数理解析研究所講究録
第 1911 巻 2014 年 18-33

18

Hasse-Weil zeta functions of$SL_{2}$-character varieties of 3-manifolds

Shinya Harada
Graduate School of Information Science and Engineering, Tokyo Institute of Technology

1 Introduction
The-character varieties of a 3-manifold is known to be a powerful tool for the study of
3-manifolds. It is showed by the work of Culler-Shalen in [4] that they can be used to constructessential surfaces in the 3-manifolds.
-character varieties are algebraic varieties definedby finite number of polynomials with rational coeﬃcients (precisely they are aﬃne algebraicsets). Later another invariant, -polynomials are defined in [3] for 3-manifolds with one cusp,-character varieties
which enabled us to study the essential surfaces coming from themore directly.
Hasse-Weil zeta functions in the title are natural generalization of the Dedekind zeta functions
for algebraic varieties over number fields. As we could see in the class number formula for the
Dedekind zeta functions and the Birth and Swinnerton-Dyer conjecture for elliptic curves, we$SL_{2}(\mathbb{C})$$SL_{2}(\mathbb{C})$$A$$SL_{2}(\mathbb{C})$expect that they inherit geometric and number theoretic properties of the varieties.
-character varieties of certain
In this note we consider Hasse-Weil zeta functions of thehyperbolic 3-manifolds and the -polynomials of torus knots, and study what kind of topological information appears in the description of the zeta functions.
$SL_{2}(\mathbb{C})$$A$

2 Hasse-Weil zeta functions of polynomials
2.1 Local zeta function
Let be a prime number and,polynomialsofpoints:$p$$f_{1},$$f_{1},$$\cdots$$\mathbb{F}_{p^{n}}$the finite field with
we denote by$f_{r}\in \mathbb{Z}[X_{1}, \cdots, X_{m}]$$p^{n}$

elements. For given finite number of-rational
the set of$V(f_{1}, \cdots , f_{r};\mathbb{F}_{t^{t}})$$\mathbb{F}_{p^{n}}$$\cdots,$ $f_{r}$$V(\mathbb{F}_{〆}):=V(f_{1}, \cdots, f_{r};\mathbb{F}_{〆})=\{(a_{1}, \cdots, a_{m})\in(\mathbb{F}_{p^{n}})^{m}|f_{1}(a_{1}, \cdots, a_{m})=\cdots=f_{r}(a_{1}, \cdots,a_{m})=0\}.$

Then the local (congruence) zeta function of

$V$

at

$p$

is defined by$Z(V, p, T):= \exp(\sum_{n=1}^{\infty}\frac{\# V(\mathbb{F}_{p^{n}})}{n}T^{n})\in \mathbb{Q}[T\Pi.$

19
Here
$\exp(T):=\sum_{n=0}^{\infty}\frac{1}{n!}T^{n}, \log(\frac{1}{1-T})=\sum_{n=1}^{\infty}\frac{1}{n}T^{n}.$Example 2.1. Consider the case $f=X\in \mathbb{Z}[X]$ and $V=V(f)$ . Then it is clear that
Therefore
and we have$V(\mathbb{F}_{p^{n}})=\{0\}.$$\# V(\mathbb{F}_{p^{n}})=1$$Z(V, p, T)= \exp(\sum_{n=1}^{\infty}\frac{1}{n}T^{n})=\frac{1}{1-T}.$

Example 2.2. When $f=X^{2}+1\in \mathbb{Z}[X]$ . In this case$\# V(f)(\mathbb{F}_{p^{n}})=\{\begin{array}{l}1, p=2,02,’ p\neq 2p\neq 2,’\}_{\frac{}{}}^{\frac{-1}{-1pp}}\{_{=-1,n\equiv 1(mod 2)}=1or(\frac{-1}{p})=-1,n\equiv 0 (mod 2) ,\end{array}$Here$( \frac{-1}{p})$is the Legendre symbol. Thus we have

$Z(V, p, T)=\{\begin{array}{ll}1/(1-T) , p=2,1/(1-T^{2})1/(1-T)^{2},’ p\neq 2p\neq 2, \}_{\frac{}{}}^{\frac{-1}{-1pp}}[Case]\end{array}$

In general it is known that $Z(V, p, T)$ is a rational function.

Theorem 2.3 (Dwork [5]). $Z(V, p, T)$ is a rational function.

Remark2.4. This is true for schemes of finite type over . In particular, $Z(V, p, T)$ is a rational
function for any open set (e.g. irreducible part of a character variety).
$\mathbb{Z}$$V$

Hence we can write down $Z(V, p, T)$ as a product of monomials as$Z(V, p, T)= \exp(\sum_{n=1}^{\infty}\frac{\# V(\mathbb{F}_{p^{\int/}})}{n}T^{n})=\frac{\prod_{i}(1-\alpha_{i}T)}{\prod_{j}(1-\beta_{j}T)}.$

where$\alpha_{i},\beta_{j}\in \mathbb{C}$are algebraic numbers. Then we have$\# V(\mathbb{F}_{p^{n}})=\sum_{j}\beta_{j}^{n}-\sum_{i}\alpha_{i}^{n} (\alpha_{j},\beta_{j}\in \mathbb{C})$.

Therefore if it is possible to compute $Z(V, p, T)$ then we can obtain. Moreover we
can also expect to study the qualitative properties oflike the Weil conjecture forsmooth projective varieties over finite fields.
TYpical examples of algebraic sets in Algebraic topology are the following two cases:$(\# V(\mathbb{F}_{p^{n}}))_{n}$$(\# V(\mathbb{F}_{p^{\prime 1}}))_{n}$20$\bullet$The set of$\bullet$$SL_{2}(\mathbb{C})$$SL_{2}(\mathbb{C})$-representations of a finitely presented group-character variety of a finitely presented group

It is not known whether the set of conjugacy classes of-representations of a finitely
presented group is an algebraic set or not. (It is true that the set of conjugacy classes of absolutely irreducible-representations of a finitely presented group is an algebraic set.) I
will list up some work in this direction. For detail, see the corresponding references.
$SL_{2}(\mathbb{F}_{\rho^{l1}})$$SL_{2}(\mathbb{F}_{p^{n}})$$\bullet$$\bullet$$\bullet$

Work of Sink [24].

Computation of the number of conjugacy classes ofgroups (Li, Xu [16, 17$SL_{2}(F_{q})$Computation of the number of conjugacy classes ofof knots in the Rolfsen’s table (Kitano,Suzuki [13]).

-representations of torus knot

$SL_{2}(\mathbb{F}_{p})$(surjective) representations

2.2 Hasse-Weil zeta function,
For given polynomials
the product of the local zeta functions as follows:$f_{1},$$\cdots$$f_{r}\in \mathbb{Z}[X_{1}, \cdots, X_{m}]$its Hasse-Weil zeta function is defined by$\zeta(V, s):=\zeta(V(f_{1}, \cdots,f_{r}), s):=\prod_{p:pnme}Z(V, p, p^{-s})$.

converges absolutely in ${\rm Re}(s)>\dim V(f_{1}, \cdots , f_{r})$ ([22]). It is conjectured to have meromorphic continuation but it is not known in general ([21]).

$\zeta(V, s)$

Example 2.5 (Riemann zeta function). If $f=X\in \mathbb{Z}[X]$ then as we see in the previous subsection we have $Z(V, p, T)=1/(1-T)$ . Therefore we have$\zeta(V, s):=\prod_{p:pnme}(1-p^{-s})^{-1}=\zeta(s)$,which is the Riemann zeta function.
In general, if is an algebraic number field (finite extension field of
is a finitely generated algebra. Hence it is written as$K$$\mathbb{Q}$), its ring of integers$0_{K}arrow\sim \mathbb{Z}[X_{1}, \cdot , X_{m}]/(f_{1}, \cdots,f_{r})$$O_{K}$.

Therefore$\zeta(V, s)=\zeta(K, s)$$:= \prod_{p:non-aeropnme}(1-N(\mathfrak{p})^{-s})^{-1}$: Dedekind zeta function

whereis the norm of the prime ideal . However the polynomials which
generate the kernel is diﬃcult to compute in general. Here we present one well-known case.
$N(\mathfrak{p})=\#(0_{K}/\mathfrak{p})$$\mathfrak{p}$21
Example 2.6 (zeta of cyclotomic polynomial). Let
namely the minimal polynomial of a primitive d-th root$\Phi_{d}(X)$$\zeta_{d}$be the d-th cyclotomic polynomial,of unity. For small , we have$d$$\Phi_{1}(X)=X-1, \Phi_{2}(X)=X+1, \Phi_{3}(X)=X^{2}+X+1, \Phi_{4}(X)=X^{2}+1.$Let$\mathbb{Q}_{d}$$\mathbb{Q}_{d}$$:=\mathbb{Q}(\zeta_{d})$is equal tobe the d-th cyclotomic field. It is well-known that the ring of integersof(cf. [29]). Hence we have
. Therefore we see that
. For instance $\zeta(X^{2}+1, s)=\zeta(\mathbb{Q}(\sqrt{-1}), s)$ .
$(9_{d}$$0_{d}arrow\sim \mathbb{Z}[X]/(\Phi_{d}(X))$$\mathbb{Z}[\zeta_{d}]$$\zeta(\Phi_{d}(X), s)=\zeta(\mathbb{Q}_{d}, s)$

3 Hasse-Weil zeta functions of hyperbolic 3-manifolds
3.1$SL_{2}\cdot$

character variety

Here we briefly review the definition of the-character variety. For details, see theoriginal paper [4] or [23].
Let $M$ be an orientable complete hyperbolic 3 manifold with finite volume. Let $X(M)$ be theset of the characters of, namely-representations of$SL_{2}(\mathbb{C})$$\pi_{1}(M)$$SL_{2}(\mathbb{C})$$X(M)$ $:=$

{characters of :$\rho$$\pi_{1}(M)arrow SL_{2}(\mathbb{C})$}.

The set $X(M)$ is known to be an (aﬃne) algebraic set over , that is, it is expressed as theset of zeros of some finite number of polynomials with coeﬃcients in . Let $X_{0}(M)$ be anirreducible component of $X(M)$ which contains the character corresponding to the holonomy
representation of $M$ , which is called a canonical component of $M$ . In general it is not known
about the dimension of $X(M)$ . However the following result is known for the dimension of$\mathbb{Q}$$\mathbb{Q}$$X_{0}(M)$.

Theorem 3.1 (Thurston [25]). if $M$ is a complete hyperbolic 3 manifold with cusp

$n$

then

$\dim X_{0}(M)=n.$There is a way to compute the defining polynomials of $X(M)$ from a given group presentationof the fundamental groupof $M$ . For an concrete example, see for instance [18] for
two-bridge knots by Riley’s method, and see [7] in the general case.
$\pi_{1}(M)$

3.2 Hasse-Weil zeta function of $M$
Now we define the Hasse-Weil zeta function of a hyperbolic 3 manifold $M$ . Since $X(M)$ is anaﬃne algebraic set over there are finitely many polynomials in. Thus we canobtain polynomials inby multiplying the above polynomials by some positive$\mathbb{Q}$$\mathbb{Z}[X_{1}, \cdots, X_{m}]$$\mathbb{Q}[X_{1}, \cdots, X_{m}]$22integer. Let $Z(X(M),p, T)$ be the local zeta function for those polynomials. Then defineby the following:$\zeta(M, s):=\prod_{p:prime}Z(X(M), p, p^{-s})$$\zeta(M, s)$.

Remark 3.2. There is an ambiguity to take the defining polynomials of $X(M)$ with integer
coeﬃcients. However we can see that it only aﬀects diﬀerence of rational functions infor$\zeta(M,s)$
finitely many primes . Thus
is defined up to rational functions infor finitely
many prime numbers . In the following examples we will abbreviate these rational functions
even for the zeta functions of polynomials with integer coeﬃcients.
$p^{-s}$$\mathbb{Q}(p^{-s})$

$p$

$p$

3.3 Hasse-Weil zeta functions of the figure 8 knot, two.bridge knots
The description of the-character variety of the figure 8 knot complement
well-known. It is defined by the polynomial$SL_{2}(\mathbb{C})$$M_{8}$

in$S^{3}$

is$(x^{2}-y-2)(y^{2}-(1+x^{2})y+2x^{2}-1)=0.$

The canonical component ofis the elliptic curve defined by $(y^{2}-(1+\nearrow)y+2x^{2}-1)=0.$
Its Weierstrass model is $E:Y^{2}=X^{3}-2X+1$ . This is an elliptic curve over whose conductor
is 40. The curve $x^{2}-y-2=0$ corresponds to the points of the reducible characters. It would
depend on the situation whether we should consider the zeta function of the whole character
variety or just the canonical component. Here we present the zeta function of the irreducible
part of $X(M_{8})$ .
$M_{8}$$\mathbb{Q}$3.3.1 Hasse.Weil zeta function of flgure 8 knot

Theorem 3.3 (H. [9]).
$\zeta(M_{8}, s)=\frac{\zeta(E,s)}{\zeta(s)\zeta(\mathbb{Q}(\sqrt{5}),s)},$At present it is diﬃcult to study the Hasse-Weil zeta functions of algebraic varieties in general.
Except the dimension case (Dedekind zeta case), elliptic curves over are the almost only
case we can study some properties of the zeta functions in the general setting. Since the above
elliptic curve has analytic rank , we can also study its special values,$0$$\mathbb{Q}$$0$$\bullet$If we consider the following completed function$\xi(M_{8}, s):=\frac{4\pi^{(3s/2)+1}}{(10\sqrt{2})^{s}\Gamma(s\int 2)^{3}}\cross\zeta(M_{8}, s)$it has a functional equation $\xi(M_{8},2-s)\xi(2-s)\xi(\mathbb{Q}(\sqrt{5}), 2-s)=-\xi(M_{8}, s)\xi(s)\xi(\mathbb{Q}(\sqrt{5}), s)$ .

23$\bullet$The special value of$\xi(M_{8}, s)$

at $s=1$ is computed as$\lim_{sarrow 1}\frac{\xi(M_{8},s)}{s-1}=-\frac{AGM(\varphi,\varphi-1)}{\sqrt{10}\log(\varphi)}.$

Here $\varphi=(\sqrt{5}+1)/2$ and $AGM(\varphi, \varphi-1)$ is the arithmetic geometric mean.

We remark that here the Hasse-Weil zeta functionmeans the Hasse-Weil zeta functionof the projective model of the aﬃne curve $E:Y^{2}=X^{3}-2X+1.$
For any Laurent polynomial
the Mahler measure of isdefined by$\zeta(E, s)$$P$$P=P(t_{1}, \cdots , t_{n})\in \mathbb{C}[t_{1}^{\pm 1}, \cdots, t_{n}^{\pm 1}]$$m(P) := \int_{0}^{1}\cdots\int_{0}^{1}\log|P(e^{2\pi t\downarrow\sqrt{-1}}, \cdots, e^{2\pi t_{n^{\sqrt{-1}}}})|dt_{1}\cdots dt_{n}.$There are some research trying to find geometric interpretation of the Mahler measures of polynomial invariants in Algebraic Topology. For instance see [1]. Since
has a functional
equation between and $2-s$ , special values at $s=1$ , 2 should be the most interesting ones.
We have some interpretation of the special values ofat $s=1$ , 2 in terms of the Mahler
measures of certain polynomials.
$\zeta(M_{8}, s)$$s$$\zeta(M_{8}, s)$$\bullet$$\bullet$$\log(\varphi^{2})=m(\Delta_{7\subset}(T))$, where$\Delta_{7C}(T)$is the Alexander polynomial o$AGM(\varphi, \varphi-1)=\frac{1}{2}(\frac{d}{dk}m(P_{k})(\sqrt{5}))^{-1}$. Here$P_{k}$$f^{(}K.$$:=x+ \frac{1}{x}+y+\frac{1}{y}-4k$

is a family of elliptic

curves $for4k\neq 0$ , 1.
Remark 3.4. The special Value at $s=2$ is as follows.
$\lim_{sarrow 2}(s-2)\xi(M_{8}, s)=\frac{75}{2\sqrt{5}\pi^{2}\mathcal{L}(E,2)}.$Hereis the completed -series of. It is numerically observed by RodriguezVillegas ([27], TABLE 4) that
. In fact this has been
would be equal toconfirmed by Mellit ([20]).
$L$$\mathcal{L}(E, s)$$\mathcal{L}_{E/Q}(2)$$E/\mathbb{Q}$$m(P_{\sqrt{-4}/4})$

3.3.2 Two-bridge knots caseWe would like to obtain similar results to the figure 8 knot case for more general families ofknots. Hence it is natural to consider the two-bridge knots case for this purpose.
It is known that the-character varieties of the two-bridge knots have dimension 1. Still
it is not so easy to study the algebraic geometric properties of their character varieties.
Macasieb, Petersen, Van Luijk [18] have studied the genus of-character curves ofcertain family of hyperbolic two-bridge knots which contains all the twist knots. In that familyof two-bridge knots, only two canonical curves have genus 1, elliptic curves.
$SL_{2}(\mathbb{C})$$SL_{2}(\mathbb{C})$24
Example 3.5. Up to $(a, b)(a, b<50)$ , In the 5 cases $(5, 2)$ $=4_{1},$ $(15, 4)$ $=7_{4},$ $(21, 8)$$(27, 8)=8_{11},$ $(45, 14)=10_{21}$ canonical curves are elliptic curves.

$=$$7_{7},$

Question 3.6. Are there finitely many elliptic curves appeared as canonical components of-character varieties of hyperbolic two-bridge knots.?
$SL_{2}(\mathbb{C})$3.4 Hasse.Weil zeta function of arithmetic$two\cdot$

bridge link

An arithmetic hyperbolic 3-manifold is a special class of hyperbolic 3-manifolds. They are
relatively simple among hyperbolic 3-manifolds, since their commensurable classes are determined by the pair of the invariant trace fields and the invariant quaternion algebras associated with the arithmetic 3-manifolds. It is also known that the smallest volume hyperbolic
3-manifolds with cusp , 1, 2 are arithmetic. For details, see [19].
In what follows, we use the notation ‘arithmetic’ in somewhat strict sense. Namely, a hyper: $\pi_{1}(M)arrow$bolic 3 manifold $M$ is arithmetic when the image of the holonomy representation
is in $SL_{2}(O)$ up to conjugacy, where is the ring of integers of some number field.
$0$$\rho_{M}$$0$$SL_{2}(\mathbb{C})$

Theorem 3.7 (Gehring, Maclachlan, Martin [6]). Arithmetic two-bridge links in the 3, sphere
are thefigure8knot, the Whitehead link $5_{1}^{2}=(8,3)$ , $6_{2}^{2}=(10,3)$ , $6_{3}^{2}=(12,5)$ .
Here we consider the canonical components of-character varieties of the Whitehead
link,
in
are
Defining polynomials of the canonical components of the Whitehead link,$SL_{2}(\mathbb{C})$$6_{2}^{2},$$6_{3}^{2}.$$\mathbb{C}^{3}$$6_{2}^{2},$$6_{3}^{2}$$f_{0}:=z^{3}-xyz^{2}+(x^{2}+y^{2}-2)z-xy,$$f_{1} :=z^{4}-xyz^{3}+(\fbox{Error::0x0000}+y^{2}-3)z^{2}-xyz+1,$$f_{2}:=z^{3}-xyz^{2}+(x^{2}+y^{2}-1)z-xy.$

These surfaces
Put$V(f)\subset \mathbb{C}^{3}$

are smooth aﬃne surfaces.

$\mathbb{P}^{2}\cross \mathbb{P}^{1}:=\{(x:y:u, z:w)|(x:y:u)\in \mathbb{P}^{2}, (z:w)\in \mathbb{P}^{1}\}.$

Consider the compactification$X(F_{i})\subset \mathbb{P}^{2}\cross \mathbb{P}^{1}$of these surfaces in$\mathbb{P}^{2}\cross \mathbb{P}^{1}$. Here$F_{0} :=u^{2}z^{3}-xyz^{2}w+(\fbox{Error::0x0000}+y^{2}-2u^{2})zw^{2}-xyw^{3},$$F_{1} :=u^{2}z^{4}-xyz^{3}w+(l+y^{2}-3u^{2})z^{2}w^{2}-xyzw^{3}+u^{2}w^{4},$$F_{2} :=u^{2}z^{3}-xyz^{2}w+(\fbox{Error::0x0000}+y^{2}-u^{2})zw^{2}-xyw^{3}.$are considered as (singular) conic bundles over by $(x:y:u,$ :$w)\mapsto(z:w)$ (Landes [14, 15], H. [8]), which enables us to compute the number of-rational
points of

The surfaces$\mathbb{P}^{1}$$X(F_{i})\subset \mathbb{P}^{2}\cross \mathbb{P}^{1}$$z$$\mathbb{F}_{q}$$F_{j}.$

25
Theorem 3.8 (H. [10]). Let$(8,3)$ ,be the canonical components of the Whitehead link
. Then we have$X_{0},$ $X_{1},$ $X_{2}$$6_{2}^{2}=(10,3),$ $6_{3}^{2}=(12,5)$$\zeta(X_{0}, s)=\zeta(\mathbb{Q}(\sqrt{2}), s-1)\zeta(s)^{2}\zeta(s-1)^{2}\zeta(s-2)$$\zeta(X_{1}, s)=\zeta(\mathbb{Q}(\sqrt{5}), s-1)^{2}\zeta(s)^{3}\zeta(s-2)$$\zeta(X_{2}, s)=\zeta(s)^{2}\zeta(s-2)$$5_{1}^{2}=$

.

.

.

Remark 3.9. Note that$\mathbb{Q}(\sqrt{2})$,$\mathbb{Q}(\sqrt{5})$,$\mathbb{Q}$for$\zeta(X_{i}, s)=\mathbb{Q}(\begin{array}{lll}\mathbb{P}^{1}- coordinatesallthe ofthedegenerate fi bers \circ fX_{i}\end{array})$

the trace fields (invariant trace fields)

Remark 3.10.

$X_{i}$$\mathbb{Q}(\sqrt{-1})$,

is birationally equivalent to$\zeta(A_{\mathbb{Q}}^{2}, s)=\zeta(s-2)$$\mathbb{Q}(\sqrt{-3})$$\mathbb{P}^{2}$and$\mathbb{Q}(\sqrt{-7})$. They are diﬀerent fromof

$5_{1}^{2},$$6_{2}^{2}$

and. However the zeta functions of$6_{3}^{2}.$$A_{Q}^{2},$$\mathbb{P}_{\mathbb{Q}}^{2}$are, $\zeta(\mathbb{P}_{\mathbb{Q}}^{2}, s)=\zeta(s)\zeta(s-1)\zeta(s-2)$ .

Recently the following result was obtained, which contains The Whitehead link,$6_{2}^{2}$

cases.

Theorem 3.11 (Tran [26]). The-character varieties of the two-bridge link $(2m, 3)(m\neq$3) have 2 irreducible components. Canonical components aredefined in terms of Chebyshev
polynomials. Their compactification have conic bundle structure.
$SL_{2}(\mathbb{C})$In general, the canonical components of thebridge link do not have a conic bundle structure over
we might expect the following.

$SL_{2}(\mathbb{C})$Question 3.12. Does the canonical component of thetwo-bridge link have a fibered structure over$\mathbb{P}^{1}$-character variety of a hyperbolic two(For examples, see [14, 15
However$SL_{2}(\mathbb{C})$-character variety of a hyperbolic$\mathbb{P}^{1_{7}}.$

3.5 Hasse.Weil zeta function of closed arithmetic 3 manifold
Here we present some examples of the Hasse-Weil zeta functions of the irreducible part ofthe-character varieties of closed arithmetic 3-manifolds with small volumes.
In the table below
means the open subset of the-character variety of $M$
consisting of irreducible characters,
is the trace fieldof $M$ , where
denotes the holonomy representation of $M$ . Note that in the examples below the character varieties$X(M)$ have dimension O.
$SL_{2}(\mathbb{C})$$X(M)_{1rr}(\mathbb{C})$$SL_{2}(\mathbb{C})$$K_{M}$$\mathbb{Q}(Tr\rho_{M}(\pi_{1}(M))$$\rho_{M}$26

The Hasse-Weil zeta functions $\zeta(X(M)_{1rr}, s)$ are equal to the Dedekind zeta functions of the trace
fields of $M$ (up tofor finitely many primes ). Thus we may expect the following.
$\mathbb{Q}(p^{-s})$$p$

Question 3.13. Let $M$ be an (arithmetic) closed 3 manifold and$\zeta(M, s)=\zeta(K_{M}, s)$$K_{M}$

the trace field of M. Then?

Moreover,$\zeta(M, s)=\zeta(M’, s)\Leftarrow\Rightarrow K_{M^{arrow}}^{\sim}K_{M’}$?

The second statement might be true since this is true for invariant trace fields as follows.
$K’$ beTheorem 3.14 (Chinburg, Hamilton, Long, Reid [2]). Let
numberfields having only$K’$$\zeta(K,s)=\zeta(K’, s)$ .
one complex place. Then and are isomorphic if and only if
$K,$

$K$

4 Hasse-Weil zeta function of -polynomials of torus knots
$A$

4.1

$A$

-polynomial of knots in$S^{3}$

Here we review some properties of the -polynomials of knots which are needed in this note.
For details, see the original paper [3].
be the canonical longitude and meridian ofand let
Let $K$ be a knot in$K$ . Let
$A$

$S^{3}$$\lambda,\mu\in\pi_{1}(S^{3}\backslash K)$$R(JC)=\{\rho:\pi_{1}(S^{3}\backslash \etaarrow SL_{2}(\mathbb{C})\}$

be the set of$SL_{2}(\mathbb{C})$-representations of$R_{U}=${$\pi_{1}(S^{3}\backslash (K)$$p\in R(7C)|\rho(\lambda),\rho(\mu)$and: upper triangular}

the subset of $R(q()$ consisting of upper triangular matrices inmap by$R(^{t}K$). Then define the eigenvalue$\xi$$\xi:R_{U}arrow \mathbb{C}^{\cross}\cross \mathbb{C}^{x}, \rho\mapsto(a_{\rho}(\lambda), a_{\rho}(\mu))$27
if$\rho(\lambda)=(^{a_{\rho}(\lambda)}0a_{\rho}(\lambda)^{-1)}b_{\rho}(\lambda),$Then we can obtain an algebraic curve in$\mathbb{C}^{2}$$\rho(\mu)=(^{a_{\rho}(\mu)}0a_{\rho}(\mu)^{-l)}b_{\rho}(\mu).$by considering$\mathbb{C}^{2}\supset\bigcup_{c}\overline{\xi(C)}.$

Here runs through all the i1Teducible components ofare curves.
whose closures $\xi(C)$ in
Since this is a plane curve, it is defined by (reduced) polynomial $A_{K}(L, M)\in \mathbb{C}[L, M]$ , namely$C$$\mathbb{C}^{2}$$R_{U}$$a$$\mathbb{C}^{2}\supset\bigcup_{c}\overline{\xi(C)}=V(A_{K}(L, M$

This polynomial is called the -polynomial of . By definition it is defined up to constant,multiplicity. Here are some properties of the -polynomials of knots in$A$$\prime\kappa$$S^{3}.$$A$$\bullet A_{7C}(L, M)\in \mathbb{Z}[L, M],$$\bullet A_{O}(L, M)=L-1,$$\bullet L-1|A\prime\kappa(L, M)$$\bullet$,$(K : non-trivia1\Rightarrow A_{K}(L, M)\neq L-1.$

4.2 Hasse-Weil zeta functions of$A\cdot$

polynomials of torus knots

be positive integers such that $(a, b)=1$ . The general form of the$(a, b)$ -torus knots $T(a, b)$ are well-known as follows.

Let$a,$ $b$$A_{a,b}(L, M)$$A$-polynomials of$:=A_{T(a,b)}(L, M)=\{\begin{array}{l}(L-1)(-1+(LM^{ab})^{2}) , if a, b>2,(L-1)(1+LM^{2(2m+1)}) , if (a, b)=(2,2m+1) .\end{array}$

Then the Hasse-Weil zeta functions of $A_{a,b}(L, M)$ are expressed as follows.
Theorem 4.1 (H.-Terashima [11]). Up to rational functions in
the zeta function $\zeta(A_{a,b}(L, M), s)$ is equal to$\mathbb{Q}(\{p^{-s}\}_{p|2ab})$$\{\begin{array}{ll}(\prod_{d|2ab}\frac{1}{\zeta(\mathbb{Q}_{d},s)}1\frac{\zeta(s-1)^{3}}{\zeta(s)^{2}}, if a, b>2,(\prod_{d|2ab.d(ab}\frac{1}{\zeta(\mathbb{Q}_{d},s)})\frac{\zeta(s-1)^{2}}{\zeta(s)}, if (a, b)=(2,2m+1) .\end{array}$

Example 4.2 (trivial knot). $\zeta(A_{O}(L, M))=\zeta(s-1)$ .

28
Example 4.3 (trefoil knot).
Example 4.4.

$\zeta(A_{2.3}(L, M), s)=\frac{\zeta(s-1)^{2}}{\zeta(s)}\cross\zeta(\mathbb{Q}_{4}, s)^{-1}\zeta(\mathbb{Q}_{12}, s)^{-1}.$$\zeta(A_{2,5}(L, M), s)=\frac{\zeta(s-1)^{2}}{\zeta(s)}\cross\zeta(\mathbb{Q}_{4}, s)^{-1}\zeta(\mathbb{Q}_{20}, s)^{-1}.$

In general the -polynomial of a knot can be divided by $L-1$ . In fact it is clear from thedefinition of the zeta function that $\zeta(A_{a.b}(L, M), s)$ is divided by $\zeta(L-1, s)=\zeta(s-1)$ . Therefore
considering reduced -polynomials divided by $L-1$ might be better in certain cases. However
when we consider the zeta function this is not the case. For each component of the -polynomialof a torus knot its zeta function has the following form:
$A$

$A$

$A$

$\zeta(L-1, s)=\zeta(s-1)$ ,$\zeta(-1+(LM^{ab})^{2}, s)=\frac{\zeta(s-1)^{2}}{\zeta(s)^{2}},$$\zeta(1+LM^{2(2m+1)}, s)=\zeta(s-1)/\zeta(s)$.

Therefore we have to consider the intersection of two components of the -polynomial to retrieve essential information $(in the$ torus knot case, invariants with respect $to a, b)$ .
$A$

4.3 Characters of the minimal model$A$

Here we explain a relation between the description of the Hasse-Weil zeta function of the-polynomial of a torus knot and the Kashaev invariant of the torus knot.
Letbe coprime positive integers and put$a,$ $b$$c(a, b)=1- \frac{6(a-b)^{2}}{ab}, h_{n,m}^{a,b}=\frac{(nb-ma)^{2}-(a-b)^{2}}{4ab}.$Minimal models are a series of infinite dimensional representations of the Virasoro algebra,
namely those are irreducible highest weight representations of the Virasoro algebra with conformal weightand central charge $c(a, b)$ where $1\leq n\leq a-1,1\leq m\leq b-1)$ . For details
we refer to [28], Chapter 7.
has a presentationof a minimal model
Especially the normalized character
[12],
(cf.
[28]):
in terms of the Dedekind -function as follows$h_{n,m}^{a.b}$$($$ch_{n.m}^{a.b}(\tau)$$\mathcal{M}(a, b)$$\eta$$ch_{n,m}^{a.b}(\tau)=\frac{\Phi_{n,m}^{a.b}(\tau)}{\eta(\tau)}, \eta(\tau)=q^{\frac{1}{24}}\prod_{k\geq 1}(1-q^{k})$Here$q=e^{2\pi i\tau}$.

. Note that$ch_{n.m}^{a,b}(\tau)=ch_{a-n,b-m}^{a,b}(\tau)=ch_{m,n}^{b.a}(\tau)=ch_{b-m,a-n}^{b,a}(\tau)$

Therefore there are $(a-1)(b-1)/2$ characters for the minimal model$\Phi_{n.m}^{a,b}(\tau)=\sum_{k=0}^{\infty}\chi_{2ab}^{(n,m)}(k)q^{\frac{k^{2}}{4ab}}.$.

$\mathcal{M}(a, b)$. Here29
The ’character’$\chi_{2ab}^{(n,m)}$:$\mathbb{Z}arrow \mathbb{C}$is defined by$\chi_{2ab}^{(n,m)}(k)=\{\begin{array}{ll}1, k\equiv\pm(nb-ma) (mod 2ab) ,-1, k\equiv\pm(nb+ma) (mod 2ab) ,0, otherwise.\end{array}$

Especially we note that$\chi_{2ab}^{(n.m)}(0)=0.$

Remark 4.5. Consider the normalized Alexander polynomial of the torus knot $T(a, b)$$\Delta_{a,b}(T^{2})=\frac{(T^{ab}-T^{-ab})(T-T^{-1})}{(T^{a}-T^{-a})(T^{b}-T^{-b})}.$(Here we consider$\Delta_{a.b}(T^{2})$just for convenience.) Then its ‘inverse’ has the following power

series expansion:$\frac{T-T^{-1}}{\Delta_{a,b}(T^{2})}=\frac{(T^{a}-T^{-a})(T^{b}-T^{-b})}{(T^{ab}-T^{-ab})}=\sum_{k\geq 0}\chi_{2ab}^{(a-1,1)}(k)T^{-k}.$

4.4 Relation with Quantum invariant
Consider the ’Eichler integral of$\Phi_{n,m}^{a,b}(\tau)$$\tilde{\Phi}_{n,m}^{a.b}(\tau)=-\frac{1}{2}\sum_{k=0}^{\infty}k\chi_{2ab}^{(n.m)}(k)q^{\frac{k^{2}}{4ab}}.$When $(n, m)=(a-1,1)$ , the following relation with the Kashaev invariant
is known.
Theorem 4.6 (Hikami-Kirillov [12]).

$\tilde{\Phi}_{a-1,1}^{a,b}(1/N)$$\tilde{\Phi}_{a-1.1}^{a,b}(1/N)$$\tilde{\Phi}_{n,m}^{a,b}(1/N)+(-iN)^{\frac{3}{2}}-$

for $Narrow\infty$ is as follows ([12]):term$\sim\sum_{k=0}^{\infty}\frac{T^{n.m}(k)}{k!}(\frac{\pi}{2abiN})^{k},$where$T^{n,m}(k)= \frac{1}{2}(-1)^{k+1}L(-2k-1,\chi_{2ab}^{(n,m)})$and$L(s,\chi_{2ab}^{(n.m)})$is the -function$L$$of\chi_{2ab}^{(n,m)}$defined by the power series$L(s, \chi_{2ab}^{(n.m)})=\sum_{k\geq 1}\frac{\chi_{2ab}^{(n.m)}(k)}{k^{s}}$for ${\rm Re}(s)>1$ , which has a meromorphic continuation to$\mathbb{C}.$.

and the description of.

The asymptotic expansion ofof $T(a, b)$$\langle T(a, b)\rangle_{N}=\tilde{\Phi}_{a-1,1}^{a,b}(1/N)\cross\exp(\frac{(ab-a-b)^{2}}{2abN}\pi i)$

We would like to see a relation between the function$\zeta(A_{a,b}(L, M), s)$$\langle T(a, b)\rangle_{N}$30
Remark 4.7. Consider $s=\log T$ for$L(s,\chi_{2ab}^{(n,m)})$. Then$\sum_{k\geq 1}\chi_{2ab}^{(n.m)}(k)T^{-\log k}=L(\log T,\chi_{2ab}^{(n,m)})$When we consider $(n, m)=(a-1,1)$ , essentiallypower series expansion of the Alexander polynomial$L(s,\chi_{2ab}^{(n.m)})$$\Delta_{a,b}(T^{2})$From the Fourier transform on finite abelian groups
characters of modulo $2ab$ as follows:$\chi_{2ab}^{(n,m)}$corresponds to the the previous.

is written in terms of the Dirichlet$\chi_{2ab}^{(n,m)}=\frac{1}{\phi(2ab)}\sum_{\chi:even}c_{\chi}(a, b, n,m)\chi,$whereis the Euler function andcharacters modulo $2ab$ that $is, \chi(-1)=1$ ), and$\phi(2ab)=\#(\mathbb{Z}/2ab\mathbb{Z})^{\cross}$$\chi$

runs through all the even Dirichlet$($$c_{\chi}(a, b, n, m)=2\overline{(\chi(nb-ma)-\chi(nb+ma))}.$

Therefore its -function is written in terms of the -functions of the even Dirichlet characters
modulo $2ab$
$L$

$L$

.

$L(s, \chi_{2ab}^{(n,m)})=\frac{1}{\phi(2ab)}\sum_{x:even}c_{\chi}(a, b, n,m)L(s,\chi)$

In general the Dedekind zeta functions of abelian number fields are written as the product ofDirichlet -functions. Especially, in the cyclotomic field case, we have$L$$\zeta(\mathbb{Q}_{d}, s)=\prod_{\chi}L(s,\chi)$

where$\chi$,

runs through all the Dirichlet characters modulo . Therefore$d$$L(s, \chi_{2ab}^{(n,m)})=\frac{1}{\phi(2ab)}\sum_{\chi:even}c_{\chi}(a, b, n, m)L(s,\chi)$.

Example 4.8. Consider the $(2, 3)$ -torus knot case. In this case, there is only one character, which is the unique even Dirichlet character modulo 12. The zeta function$\zeta(A_{2.3}(L, M), s)$ is expressed as$\chi_{12}^{(1.1)}=\chi_{12}^{(1,2)}$$\zeta(A_{2,3}(L, M), s)=\frac{\zeta(s-1)^{2}}{\zeta(s)}\cross\zeta(\mathbb{Q}_{4}, s)^{-1}\zeta(\mathbb{Q}_{12}, s)^{-1}.$

has 4 characters, the Dedekind zeta function
product of the Dirichlet -functions

Since$(\mathbb{Z}/12\mathbb{Z})^{\cross}$$\zeta(\mathbb{Q}_{12}, s)$is decomposed into the$L$$\zeta(\mathbb{Q}_{12}, s)=\zeta(s)L(s,\chi_{12}^{(1.1)})L(s,\chi_{2})L(s,\chi_{3})$whereare the other two odd Dirichlet characters ofessentially corresponds to the Alexander polynomial$\chi_{2},\chi_{3}$$(\mathbb{Z}/12\mathbb{Z})^{x}$$\Delta_{2,3}(T^{2})$.

,. The -function$L$$L(s,\chi_{12}^{(1.1)})$

31
Question 4.9. Is there a topological interpretation for$\tilde{\Phi}_{n,m}^{a,b}(\tau)$and$L(s,\chi_{2ab}^{(n,m)})$for $(n, m)\neq$$(a-1,1)$ ?

Question 4.10. Is there a direct relation between$\langle T(a, b)\rangle_{N}$(or colored Jones polynomial of$T(a, b))$ and $L(s,\chi_{2ab}^{(n.m)})$ $(or other$ components $of\zeta(A_{a,b}(L, M),$ $s)$ )?

Acknowledgments
I would like to thank Professor Tomotada Ohtsuki for inviting me the workshop “Intelligenceof Low-dimensional Topology 2014”’ and giving the opportunity to write this article. The author
is supported by the JSPS Fellowships for Young Scientists.

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