1.34.
Then we have

$$ CS_{V_S}^{x'_S}(\tilde{\rho}) - CS_{V_S}^{x_S}(\tilde{\rho}) = \delta_S^{x_S,x'_S}(\tilde{{\rm res}}_S(\tilde{\rho})).$$

Proof. This follows from Proposition 5.1.2. (1) and Lemma 1.1.4. $\;\; \Box$

For $\rho_S \in {\cal F}_S$, we define the subset ${\cal F}_{V_S}(\rho_S)$ of ${\cal F}_{V_S}$ by

$$ {\cal F}_{V_S}(\rho_S) := \{ \tilde{\rho} \in {\cal F}_{V_S} \, | \, \tilde{{\rm res}}_S(\tilde{\rho}) = \rho_S \}.$$

We then define the arithmetic Dijkgraaf-Witten invariant $Z_{V_S}(\rho_S)$ of $\rho_S$ with respect to $x_S$ by

$$ Z_{V_S}^{x_S}(\rho_S) := \sum_{\tilde{\rho} \in F_{V_{S} }(\rho_{S})} {\zeta_N}^{ CS^{x_{{S}}}_{ V_{S} }(\tilde{\rho}) }.$$

Theorem 5.1.4. (1) $Z_{V_S}^{x_S}(\rho_S)$ is independent of the choice of $\tilde{\beta}_{\rho_{\frak{p}_i}}$.

(2) We have

$$ Z_{V_S}^{x_S} \in {\cal H}_S^{x_S}.$$

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

(2) This follows from Proposition 5.1.2. (2). $\;\; \Box$

We call $Z_{V_S}^{x_S}$ the arithmetic Dijkgraaf-Witten partition function) for $V_S$ with respect to $x_S$.

Proposition 5.1.5. For sections $x_S, x'_S \in \Gamma({\cal F}_S, {\cal L}_S)$ we see that

$$ \Theta^{x_S,x'_S} (Z_{V_S}^{x_S})=Z_{V_S}^{x'_S}. $$

Proof. This follows from Proposition 5.1.3. $\;\; \Box$

By the identification (3.1.3), $Z_{V_S}^{x_S}$ defines the element $Z_{V_S}$ of ${\cal H}_S$ which is independent of the choice of $x_S.$ We call it the arithmetic Dijkgraaf-Witten partition function for $V_S$.

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

$$  CS_{V_S^{*}}^{x_S} = - CS_{V_S}^{x_S}, \;\; Z_{V_S^{*}}^{x_S}(\rho_S) = \sum_{\rho \in {\cal F}_{V_S}} \zeta_N^{- CS_{\overline{X}_S}^{x_S}(\rho)}.	  \leqno{(5.1.6)}$$

5.2. Gluing formulas for arithmetic Chern-Simons invariants and gluing formulas for arithmetic Dijkgraaf-Witten partition functions.  Let $S_1$ and $S_2$ be disjoint sets of finite primes of $k$, where $S_1$ may be empty and $S_2$ is non-empty. We assume that any prime dividing $N$ is contained in $S_2$ if $S_1$ is empty and that any prime dividing $N$ is contained in $S_1$ if $S_1$ is non-empty. We let $S := S_1 \sqcup S_2$. We may think of $\overline{X}_{S_1}$ as the space obtained by gluing $\overline{X}_S$ and $V_{S_2}^{*}$ along $\partial V_{S_2}$. Let  $\eta_S : \Pi_S \rightarrow \Pi_{S_1}$, $\iota_{\frak{p}} : \Pi_{\frak{p}} \rightarrow \Pi_S$, $v_{\frak{p}} : \Pi_{\frak{p}} \rightarrow \tilde{\Pi}_{\frak{p}}$, and   $u_{\frak{p}} :  \tilde{\Pi}_{\frak{p}} \rightarrow \Pi_{S_1}$ be the natural homomorphisms, where $\frak{p} \in S_2$,  so that  we have  $\eta_S \circ \iota_{\frak{p}}= u_{\frak{p}} \circ v_{\frak{p}}$ for $\frak{p} \in S_2$.

(64,10)(2, -1)16
(64, 15)(2,1)16

Let  $\boxplus  : {\cal{L}}_{S_1} \times {\cal{L}}_{S_2} \rightarrow \cal{L}_{S}$ be the map defined as in (4.3.1) and (4.3.2).  Now we have the following decomposition formula.

Theorem 5.2.1 (Decomposition formula). For $\rho \in {\rm Hom}_{\rm cont}(\Pi_{S_1}, G)$, we have

$$CS_{\overline{X}_{S_1}}(\rho) \boxplus CS_{V_{S_2}}((\rho \circ u_{\frak{p}})_{\frak{p} \in S_2} ) = CS_{\overline{X}_S}(\rho \circ \eta_S).  $$

Proof. Case that $S_1= \emptyset$. 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 $\zeta_N \in \mu_N$,

$$H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \cong {\rm Hom}_{X_S}(\mathbb{Z}/N \mathbb{Z}, \mathbb{G}_{m, X_S} )^{\sim} \cong \mu_N(k)^{\sim} \cong \mathbb{Z}/N \mathbb{Z}, $$

$$H^3(\overline{X}_k,\mathbb{Z}/N \mathbb{Z}) \cong {\rm Hom}_{\overline{X}_k}(\mathbb{Z}/N \mathbb{Z}, \mathbb{G}_{m, \overline{X}_k} )^{\sim} \cong \mu_N(k)^{\sim} \cong \mathbb{Z}/N \mathbb{Z}, $$

where $\mathbb{G}_{m, X_S}$ (resp. $\mathbb{G}_{m, \overline{X}_k}$) is the  sheaf of units on $X_S$ (resp. $\overline{X}_k$) and $(-)^{\sim}$ is given by ${\rm Hom}(-,\mathbb{Q}/\mathbb{Z})$. We denote the isomorphisms above by ${\rm inv}' :H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \rightarrow  \mathbb{Z}/N \mathbb{Z}$ and  ${\rm inv} :H^3(\overline{X}_k,\mathbb{Z}/N \mathbb{Z}) \rightarrow \mathbb{Z}/N \mathbb{Z}$. Now we recall the definition of $H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$ ([Mil; p.165]). We define the complex $C_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z})$ by

$$C^n_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z}) :=C^n(\Pi_S,\mathbb{Z}/N \mathbb{Z}) \times  \prod_{\frak{p}\in S} C^{n-1} (\Pi_{\frak{p}} ,\mathbb{Z}/N\mathbb{Z}), $$

$$ d(a ,(b_\frak{p}) ) := (da, ({\rm Res}_\frak{p} (a) - d b_\frak{p})), $$

where $a \in C^n(\Pi_S,\mathbb{Z}/N \mathbb{Z})$ and $ (b_\frak{p}) \in \prod_{\frak{p}\in S} C^{n-1} (\Pi_{\frak{p}} ,\mathbb{Z}/N\mathbb{Z}).$ $H^n_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$ is defined by

$$ H^n_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) := H^n(C^{*}_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z})).$$
Then we can describe ${\rm inv}' :H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \rightarrow  \mathbb{Z}/N \mathbb{Z}$ as follows. Let $[(a,(b_{\frak{p}}))] \in  H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$. Since $da=0$ and $H^3(\Pi_S,\mathbb{Z}/N \mathbb{Z})=0$, there is a cochain $b\in C^2 (\Pi_S,\mathbb{Z}/N \mathbb{Z})$ such that $db=a$. Then we have

$${\rm inv}'  ([(a,(b_\frak{p})]) = {\displaystyle \sum_{\frak{p} \in S} {\rm inv}_\frak{p} ([{\rm Res}_\frak{p} (b) -b_\frak{p} ])}, $$

where ${\rm inv}_\frak{p} : H^2(\Pi_\frak{p},\mathbb{Z}/N \mathbb{Z}) \rightarrow \mathbb{Z}/N \mathbb{Z}$ 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 $\tilde{\Pi}_k$ denotes the modified étale fundamental group of $\overline{X}_k$. Let $j_3 : H^3(\tilde{\Pi}_k , \mathbb{Z}/ N \mathbb{Z}) \rightarrow H^3(\overline{X}_k, \mathbb{Z}/N \mathbb{Z})$ be the natural homomorphism induced by the modified Hochschild-Serre spectral sequence ([H; Corollary 2.2.8]). We describe the image of the cohomology class $[c \circ \rho ] \in H^3(\Pi_k,\mathbb{Z}/ N\mathbb{Z})$ by the composed map

$${\rm inv'}^{-1} \circ {\rm inv} \circ j_3 : H^3(\tilde{\Pi}_k , \mathbb{Z}/ N \mathbb{Z}) \rightarrow H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}).$$

Since $c \circ (\rho \circ \eta_S) \in Z^3(\Pi_S,\mathbb{Z}/ N \mathbb{Z})$ and $H^3(\Pi_S,\mathbb{Z}/N \mathbb{Z})=0$, there exists a cochain $\beta_{\rho \circ \eta_S} \in C^2(\Pi_S,\mathbb{Z}/N \mathbb{Z})$ such that $d \beta_{\rho \circ \eta_S} =c \circ (\rho \circ \eta_S).$  We note that $d {\rm Res}_\frak{p} (\beta_{\rho \circ \eta_S})=d (\beta_{\rho \circ \eta_S} \circ \iota_\frak{p}) =c \circ \rho \circ u_\frak{p} \circ v_{\frak{p}}$.  Since $c \circ (\rho \circ u_\frak{p}) \in Z^3(\tilde{\Pi}_\frak{p},\mathbb{Z}/ N \mathbb{Z})$ and $H^3(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})=H^2(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})=0$, there  exists a cochain $\tilde{\beta}_{\rho \circ u_\frak{p}} \in C^2(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})$ such that $d \tilde{\beta}_{\rho \circ u_\frak{p}} =c \circ (\rho \circ u_\frak{p})$. We set $\beta_{\rho \circ u_{\frak{p}} \circ v_{\frak{p}}} := \tilde{\beta}_{\rho \circ u_\frak{p}} \circ v_\frak{p} \in C^2(\Pi_\frak{p},\mathbb{Z}/N \mathbb{Z}).$ So we have $d \beta_{\rho \circ u_\frak{p} \circ v_{\frak{p}} } = c \circ (\rho \circ u_\frak{p} \circ v_{\frak{p}}).$ Then we obtain

$$ ({\rm inv'}^{-1} \circ {\rm inv} \circ j_3)([c \circ \rho])=[(c \circ (\rho \circ \eta_S),(\beta_{\rho \circ u_\frak{p}}))].$$

We see that $[{\rm Res}_\frak{p} (\beta_{\rho \circ \eta_S})], [\beta_{\rho \circ u_\frak{p} \circ v_{\frak{p}} }] \in {\cal L}_\frak{p} (\rho \circ u_\frak{p} \circ v_{\frak{p}} )$. Thus we obtain

\begin{eqnarray}
CS_{\overline{X}_k}(\rho)&=& ({\rm inv} \circ j_3)([c\circ \rho]) \nonumber \\
&=&  ({\rm inv'} \circ {\rm inv'}^{-1} \circ {\rm inv} \circ j_3)([c \circ \rho]) \nonumber  \\
&=& {\rm inv'} ([(c \circ (\rho \circ \eta_S),(\beta_{\rho \circ u_\frak{p} \circ v_{\frak{p}} }))]) \nonumber \\
&=& {\displaystyle \sum_{\frak{p} \in S} {\rm inv}_\frak{p} ([{\rm Res}_\frak{p} (\beta_{\rho \circ \eta_S}) -\beta_{\rho \circ u_\frak{p} \circ v_{\frak{p}} }) ])} \nonumber \\
&=& CS_{\overline{X}_S}(\rho \circ \eta_S)-CS_{V_S}( {(\rho \circ u_\frak{p})}_{\frak{p} \in S}). \nonumber
\end{eqnarray}

Case that $S_1 \neq \emptyset$. Let $\beta_\rho \in C^2(\Pi_{S_1} ,\mathbb{Z} / N \mathbb{Z})$ be a cochain such that $d \beta_\rho=c \circ \rho$. We have $d (\beta_\rho \circ \eta_S)=c \circ (\rho \circ \eta_S)$ and $d (\beta_\rho \circ \iota_\frak{p})=c \circ (\rho \circ \iota_\frak{p})$ for $\frak{p} \in S_2$. So we obtain

\begin{eqnarray}
CS_{\overline{X}_{S_1}}(\rho)+CS_{V_{S_2}}( {(\rho \circ u_\frak{p})}_{\frak{p} \in S_2}) &=& [(\beta_\rho \circ \eta_S \circ \iota_\frak{p})_{ \frak{p} \in S_1} ]+[(\beta_\rho \circ u_\frak{p} \circ v_\frak{p})_{ \frak{p} \in S_2}] \nonumber \\
&=& [(\beta_\rho \circ u_\frak{p} \circ v_{\frak{p}})_{ \frak{p} \in S}] \nonumber  \\
&=& [(\beta_\rho \circ \eta_S \circ \iota_\frak{p})_{ \frak{p} \in S}] \nonumber \\
&=& CS_{\overline{X}_S}({\rho \circ \eta_S}). \;\;\;\; \Box \nonumber
\end{eqnarray}

Let $x_{S_i} \in \Gamma({\cal F}_{S_i}, {\cal{L}}_{S_i}) \ (i=1,2))$ be any sections.  We define the section $x_S \in \Gamma({\cal F}_S, \cal{L}_S)$ by

$$x_S(\rho_{S_1}, \rho_{S_2}) := x_{S_1}(\rho_{S_1}) + x_{S_1}(\rho_{S_2}). $$
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 $\mathbb{Z}/ N \mathbb{Z}$.

$$CS^{x_{S_1}}_{\overline{X}_{S_1}}(\rho)+CS^{x_{S_2}}_{V_{S_2}}((\rho \circ u_{\frak{p}})_{\frak{p} \in S_2} ) = CS^{x_{S}}_{\overline{X}_S}(\rho \circ \eta_S).  $$

We consider the situation that we obtain the space $\overline{X}_{S_1}$  by gluing $\overline{X}_S$ and $V_{S_2}^*$ along $\partial V_{S_2}$.  We define the pairing $<\ ,\ > : {\cal H}^{x_S}_{S} \times {\cal H}^{x_{S_2}}_{{S_2}^*} \rightarrow {\cal H}^{x_{S_1}}_{S_1} $ by

$$<\theta_S , \theta_{{S_2}^*}>(\rho_{S_1}) := \sum_{\rho_{S_2} \in {\cal F}_{S_2}} \theta_S(\rho_{S_1},\rho_{S_2}) \theta_{{S_2}^*}(\rho_{S_2}) \leqno{(5.2.3)}$$

for $\theta_S \in {\cal H}_S^{x_S}, \theta_{S_2^*} \in {\cal H}_{S_2^*}^{x_{S_2}}$ and $\rho_{S_1} \in {\cal F}_{S_1}$.   This induces the pairing  $<\ ,\ > : {\cal H}_{S} \times {\cal H}_{{S_2}^*} \rightarrow H_{S_1}$ by (3.1.2).  Now we prove the following gluing formula.

Theorem 5.2.4 (Gluing formula). Notations being as above, We have the following equality

$$<Z_{{\overline{X}}_S}, Z_{V_{S_2}^*}  > \; = \; Z_{{\overline{X}}_{S_1}}. $$

Proof. We show the equality

$$<Z^{x_{S}}_{\overline{X}_S}, Z^{x_{S_1}}_{V_{S_2}^*}  > \; = \; Z^{x_{S_1}}_{{\overline{X}}_{S_1}}$$

for any sections $x_{S_i}  \in \Gamma({\cal F}_{S_i}, {\cal{L}}_{S_i}) \ (i=1,2)$. Noting (5.1.6), we have

\begin{eqnarray}
<Z^{x_{S}}_{{\overline{X}}_S}, Z^{x_{S_1}}_{V_{S_2}^* } > (\rho_{S_1})  &=&  \sum_{\rho_{S_2} \in F_{S_2}} \Bigl( \sum_{\rho' \in F_{ {\overline{X}}_S }(\rho_{S_1}, \rho_{S_2})} {\zeta_N}^{ CS^{x_{S}}_{ \overline{X}_{S} }(\rho') } \Bigr)  \Bigl(\sum_{\tilde{\rho} \in F_{V_{S_2} }(\rho_{S_2})} {\zeta_N}^{ -CS^{x_{{S_2}}}_{ V_{S_2} }(\tilde{\rho}') } \Bigr) \nonumber \\
&=& \sum_{\rho_{S_2} \in F_{S_2}} \Bigl( \sum_{ (\rho', \tilde{\rho}) \in F_{ {\overline{X}}_S }(\rho_{S_1}, \rho_{S_2}) \times F_{V_{S_2} }(\rho_{S_2})} {\zeta_N}^{ CS^{x_{S}}_{ \overline{X}_{S} }(\rho')  -CS^{x_{{S_2}}}_{ V_{S_2} }(\tilde{\rho})} \Bigr)   \nonumber
\end{eqnarray}

for $\rho_{S_1} \in {\cal F}_{S_1}$. We define the map

$$ \chi (\rho_{S_1}) : {\cal F}_{ {\overline{X}}_{S_1} }(\rho_{S_1}) \rightarrow \underset{ \rho_{S_2} \in {\cal F}_{S_2} } {\bigsqcup} \Bigl( {\cal F}_{ {\overline{X}}_S }(\rho_{S_1}, \rho_{S_2}) \times {\cal F}_{V_{S_2} }(\rho_{S_2}) \Bigr)$$

$$ \chi (\rho_{S_1})(\rho_1) = (\rho_1 \circ \eta_S, (\rho_1 \circ u_\frak{p})_{\frak{p} \in S_2} )$$

for $\rho_1 \in {\cal F}_{\overline{X}_{S_1}}(\rho_{S_1})$.  In order to obtain the required statement by Corollary 5.2.2, it suffices to show that  $\chi (\rho_{S_1})$ is bijective. (Though this may be seen by noticing that $\Pi_{S_1}$ is the push-out of the maps $\iota_{\frak{p}}$ and $v_{\frak{p}}$ ($\Pi_{S_1}$ is the amalgamated product of $\Pi_S$ and $\tilde{\Pi}_k$ along $\Pi_{\frak{p}}$) for $S_2 = \{ \frak{p} \}$, we give here a straightforward proof.)

$\chi(\rho_{S_1})$ is injective: Suppose $\chi(\rho_{S_1})(\rho_1) = \chi(\rho_{S_1})(\rho_1')$ for $\rho_1, \rho_1' \in F_{ {\overline{X}}_{S_1} }(\rho_{S_1})$. Then $\rho_1 \circ \eta_S = \rho_1' \circ \eta_S$. Since $\eta_S$ is surjective, $\rho_1 = \rho_1'$.

$\chi (\rho_{S_1})$ is surjective: Let $(\rho, (\tilde{\rho}_{\frak{p}})_{\frak{p} \in S_2}) \in F_{ {\overline{X}}_S }(\rho_{S_1}, \rho_{S_2}) \times F_{V_{S_2} }(\rho_{S_2})$. Then we have

$${\rm res}_{S_1}(\rho) = \rho_{S_1},   {\rm res}_{S_2}(\rho) = \rho_{S_2}, \tilde{\rm res}_{S_2}( (\tilde{\rho}_{\frak{p}})_{\frak{p} \in S_2}) = \rho_{S_2}. $$

Since $\tilde{\rm res}_{\frak{p}}(\tilde{\rho}_{\frak{p}})$ is unramified representation of $\Pi_{\frak{p}}$ for $\frak{p} \in S_2$, $\rho$ is unramified over $S_2$. Therefore there is $\rho_1 \in {\cal F}_{\overline{X}_{S_1}}$ such that $\rho = \rho_1 \circ \eta_S$. Since we see that

$$ \rho_1 \circ u_{\frak{p}} \circ v_{\frak{p}} = \rho_1 \circ \eta_S \circ \iota_{\frak{p}} = \rho \circ \iota_{\frak{p}} = \tilde{\rho}_{\frak{p}} \circ v_{\frak{p}}$$

for $\frak{p} \in S_2$ and $v_{\frak{p}}$ is surjective, we have $\rho_1 \circ u_{\frak{p}} = \tilde{\rho}_{\frak{p}}$ for $\frak{p} \in S_2$. Hence $\chi(\rho_{S_1})(\rho_1) = (\rho, (\tilde{\rho}_{\frak{p}})_{\frak{p} \in S_2})$ and so $\chi(\rho_{S_1})$ is surjective. $\;\; \Box$

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H. Hirano:

Graduate School of Mathematics, Kyushu University,

744, Motooka, Nishi-ku, Fukuoka  819-0395, Japan.

e-mail<EMAIL_ADDRESS>
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_ADDRESS>
M. Morishita:

Graduate School of Mathematics, Kyushu University,

744, Motooka, Nishi-ku, Fukuoka  819-0395, Japan.

e-mail<EMAIL_ADDRESS>
1. Preliminaries on torsors and group cochains

2. Classical theory

2.1. Arithmetic prequantization bundles and arithmetic Chern-Simons 1-cocycles

2.2. Arithmetic Chern-Simons functionals

3. Quantum theory

3.1. Arithmetic quantum spaces

3.2. Arithmetic Dijkgraaf-Witten partition functions

4. Some basic and functorial properties

4.1 Change of the 3-cocycle $c$

4.2. Change of number fields

4.3. The case that $S$ is empty

4.4. Disjoint union of finite set of primes and reversing the orientation of $\partial V_S$

5. Decomposition and gluing  formulas

5.1. Arithmetic Chern-Simons functionals and arithmetic Dijkgraaf-Witten partition functions for $V_S$

5.2. Decomposition formulas for arithmetic Chern-Simons invariants and gluing formulas for arithmetic Dijkgraaf-Witten invariants

Example 3.1.2  For the case that $G$ is an abelian group, we can give a formula of the dimension of ${\cal H}_S$ as follows. By (1.2.2), $G$ acts on the cocycle $c$ trivially. By the construction of the Chern-Simons 1-cocycle (2.1.5) $\sim$ (2.1.9), $\lambda_{\frak{p}} = 0$ for any $\frak{p}$ and so $\lambda_S = 0$. By (2.1.1) and (2.2.1), $G$ acts on ${\cal F}_S$ trivially, and  $G$ acts on  the arithmetic quantum $F$-line bundle $L_S = {\cal F}_S \times \mathbb{Z}/N\mathbb{Z}$ trivially by (2.1.). So, by (3.1.1), the arithmetic quantum space ${\cal H}_S$ is just the set of maps from ${\cal F}_S$ to $F$:

$$ {\cal H}_S = {\rm Map}({\cal F}_S, F). \leqno{(3.1.2.1)}$$

Theorem 3.1.2.2. Let us write $G = \displaystyle{ \bigoplus_{j=1}^{t} \mathbb{Z}/m_j\mathbb{Z} }$, where $m_j$'s are  positive integers. Assume that eacch $m_j$ divides $N$. Then we have

$$ \dim {\cal H}_S = \sum_{i=1}^r \sum_{j=1}^t \#(k_{\frak{p}_i}^{\times}/(k_{\frak{p}_i}^{\times})^{m_j}).$$

Proof. By (3.2.1), we have

$$ \dim {\cal H}_S = \# {\cal F}_S = \sum_{i=1}^r \# {\rm Hom}_{\rm cont}(\Pi_{\frak{p}_i}, G).$$

By Tate's local duality ([S1; $\S 5$, Théorème 2], [NSW; Theorem 7.2.6]) and Kummer theory, we have

$$ \begin{array}{ll} {\rm Hom}_{\rm cont}(\Pi_{\frak{p}_i}, G) & = \displaystyle{ \bigoplus_{j=1}^t H^1(\Pi_{\frak{p}_i}, \mathbb{Z}/m_j\mathbb{Z}) }\\
& \simeq \displaystyle{ \bigoplus_{j=1}^t H^1(k_{\frak{p}_i}, \mu_{m_j})^{*} }\\
& \simeq \displaystyle{ \bigoplus_{j=1}^t (k_{\frak{p}_i}^{\times}/(k_{\frak{p}_i}^{\times})^{m_j})^{*}, }
\end{array}

where $*$ means the Pontryagin dual and $\mu_{m_j}$ denotes the group of $m_j$-th roots of unity. Hence we obtain the assertion. $\;\; \Box$

In this subsection, we give an extension of the "decomposition formula" by Kim and his collaborators ([CKKPY]). Let $S_1$ and $S_2$ be disjoint sets of finite primes of $k$, where $S_1$ may be empty and $S_2$ is non-empty. We assume that when $S_i$ is non-empty, any prime dividing $N$ is contained in $S_i$. We let $S := S_1 \sqcup S_2$. We may think of $\overline{X}_{S_1}$ as the space obtained by gluing $\overline{X}_S$ and $V_{S_2}$ along $\partial V_{S_2}$. Let  $\varpi_S : \Pi_S \rightarrow \Pi_{S_1}$, $\varpi_{\frak{p}} :  \pi_1^{  {\rm \acute{e}t} }(V_{\frak{p}}) \rightarrow \Pi_{S_1}$, $u_{\frak{p}} : \Pi_{\frak{p}} \rightarrow \Pi_S$, and  $v_{\frak{p}} : \Pi_{\frak{p}} \rightarrow \pi_1^{  {\rm \acute{e}t} }(V_{\frak{p}})$ be natural homomorphisms such that $\varpi_S \circ u_{\frak{p}}=\varpi_{\frak{p}} \circ v_{\frak{p}} $. For $ \frak{p} \in S_2$, we define $w_{\frak{p}} : \Pi_{\frak{p}} \rightarrow \Pi_S$ by $w_{\frak{p}} :=  \varpi_S \circ u_{\frak{p}}=\varpi_{\frak{p}} \circ v_{\frak{p}}$.Then we have the following gluing formula.

Theorem 3.1.4 For $\rho \in {\rm Hom}_{\rm cont}(\Pi_{S_1}, G)$ and a fixed section $x:  {\cal F}_S \rightarrow {\cal L}_S$, we have

$$CS_{\overline{X}_{S_1}}(\rho) = CS_{\overline{X}_S}(\rho \circ \varpi_S) - CS_{V_{S_2}}((\rho \circ \varpi_{\frak{p}})_{\frak{p} \in S_2} ).  $$

Proof. Case that $S_1= \emptyset$. By the Artin–Verdier Duality for compact support étale cohomologies ([Ml; Chapter II. Theorem 3.1]) and modified étale cohomologies ([Bi; Theorem 5.1]), we have the following isomorphisms for a fixed $\zeta_N \in \mu_N$,

$$H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \cong {\rm Hom}_{X_S}(\mathbb{Z}/N \mathbb{Z},G_{m, X_S} )^{\sim} \cong \mu_N(k)^{\sim} \cong \mathbb{Z}/N \mathbb{Z}, $$

$$H^3(\overline{X}_k,\mathbb{Z}/N \mathbb{Z}) \cong {\rm Hom}_{\overline{X}_k}(\mathbb{Z}/N \mathbb{Z},G_{m, \overline{X}_k} )^{\sim} \cong \mu_N(k)^{\sim} \cong \mathbb{Z}/N \mathbb{Z}, $$

where $G_{m, X_S}$ (resp. $G_{m, \overline{X}_k}$) is the  sheaf of units on $X_S$ (resp. $\overline{X}_k$) and $(-)^{\sim}$ is given by ${\rm Hom}(-,\mathbb{Q}/\mathbb{Z})$. We denote the isomorphisms above by ${\rm inv}' :H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \rightarrow  \mathbb{Z}/N \mathbb{Z}$ and  ${\rm inv} :H^3(\overline{X}_k,\mathbb{Z}/N \mathbb{Z}) \rightarrow \mathbb{Z}/N \mathbb{Z}$. Now we recall the definition of $H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$ ([Ml; p.165]). We define the complex $C_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z})$ by

$$C^n_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z}) :=C^n(\Pi_S,\mathbb{Z}/N \mathbb{Z}) \times  \prod_{\frak{p}\in S} C^{n-1} (\Pi_{\frak{p}} ,\mathbb{Z}/N\mathbb{Z}), $$

$$ d(a ,(b_\frak{p}) ) := (da, ({\rm Res}_\frak{p} (a) - d b_\frak{p})), $$

where $a \in C^n(\Pi_S,\mathbb{Z}/N \mathbb{Z})$ and $ (b_\frak{p}) \in \prod_{\frak{p}\in S} C^{n-1} (\Pi_{\frak{p}} ,\mathbb{Z}/N\mathbb{Z}).$ $H^n_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$ is defined by

$$ H^n_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) := H^n(C^{*}_{comp} (\Pi_S,\mathbb{Z}/N \mathbb{Z})).$$
Then we can describe ${\rm inv}' :H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}) \rightarrow  \mathbb{Z}/N \mathbb{Z}$ as follows. Let $[(a,(b_{\frak{p}}))] \in  H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z})$. Since $da=0$ and $H^3(\Pi_S,\mathbb{Z}/N \mathbb{Z})=0$, there is a cochain $b\in C^2 (\Pi_S,\mathbb{Z}/N \mathbb{Z})$ such that $db=a$. Then we have

$${\rm inv}'  ([(a,(b_\frak{p})]) = {\displaystyle \sum_{\frak{p} \in S} {\rm inv}_\frak{p} ([{\rm Res}_\frak{p} (b) -b_\frak{p} ])}, $$

where ${\rm inv}_\frak{p} : H^2(\Pi_\frak{p},\mathbb{Z}/N \mathbb{Z}) \rightarrow \mathbb{Z}/N \mathbb{Z}$ 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 $b$.
Let $j_3 : H^3(\Pi_k , \mathbb{Z}/ N \mathbb{Z}) \rightarrow H^3(\overline{X}_k, \mathbb{Z}/N \mathbb{Z})$ be the natural homomorphism induced by the modified Hochschild-Serre spectral sequence ([H; Corollary 2.2.8]). We describe the image of the cohomology class $[c \circ \rho ] \in H^3(\Pi_k,\mathbb{Z}/ N\mathbb{Z})$ by the composed map

$${\rm inv'}^{-1} \circ {\rm inv} \circ j_3 : H^3(\Pi_k , \mathbb{Z}/ N \mathbb{Z}) \rightarrow H^3_{\rm comp}(X_S,\mathbb{Z}/N \mathbb{Z}).$$

Since $c \circ (\rho \circ \varpi_S) \in Z^3(\Pi_S,\mathbb{Z}/ N \mathbb{Z})$ and $H^3(\Pi_S,\mathbb{Z}/N \mathbb{Z})=0$, there exists a cochain $\beta_{\rho \circ \varpi_S} \in C^2(\Pi_S,\mathbb{Z}/N \mathbb{Z})$ such that $d \beta_{\rho \circ \varpi_S} =c \circ (\rho \circ \varpi_S).$  We note that $d {\rm Res}_\frak{p} (\beta_{\rho \circ \varpi_S})=d (\beta_{\rho \circ \varpi_S} \circ u_\frak{p}) =c \circ \rho \circ w_\frak{p} $.  Since $c \circ (\rho \circ \varpi_\frak{p}) \in Z^3(\tilde{\Pi}_\frak{p},\mathbb{Z}/ N \mathbb{Z})$ and $H^3(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})=H^2(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})=0$, there uniquely exists a cochain $\tilde{\beta}_{\rho \circ \varpi_\frak{p}} \in C^2(\tilde{\Pi}_\frak{p},\mathbb{Z}/N \mathbb{Z})$ such that $d \tilde{\beta}_{\rho \circ \varpi_\frak{p}} =c \circ (\rho \circ \varpi_\frak{p})$. We set $\beta_{\rho \circ w_\frak{p}} := \tilde{\beta}_{\rho \circ \varpi_\frak{p}} \circ v_\frak{p} \in C^2(\Pi_\frak{p},\mathbb{Z}/N \mathbb{Z}).$ So we have $d \beta_{\rho \circ w_\frak{p}} = c \circ (\rho \circ w_\frak{p}).$ Then we have

$$ ({\rm inv'}^{-1} \circ {\rm inv} \circ j_3)([c \circ \rho])=[(c \circ (\rho \circ \varpi_S),(\beta_{\rho \circ w_\frak{p}}))].$$

We see that $[{\rm Res}_\frak{p} (\beta_{\rho \circ \varpi_S})], [\beta_{\rho \circ w_\frak{p}}] \in {\cal L}_\frak{p} (\rho \circ w_\frak{p})$. Thus we obtain

\begin{eqnarray}
CS_{\overline{X}_k}(\rho)&=& ({\rm inv} \circ j_3)([c\circ \rho]) \nonumber \\
&=&  ({\rm inv'} \circ {\rm inv'}^{-1} \circ {\rm inv} \circ j_3)([c \circ \rho]) \nonumber  \\
&=& {\rm inv'} ([(c \circ (\rho \circ \varpi_S),(\beta_{\rho \circ w_\frak{p}}))]) \nonumber \\
&=& {\displaystyle \sum_{\frak{p} \in S} {\rm inv}_\frak{p} ([{\rm Res}_\frak{p} (\beta_{\rho \circ \varpi_S}) -\beta_{\rho \circ w_\frak{p}}) ])} \nonumber \\
&=& CS_{\overline{X}_S}(\beta_{\rho \circ \varpi_S})-CS_{V_S}( {(\rho \circ \varpi_\frak{p})}_{\frak{p} \in S}). \nonumber
\end{eqnarray}

Case that $S_1 \neq \emptyset$. Let $\beta_\rho \in C^2(\Pi_{S_1} ,\mathbb{Z} / N \mathbb{Z})$ be a cochain such that $d \beta_\rho=c \circ \rho$. We have $d (\beta_\rho \circ \varpi_S)=c \circ (\rho \circ \varpi_S)$ and $d (\beta_\rho \circ \varpi_\frak{p})=c \circ (\rho \circ\varpi_\frak{p})$ for $\frak{p} \in S_2$. So we obtain

\begin{eqnarray}
CS_{\overline{X}_{S_1}}(\rho)&=& \sum_{\frak{p} \in S_1} \varphi_{x({\rm res}_\frak{p}(\rho))}({\rm Res}_\frak{p} (\beta_\rho)) \nonumber \\
&=& \sum_{\frak{p} \in S_1} \varphi_{x(\rho \circ w_\frak{p})}(\beta_\rho \circ w_\frak{p}) \nonumber  \\
&=& \sum_{\frak{p} \in S} \varphi_{x(\rho \circ \varpi_S \circ u_{\frak{p}} )}(\beta_\rho \circ \varpi_S \circ u_{\frak{p}}) - \sum_{\frak{p} \in S_2} \varphi_{x(\rho \circ \varpi_{\frak{p}} \circ v_{\frak{p}} )}(\beta_\rho \circ \varpi_{\frak{p}} \circ v_{\frak{p}}) \nonumber \\
&=& \sum_{\frak{p} \in S} \varphi_{x({\rm res}_\frak{p}( \rho \circ \varpi_S ) )}( {\rm Res}_\frak{p} ( \beta_\rho \circ \varpi_S )) - \sum_{\frak{p} \in S_2} \varphi_{x( {\rm res}_\frak{p} (\rho \circ \varpi_{\frak{p}} ) )}({\rm Res}_\frak{p} ( \beta_\rho \circ  \varpi_{\frak{p}} )) \nonumber \\
&=& CS_{\overline{X}_S}(\beta_{\rho \circ \varpi_S})-CS_{V_{S_2}}( {(\rho \circ \varpi_\frak{p})}_{\frak{p} \in S_2}).  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Box \nonumber
\end{eqnarray}

Proposition 4.1.1. (1) There is a map $\kappa_{S}^{c,c'} : {\cal F}_{S} \rightarrow \mathbb{Z}/N\mathbb{Z}$ such that

$$ \lambda_{S}^{c'}(g) - \lambda_{S}^{c}(g) = g.\kappa_{S}^{c,c'} - \kappa_{S}^{c,c'}$$

for any $g \in G$. So the cohomology class $[\lambda_{S}^{c}]$ depends on the cohomology class $[c]$. If $[c] = 0$, then $[\lambda_{\frak{p}}] = 0$.

(2) ${\cal L}_S^{c'}$ (resp $L_S^{c'}$) is isomorphic to ${\cal L}_S^{c}$ (resp $L_S^{c}$) as $G$-equivariant principal $\mathbb{Z}/N\mathbb{Z}$-bundles over ${\cal F}_S$.

Proof. (1) It suffices to show the assertion for the case that $S$ consists of a single $\frak{p}$. There is $b \in C^2(G, \mathbb{Z}/N\mathbb{Z})$ such that  $c' - c = db$.   Since we have

$$ d(\alpha+b \circ \rho_{\frak{p}}) = d\alpha + db \circ \rho_{\frak{p}} = c\circ \rho_{\frak{p}} + (c' -c)\circ \rho_{\frak{p}} = c' \circ \rho_{\frak{p}}$$

for $\alpha \in {\cal L}_{\frak{p}}^{c}(\rho_{\frak{p}})$, we  have the isomorphism of $\mathbb{Z}/N\mathbb{Z}$-torsors

$$ \Phi_{\rho_{\frak{p}}} : {\cal L}_{\frak{p}}^{c}(\rho_{\frak{p}}) \stackrel{\sim}{\longrightarrow} {\cal L}_{\frak{p}}^{c'}(\rho_{\frak{p}}); \;\; \alpha \mapsto \alpha + b \circ \rho_{\frak{p}}. $$

Let $f_{\frak{p}}'(g,\rho_{\frak{p}}) : {\cal L}_{\frak{p}}^{c'}(\rho_{\frak{p}}) \stackrel{\sim}{\rightarrow} {\cal L}_{\frak{p}}^{c'}(\rho_{\frak{p}}.g)$ be the isomorphism of $\Mathbb{Z}/N\mathbb{Z}$-torsors defined as in (2.1.9), using $c'$.
Let $x_{\frak{p}} : {\cal F}_{\frak{p}} \rightarrow {\cal L}_{\frak{p}}$ be a section which yields $\lambda_{\frak{p}}^c$. Since a different choice of the section changes $\lambda_{\frak{p}}$ to a cohomologous cocycle by Proposition 2.1.13, in order to define $\lambda_{\frak{p}}^{c'}$, we may use the section $x'_{\frak{p}}$ so that

$$x'_{\frak{p}}(\rho_{\frak{p}}) = \Phi_{\rho_{\frak{p}}}(x_{\frak{p}}(\rho_{\frak{p}})) \leqno{(4.1.1.1)}$$

for $\rho_{\frak{p}} \in {\cal F}_{\frak{p}}$.  For $\rho_{\frak{p}} \in {\cal F}_{\frak{p}}$, there is uniquely $\kappa_{\frak{p}}(\rho_{\frak{p}}) = \kappa_{\frak{p}}^{c,c'}(\rho_{\frak{p}} \in \mathbb{Z}/N\mathbb{Z}$ such that  the following commutative diagram holds:

$$   \begin{array}{rcl}
{\cal L}_{\frak{p}}(\rho_{\frak{p}})  &\stackrel{\Phi_{\rho_{\frak{p}}}}{\longrightarrow}  & {\cal L}'_{\frak{p}}(\rho_{\frak{p}}) \\
\varphi_{x(\rho_{\frak{p}})} \downarrow \;\; & & \;\; \downarrow \varphi_{x'(\rho_{\frak{p}})} \\
\mathbb{Z}/N\mathbb{Z} & \stackrel{f_{\kappa_{\frak{p}}(\rho_{\frak{p}})}}{\longrightarrow} & \mathbb{Z}/N\mathbb{Z}.
\end{array}

We then have the following commutative diagram for $g \in G, \rho_{\frak{p}} \in {\cal F}_{\frak{p}}$:

(0,38)${\cal L}_{\frak{p}}(\rho_{\frak{p}}.g)$
(50,38)${\cal L}'_{\frak{p}}(\rho_{\frak{p}}.g)$
(40,58)${\cal L}_{\frak{p}}(\rho_{\frak{p}})$
(88,58)${\cal L}'_{\frak{p}}(\rho_{\frak{p}})$

By the commutativity of the upper square diagram,  we have

$$ \lambda_{\frak{p}}^{c}(g, \rho_{\frak{p}}) + \kappa_{\frak{p}}^{c,c'}(\rho_{\frak{p}}.g) = \lambda_{\frak{p}}^{c'}(g, \rho_{\frak{p}}) + \kappa_{\frak{p}}^{c,c'}(\rho_{\frak{p}})$$

and hence $\lambda{\frak{p}}^{c'}(g) - \lambda_{\frak{p}}^{c}(g) = g.\kappa_{\frak{p}}^{c,c'} - \kappa_{\frak{p}}^{c,c'}$. Hence $[\lambda_{\frak{p}}^c]$ depends on the class $[c]$.

When $c = 0$, ${\cal L}_{\frak{p}}^c(\rho_{\frak{p}}) = H^2(\Pi_{\frak{p}}, \mathbb{Z}/N\mathbb{Z})$, the trivial $\mathbb{Z}/N\mathbb{Z}$-torsor for all $\rho_{\frak{p}} \in {\cal F}_{\frak{p}}$. By (2.1.9) and (2.1.11), we can take $h_g = 0$ and so $f_{\frak{p}}^c(g, \rho_{\frak{p}}) = {\rm id}_{H^2(\Pi_{\frak{p}},\mathbb{Z}/N\mathbb{Z})}$ for all $g \in G$ and $\rho_{\frak{p}} \in {\cal F}_{\frak{p}}$. Hence $\lambda_{\frak{p}}^c = 0$
by (2.1.10) and (2.1.11). Since $[\lambda_{\frak{p}}^c]$ depends only on $[c]$ as proved above, $[\lambda_{\frak{p}}^c]=0$ if $[c]=0$.

(2) The assertion follows from Proposition 2.1.16.  $\;\; \Box$.

Let $CS^{c}_{\overline{X}_S}$ (resp. $CS^{c'}_{\overline{X}_S}(\rho)$) be the arithmetic Chern-Simons functionals, ${\cal H}_S^c$ (resp. ${\cal H}_{S}^{c'}$) the arithmetic quantum space and ${\cal Z}_{\overline{X}_S}^c$ (resp. ${\cal Z}_{\overline{X}_S}^{c'}$) the arithmetic Dijkgraaf-Witten functions,  by using $c$ and a section $x_{\frak{p}_i}$ (resp. $c'$ and the above $x'_{\frak{p}_i}$).

Proposition 4.1.2. (1) We have

$$ CS^{c'}_{\overline{X}_S}(\rho) - CS^{c}_{\overline{X}_S}(\rho) = \kappa_S^{c,c'}({\rm res}_S(\rho)). $$

(2) There is an isomorphism of $F$-vector spaces

$$\Psi_S^{c,c'} : {\cal H}_S^c \stackrel{\sim}{\longrightarrow} {\cal H}_S^{c'}$$

such that

$$ \Psi_S^{c,c'}({\cal Z}_{\overline{X}_S}^c) = {\cal Z}_{\overline{X}_S}^{c'}.$$

Proof.  (1) It can be proved in the manner similar to Proposition 2.2.5 (2).

(1) It is also proved as in Proposition 3.1.5 by defining $\Psi_S^{c,c'}(\rho) := \rho \zeta^{-\kappa_S}$ for $\rho \in {\cal F}_{\overline{X}_S}$. $\;\; \Box$

Z. Fiedorowicz and S. Priddy, Homology of Classical Groups over Finite Fields and their Associated Infinite Loop Spaces, SLN 674, Berlin-Heidelberg-New York 1978.
D. Quillen, On the cohomology add K-theory of the genera] linear group over a finite field. Ann. of Math. 96, 552–586 (1972).
E. M. Friedlander, Computations of K-theory of finite fields. Topology 15, 87–109 (1976).
O. Kroll, The cohomology of the finite general linear group, J. Pure and Applied algebra, 54, 1988.

For  Chern-Simons TQFT with compact connected gauge group, we consult [A2], [Ko], [Wi].

Let $T$ be an $A$-torsor and let $B$ be a subgroup of $A$. We then note that the quotient set $T/B$ is an $A/B$-torsor by the obvious manner.
Next, for a positive integer $r$,  let $\frak{s} : A^r \rightarrow A$
be the homomorphism defined by $\frak{s}((a_1,\dots,a_r)) := a_1 + \cdots + a_r$ ($a_i \in A$),
which induces the isomorphism $A^r/{\rm Ker}(\frak{s}) \simeq A$. Thus,
for an $A^r$-torsor $T$,
the quotient set $\Sigma(T) := T/{\rm Ker}(\frak{s})$ is naturally an $A$-torsor. A morphism of $A^r$-torsors
$f : T \rightarrow T'$ induces in the obvious manner a morphism of $A$-torsors
$\Sigma(f) : \Sigma(T) \rightarrow \Sigma(T')$. Hence we have the

$$ \Sigma : \frak{Tors}(A^r) \longrightarrow \frak{Tors}(A). \leqno{(1.1.4)} $$

Dedicated to the memory of Professor Hiroo Miki