数理解析研究所講究録
第 1715 巻 2010 年 64-74

64

DIFFERENCES OF THE SELBERG TRACE FORMULA AND SELBERG
TYPE ZETA FUNCTIONS FOR HILBERT MODULAR SURFACES
九州大学・数理学研究院 権 寧魯
(KYUSHU UNIV.)YASURO GON

ABSTRACT. We study analytic properties of a certain kind of Selberg type zeta functions
attached to Hilbert modular surfaces. The method is based on considering the differencesamong the Selberg trace formula with several weights.

1. INTRODUCTION
In this article, we consider Selberg type zeta functions attached to the Hilbert modular group of a real quadratic field. First of all, we recall the definition of Selberg zeta
and
SL 2,
function for a comapct Riemann surface. Let $G=$ PSL 2,$\mathbb{H}=\{z\in \mathbb{C}|{\rm Im} z>0\}$ be the upper half plane.
Then acts on by the fractional. Let be a co-compact torsion-free discrete subgroup of
linear transformation
is a compact Riemann surface of genus $g\geq 2$ .
, then the quotient space$(\gamma)|>2$, then the centralizer of in is infinite cyclic
is hyperbolic, that is trLet
and is conjugate in to$($$\mathbb{R})=$$G$$g.z= \frac{az+b}{cz+d}$

$G$

$($

$\mathbb{R})/\{\pm I\}$$\mathbb{H}$$\Gamma$$X=\Gamma\backslash \mathbb{H}$$\Gamma$$\gamma\in\Gamma$$\gamma$$G$$\gamma$$\gamma\sim(\begin{array}{ll}N(\gamma)^{1/2} 00 N(\gamma)^{-1/2}\end{array})$

with $N(\gamma)>1$ .

Put Prim(F) be the set of F-conjugacy classes of the primitive hyperbolic elements in .
(i.e, not a power of other hyperbolic elements)
The Selberg zeta function for (or $X$ ) is defined by the following Euler product:$\Gamma$$\Gamma$$Z_{\Gamma}(s):= \prod_{p\in Prim(\Gamma)}\prod_{k=0}^{\infty}(1-N(p)^{-(k+s)})$Selberg proved the following theorem on$Z_{\Gamma}(s)$for ${\rm Re}(s)>1$ .

:(1)Theorem 1.1 (Selberg 1956, [14]).
defined for ${\rm Re}(s)>1$ extends meromorentire).
(actually
phically over(2)
has zeros at $s=-k(k\in N)$ of order $(2g-2)(2k+1)$ ,at $s=0$ of order $2g-1$ and at $s=1$ of order 1 : trivial zeros.
: nontrivial zeros.
(3)
has zeros at$Z_{\Gamma}(s)$$\mathbb{C}$$Z_{\Gamma}(s)$$Z_{\Gamma}(s)$$s= \frac{1}{2}\pm ir_{n}$Date: July 1, 2010.
Key words and phrases. Hilbert modular surface; Selberg zeta function.
2000 Mathematics Subject Classification. llM36,llF72.

65

actingHere, $\{\lambda_{n}=1/4+r_{n}^{2}\}$ is the eigenvalues of the Laplacianon. This theorem is proved by using the Selberg trace formula for the compact$\triangle_{0}=-y^{2}(\partial^{2}=+=\partial\partial^{2}y)$$L^{2}(\Gamma\backslash \mathbb{H})$Riemann surface.
This zeta function$\Gamma\backslash \mathbb{H}$$Z_{\Gamma}(s)$satisfies the following functional equation:

Theorem 1.2 (Functional equation by Selberg 1956, [14]).
$Z_{\Gamma}(1-s)=Z_{\Gamma}(s) \exp(-4(g-1)\pi\int_{0}^{s-\frac{1}{r2}}\tan(\pi r)dr)$.

The above functional equation is rewritten to a symmetric functional equation by using
the double gamma function.
$\hat{Z}_{\Gamma}(1-s)=\hat{Z}_{\Gamma}(s):=Z_{\Gamma}(s)(\Gamma_{2}(s)\Gamma_{2}(s+1))^{2g-2}$Here, $\Gamma_{2}(z)=\exp(\zeta_{2}’(0, z))$ is the double gamma function and $\zeta_{2}(s, z)=\sum_{n,m\geq 0}(n+m+$
is the double Hurwitz zeta function. The theory of Selberg zeta functions for locallysymmetric spaces of rank one is evolved by Gangolli [4] (compact case) and Gangolli-Warner[5] (noncompact case). Multiple gamma functions also appear in functional equation forthese Selberg zeta functions. We refer to [11] for multiple gamma functions and [6], [7] forgamma factors of Selberg zeta functions of rank one locally symmetric spaces. Therefore,
our concern is “Selberg type zeta functions” for higher $mnk$ locally symmetric spaces such
as Hilbert modular varieties etc.
In this article, we consider the following problems:.
(1) Construct Selberg type zeta functions for
PSL 2,.
PSL 2,(2) Study analytic properties of the above Selberg type zeta functions forIn the next section, we introduce Selberg type zeta functions for Hilbert modular surfaces
and study analytic properties of them.
$z)^{-s}$$\Gamma\subset$$\mathbb{R})^{2}$$($$\Gamma\subset$$($$\mathbb{R})^{2}$

2. SELBERG TYPE ZETA FUNCTIONS FOR HILBERT MODULAR SURFACES

2.1. Notation and definition. Let
be a real quadratic field with class number one$K$
be thebe the discriminant of $K$ andandbe the ring of integers of . Put$a’=\sigma(a)$$K$
forby and put
fundamental unit of . We denote the generator of Gal$K/\mathbb{Q}$$\epsilon>1$$D$$\mathcal{O}_{K}$$(K/\mathbb{Q})$$a\in K$

. We also put$\gamma’=(\begin{array}{ll}a’ b’c d’\end{array})$for$\gamma=(\begin{array}{ll}a bc d\end{array})\in$

PSL 2,$($$\sigma$$\mathcal{O}_{K})$.

PSL(2,
be the Hilbert modular group.
Let
acts on theand
It is known thatis an irreducible discrete subgroup of PSL 2,
by linear fractional transformation every component.
product of two upper half planes, i.e.
-inequivalent parabolic fixed point.
have only one cusp
is called the Hilbert modular surface.
Letbe hyperbolic-elliptic, i.e, tr $(\gamma)|>2$ and tr $(\gamma’)|<2$ . Then theincentralizer of hyperbolic-elliptic
is infinite cyclic.
$m\geq6$.
Fix an even integer$\Gamma_{K}=\{(\gamma,$ $\gamma’)|\gamma\in$$\mathcal{O}_{K})\}$$($$\Gamma_{K}$$\mathbb{R})^{2}$$\Gamma_{K}$$\mathbb{H}^{2}$$(\infty, \infty)$$\Gamma_{K}$$X_{K}=\Gamma_{K}\backslash \mathbb{H}^{2}$$\Gamma_{K}$$(\gamma, \gamma’)\in\Gamma_{K}$$(\gamma, \gamma’)$$\Gamma_{K}$66

Definition 2.1 (Selberg type zeta function for$\Gamma_{K}$).

$Z_{K}(s;m):= \prod_{(p,p’)}\prod_{k=0}^{\infty}(1-e^{i(m-2)\omega}N(p)^{-(k+s)})^{-\kappa}$

for ${\rm Re}(s)\gg 0$

Here, $(p,p’)$ run through the set of primitive hyperbolic-elliptic
to, and $(p,p’)$ is conjugate in PSL 2,-conjugacy classes of$\mathbb{R})^{2}$$($$\Gamma_{K}$$\Gamma_{K}$$(p,p’)\sim((N(p)^{1/2}0$$N(p)^{-1/2}0),$$(\begin{array}{ll}cos\omega -sin\omegasin\omega cos\omega\end{array}))$.

. We define the smallest natural number such
Here, $N(p)>1,$ $\omega\in(0, \pi)$ andis the Dedekind zeta function, whereandthat.
is the set of the orders of primitive elliptic elements inof $K$ and$\kappa$$\omega\not\in\pi \mathbb{Q}$$2\kappa\zeta_{K}(-1)\in \mathbb{Z}$$\zeta_{K}(s)$$\kappa\nu_{j}^{-1}\in \mathbb{Z}(1\leq j\leq N)$$\Gamma_{K}$$\{\nu_{1}, \nu_{2}, \cdots, \nu_{N}\}$2.2. Analytic properties of $Z_{K}(s;m)$ . Our main theorems on analytic properties of$Z_{K}(s;m)$ are followings.
Theorem 2.2. For an even integer $m\geq 6_{f}Z_{K}(s;m)$ a priori defined for ${\rm Re}(s)>1$ has ameromorphic extension over the complex plane .
$\mathbb{C}$

Theorem 2.3. $Z_{K}(s, m)$ has the following aessential” zeros and poles at: zeros$j=0,1,2,$: poles$k=0,1,2,$$\bullet$$s= \frac{1}{2}\pm i\rho_{j}$$\bullet$$s= \frac{1}{2}\pm i\mu_{k}$$\cdots$$\cdots$

Here,$\bullet\{\frac{1}{4}+\rho_{j}^{2}|j=0,1,2, \cdots\}=Spec(\Delta_{0}^{(1)}|_{Ker(\Lambda_{m}^{(2)})})$$\bullet$$\{\frac{1}{4}+\mu_{k}^{2}|k=0,1,2, \cdots\}=Spec(\Delta_{0}^{(1)}|_{Ker(\Lambda_{m-2}^{(2)})})$

acting on “Hilbert-Maass foms” of weight
are the sets of eigenvalues of the Laplacian$(0, m)$ or $(0, m-2)$ andare “Maass operators”.
$\Delta_{0}^{(1)}$$\Lambda_{m}^{(2)},$$\Lambda_{m-2}^{(2)}$2.3. Functional equation of $Z_{K}(s;m)$ . $Z_{K}(s, m)$ has another series of zeros and poles
and thecoming from the identity, elliptic, “type 2 hyperbolic” conjugacy classes ofscattering terms. (See Definition 3.2 for type 2 hyperbolic element.)$\Gamma_{K}$Theorem 2.4. $Z_{K}(s, m)$ satisfies the following functional equation$\hat{Z}_{K}(s;m)=\hat{Z}_{K}(1-s;m)$

Here the completed zeta function$\hat{Z}_{K}(s, m)$.

is given by$\hat{Z}_{K}(s;m):=Z_{K}(s;m)(Z_{id}(s)Z_{el1}(s)Z_{sct/hyp2}(s))^{\kappa}$

67

with
$Z_{id}(s):=(\Gamma_{2}(s)\Gamma_{2}(s+1))^{2\zeta_{K}(-1)}$$Z_{e11}(s):= \prod_{j=1}^{N}\prod_{l=0}^{\nu_{j}-1}\Gamma(_{\nu_{j}}^{\underline{s}}\pm\iota)^{\frac{\nu-1-\alpha_{l}(mj)-\overline{\alpha_{l}}(mj)}{\nu_{j}}}$$Z_{sct/hyp2}(s):= \zeta_{\epsilon}(s+\frac{m}{2}-1)\zeta_{\epsilon}(s+\frac{m}{2}-2)^{-1}$

and
Here,
is the set of the orders of primitive elliptic elements inwill be given in the next subsection.
the definition of$:=(1-\epsilon^{-2s})^{-1}$ andWe defineis the fundamental unit of $K$ . The zeros and poles$Z_{id}(s),$ofand $Z_{sct/hyp2}(s)$ are easily calculated. Therefore, all zeros and poles of$Z_{K}(s;m)$ are determined.
These analytic properties and functional equation of $Z_{K}(s;m)$ are obtained by using
the “differences” of the Selberg trace formula for Hilbert modular surfaces. In the next
subsection, we introduce and investigate the Selberg trace formula for our case and their$\{\nu_{1}, \nu_{2}, \cdots, \nu_{N}\}$$\alpha_{l}(m,j),$$\Gamma_{K}$$\overline{\alpha_{l}}(m,j)\in\{0,1, \cdots, \nu_{j}-1\}$$\zeta_{\epsilon}(s)$$\epsilon$$Z_{e11}(s)$differences.

3. DIFFERENCES OF THE SELBERG TRACE FORMULA FOR HILBERT MODULAR
SURFACES

3.1. Notation. Let $G=$ PSL$A2)\in \mathbb{H}^{2}$ .
byacts onif
itisdiscrete subgroupnot commensurable with any direct productsubgroups of PSL 2, .
We have classification of the elements of irreducible .
(1) $\gamma=(I, I)$ is the identity(2)
is hyperbolictrand tr(3)
is elliptictrand tr$(2, \mathbb{R})^{2}=(SL(2, \mathbb{R})/\{\pm I\})^{2}$$G$$\mathbb{H}^{2}$$($is called irreducibleof two discrete$\Gamma\subset G$$(g_{1}, g_{2}).(z_{1}, z_{2})=(_{cz_{1}+d_{1}c_{2}z_{2}\vec{+d_{2}}}^{az+baz+b} \bigcup_{1},$$\Gamma_{1}\cross\Gamma_{2}$$\mathbb{R})$$\Gamma$$\gamma=(\gamma_{1}, \gamma_{2})$$\Leftrightarrow|$$(\gamma_{2})|>2$$(\gamma_{1})|>2$$\Leftrightarrow|$$\gamma=(\gamma_{1}, \gamma_{2})$$(\gamma_{2})|<2$$(\gamma_{1})|<2$

is hyperbolic-elliptic
tr $(\gamma_{1})|>2$ and tr $(\gamma_{2})|<2$
is elliptic-hyperbolic
tr $(\gamma_{1})|<2$ and tr $(\gamma_{2})|>2$
is parabolic
tr $(\gamma_{2})|=2$
tr
Note that there are no other types in . (parabolic-elliptic etc.) (Cf. Shimizu [16])We consider the Hilbert modular group,
(4)
(5)
(6)

$\gamma=(\gamma_{1}, \gamma_{2})$$\Leftrightarrow|$$\gamma=(\gamma_{1}, \gamma_{2})$$\Leftrightarrow|$$\gamma=(\gamma_{1}, \gamma_{2})$$\Leftrightarrow|$$(\gamma_{1})|=|$$\Gamma$$\Gamma_{K}$$\Gamma_{K}$$:=\{(\gamma, \gamma’)=((\begin{array}{ll}a bc d\end{array}),$$(\begin{array}{ll}a^{/} b’c^{/} d\end{array}))|(\begin{array}{ll}a bc d\end{array})\in$

is an irreducible discrete subgroup of $G=$ PSL 2,$(\infty, \infty)$$($$\mathbb{R})^{2}$

PSL 2,$($$\mathcal{O}_{K})\}$.

with the only one cusp oo:$=$.

Lemma 3.1 (Stabilizer of the cusp $\infty=(\infty,$
is given by$\Gamma_{\infty}:=\{$$((u0$$u^{-1}\alpha),$$\infty)$). The stabilizer of $\infty=(\infty, \infty)$ in$(\begin{array}{ll}u^{/} \alpha’0 u^{f-1}\end{array}))|u\in \mathcal{O}_{K}^{\cross},$$\alpha\in \mathcal{O}_{K}\}$.

$\Gamma_{K}$68Definition 3.2 (Types of hyperbolic elements). For a hyperbolic element , we define thatis type 1 hyperbolic whose all fixed points are not fixed by parabolic elements.
not type 1 hyperbolic.
is type 2 hyperbolic$\gamma$$\Leftrightarrow$$\bullet$$\gamma$$\Leftrightarrow$$\bullet$$\gamma$Lemma 3.3. Any type 2 hyperbolic elements of$\Gamma_{K}$are conjugate to an element of$\{\gamma_{k,\alpha}=(\begin{array}{ll}\epsilon^{k} \alpha 0 \epsilon^{-k}\end{array})|k\in N,$

in$\Gamma_{K}$. The centmlizer of$\gamma_{k,\alpha}$in$\Gamma_{K}$$\alpha\in \mathcal{O}_{K}\}$is an infinite cyclic group.

withas
By the above lemma, we may take a generator of the centralizer$k_{0}\in N$ and.
. We also write as-conjugacy classes
be a complete system of representatives of theLet$\nu_{N}(\nu\inN,\nu\geq2)$
the orders ofdenote.
of primitive elliptic elements oftois conjugate in PSL 2,. We may assume that
$Z_{\Gamma_{K}}(\gamma_{k,\alpha})$$R_{1},$ $R_{2},$$\cdots,$$\Gamma_{K}$$R_{N}$$\Gamma_{K}$$R_{1},$ $R_{2},$$\cdots,$$R_{N}$$R_{j}\sim$$\gamma_{k_{0},\beta}$$k_{0}(\gamma_{k,\alpha})$$k_{0}$$\beta\in \mathcal{O}_{K}$$\nu_{2},$$\nu_{1},$$\cdots,$$(( \cos\sin\frac{\frac{\pi}{\nu_{j}\pi}}{\nu_{j}}$$- \sin\frac{\pi}{\nu_{j}}\cos\frac{\pi}{\nu_{j}}),$$\mathbb{R})^{2}$$($$R_{j}$$-\sin^{\llcorner^{\pi}}\cos_{\nu^{\frac{\underline t\pi\nu}{j}}}^{t^{j}}\lrcorner))$$( \cos_{\lrcorner}^{t_{\llcorner}}\sin^{t}\frac{\nu j\pi\pi}{\nu_{j}}$,$(t_{j}, \nu_{j})=1$

For even natural number $m\geq 4$ and $l\in\{0,1, \cdots, \nu_{j}-1\}$ , we define$\{0,1, \cdots, \nu_{j}-1\}$ by$l+t_{j}( \frac{m-2}{2})\equiv\alpha_{l}(m,j)$$(mod \nu_{j})$$l-t_{j}( \frac{m-2}{2})\equiv\overline{\alpha_{l}}(m,j)$$(mod \nu_{j})$.

$\alpha_{l}(m,j),$$\overline{\alpha_{l}}(m,j)\in$-conjugacy classes,, type 1 hyperbolicandWe denote by-conjugacy classes, elliptic-hyperbolic-conjugacy classes, hyperbolic-elliptic
elliptic
respectively.
-conjugacy classes of-conjugacy classes and type 2 hyperbolic$\Gamma_{H1},$$\Gamma_{HE},$$\Gamma_{E},$$\Gamma_{EH}$$\Gamma_{K}$$\Gamma_{H2}$$\Gamma_{K}$$\Gamma_{K}$$\Gamma_{K}$$\Gamma_{K}$$\Gamma_{K}$3.2. Selberg trace formula for Hilbert modular surfaces. Fix the weightfor
PSL 2, $\mathbb{R})(j=1,2)$ . Let. Set the automorphic factor$(m_{1}, m_{2})\in$$j_{\gamma}(z_{j})= \frac{cz_{j}+d}{|cz_{j}+d|}$$(2\mathbb{Z}_{\geq 0})^{2}$$\Delta_{m_{j}}:=-y_{j}^{2}(_{\overline{\partial}\partial y_{j}}\pi_{x_{j}}+=)+im_{j}y_{j^{\frac{\partial}{\partial x_{j}}}}$$\gamma\in$$($$(j=1,2)$ .

Definition 3.4.
$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2})):=\{f:\mathbb{H}^{2}arrow \mathbb{C},$(i)$C^{\infty}|$$f((\gamma, \gamma’)(z_{1}, z_{2}))=j_{\gamma}(z_{1})^{m_{1}}j_{\gamma’}(z_{2})^{m}2f(z_{1}, z_{2})$(ii) ョ

$(\lambda^{(1)}, \lambda^{(2)})\in \mathbb{R}^{2}$$\Delta_{m_{1}}^{(1)}f(z_{1}, z_{2})=\lambda^{(1)}f(z_{1}, z_{2})$$(iii)||f||^{2}= \int_{\Gamma_{K}\backslash \mathbb{H}^{2}}f(z)\overline{f(z)}d\mu(z)<\infty.\}$Here,$d \mu(z)=\frac{dx_{1}dy_{1}}{y_{1}^{2}}\frac{dx2dy_{2}}{y_{2}^{2}}$$\forall(\gamma,\gamma’)\in\Gamma_{K}$for$z=(z_{1}, z_{2})\in \mathbb{H}^{2}$.

,$\triangle_{m}^{(2)}2f(z_{1}, z_{2})=\lambda^{(2)}f(z_{1}, z_{2})$

69

Let

$(m_{1}, m_{2})\in(2\mathbb{Z}_{\geq 0})^{2},$${\rm Re}(s_{1}),$${\rm Re}(s_{2})\gg 0$$z=(z_{1}, z_{2})=(x_{1}+iy_{1}, x_{2}+iy_{2})\in \mathbb{H}^{2}$, and$(s_{1}, s_{2})\in \mathbb{C}^{2}$$E_{(m_{1},m)}2(z, s_{1}, s_{2}):= \sum_{\gamma\in\Gamma_{\infty}\backslash \Gamma_{K}}\frac{y_{1}^{s_{1}}}{|cz_{1}+d|^{2s_{1}}}\frac{y_{2}^{s_{2}}}{|c’z_{2}+d’|^{2s}2}\frac{|cz_{1}+d|^{m1}}{(cz_{1}+d)^{m1}}\frac{|c’z_{2}+d’|^{m}2}{(cz_{2}+d’)^{m}2}$

Definition 3.5 (with. We define,of Eisenstein series). Forwith ${\rm Re}(s)>1$ and$I^{\urcorner}\{alni1\}^{r}$$(x_{1}+iy_{1}, x_{2}+iy_{2})\in \mathbb{H}^{2},$$s\in \mathbb{C}$$(m_{1}, m_{2})\in(2\mathbb{Z}_{\geq 0})^{2},$$k\in \mathbb{Z}$.

$z=(z_{1}, z_{2})=$

we define$E_{(m_{1},m)}2(z, s;k):=E_{(m_{1},m_{2})}(z,$ $s+ \frac{\pi ik}{2\log\epsilon},$ $s- \frac{\pi ik}{2\log\epsilon})$.

Proposition 3.6. For ${\rm Re}(s)>1$ , the Eisenstein series $E_{(m_{1},m)}2(z, s;k)$ is absolutely convergent and$E_{(m1m)}2(\gamma z, s;k)=E_{(mm2}1,)(z, s;k)$for any$\gamma\in\Gamma_{K}$. $E_{(m_{1},m_{2})}(z, s;k)$ is a common eigenfunction ofand$\Delta_{m_{1}}^{(1)}$$\triangle_{m2}^{(2)}$.

Proposition 3.7. We have a direct sum decomposition:$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))=L_{d\iota s}^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))\oplus L_{\omega n}^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))$and there is an orthonorm

$al$

basis

$\{\phi_{j}\}_{j=0}^{\infty}$

To subtract continuous spectrum on.
determinantof$L_{dis}^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))$$L_{con}^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))$.

, we introduce the scattering$\varphi_{(m_{1},m)}2(s, k)$

Proposition 3.8. The constant $tem$ of $E_{(m_{1},m)}2(z, s;k)$ is given by$y_{1}^{s+\frac{\pi 1k}{2\log\epsilon}}y_{2^{+\varphi(m_{1},m_{2})(s,k)y_{1}^{1-s-\frac{\pi\cdot k}{2\log e}}y_{2}^{1-s+\frac{\pi\cdot k}{2\log\epsilon}}}}^{s-\frac{\pi ik}{2\log\epsilon}}$with$\varphi_{(m_{1},m_{2})}(s, k)=\frac{(-1)^{\frac{m+m}{2}}\pi}{2\sqrt{D}}\frac{L(2s-1,\chi_{-k})}{L(2s,\chi_{-k})}\frac{\Gamma(s+\frac{\pi ik}{2\log\epsilon}-\frac{1}{2})\Gamma(s+\frac{\pi ik}{2\log\epsilon})}{\Gamma(s+\frac{\pi ik}{2\log\epsilon}+\frac{m}{2}L)\Gamma(s+\frac{\pi ik}{2\log\epsilon}-\frac{m}{2}1)}$$\cross\frac{\Gamma(s-\frac{\pi ik}{2\log\epsilon}-\frac{1}{2})\Gamma(s-\frac{\pi ik}{2\log\epsilon})}{\Gamma(s-\frac{\pi ik}{2\log\epsilon}+-4)\Gamma(s-\frac{\pi ik}{2\log\epsilon}--2)}$

for$k\in \mathbb{Z}$

Let
that

, where $L(s, \chi_{-k})$ is defined by $L(s, \chi_{-k})$$\{\phi_{j}\}_{j=0}^{\infty}$

be an orthonormal basis of$\triangle_{m_{1}}^{(1)}\phi_{j}=\lambda_{j}^{(1)}\phi_{j}$Put $Spec(m_{1}, m_{2})$$:= \sum_{(c)\subset \mathcal{O}_{K}}|\frac{c}{c}|^{-}\frac{0\epsilon\pi k}{g}N(c)^{-s}$$L_{dis}^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(m_{1}, m_{2}))$and$:=\{(r_{j}^{(1)}, r_{j}^{(2)})\}_{j=0}^{\infty}\subset \mathbb{R}^{2}$$\triangle_{m_{2}}^{(2)}\phi_{j}=\lambda_{j}^{(2)}\phi_{j}$and.

$(\lambda_{j}^{(1)}, \lambda_{j}^{(2)})\in \mathbb{R}^{2}$.

. (discrete subset), where, we write. $(l=1,2)$ .
Now we can state on the Selberg trace formula,$\frac{1}{4}+(r_{j}^{(l)})^{2}$$\bullet$$h(r_{1}, r_{2})=h(\pm r_{1}, \pm r_{2})$such: test function (satisfying certain analytic conditions)$\lambda_{j}^{(l)}=$70: the Fourier transform of
Hereafter, we assume that $h(r_{1}, r_{2})=h_{1}(r_{1})h_{2}(r_{2})$ and also write $g(u_{1}, u_{2})=g_{1}(u_{1})g_{2}(u_{2})$ .
$\bullet$$g(u_{1}, u_{2})$$h$$:= \overline{4}\pi\nabla 1\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}2$

Theorem 3.9 (Selberg trace formula forand put
be an even function in
Then we have,$C_{c}^{\infty}(\mathbb{R}^{2})$$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0,$$h(r_{1}, r_{2})$

$m))$

with

$m\in 2\mathbb{Z}_{\geq 0}$). Let$g(u_{1}, u_{2})$.

$:= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}2$$\sum_{j=0}^{\infty}h(r_{j}^{(1)}, r_{j}^{(2)})-\frac{1}{4\pi}\sum_{k\in \mathbb{Z}}\int_{\mathbb{R}}h(r+\frac{\pi k}{2\log\epsilon}, r-\frac{\pi k}{2\log\epsilon})\frac{\varphi_{(0,m)}’}{\varphi_{(0,m)}}(\frac{1}{2}+ir, k)dr$$+ \frac{1}{4}h(0,0)\varphi_{(0,m)}(\frac{1}{2},0)$$= \frac{vol(\Gamma_{K}\backslash \mathbb{H}^{2})}{16\pi^{2}}\int\int_{\mathbb{R}^{2}}\frac{\frac{\partial^{2}}{\partial u_{1}\partial u2}g(u_{1},u_{2})}{\sinh(u_{1}/2)\sinh(u_{2}/2)}e^{-\frac{m}{2}u_{2}}du_{1}du_{2}$$+ \sum_{(\gamma,\gamma)\in\Gamma_{H1}}\frac{vo1(\Gamma_{\gamma}\backslash G_{\gamma})g(\log N(\gamma),\log N(\gamma’))}{(N(\gamma)^{1/2}-N(\gamma)^{-1/2})(N(\gamma’)^{1/2}-N(\gamma^{f})^{-1/2})}$$+ \sum_{R(\theta_{1},\theta_{2})\in\Gamma_{E}}\frac{-e^{-i\theta_{l}+i(m-l)\theta_{2}}}{16\nu_{R}\mathfrak{X}n\theta_{1}\mathfrak{X}n\theta_{2}}\int\int_{\mathbb{R}^{2}}g(u_{1}, u_{2})e^{-\lrcorner^{u_{2^{+\frac{(m-1)}{2}u2}}}}\prod_{j=1}^{2}[\frac{e^{u_{j}}-e^{2i\theta_{j}}}{\cosh u_{j}-\cos 2\theta_{j}}]du_{1}du_{2}$$+ \sum_{(\gamma,\omega)\in\Gamma_{HE}}\frac{\log N(\gamma_{0})ie^{i(m-1)\omega}}{N(\gamma)^{1/2}-N(\gamma)^{-1/2}4\sin\omega}\int_{-\infty}^{\infty}g(\log N(\gamma), u)e^{\frac{m-1}{2}u}[\frac{e^{u}-e^{2i\omega}}{\cosh u-\cos 2\omega}]du$$+ \sum_{(\omega’,\gamma)\in\Gamma_{EH}}\frac{\log N(\gamma_{0}’)ie^{-ud’}}{N(\gamma’)^{1/2}-N(\gamma’)^{-1/2}4\sin\omega’}\int_{-\infty}^{\infty}g(u, \log N(\gamma’))e^{\frac{-1}{2}u}[\frac{e^{u}-e^{2i\omega’}}{\cosh u-\cos 2\omega’}]du$$+[2 \sqrt{D}A_{0}-4\log\epsilon(\log 2+C_{E})]g(0,0)+\log\epsilon\int_{-\infty}^{\infty}[g(u, 0)+g(0, u)]du$$- \frac{\log\epsilon}{2\pi^{2}}\iint_{\mathbb{R}^{2}}[\frac{\Gamma’}{\Gamma}(1+ir_{1})+\frac{\Gamma’}{\Gamma}(1+ir_{2})]h(r_{1}, r_{2})dr_{1}dr_{2}$$+2 \log\epsilon\int_{0}^{\infty}\frac{g(0,u)}{e^{u/2}-e^{-u/2}}[1-\cosh\frac{m}{2}u]du$$-4 \log\epsilon\sum_{k=1\gamma_{k},\alpha}^{\infty}\sum_{\in\Gamma_{H2}}\frac{k_{0}(\gamma_{k,\alpha})\log(N(\alpha,\epsilon^{k}-\epsilon^{-k}))}{|N(\epsilon^{k}-\epsilon^{-k})|}g(2k\log\epsilon, 2k\log\epsilon)$$+2 \log\epsilon\sum_{k=1}^{\infty}\int_{2k\log\epsilon}^{\infty}[g(u, 2k\log\epsilon)+g(2k\log\epsilon, u)]\frac{\cosh(u/2)}{\sinh(u/2)+\sinh(k\log\epsilon)}du$$+2 \log\epsilon\sum_{k=1}^{\infty}\int_{2k\log\epsilon}^{\infty}g(2k\log\epsilon, u)\frac{1-\cosh(m(u/2-k\log\epsilon))}{\sinh(u/2-k\log\epsilon)}du$

.

Here,
is the constant term of the Laurent expansion ofat $s=1$ andEuler constant. The case of $(0, m)=(0,0)$ is proved by Zograf [17] and Efrat [1].
$A_{0}$$\zeta_{K}(s)$$C_{E}$

is the71

Next we consider the following Maass operator$\Lambda_{m}^{(2)}:=iy_{2}\frac{\partial}{\partial x_{2}}-y_{2}\frac{\partial}{\partial y_{2}}+\frac{m}{2}:L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m))arrow L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m-2))$

Let$\{\frac{1}{4}+\rho_{j}^{2}\}_{j=0}^{\infty}:=Spec(\triangle_{0}^{(1)}|_{Ker(\Lambda_{m}^{(2)})})$

and recall that$Ker(\Lambda_{m}^{(2)})=L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};$i.e.

$\lambda^{(2)}=\frac{m}{2}(1-\frac{m}{2})$.

$(*, \frac{m}{2}(1-\frac{m}{2}))$, $(0, m))$ .

-eigenspace.

Theorem 3.10 (Differel $1(es$ of STF for
and $m\geq 4$ . We have$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0,$$m))-L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0,$.

$m-2)))$ Let$m\in 2\mathbb{N}$$\sum_{j=0}^{\infty}h_{1}(\rho_{j})h_{2}(\frac{i(m-1)}{2})$$=(m-1)h_{2}( \frac{i(m-1)}{2})\frac{vol(\Gamma_{K}\backslash \mathbb{H}^{2})}{16\pi^{2}}\int_{-\infty}^{\infty}r_{1}h_{1}(r_{1})\tanh(\pi r_{1})dr_{1}$$+ \sum_{R(\theta_{1},\theta_{2})\in\Gamma_{E}}\frac{ie^{i(m-1)\theta_{2}}}{8\nu_{R}\sin\theta_{1}\sin\theta_{2}}h_{2}(\frac{i(m-1)}{2})\int_{-\infty}^{\infty}\frac{\cosh((\pi-2\theta_{1})r_{1})}{\cosh\pi r_{1}}h_{1}(r_{1})dr_{1}$$+ \sum_{(\gamma,\omega)\in\Gamma_{HE}}\frac{\log N(\gamma_{0})}{N(\gamma)^{1/2}-N(\gamma)^{-1/2}}g_{1}(\log N(\gamma))\frac{ie^{i(m-1)\omega}}{2\sin\omega}h_{2}(\frac{i(m-1)}{2})$$- \log\epsilon g_{1}(0)h_{2}(\frac{i(m-1)}{2})-2\log\epsilon h_{2}(\frac{i(m-1)}{2})\sum_{k=1}^{\infty}g_{1}(2k\log\epsilon)\epsilon^{-k(m-1)}$.

We write the above formula as $L(m)-L(m-2)$ for $m\geq 4$ .
$($$(L(m)-L(m-2))h_{2}( \frac{i(m-1)(m-1)2}{2})^{-1}-(L(m-L(m-4))h_{2}(\frac{i(m-3)CO}{2})^{-1}Weassumethath_{2}(\frac{i}{})\neq 0andh_{2}(\frac{i(m-3)}{2^{2})-})\neq 0.Nextwensider$

Theorem 3. $1I$ (Double differences of STF for$m\geq 6$ . We have$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0,$$m))$

for $m\geq 6)$ :). Let$\sum_{j=0}^{\infty}h_{1}(\rho_{j})-\sum_{k=0}^{\infty}h_{1}(\mu_{k})=\frac{vol(\Gamma_{K}\backslash \mathbb{H}^{2})}{8\pi^{2}}\int_{-\infty}^{\infty}rh_{1}(r)\tanh(\pi r)dr$$- \sum_{R(\theta_{1},\theta_{2})\in\Gamma_{E}}\frac{e^{i(m-2)\theta_{2}}}{4\nu_{R}\sin\theta_{1}}\int_{-\infty}^{\infty}\frac{\cosh((\pi-2\theta_{1})r)}{\cosh\pi r}h_{1}(r)dr$$- \sum_{(\gamma,\omega)\in\Gamma_{HE}}\frac{\log N(\gamma_{0})}{N(\gamma)^{1/2}-N(\gamma)^{-1/2}}g_{1}(\log N(\gamma))e^{i(m-2)\omega}$$-2 \log\epsilon\sum_{k=1}^{\infty}g_{1}(2k\log\epsilon)(\epsilon^{-k(m-1)}-\epsilon^{-k(m-3)})$.

$m\in 2\mathbb{N}$

and

72

4. PROOF OF THEOREMS 2.2, 2.3 AND 2.4
4.1. Test function. Theorem 3.9, the Selberg trace formula forwhich satisfies the following condition:
for the test function$L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m))$, holds$h(r_{1}, r_{2})$(1) $h(\pm r_{1}, \pm r_{2})=h(r_{1}, r_{2})$ ,for some $\delta>0$ ,(2) is analytic in the domain
I(3) $h(r_{1}, r_{2})=O((1+|r_{1}|^{2}+|r_{2}|^{2})^{-2-\delta})$ for some $\delta>0$ in this domain.
$h$$|{\rm Im}(r_{1})$$< \frac{1}{2}+\delta,$$|{\rm Im}(r_{2})|< \frac{m-1}{2}+\delta$

Let us consider the following test function: Firstly, we fix real numbers${\rm Re}(s)>1$ , We set. For$\beta_{1},$$\beta_{2}\geq 2$,$s\in \mathbb{C},$$\beta_{1}\neq\beta_{2}$$h_{1}(r):= \frac{((\beta_{1}^{2}-(s-\frac{1}{2})^{2})((\beta_{2}^{2}-(s-\frac{1}{2})^{2})}{(r^{2}+(s-\frac{1}{2})^{2})(r^{2}+\beta_{1}^{2})(r^{2}+\beta_{2}^{2})}=\frac{1}{r^{2}+(s-\frac{1}{2})^{2}}+\frac{c_{1}(s)}{r^{2}+\beta_{1}^{2}}+\frac{c_{2}(s)}{r^{2}+\beta_{2}^{2}}$with$c_{1}(s)= \frac{(s-\frac{1}{2})^{2}-\beta_{2}^{2}}{\beta_{2}^{2}-\beta_{1}^{2}}$,$c_{2}(s)=- \frac{(s-\frac{1}{2})^{2}-\beta_{1}^{2}}{\beta_{2}^{2}-\beta_{1}^{2}}$.

(See [13] for this type test functions.) Then the Fourier transform of$h_{1}$

is given by$g_{1}(u)= \frac{1}{2\pi}\int_{-\infty}^{\infty}h_{1}(r)e^{-iru}dr=\frac{1}{2s-1}e^{-(s-\frac{1}{2})|u|}+\frac{c_{1}(s)}{2\beta_{1}}e^{-\beta_{1}|u|}+\frac{c_{2}(s)}{2\beta_{2}}e^{-\beta_{2}|u|}$Secondly, we takeandsatisfies
such that its Fourier inverse transform. Then we can easily check that our test functionistheabove sufficient condition for Theorem 3.9.
satisfies$h_{2}(r)$$g_{2}(u)\in C_{c}^{\infty}(\mathbb{R})$$h_{2}( \frac{i(m-1)}{2})\neq 0$$h(r_{1}, r_{2})$.

$h_{2}( \frac{i(m-3)}{2})\neq 0$$:=\kappa h_{1}(r_{1})h_{2}(r_{2})$$(\kappa$

defined in Definition 2.1.)
Finally, we consider Theorem 3.11, the double difference of the Selberg trace formula,for the above our test function.
is given by
We recall that$\{\rho_{j}\}$$\{\frac{1}{4}+\rho_{j}^{2}\}_{j=0}^{\infty}=Spec(\Delta_{0}^{(1)}|_{Ker(\Lambda_{m}^{(2)})})$

with

.

$\Lambda_{m}^{(2)}:L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m))arrow L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m-2))$

Note that$Ker(\Lambda_{m}^{(2)})=L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(*, \frac{m}{2}(1-\frac{m}{2})),$

i.e.

$\lambda^{(2)}=\frac{m}{2}(1-\frac{m}{2})$And$\{\mu_{k}\}$$(0, m))$.

-eigenspace.

is given by$\{\frac{1}{4}+\mu_{k}^{2}\}_{k=0}^{\infty}=Spec(\Delta_{0}^{(1)}|_{Ker(\Lambda_{m-2}^{(2)})})$

with$\Lambda_{m-2}^{(2)}:L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m-2))arrow L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(0, m-4))$.

Note that$Ker(\Lambda_{m-2}^{(2)})=L^{2}(\Gamma_{K}\backslash \mathbb{H}^{2};(*, \frac{m-2}{2}(2-\frac{m}{2})),$$(0, m-2))$ .

73

i.e.

$\lambda^{(2)}=\frac{m-2}{2}(2-\frac{m}{2})$-eigenspace.

Theorem 4.1 (DD-STF for the above test function$h_{1}$

and$h_{2}$).

$\kappa\sum_{j=0}^{\infty}[\frac{1}{\rho_{j}^{2}+(s-\frac{1}{2})^{2}}+\sum_{l=1}^{2}\frac{c_{l}(s)}{\rho_{j}^{2}+\beta_{l}^{2}}]-\kappa\sum_{k=0}^{\infty}[\frac{1}{\mu_{k}^{2}+(s-\frac{1}{2})^{2}}+\sum_{l=1}^{2}\frac{c_{l}(s)}{\mu_{k}^{2}+\beta_{l}^{2}}]$$=2 \kappa\zeta_{K}(-1)\sum_{k=0}^{\infty}[\frac{1}{s+k}+\sum_{l=1}^{2}\frac{c_{l}(s)}{\beta_{l}+\frac{1}{2}+k}]$$+ \frac{1Z_{K}’(s)}{2s-1Z_{K}(s)}+\sum_{l=1}^{2}\frac{c_{l}(s)}{2\beta_{l}}\frac{Z_{K}’(\frac{1}{2}+\beta_{l})}{Z_{K}(\frac{1}{2}+\beta_{l})}+\frac{\kappa Z_{ell}’(s)}{2s-1Z_{ell}(s)}+\sum_{l=1}^{2}\frac{\kappa c_{l}(s)Z_{ell}’(\frac{1}{2}+\beta_{l})}{2\beta_{l}Z_{dl}(\frac{1}{2}+\beta_{l})}$.

$+ \frac{\kappa d}{2s-1ds}\log\{\frac{(1-\epsilon^{-(2s+m-4)})}{(1-\epsilon^{-(2s+m-2)})}\}+\sum_{l=1}^{2}\frac{\kappa c_{l}(s)d}{2\beta_{l}d\beta_{l}}\log\{\frac{(1-\epsilon^{-(2\beta_{l}+m-3)})}{(1-\epsilon^{-(2\beta_{l}+m-1)})}\}$

Note that $c_{l}(1-s)=c_{l}(s)(l=1,2),$ $c_{1}(s)+c_{2}(s)=-1$ and
By using the above formula, we can easily obtain Theorems 2.2, 2.3 and 2.4.

$\kappa\frac{vol(\Gamma_{K}\backslash \mathbb{H}^{2})}{4\pi^{2}}=2\kappa\zeta_{K}(-1)\in$

N.

4.2. Final remark. We remark that scattering and type 2 hyperbolic components of$Z_{K}(s;m)$ are local Selberg zeta functions for PSL 2,:$($$\mathbb{Z})$$Z_{sct/hyp2}(s)= \zeta_{\epsilon}(s+\frac{m}{2}-1)\zeta_{\epsilon}(s+\frac{m}{2}-2)^{-1}$with$\zeta_{\epsilon}(s)=(1-\epsilon^{-2s})^{-1}$is the fundamental unit of $K$ .
Let
PSL 2, . The Selberg (Ruelle) zeta function forHere,$\epsilon$$\Gamma=$$($$\mathbb{Z})$$\zeta_{\Gamma}(s)$$:= \prod_{p\in Prim(\Gamma)}(1-N(p)^{-s})^{-1}$then$\Gamma$

is given by$\zeta_{\Gamma}(s)=\prod_{K}(1-\epsilon(K)^{-2s})^{-h(K)}$

where, $K$ run through “all” real quadratic fields overfundamental unit and the class number of $K$ .

$\mathbb{Q}$and$\epsilon(K)$,and $h(K)$ are the

REFERENCES[1] I. Ehat, The Selberg trace formula for. Mem. Amer. Math. Soc. 65 (1987), no. 359, iv $+111$$PSL_{2}(\mathbb{R})^{n}$pp.
[2] J. Fischer, An approach to the Selberg trace formula via the Selberg zeta-function. Lecture Notes inMathematics, 1253. Springer-Verlag, Berlin, 1987. iv $+I84$ pp.
[3] E. Freitag, Hilbert modular forms. Springer-Verlag, Berlin, 1990. viii $+250$ pp.
[4] R. Gangolli, Zeta functions of Selberg‘s type for compact space forms of symmetric spaces of rank
one. Illinois J. Math. 21 (1977), no. 1, 1-41.
[5] R. Gangolli and G. Warner, Zeta functions of Selberg‘s type for some noncompact quotients of symmetric spaces of rank one. Nagoya Math. J. 78 (1980), 1-44.
[6] Y. Gon, Gamma factors of Selberg zeta functions and functional equation of Ruelle zeta functions.
Math. Ann. 308 (1997), no. 2, 251-278.

74[7] Y. Gon and J. Park, The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps.
Math. Ann. 346 (2010), no. 3, 719-767.
[8] D. A. Hejhal, The Selberg trace formula for PSL(2, . Vol. 1. Lecture Notes in Mathematics, 548.
Springer-Verlag, Berlin-New York, 1976. vi $+516$ pp.
[9] D. A. Hejhal, The Selberg trace formula for PSL(2, . Vol. 2. Lecture Notes in Mathematics, 1001.
Springer-Verlag, Berlin-New York, 1983. viii $+806$ pp.
[10] D. Joyner, On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp.
Math. Z. 203 (1990), 59-104.
[11] N. Kurokawa and S. Koyama, Multiple sine functions. Forum Math. 15 (2003), no. 6, 839-876.
[12] D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of PSL(2, . J.
Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 283-313.
[13] L. B. Parnovskii, The Selberg trace formula and Selberg zeta-function for cocompact discrete subgroups of $SO_{+}(1,$ n), Funkts. Anal. Prilozh., 26 (1992), no. 3, 55-64.
[14] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces withapplications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20 (1956), 47-87.
[15] A. Selberg, Harmonic analysis. Collected papers. Vol. I. Springer-Verlag, (Berlin, 1989), 626-674.
[16] H. Shimizu, On discontinuous groups operating on the product of the upper half planes. Ann. of Math.
(2) 77 (1963), 33-71.
[17] P. G. Zograf, Selberg trace formula for the Hilbert modular group of a real quadratic algebraic number
field. J. Sov. Math. 19 (1982), no. 6, 1637-1652.
$E$-mad address: ygonOmath. kyushu-u. ac. jp$\mathbb{R})$$\mathbb{R})$$\mathbb{R})$

FACULTY OF MATHEMATICS, KYUSHU UNIVERSITY, MOTOOKA, FUKUOKA 819-0395, JAPAN