数理解析研究所講究録
第 1571 巻 2007 年 23-30

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Chern-Simons variation and Hida-Mazur theory
Masanori Morishita (Kyushu Univ.)
Yuji Terashima (Tokyo Inst. of Tech.)

Chern-Simons invariants over the
We would like to discuss the variation of
deformation space of hyperbolic structures on a knot complement in analogy with HidaMazur theory on the deformation of Galois representations and modular p-adic L-functions.
The motivation and idea are coming from the analogy between knot theory and number
theory, and so let us start to recall the basic analogies between knots and primes.
$SL_{2}(\mathbb{C})$

1. Analogies
knot
$K$

:

$Spec(\mathbb{F}_{p})=K(\hat{\mathbb{Z}}, 1)\mapsto Spec(\mathbb{Z})\cup t\infty$$S^{1}=K(\mathbb{Z}, 1)\mapsto S^{3}=\mathbb{R}^{3}\cup\{\infty\}$

tube neighborhood

prime$rightarrow$$V_{K}$

p.adic integers
p-adic numbers$rightarrow$$\partial V_{K}$$1arrow\langle m_{k}\ranglearrow D_{K}arrow\langle l_{K}\ranglearrow 1$$l_{K}$$Spec(\mathbb{Z}_{p})$$Spec(\mathbb{Q}_{p})$$D_{p}=\pi_{1}^{6t}(Spec(\mathbb{Q}_{p}))$$D_{K}=\pi_{1}(\partial V_{K})$$m\kappa$}$1arrow I_{p}arrow D_{p}arrow(\sigma_{p}\rangle$ $arrow 1$$rightarrow$: longitude of $K$: meridian of K

: Robenius over: inertia gr. over$\sigma_{p}$$I_{p}$$[m_{K}, l_{K}]=1$$\tau_{p}$Here,denotes the maximal tame quotient of
quotient of a monodromy over .
$I_{p}^{l}$$p,$: monodromy over$I_{p^{i}}$

$p,$

$p$

$I_{p}^{t}=\langle\tau_{p}\rangle$$\tau_{p}^{p-1}[\tau_{p}, \sigma_{p}]=1$

In general, we call an element of a$p$

$I_{p}$$X_{K}=S^{3}\backslash K$$rightarrow$$knotgroupG_{K}=\pi_{1}(X_{K})$infinite cyclic cover$X_{p}=Spec(\mathbb{Z}[1/p])$$primegroupG_{p}=\pi_{1}^{\acute{e}t}(X_{p})$$rightarrow$$\mathbb{Z}_{p}$-cover

$X_{K}^{\infty}arrow X_{K}$$x_{p}\infty=Spec(\mathbb{Z}[1/p, \prime^{\infty}\eta_{1})arrow X_{p}$$Ga1(X_{K}^{\infty}/X_{K})=\langle\alpha\rangle$$Ga1(X_{p}^{\infty}/X_{p})=\langle\gamma\rangle$

Alexander $module/polynomial$$rightarrow$analytic $tor8ion=Reidemeister$ torsion

Iwasawa $module/polynomial$$I_{p}(T)=\det(T-(\gamma-1)|H_{1}(X_{p}^{\infty}))$$\Delta_{K}(t)=\det(t-\alpha|H_{1}(X_{K}^{\infty}))$$rightarrow$

Iwasawa main conjecture

For more analogies and details, we refer to [Mo]. In fact, there are close analogies between
Alexander-Fox theory and Iwasawa theory. From the variational point of view, one sees
that:

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( $resp$ .
-coverconsidering the infinite cyclic coverequivalenttois-representations of the knot groupconsidering the deformation ofgroup ).
$X_{K}^{\infty}arrow X_{K}$$X_{p}^{\infty}arrow X_{p}$$\mathbb{Z}_{p}$$GL_{1}$$G_{K}$)(resp. prime$G_{p}$

In number theory, there is a non-abelian generalization of the classical Iwasawa theory
along this variational viewpoint, due to mainly H. Hida and B. Mazur, namely the de-representations ofand the theory of associated arithmetic
formation theory ofinvariants. Our motivation was to find an analogue of Hida-Mazur theory in the context
of knot theory, which would be a natural non-abelian generalization of Alexander-Fox
theory:$GL_{n}$$G_{p}$

Alexander-Fox theory

Iwasawa theory$rightarrow$

$||$

$||$

deformation theory of l-dim. repr’s ofand associated arithmetic invariants

deformation theory of l-dim. repr’s ofand associated topological invariants$G_{K}$$G_{p}$$\downarrow$$\downarrow$Hida-Mazur theory$rightarrow$II

deformation theory of n-dim. repr’s ofand associated arithmetic invariants$G_{p}$

2. Deformation of hyperbolic structures on a knot complement and of
modular Galois representations

We start to recall some general notions in group representations.
knot: For a knot

$K\subset S^{3}$

with$G_{K}$$:=\pi_{1}(S^{3}\backslash K)$

and $n\geq 1$ , we set$X_{K}^{n}=Hom(G_{K}, GL_{n}(\mathbb{C}))//GL_{n}(\mathbb{C})$$:=Hom_{C-alg}((R_{K}^{n})^{G_{K}},\mathbb{C})$,acts
denotes the tautological n-dimensional representation ring on which
where
and
representation
tautologicalvia
theconjugationby theis a complex affine variety, called thestands for the invariant subring. The set.
representationsofcharacter variety of n-dimensional$G_{K}$$R_{K}^{n}$$G_{K}arrow GL_{\mathfrak{n}}(R_{K}^{n}))$$\mathfrak{X}_{K}^{n}$$G_{K}$$(R_{K}^{n})^{G_{K}}$25prime: For a prime $Spec(F_{p})$ , the prime groupis profinite and hence the naive analogueof the character variety does not provide a good moduli. Thus, following Mazur ([Mal]),we consider “infinitesimal deformations” of a given residual representation$G_{p}$$\overline{\rho}:G_{p}arrow GL_{n}(F_{p})$Namely, the pair $(R, \rho)$ is called a deformation of$\overline{\rho}$.

if$\{\begin{array}{l}\bullet RR/m_{R}=\mathbb{F}_{p}\bullet\rho:G_{p}arrow GL_{n}(R)\rho m_{R}=\overline{\rho}\end{array}$In the rest of this note, we assume for simplicity that $p>2$ andducible. A fundamental theorem by Mazur is:mere

is a universal deformation
Theorem 1 ([Ma]).
is obtained up to a certain conjugacy via amation of$(R_{p}^{\mathfrak{n}}, \rho_{p}^{n})$$(R, \rho)$$1*arrow R$$\mathbb{Z}_{p}$.

We then define the universal deformation space$X_{p}^{n}(\overline{\rho})$$:=Hom$ろー$X_{p}^{n}(\overline{\rho})$of$\overline{\rho}$$\overline{\rho}$

is absolutely irre-of so that any defor$\overline{\rho}$-dgebra homomorphismby$alg(R_{p}^{n},\mathbb{C}_{P})$wherestands for the p-adic completion of an algebraic closure of , andfor a:regarded as a rigid analytic space. We write$\mathbb{Q}_{p}$$\mathbb{C}_{p}$$\rho_{\varphi}$

We will discuss analogies between
for the cases of $n=1$ and 2.

$X_{K}^{\mathfrak{n}}$and$:=\varphi\circ\rho_{p}^{n}$$X_{p}^{n}(\beta)$$G_{p}arrow GL_{n}(\mathbb{C}_{p})$$\mathfrak{X}_{p}^{\mathfrak{n}}(\overline{\rho})$is$\varphi\in \mathfrak{X}_{p}^{\mathfrak{n}}(\overline{\rho})$.

and some invariants defined on them-theory is simply a restatement of the analogy between Alexander-Fox
The
theory and Iwasawa theory:$n=1$ :

$GL_{1}$$R_{p}^{1}=\hat{\Lambda}=\mathbb{Z}_{p}[[T]]$$R_{K}^{1}=\Lambda_{C}=\mathbb{C}[t^{\pm 1}]$

$X_{K}^{1}\simeq \mathbb{C}^{x}$$rightarrow$$\mathfrak{X}_{p}^{1}(\overline{\rho})\simeq D_{p}^{1}=\{z\in \mathbb{C}_{p}||z|_{p}<1\}$}$\chi’arrow\chi(\alpha)$$\varphi\succ$$\rho_{\varphi}(\gamma)-1$$(Ga1(X_{K}^{\infty}/X_{K})=\langle\alpha\rangle=\mathbb{Z})$$(Ga1(\mathbb{Q}^{\infty}/\mathbb{Q})=\langle\gamma\rangle=\mathbb{Z}_{p})$:invariants on
twisted Alexander poly. (analytic torsion)
for a repr.

:
invariants on
twisted Iwasawa poly. (p-adic L-function)
for a repr.
$\mathfrak{X}_{p}^{1}(\overline{\rho})$$\mathfrak{X}_{K}^{1}$$\rho:G_{K}arrow GL_{n}(\mathbb{C})$$\Delta_{K,\rho}(t)(\tau_{K,\rho})$describes the variation of$H^{1}(G_{K},\rho\otimes\chi),$$\chi\in \mathfrak{X}_{K}^{1}$$rightarrow$$\rho:G_{p}arrow GL_{n}(\mathbb{Z}_{p})$$I_{p,\rho}(T)(L_{p}(\rho, s))$describes the variation of$Se1(G_{p},\rho\otimes\rho_{\varphi}),$$\varphi\in X_{p}^{1}(\overline{\rho})$26Here, $Se1(G_{p}, M)$ denotes the Selmer group for awith a local condition).

$G_{p}$-module $M$ (a subgroup of $H^{1}(G_{p}, M)$$n=2$ : The $GL_{2}$ -theory is concerned with hyperbolic geometry and Chern-Simons gauge

theory in the knot side and Hida-Mazur theory andside.

gauge theory in the prime$\gamma adic$

has a trivial center and any
knot: We assume that $K$ is a hyperbolic knot. Since$G_{K}arrow
PGL_{2}(\mathbb{C})=PSL_{2}(\mathbb{C})$, we maycanbeliftedtorepresentation-representations without losing generality. So, we set
consider only
$G_{K}$$G_{K}arrow SL_{2}(\mathbb{C})$$SL_{2}(\mathbb{C})$$\mathfrak{X}_{K}^{2}$

Note that the restriction ofsentation:$:=Hom_{gr}(G_{K}, SL_{2}(\mathbb{C}))//SL_{2}(\mathbb{C})$$[\rho]\in X_{K}^{2}$

to$D_{K}$.

is conjugate to an upper triangular repre-$\rho|_{D_{K}}\simeq(^{x_{0}}\rho\chi_{\rho}^{-1)}*$be a lift of the holonomy representation associated to the hyperbolic structure on
containing. The following
be the irreducible component ofand lettheorem was shown by W. Thurston:Let$\rho^{o}$$S^{3}\backslash K$$X_{K}^{2}$$\mathfrak{X}_{K}^{2,0}$:Theorem 2 ([T]). The map. In particular,
in a neighborhood of$X_{K}^{2,0}arrow \mathbb{C}$$\Phi_{K}$$[\rho^{o}]$Let $m$ and be functions on$l$$\mathfrak{X}_{K}^{2,0}$

defined by$X_{K}^{2,a}$$[\rho^{o}]$$\Phi_{K}([\rho])$$:=tr(\rho(m_{K}))$is a complex algebraic curve.

defined by $m(\rho)=\chi_{\rho}(m_{K})$ and$\chi_{\rho}(l_{K})$

Theorem 3 ([NZ]). Let $x:=\log m(\rho)(\log m(\rho^{o})=0)$ and suppose
we have$\frac{dl}{dx}|_{x=0}=\frac{1}{2}\frac{\log q}{2\pi\sqrt{-1}}$

is bianalytic

respectively.

$\partial V_{K}=\mathbb{C}^{x}/q^{l}$. Then.

prime: We assume that is a mod representation associated to an ordinary modular
elliptic curve $E$ over which corresponds to an ordinary Hecke eigenform of weight. Here a representation:
mod $p=\rho_{f}$ mod2:$\rho:G_{p}arrow GL_{2}(A)$ is called ordinary if the restriction oftois conjugate to an upper
triangular representation:$p$$\overline{\rho}$$f$$\mathbb{Q}$$\overline{\rho}=\rho_{E}$$p,$$G_{p}arrow GL_{2}(\mathbb{Z}_{p})$$\rho_{E}=\rho_{f}$$\rho$$\rho|_{D_{p}}\simeq(x_{0}\rho,1$$\chi_{\rho 2}*,$)$D_{p}$$\chi_{\rho,1}|_{I_{p}}=1$.

Compared with the knot case, it is natural to impose the ordinary condition to deforma-tions of$\overline{\rho}$.

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We have the following fundamental:
Theorem 4 (1) ([H1,2]). There is a universal ordinary modular deformation(
is called the p-adic Hecke-Hida ring) of such that any ordinary modular deformation $(R,\rho)$ of is obtained via a -algebra homomorphism.
(2) ([Ma]). There is a $univer8al$ ordinary deformation
suchthat any orof$(R,\rho)$
dinary deformation.
of is obtained ma a -algebra homomomphism$(R_{p}^{2,0.m},\rho_{p}^{2,0.m})$$R_{p}^{2,0.m}$$\overline{\rho}$$R_{p}^{2,0.m}arrow R$$\mathbb{Z}_{p}$$\overline{\rho}$$(R_{p)}^{2,0}\rho_{p}^{2,0})$

By the universality of$(R_{p}^{2,0},\rho_{p}^{2,0})$$\overline{\rho}$$R_{p}^{2,0}arrow R$$\mathbb{Z}_{p}$$\overline{\rho}$, we have a$\mathbb{Z}_{p}$-algebra homomorphism$R_{p}^{2,0}arrow R_{p}^{2,0.m}$

which we assume in the following to be an isomorphism. In fact, this assumption issatisfied under a mild condition owing to the works of A. Wiles etc.

We then define the universal ordinary deformation space of$\mathfrak{X}_{p}^{2,0}(\overline{\rho})$$:=Hom$ろー$\overline{\rho}$by$alg(R_{p}^{2,0}, \mathbb{C}_{P})$which may be regarded as an (infinitesimal) analog of
2, we have:$X_{K}^{2,0}$. As an analogue of Theorem

Theorem 5. Take an element
which is mapped to a generator ofis the unique -extension of . The map:
wheredefined by.
is bianalytic in aneighborhood of
where$Ga1(\mathbb{Q}^{\infty}/\mathbb{Q})$$\gamma\in I_{p}$$\mathbb{Q}^{\infty}$$\Phi_{p}$$\mathbb{Q}$$\mathbb{Z}_{p}$$tr(\rho_{\varphi}(\gamma))$$\varphi_{f}$Remark. The analogy between the structures of
by Kazuhiro Fujiwara.

$\mathfrak{X}_{p}^{2,0}arrow \mathbb{C}_{p}$$\Phi_{p}(\varphi)$$:=$$\varphi_{f}\circ\rho_{p}^{2,0}=\rho_{f}$$X_{K}^{2,0}$and$\mathfrak{X}_{p}^{2,0}(\rho)$

was first pointed out

The following theorem by Greenberg and Stevens may be seen as an analog of NeumannZagier’s theorem 3.
Theorem 6 ([GS]). Suppose that $E$ is split multiplicative at$\rho_{p}^{2,0}|_{D_{p}}\simeq(^{x_{0}}1\chi_{2}*)$and set$a_{p}:=\chi_{1}(\sigma_{p})$:$\mathfrak{X}_{p}^{2,0}arrow \mathbb{C}_{p}$, and let$\frac{da_{p}}{d\rho}|_{\rho=2}=-\frac{1}{2}\frac{\log_{p}(q)}{ord_{p}(q)}$$p$. Write$\chi_{1}|_{I_{p}}=1$$E(\mathbb{C}_{p})=\mathbb{C}_{p}^{x}/q^{\mathbb{Z}}$. Then we have

( $=Mazur- Tate- Teitelbaum’ s\mathcal{L}$-invariant).

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Next, we discuss some analogies between invariants defined onprime: A typical invariant on([GS]). Geometrically,of modular symbols:

and$X_{p}^{2,0}(\rho)$.

$s\in$
is a p-adic modular L-function
is given as a section of a rigid analytic line bundle$L_{p}(\rho, s),$$\mathfrak{X}_{p}^{2,0}(\overline{\rho})$$L_{p}(\rho, s)$$\overline{\mathbb{Z}_{p}}$$X_{K}^{2,0}$$\rho\in \mathfrak{X}_{p}^{2,0},$$\mathfrak{L}_{p}$is a section ( fixed)$s$$\mathfrak{X}_{p}^{2,0}\mathfrak{L}_{p}\downarrow L_{p}(\rho, s)$

We note that the value $L_{p}(\rho, 0)=L_{p}(f_{\rho},0)$ at $s=0(f_{\rho}$ being a modular form correbeing, where :sponding to ) is given bya modular curve) is the p-adic regulator.
$r_{p}(\{u, v\})(\omega)\cdot c$$\rho$

knot: Take a small affine open

$X\subset X_{K}^{2,0}$$L_{K}(\rho)$be the$SL_{2}(\mathbb{C})$$K_{2}(C_{\rho})arrow H_{DR}^{1}(C_{\rho}/\mathbb{Q}_{p})(C_{\rho}$$r_{p}$

containing$\rho^{o}$if necassary, and let$:=-2\pi^{2}CS(\rho)+\sqrt{-1}Vol(\rho)$

Chem-Simons invariant. Our theorem is

Theorem 7. There is a holomorphic line bundle
is given by a flat section:such that$\mathfrak{L}_{K}$with holomorphic connection on$X$$L_{K}(\rho)$$\mathfrak{L}_{K}\mathfrak{X}\downarrow$$L_{K}(\rho)$

is a flat section

For the construction of , we apply S. Bloch’s geometric construction of a tame symbol([B1]). Let $H$ be the 3 $x3$ Heisenberg group:$\mathfrak{L}_{K}$$H(R)$ $:=\{(\begin{array}{lll}1 a c0 l b0 0 1\end{array})|a,b,c\in R\}$ ( $R$

The complex manifold
map$P:=H(\mathbb{Z})\backslash H(\mathbb{C})$$Parrow \mathbb{C}^{x}\cross \mathbb{C}^{x}$;$(\begin{array}{lll}l a c0 1 b0 0 l\end{array})$: a commutative ring)is a principal$\mathbb{C}^{x}$-bundle overby the}$arrow(\exp(2\pi\sqrt{-1}a),\exp(2\pi\sqrt{-1}b))$and l-form $\theta=dc$ –adb gives a connection on . Let $T(l, m^{2})$ :holomorphic map defined by $T(l,m^{2})(\rho)$ $:=(l(\rho), m^{2}(\rho))$ and define$P$$\mathfrak{L}_{K}$$\mathbb{C}^{x}\cross \mathbb{C}^{x}$$:=T(l,m^{2})^{s}(P,\theta)$.

$\mathfrak{X}arrow \mathbb{C}^{x}x\mathbb{C}^{x}$

be a

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Then the flat section is given by$S( \rho^{o})+\log l(\rho^{o})\int_{\rho^{o}}^{\rho}d$log$m^{2}+ \int_{\rho^{o}}^{\rho}d$log $ld$ log $m^{2}=L_{K}(\rho)$ .

, we give a cohomological interpretation
To see whyis seen as an analog ofbe the Beilinson regulator.
Let
:
the
construction.
of
above(SC, R) where
We consider the natural map :is interpreted
stands for the Deligne cohomology ([Brj). It is known thatas the group of isomorphism classes of holomorphic line bundlee on with holomorphic. Then we canis regarded as an element ofconnection ([ibid]), and soshow that$L_{p}(\rho, 0)$$L_{K}(\rho)$$r_{\infty}$$\iota$$K_{2}(X)arrow H^{1}(\mathfrak{X},\mathbb{R})$$H_{D}^{2}(X, \mathbb{Z}(2))arrow H_{D}^{2}(X, \mathbb{R}(2))rightarrow H^{1}$$H_{D}^{*}$$H^{2}(\mathfrak{X}, \mathbb{Z}(2))$

ec$H^{2}(\mathfrak{X}, \mathbb{Z}(2))$$\mathfrak{L}_{K}$$\iota(\mathfrak{L}_{K})=r_{\infty}(\{l,m^{2}\})$

which is reminiscient of the connection between$L_{p}(\rho, 0)$

and the$\Psi$

adic regulator.

so thatoverRemark. Kirk and Klassen ([KK]) also constructed a line bundle
andis regarded as a section. Though we have not seen the connection between
yet, our construction using Deligne cohomology seems to be natural conceptually.
$E_{K}$$X_{K}^{2,0}$$E_{K}$$\mathfrak{L}_{K}(\rho)$$\mathfrak{L}_{K}$Now, compared with the prime side, we may expect that

there should be a 2-variable L-function
would be a dominant term (special value) of$L_{K}(\rho, s),$$\rho\in X,$ $s\in \mathbb{C}$$L_{K}(\rho, s)$

at $s=0$ .

such that$L_{K}(\rho)$be the hyperbolic deformation ofHere is a candidate for such a L-function. Let$M=S^{3}\backslash K$ with holonomy. Then is a spin manifold with $S\dot{\mu}n(3)=SU(2)-$. Let be the corresponding Dirac operator acting
principal bundle
and we define the spectral zeta function byon$M_{\rho}$$M_{\rho}$$\rho$$S\dot{\mu}n(M_{\rho})arrow M_{\rho}$$D_{\rho}$$C^{\infty}(S\dot{\mu}n(M_{\rho})\otimes(\mathbb{C}^{2})_{\rho})$$L_{K}( \rho, s):=\sum_{\lambda}\pm(\pm\lambda)^{\epsilon},$$\pm=sign(Re(\lambda)),$ ${\rm Re}(s)>>0$

may not be self-adjoint (though its
run over eigenvalues of . Note that
where
symbol is self-adjoint) and so may be imaginary. For a closed hyperbolic 3-manifold,is continued as a meromorphic function
Jones-Westbury ([JW],[Y]) showed thatto and the equality$\lambda’ s$$D_{\rho}$$D_{\rho}$$\lambda$$L_{K}(\rho, s)$$\mathbb{C}$$L_{K}(\rho)=2\pi^{2}L_{K}(\rho,0)$.

It is desirable to extend this equality for a non-closed hyperbolic 3-manifold.
Finally, we note the following

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Theorem 8 (cf. [MT]).
defined by:ec$V=\mathbb{Z}^{3}$$L_{K}(\rho)$gives a variation of mixed Hodge structure (V,$W_{*},$ $F^{*}$

) on

,

$V=W_{0}\supset W_{-1}=\mathbb{Z}v_{2}\oplus \mathbb{Z}v_{3}\supset W_{-2}=\mathbb{Z}v_{3}$

where

log
$l(\rho^{o})01$$(\begin{array}{l}v_{l}v_{2}v_{3}\end{array}):=(100$Here$\{e_{1}, e_{2}, e_{3}\}$$W_{-2}=Z(2)$.

$\log Sm_{1}^{2}((\rho^{o}))(001\int_{\rho^{o}}^{\rho}d\log l01\int_{\rho^{o}}^{\rho}d\log ld\log m^{2}\int_{\rho^{o_{1}}}^{\rho}d\log m^{2})((2\pi e_{1}\sqrt{-1})e_{2})$is a standard basis of , and $W_{0}/W_{-1}=\mathbb{Z}(0),$ $W_{-1}/W_{-2}=\mathbb{Z}(1)$ ,$V$$V=F^{-2}\supset F^{-1}=\mathbb{Z}e_{1}\oplus \mathbb{Z}e_{2}\supset F^{0}=\mathbb{Z}e_{1}$$\nabla v:=dv-v(\begin{array}{llll}1 dlogl 0 0 1 dlog m^{2}0 0 1 \end{array})$

with$\nabla P^{-1}\subset\Omega^{1}\otimes F^{i}$so that.

References[B1] S. Bloch, The dilogarithm and extensions of Lie algebras, Algebraic K-theory, Evanston 1980
(Proc. Conf., Northwestem Univ., Evanston, $m.$ , 1980), pp. 1-23, Lecture Notes in Math., 854,
Springer, Berlin-New York, 1981.
[Br] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress inMathematics, 107, $Birkh\ddot{a}u8er$ Boston, Inc., Boston, MA, 1993.
L-function8 and adic periods of modular forms, Invent. Math.
[GS] R. Greenberg, G. Stevens,
111 (1993), no. 2, 407-447.
[H1] H. Hida, Iwasawa modules attached to conyuences of cusp forms, Ann. Sci. \’E\infty le Norm. Sup.
(4) 19, (1986), 231-273.
[H2] H. Hida, Galois representations into $GL_{2}(Z_{p}[[X]])$ attached to ordinary cusp forms, Invent. Math.
85, (1986),.
[JW] J. Jones, B. Westbury, Algebraic K-theory, homology spheres, and the -invariant, Topology 34(1995), no. 4, 929-957.
[KK] P. Kirk, E. Klassen, Chea-Simons invariants of 3-manifolds decomposed along tori and the circle
bundle over the representation space of . Comm. Math. Phys. 153, (1993), no. 3, 521-557.
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[Ma] B. Mazur, Deforming Galois representations, In: $Galoi8$ groups over $Q$ , Math. Sci. Res.
Publ., 16, Springer, (1989), 385-437.
[Mo] M. Morishita, Analogies between knots and primes, 3-manifolds and number ringa, to appear in
Sugaku Expositions, AMS. (Analogies between prime numbers and knots, (Japanese) Sugaku 58,), no. 1,] .
[MT] M. Morishita, Y. Terashima, $G\infty metry$ of polysymbols, preprint,[NZ] W. Neumann, D. Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3,$p\cdot adic$$\gamma$$54\triangleright 613$$\eta$$T^{2}$$\bm{t}_{8}t$$(2\alpha)6)$$4\alpha 63.$$2\alpha$$6$307-332.

[T] W. Thurston, The geometry and topology of 3-manifolds, Lect. Note, Princeton, 1977.
[ T. Yoshida, The -invuiant of hyperbolic 3-manifolds, Invent. Math. 81, (1985), no. 3, $47\succ 514$$\eta$$\eta$.

M.M: Graduate School of Mathematics, Kyushu $Univer8ity,$ $6- 10- 1$ , Hakozaki, Higashi-ku, 1 kuoka, 8128581 Japan. email: morisitaOmath.kyushu-u.ac.jp
Y.T: Department of Mathematics, Tokyo Institute of Ibchnology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo: teraOmath.titech.ac.jp152-8551, Japan.
$e- m\dot{u}1$