Determinants of Laplacians on Hilbert modular surfaces

0.22 MB·

by  Yasuro Gon

DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR
SURFACES
YASUROGON
7
1
0 Abstract. WestudyregularizeddeterminantsofLaplaciansactingonthespaceofHilbert-
2 Maass forms for the Hilbert modular group of a real quadratic field. We show that these
determinantsare described by Selberg typezeta functions introduced in [4, 5].
n
a
J

3
2
1. Introduction

]
T Determinants of the Laplacian ∆ acting on the space of Maass forms on a hyperbolic
N Riemann surface X are studied by many authors. (See, for example [13, 2, 8, 9].) It is known
. that the determinants of ∆ are described by the Selberg zeta function (cf. [14]) for X.
h
t On the other hand, two Laplacians ∆(1), ∆(2) act on the space of Hilbert-Maass forms on
a
the Hilbert modular surface X of a real quadratic field K. For this reason, it seems that
m K
there are no explicit formulas for “Determinants of Laplacians” on X until now. In this
[ K
article we consider regularized determinants of the first Laplacian ∆(1) acting on its certain

1 subspaces V(2), indexed by m 2N. We show that these determinants are described by
v m
∈
0 Selberg type zeta functions for XK introduced in [4, 5].
8 Let K/Q be a real quadratic field with class number one and be the ring of integers of
K
3 O
K. Put D be the discriminant of K and ε> 1 be the fundamental unit of K. We denote the
6
a b
0 generator of Gal(K/Q) by σ and put a := σ(a) for a K. We also put γ = ′ ′ for
. ′ ∈ ′ c′ d′
1
a b (cid:16) (cid:17)
0
γ = PSL(2, ). Let Γ = (γ,γ ) γ PSL(2, ) be the Hilbert modular
7 c d ∈ OK K { ′ | ∈ OK }
1 group(cid:16)of K. (cid:17)It is known that Γ is a co-finite (non-cocompact) irreducible discrete subgroup
K
:
v of PSL(2,R) PSL(2,R) and Γ acts on the productH2 of two copies of the upperhalf plane
K
Xi H by compon×ent-wise linear fractional transformation. ΓK have only one cusp ( , ), i.e.
∞ ∞
Γ -inequivalent parabolic fixed point. X := Γ H2 is called the Hilbert modular surface.
r K K K
a Let(γ,γ ) Γ behyperbolic-elliptic, i.e, tr(γ)\ > 2and tr(γ ) < 2. Thenthecentralizer
′ K ′
∈ | | | |
of hyperbolic-elliptic (γ,γ ) in Γ is infinite cyclic.
′ K
Definition 1.1 (Selberg type zeta function for Γ with the weight (0,m)). For an even
K
integer m 2, we define
≥
∞ 1
(1.1) Z (s) := 1 ei(m 2)ωN(p) (n+s) − for Re(s) > 1.
m − −
−
(p,p′)Y∈PΓHEnY=0(cid:16) (cid:17)
Date: January 24, 2017.
Key words and phrases. Hilbert modular surface; Selberg zeta function; Regularized determinant.
2010 Mathematics Subject Classification. 11M36, 11F72, 58J52
This work was partially supported by JSPS Grant-in-Aidfor ScientificResearch (C) no. 26400017.
1
2 Y.GON
Here, (p,p) run through the set of primitive hyperbolic-elliptic Γ -conjugacy classes of
′ K
Γ , and (p,p) is conjugate in PSL(2,R)2 to
K ′
N(p)1/2 0 cosω sinω
(p,p′)∼ 0 N(p) 1/2 , sinω −cosω .
−
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
Here, N(p) > 1, ω (0,π) and ω / πQ. The product is absolutely convergent for Re(s)> 1.
∈ ∈
Analytic properties of Z (s) are known.
m
Theorem 1.2 ([5, Theorems5.3and6.5]). For an even integer m 2, Z (s) a priori defined
m
≥
for Re(s)> 1 has a meromorphic extension over the whole complex plane.
In this article, we also consider “the square root of Z (s)”.
2
Definition 1.3 ( Z (s)).
2
p ∞ 1/2
Z (s):= 1 N(p) (n+s) −
2 −
−
(1.2) p (p,p′)Y∈PΓHEnY=0(cid:16) (cid:17)
1 ∞ 1 N(p)−ks
= exp for Re(s)> 1.
2 k1 N(p) k
(cid:16) (Xp,p′)Xk=1 − − (cid:17)
By [5, Theorem 6.5] and the fact that the Euler characteristic of X is even (See Lemma
K
2.2), we see that d logZ (s) has even integral residues at any poles. Therefore, we find that
ds 2
Z (s) has a meromorphic continuation to the whole complex plane.
2
1
pLet us introduce the completed Selberg type zeta functions Z22(s) and Zm(s) (m ≥ 4),
which are invariant under s 1 s. (See [5, Theorems 5.4 and 6.6].)
→ −
b b
Definition 1.4 (Completed Selberg zeta functions).
1 1 1 1 1
(1.3) Z2(s) := Z (s)Z2(s)Z2 (s;2)Z2 (s;2)Z2 (s;2)
2 2 id ell par/sct hyp2/sct
with p
Zi21d(s)b:= Γ2(s)Γ2(s+1) ζK(−1), Ze12ll(s;2) := N νj−1Γ sν+jl νj−2ν1j−2l,
j=1 l=0
(cid:0) (cid:1) Y Y (cid:0) (cid:1)
1 1
Z2 (s;2) := ε s, Z2 (s;2) := ζ (s).
par/sct − hyp2/sct ε
(1.4) Z (s):= Z (s)Z (s)Z (s;m)Z (s;m) (m 4)
m m id ell hyp2/sct
≥
with
b
N νj−1 νj−1−αl(m,j)−αl(m,j)
Zid(s) := Γ2(s)Γ2(s+1) 2ζK(−1), Zell(s;m) := Γ sν+jl νj ,
j=1 l=0
(cid:0) (cid:1) Y Y (cid:0) (cid:1)
m m 1
Z (s;m) := ζ s+ 1 ζ s+ 2 − .
hyp2/sct ε 2 − ε 2 −
Here, Γ (s) is the double Gamma functi(cid:16)on (for defi(cid:17)niti(cid:16)on, we refer(cid:17)to [10] or [3, Definition
2
4.10, p. 751]), the natural numbers ν ,ν ,...,ν are the orders of the elliptic fixed points in
1 2 N
X and the integers α (m,j),α (m,j) 0,1,...,ν 1 are defined in (2.1), ζ (s) is the
K l l j K
∈ { − }
Dedekind zeta function of K, ζ (s):= (1 ε 2s) 1 and ε is the fundamental unit of K.
ε − −
−
DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES 3
Let m 2N. We recall that two Laplacians
∈
∂2 ∂2 ∂2 ∂2 ∂
(1.5) ∆(1) := y2 + , ∆(2) := y2 + +imy
0 − 1 ∂x2 ∂y2 m − 2 ∂x2 ∂y2 2∂x
1 1 2 2 2
(cid:16) (cid:17) (cid:16) (cid:17)
areactingonL2 (Γ H2;(0,m)), thespaceofHilbert-MaassformsforΓ withweight(0,m).
dis K\ K
(See Definition 2.6.) We consider a certain subspace of L2 (Γ H2;(0,m)) given by
dis K\
m m
(1.6) V(2) = f(z ,z ) L2 (Γ H2;(0,m)) ∆(2)f = 1 f .
m 1 2 ∈ dis K\ m 2 − 2
n (cid:12) o
The set of eigenvalues of ∆(1) are enumerated a(cid:12)s (cid:0) (cid:1)
0 Vm(2) (cid:12)
0 < λ(cid:12)(m) λ (m) λ (m)
0(cid:12) ≤ 1 ≤ ··· ≤ n ≤ ···
Let s be a fixed sufficiently large real number. We consider the spectral zeta function by
using these eigenvalues.
∞ 1
(1.7) ζ (w,s) = (Re(w) 0).
m w
λ (m)+s(s 1) ≫
n=0 n −
X
We can show that ζm(w,s) is holom(cid:0)orphic at w = 0. ((cid:1)See Proposition 4.3.)
(1)
Let us define the regularized determinants of the Laplacian ∆ .
0 Vm(2)
Definition 1.5 (Determinants of restrictions of ∆(1)). Let m 2N(cid:12)(cid:12). For s 0, define
0 ∈ ≫
(1) ∂ ∞ 1
(1.8) Det ∆ +s(s 1) := exp .
(cid:16) 0 (cid:12)Vm(2) − (cid:17) (cid:16)−∂w(cid:12)w=0nX=0 λn(m)+s(s−1) w(cid:17)
(cid:12)
(cid:12) (1) (cid:12) (cid:0) (cid:1)
We see later that Det ∆ +s(s 1) can beextended to an entire function of s. (See
0 Vm(2) −
Corollary 1.7.) (cid:16) (cid:12) (cid:17)
Our main theorem is as fol(cid:12)lows.
Theorem 1.6 (Main Theorem). Let (cid:3) := ∆(1) for m 2N. We have the following
m 0 Vm(2) ∈
determinant expressions of the completed Selberg type zeta functions.
(cid:12)
(1) Z212(s)= e(s−12)2ζK(−1)+C2 Det (cid:3)s2(s+s(1s)−1(cid:12)) .
s(s(cid:0) 1) −Det (cid:3) +(cid:1) s(s 1)
(2) Zb4(s) = e2(s−12)2ζK(−1)+C4 −Det·(cid:3) +s4(s 1)− .
2 (cid:0) (cid:1)
−
Det (cid:3) +s(s 1)
(3) Fbor m ≥6, Zm(s) = e2(s−21)2ζK(−1)+(cid:0)CmDet (cid:3) m (cid:1)+s(s− 1) .
(cid:0) m 2 (cid:1)
− −
Here, the constants C are given by
bm (cid:0) (cid:1)
1 N ν2 1
C = logε+ j − logν ,
2 j
−2 12ν
j
j=1
X
N ν2 1 12α (m,j) ν α (m,j)
C = j − − 0 j − 0 logν (m 4),
m j
6ν ≥
j(cid:8) (cid:9)
j=1
X
the natural numbers ν ,ν ,...,ν are the orders of the elliptic fixed points in X and the
1 2 N K
integers α (m,j) 0,1,...,ν 1 are defined in (2.1).
0 j
∈ { − }
4 Y.GON
We know the following Weyl’s law:
Nm+(T) := #{j|λj(m) ≤ T}∼ (m2−1) ·ζK(−1)·T (T → ∞).
(See [5, Theorem 6.11].) Therefore, we may say that Z (s) (m 4) have “more” zeros than
m
≥
poles.
We have several corollaries from Theorem 1.6 by direct calculation.
Corollary 1.7. Let (cid:3) = ∆(1) for m 2N. For m 2N, we have
m 0 Vm(2) ∈ ∈
(1) Det (cid:3)2+s(s−1) = s((cid:12)(cid:12)s−1)e−(s−12)2ζK(−1)−C2Z212(s).
(2) Det(cid:0)(cid:3)m + s(s − 1(cid:1)) = e−(m−1)(s−12)2ζK(−1)−(C2+bC4+···+Cm)Z212(s)Z4(s)···Zm(s) for
m 4.
≥(cid:0) (cid:1)
b b b
It follows from the above corollary that Det (cid:3) +s(s 1) (m 2N) can be extended to
m
− ∈
entire functions of s.
(cid:0) (cid:1)
By putting s= 1 in the above, we have
Corollary 1.8. For m 2N, we have
∈
(1) Det((cid:3)2) = e−14ζK(−1)−C2Ress=1Z212(s).
((32)) DDeett(((cid:3)(cid:3)m4))==ee−−43mζK4−1(−ζK1)(−−(1C)−2+(CC24+)CR4e+bs·s·=·+1CZbm221)(Rs)es·sZ=b41′(Z12)12.(s)·Z4′(1)·Z6(1)···Zm(1)
for m 6.
≥
b b b b
Here, (cid:3) = ∆(1) for m 2N.
m 0 Vm(2) ∈
(cid:12)
(cid:12) 2. Preliminaries
We fix the notation for the Hilbert modular group of a real quadratic field in this section.
We also recall the definition of Hilbert-Maass formsfor the Hilbertmodulargroup and review
“Differences of the Selberg trace formula”, introduced in [5], which play a crucial role in this
article.
2.1. Hilbert modular group of a real quadratic field. LetK/Q beareal quadraticfield
with class number one and be the ring of integers of K. Put D be the discriminant of
K
O
K and ε> 1 be the fundamental unit of K. We denote the generator of Gal(K/Q) by σ and
a b a b
put a := σ(a) for a K. We also put γ = ′ ′ for γ = PSL(2, ).
′ ∈ ′ c′ d′ c d ∈ OK
(cid:16)2 (cid:17) (cid:16) (cid:17)
Let G be PSL(2,R)2 = SL(2,R)/ I and H2 be the direct product of two copies of
{± }
the upper half plane H := (cid:16)z C Im(z) >(cid:17)0 . The group G acts on H2 by
{ ∈ | }
a z +b a z +b
g.z = (g ,g ).(z ,z ) = 1 1 1, 2 2 2 H2
1 2 1 2
c z +d c z +d ∈
(cid:18) 1 1 1 2 2 2(cid:19)
a b a b
for g = (g ,g ) = ( 1 1 , 2 2 ) and z = (z ,z ) H2.
1 2 c1 d1 c2 d2 1 2 ∈
A discrete subgr(cid:16)oup Γ G(cid:17)i(cid:16)s called irr(cid:17)educible if it is not commensurable with any direct
⊂
productΓ Γ oftwo discretesubgroupsofPSL(2,R). We haveclassification oftheelements
1 2
×
of irreducible Γ.
DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES 5
Proposition 2.1 (Classification of the elements). Let Γ be an irreducible discrete subgroup
of G. Then any element of Γ is one of the followings.
(1) γ = (I,I) is the identity
(2) γ = (γ ,γ ) is hyperbolic tr(γ ) > 2 and tr(γ ) > 2
1 2 1 2
⇔ | | | |
(3) γ = (γ ,γ ) is elliptic tr(γ ) < 2 and tr(γ ) < 2
1 2 1 2
⇔ | | | |
(4) γ = (γ ,γ ) is hyperbolic-elliptic tr(γ ) > 2 and tr(γ ) < 2
1 2 1 2
⇔ | | | |
(5) γ = (γ ,γ ) is elliptic-hyperbolic tr(γ ) < 2 and tr(γ ) > 2
1 2 1 2
⇔ | | | |
(6) γ = (γ ,γ ) is parabolic tr(γ ) = tr(γ ) = 2
1 2 1 2
⇔ | | | |
Note that there are no other types in Γ. (parabolic-elliptic etc.)
Let us consider the Hilbert modular group of the real quadratic field K with class number
one,
a b a b a b
Γ := (γ,γ ) = , ′ ′ PSL(2, ) .
K ′ c d c′ d′ c d ∈ OK
n (cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)(cid:12)(cid:16) (cid:17) o
It is known that Γ is an irreducible discrete subg(cid:12)roup of G = PSL(2,R)2 with the only
K (cid:12)
one cusp := ( , ), i.e. Γ -inequivalent parabolic fixed point. X = Γ H2 is called
K K K
∞ ∞ ∞ \
the Hilbert modular surface.
We have a lemma about the Euler characteristic of the Hilbert modular surface X .
K
Lemma 2.2. Let E(X ) be the Euler characteristic of the Hilbert modular surface X =
K K
Γ H2. Then we have E(X ) 2N.
K K
\ ∈
Proof. By noting the formula E(XK)= 2ζK(−1)+ Nj=1 νjν−j1 (see (2), (4) on [7, pp.46-47]),
E(X ) is a positive integer. Let Y and Y be the non-singular algebraic surfaces resolved
K K K− P
singularities, in the canonical minimal way, of compactifications of Γ H2 and Γ (H H )
K K −
\ \ ×
respectively. Here H is the lower half plane. Let χ(Y ) and χ(Y ) be the arithmetic genera
− K K−
of Y and Y respectively. By the formulas (12) and (14) on [7, p.48], we have
K K−
E(X ) = 2 χ(Y )+χ(Y ) .
K K K−
We complete the proof. (cid:0) (cid:1) (cid:3)
We fix the notation for elliptic conjugacy classes in Γ . Let R ,R , ,R be a complete
K 1 2 N
···
system of representatives of the Γ -conjugacy classes of primitive elliptic elements of Γ .
K K
ν ,ν , ,ν (ν N, ν 2) denote the orders of R ,R , ,R . We may assume that R
1 2 N j j 1 2 N j
··· ∈ ≥ ···
is conjugate in PSL(2,R)2 to
cos π sin π cos tjπ sin tjπ
R νj − νj , νj − νj , (t ,ν ) = 1.
j ∼ sin π cos π sin tjπ cos tjπ j j
(cid:16)(cid:16) νj νj (cid:17) (cid:16) νj νj (cid:17)(cid:17)
For even natural number m 2 and l 0,1, ,ν 1 , we define α (m,j), α (m,j)
j l l
≥ ∈ { ··· − } ∈
0,1, ,ν 1 by
j
{ ··· − }
t (m 2)
j
l+ − α (m,j) (mod ν ),
l j
2 ≡
(2.1)
t (m 2)
j
l − α (m,j) (mod ν ).
l j
− 2 ≡
We divide hyperbolic conjugacy classes of Γ into two subclasses according to their types.
K
Definition 2.3 (Types of hyperbolic elements). For a hyperbolic element γ, we define that
6 Y.GON
(1) γ is type 1 hyperbolic whose all fixed points are not fixed by parabolic elements.
⇔
(2) γ is type 2 hyperbolic not type 1 hyperbolic.
⇔
We denote by Γ , Γ , Γ , Γ and Γ , type 1 hyperbolic Γ -conjugacy classes, el-
H1 E HE EH H2 K
liptic Γ -conjugacy classes, hyperbolic-elliptic Γ -conjugacy classes, elliptic-hyperbolic Γ -
K K K
conjugacy classes and type 2 hyperbolic Γ -conjugacy classes of Γ respectively.
K K
2.2. The space of Hilbert-Maass forms. Fix the weight (m ,m ) (2Z)2. Set the auto-
1 2
∈
morphic factor j (z ) = czj+d for γ PSL(2,R) (j = 1,2).
γ j |czj+d| ∈
Let ∆(j) := y2( ∂2 + ∂2 )+im y ∂ (j = 1,2) be the Laplacians of weight m for the
mj − j ∂x2j ∂yj2 j j∂xj j
variable z .
j
Let us define the L2-space of automorphic forms of weight (m ,m ) with respect to the
1 2
Hilbert modular group Γ .
K
Definition 2.4 (L2-space of automorphic forms of weight (m ,m )).
1 2
L2(Γ H2; (m ,m )) := f: H2 C, C
K 1 2 ∞
\ →
(i)f((γ,γ′)(z1,z2)) = jγ(zn1)m1jγ′(z2)m2f(z1(cid:12)(cid:12),z2) (γ,γ′) ΓK
(cid:12) ∀ ∈
(ii) (λ(1),λ(2)) R2 ∆(1) f(z ,z ) = λ(1)f(z ,z ), ∆(2) f(z ,z )= λ(2)f(z ,z )
∃ ∈ m1 1 2 1 2 m2 1 2 1 2
(iii) f 2 = f(z)f(z)dµ(z) < .
|| || ∞
ZΓK\H2 o
Here, dµ(z) = dx1dy1dx2dy2 for z = (z ,z ) H2.
y2 y2 1 2 ∈
1 2
Then, it is known that
Proposition 2.5. Let L2 (Γ H2; (m ,m )) be the subspace of the discrete spectrum of the
dis K 1 2
\
Laplacians and L2con(ΓK H2; (m1,m2)) be the subspace of the continuous spectrum. Then,
\
we have a direct sum decomposition :
L2(Γ H2; (m ,m )) = L2 (Γ H2; (m ,m )) L2 (Γ H2; (m ,m ))
K 1 2 dis K 1 2 con K 1 2
\ \ ⊕ \
and there is an orthonormal basis φ of L2 (Γ H2; (m ,m )).
{ j}∞j=0 dis K\ 1 2
Definition 2.6 (Hilbert Maass forms of weight (m ,m )). Let (m ,m ) (2Z)2. We call
1 2 1 2
∈
L2 (Γ H2; (m ,m ))
dis K\ 1 2
the space of Hilbert Maass forms for Γ of weight (m ,m ).
K 1 2
Let φ be an orthonormal basis of L2 (Γ H2; (m ,m )) and (λ(1),λ(2)) R2 such
{ j}∞j=0 dis K\ 1 2 j j ∈
that
∆(1)φ = λ(1)φ and ∆(2)φ = λ(2)φ .
m1 j j j m2 j j j
We write λ(l) = 1 +(r(l))2 and r(i) are defined by
j 4 j j
λ(l) 1 if λ(l) 1,
(2.2) r(l) := j − 4 j ≥ 4
j qi 1 λ(l) if λ(l) < 1,
 4 − j j 4
q
for l = 1,2.

DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES 7
2.3. Double differences of the Selberg trace formula. Let m be an even integer. We
studied and derived the full Selberg trace formula for L2(Γ H2; (0,m)) in [5]. (See [5, The-
K
\
orem 2.22].) Let h(r ,r )bean even “test function”which satisfy certain analytic conditions.
1 2
Roughly speaking, [5, Theorem 2.22] is as follows.
∞ (1) (2)
h(r ,r ) = I(h)+IIa(h)+IIb(h)+III(h).
j j
j=0
X
Here, the right hand side is a sum of distributions of h contributed from several conjugacy
classes of Γ and Eisenstein series for Γ . Assuming that the test function h(r ,r ) is a
K K 1 2
product of h (r ) and h (r ), we derived “differences of STF”([5, Theorem 4.1]) and “double
1 1 2 2
differences of STF” ([5, Theorem 4.4]). We explain for this.
Let us consider the subspace of L2 (Γ H2;(0,m)) given by
dis K\
m m
V(2) = f L2 (Γ H2;(0,m)) ∆(2)f = 1 f .
m ∈ dis K\ m 2 − 2
n (cid:12) o
(cid:12) (cid:0) (cid:1)
(cid:12)
Let h (r) be an even function, analytic in Im(r)< δ for some δ > 0,
1
h (r)= O((1+ r 2) 2 δ)
1 − −
| |
for some δ > 0 in this domain. Let g1(u) := 21π ∞ h1(r)e−irudr. Then we have
−∞
R
Proposition 2.7 (Double differences of STF for L2 Γ H2; (0,2) ). Let m = 2. We have
K
\
(cid:0) (cid:1)
∞ i
h ρ (2) h
1 j 1
− 2
Xj=0 (cid:16) (cid:17) (cid:16) (cid:17)
vol(Γ H2)
K ∞
= \ rh (r)tanh(πr)dr
16π2 1
Z−∞
ie−iθ1 ∞ g1(u)e−u/2 eu −e2iθ1 du
− 8ν sinθ coshu cos2θ
R(θ1X,θ2)∈ΓE R 1 Z−∞ (cid:20) − 1(cid:21)
1 logN(γ0)g1(logN(γ)) logεg (0) 2logε ∞ g (2klogε)ε k.
1 1 −
− 2 N(γ)1/2 N(γ) 1/2 − −
−
(γ,ωX)∈ΓHE − Xk=1
Here, λ (2) = 1/4+ρ (2)2 is the set of eigenvalues of the Laplacian ∆(1) acting on V(2).
{ j j }∞j=0 0 2
Proof. See [5, Corollary 6.3]. (cid:3)
8 Y.GON
Proposition 2.8 (Double differences of STF for L2 Γ H2; (0,m) ). Let m 2N and
K
\ ∈
m 4. We have
≥ (cid:0) (cid:1)
∞ ∞ i
h ρ (m) h ρ (m 2) +δ h
1 j 1 j m,4 1
− − 2
Xj=0 (cid:16) (cid:17) Xj=0 (cid:16) (cid:17) (cid:16) (cid:17)
vol(Γ H2)
K ∞
= \ rh (r)tanh(πr)dr
8π2 1
Z−∞
ie−iθ1ei(m−2)θ2 ∞ g1(u)e−u/2 eu −e2iθ1 du
− 4ν sinθ coshu cos2θ
R(θ1X,θ2)∈ΓE R 1 Z−∞ (cid:20) − 1(cid:21)
logN(γ )
0 g (logN(γ))ei(m 2)ω
1 −
− N(γ)1/2 N(γ) 1/2
−
(γ,ωX)∈ΓHE −
∞
2logε g (2klogε) ε k(m 1) ε k(m 3) .
1 − − − −
− −
Xk=1 (cid:16) (cid:17)
Here, λ (q) = 1/4+ρ (q)2 is the set of eigenvalues of the Laplacian ∆(1) acting on
{ j j }∞j=0 0
(2)
V (q = m,m 2).
q
−
Proof. See [5, Theorem 4.4] and [5, (5.3)].
(cid:3)
3. Asymptotic behavior of the completed Selberg zeta functions
1
We have to know the asymptotic behavior of the completed Selberg zeta functions Z2(s)
2
and Z (s) (m 4) when s , to prove Main Theorem (Theorem 1.6). We calculate their
m
≥ → ∞
asymptotic behavior in this section. b
b
Lemma 3.1 (Stirling’s formula for Γ2(z)). Let Γ2(z) := exp ∂∂s s=0 ∞m,n=0(m+n+z)−s
be the double Gamma function. Then we have
(cid:0) (cid:12) P (cid:1)
(cid:12)
3 z2 1
(3.1) logΓ (z+1) = z2 logz+o(1) (z ).
2
4 − 2 − 12 → ∞
(cid:16) (cid:17)
Proof. Let G(z) be the Barnes G-function defined by (See [1, p.268].)
z z+z2(1+γ) ∞ z k z+z2
G(z+1) = (2π)2e− 2 1+ e− k2 .
k
kY=1(cid:26)(cid:16) (cid:17) (cid:27)
Here, γ = Γ(1) is the Euler constant. By using the relation (See [12, Proposition 4.1].)
′
−
Γ2(z) = eζ′(−1)(2π)z−21G(z)−1,
and the asymptotic formula (See [1, p.269].)
z 3 z2 1
logG(z+1) = log(2π)+ζ ( 1) z2+ logz+o(1) (z ),
′
2 − − 4 2 − 12 → ∞
(cid:16) (cid:17)
we have the desired formula. (cid:3)
DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES 9
Lemma 3.2 (Asymptotics of the identity factors). We have
1 3 1
(3.2) logZ2(s)= ζ ( 1) s2 s s2 s+ logs +o(1) (s ),
id K − 2 − − − 3 → ∞
(cid:26) (cid:27)
(cid:16) (cid:17)
3 1
(3.3) logZ (s)= 2ζ ( 1) s2 s s2 s+ logs +o(1) (s ).
id K
− 2 − − − 3 → ∞
(cid:26) (cid:27)
(cid:16) (cid:17)
Proof. By Definition 1.4,
1
logZ2(s) = ζ ( 1) logΓ (s)+logΓ (s+1)
id K − 2 2
(cid:16) (cid:17)
1
and Lemma 3.1, we have the desired (3.2). We see that the relation logZ (s) = 2logZ2(s)
id id
implies (3.3). It completes the proof. (cid:3)
Lemma 3.3 (Asymptotics of the elliptic factors). We have
(3.4) logZ21 (s;2) = N νj2−1log s +o(1) (s ),
ell − 12ν ν → ∞
j j
j=1
X
(3.5) logZ (s;m) = N νj2−1−12α0(m,j) νj −α0(m,j) log s +o(1) (s )
ell
− 6ν ν → ∞
j(cid:8) (cid:9) j
j=1
X
for m 2N and m 4. Here α (m,j) 0,1,...,ν 1 are defined in (2.1).
0 j
∈ ≥ ∈ { − }
Proof. We use Stirling’s formula of Γ(z). (See [11, p.12].)
1 1
logΓ(z) = z logz z+ log(2π)+o(1) (z ).
− 2 − 2 → ∞
(cid:16) (cid:17)
By Definition 1.4,
N νj−1 ν 1 α (m,j) α (m,j) s+l
j l l
logZ (s;m) = − − − logΓ .
ell
ν ν
j j
Xj=1 Xl=0 (cid:16) (cid:17)
10 Y.GON
We see that α (m,j) 0 l ν 1 = α (m,j) 0 l ν 1 = 0,1,2,...,ν 1 for
l j l j j
{ | ≤ ≤ − } { | ≤ ≤ − } { − }
each j. Thus we have νl=j−01 νj −1−αl(m,j)−αl(m,j) = 0, and find that
νj−1ν 1 α (m,jP) α ((cid:0)m,j) s+l (cid:1)
j l l
− − − logΓ
ν ν
j j
Xl=0 (cid:16) (cid:17)
νj−1ν 1 α (m,j) α (m,j) s+l 1 s+l s+l 1
j l l
= − − − log + log(2π) +o(1)
ν ν − 2 ν − ν 2
Xl=0 j (cid:26)(cid:16) j (cid:17) (cid:16) j (cid:17) j (cid:27)
νj−1ν 1 α (m,j) α (m,j) s+l 1 l l
l l
= − − − log(s+l) logν +o(1)
j
ν ν − 2 − ν − ν
Xl=0 j (cid:26)(cid:16) j (cid:17) j(cid:27)
νj−1ν 1 α (m,j) α (m,j) s 1 l s
j l l
= − − − logs+ log +o(1)
ν ν − 2 ν ν
Xl=0 j (cid:26)(cid:16) j (cid:17) j j(cid:27)
νj−1ν 1 α (m,j) α (m,j) l s
j l l
= − − − log +o(1)
ν · ν ν
j j j
l=0
X
(ν 1)2 s νj−1α (m,j)+α (m,j) l s
j l l
= − log log +o(1) (s ).
2ν ν − ν · ν ν → ∞
j j j j j
l=0
X
By (2.1), we can check that
α (m,j)+l (0 l ν α (m,j) 1)
0 j 0
α (m,j) = ≤ ≤ − − ,
l
(α0(m,j) νj +l (νj α0(m,j) l νj 1)
− − ≤ ≤ −
hence we calculate further,
νj−1lα (m,j) νj−α0(m,j)−1 l α (m,j)+l νj−1 l α (m,j) ν +l
l 0 0 j
= + −
ν2 ν2 ν2
Xl=0 j Xl=0 (cid:0) j (cid:1) l=νj−Xα0(m,j) (cid:0) j (cid:1)
(ν 1)(2ν 1) α (m,j) α (m,j) ν
j j 0 0 j
= − − + − .
6ν ν
j (cid:0) j (cid:1)
By noting α (m,j) α (m,j) ν = α (m,j) α (m,j) ν , we have
0 0 j 0 0 j
− −
νj−1ν 1 α(cid:0) (m,j) α (m(cid:1),j) s(cid:0)+l (cid:1)
j l l
− − − logΓ
ν ν
j j
Xl=0 (cid:16) (cid:17)
(ν 1)2 s νj−1α (m,j)+α (m,j) l s
j l l
= − log log +o(1)
2ν ν − ν · ν ν
j j j j j
l=0
X
(ν 1)2 s (ν 1)(2ν 1) α (m,j) α (m,j) ν s
j j j 0 0 j
= − log 2 − − + − log +o(1)
2ν ν − 6ν ν ν
j j j (cid:0) j (cid:1) j
n o
= νj2−1−12α0(m,j) νj −α0(m,j) log s +o(1) (s ).
− 6ν ν → ∞
j(cid:8) (cid:9) j