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.

.. .

.

.Selberg type zeta functions for the Hilbert modular

group of a real quadratic field

Yasuro GON

Faculty of MathematicsKyushu University

April 23, 2011

Yasuro GON (Kyushu Univ.) Selberg type zeta functions April 23, 2011 1 / 32

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Contents

.. .1 Introduction

.. .2 Selberg type zeta functions for the Hilbert modular group of a real quadraticfield

.. .3 What is the differences of the Selberg trace formula ?

.. .4 Differences of the Selberg trace formula for compact Riemann surfaces

.. .5 Differences of the Selberg trace formula for the Hilbert modular group


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Introduction (1)

G := PSL(2,R

) = SL(2,R

)/{I}H := {z C | Im z > 0}, G acts on H by g.z := az+bcz+d H

G : co-compact torsion-free discrete subgroup

X := \G/K is a compact Riemann surface of genus g 2

.

.. . . . Let is hyperbolic |tr()| > 2. the centralizer of in is infinite cyclic and is conjugate in G to

N()1/2 0

0 N()

1/2 with N() > 1.

Prim() := the set of -conjugacy classes of the primitive hyperbolic elementsin . (i.e, not a power of other hyperbolic elements)


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Introduction (2)

The Selberg zeta function for (or X) is defined by

Z(s) :=

pPrim()

k=0

1 N(p)(k+s)

for Re(s) > 1.

.Theorem (Selberg 1956).

.. .

.

.

...1 Z(s) defined for Re(s) > 1 extends meromorphically overC (actually entire)...2 Z(s) has zeros at s = k (k N) of order (2g 2)(2k + 1),at s = 0 of order 2g 1 and at s = 1 of order 1 : trivial zeros...3Z(s) has zeros at s =

1

2 irn : nontrivial zeros

Here, {n = 1/4 + r2n} is the eigenvalues of the Laplacian 0 = y

2( 2

x2 +2

y2 )

acting on L2(\H). The theorem is proved by using the Selberg trace formula.


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Introduction (3)

.Theorem (Functional equation).

.. .

.

.

Z(1 s) = Z(s)exp

4(g 1)

s 120

r tan(r) dr

by Selberg 1956. We have also

Z(1 s) =

Z(s) := Z(s)

(2(s)2(s + 1)

)2g2.

2(z) := exp(2(0, z)) with 2(s, z) =

n,m0(n + m + z)

s

: the double function.Problem.

.. .

.

.

Generalize Selbergs Theorem for PSL(2,R) Theorem for PSL(2,R)2. Construct Selberg type zeta functions for PSL(2,R)2. Study analytic properties of the above Selberg type zeta functions for PSL(2,R)2.


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Selberg type zeta functions for the

Hilbert modular group of a realquadratic field


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Notation

K/Q : a real quadratic field with class number one

: the generator of Gal(K/Q)a := (a) for a K

OK : the ring of integers of K

:= a b

c d for = a bc d PSL(2, OK)

K := {(, ) | PSL(2, OK)} : the Hilbert modular group

K PSL(2,R)2 : an irreducible discrete subgroup

K acts on H2 (product of two upper half planes) by linear fractional

transformationK have only one cusp (, ) (K-inequivalent parabolic fixed point)

XK := K\H2 : the Hilbert modular surface

Let (, ) K be hyperbolic-elliptic, i.e, |tr()| > 2 and |tr()| < 2


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the centralizer of hyperbolic-elliptic (, ) in K is infinite cyclic. Fix m 4 : even integer.Definition (Selberg type zeta function for K).

.. .

.

.ZK(s; m) :=

(p,p)

k=0

1 ei(m2) N(p)(k+s)

for Re(s) 0

Here, (p,p) run through the set of primitive hyperbolic-elliptic K-conjugacyclasses of K , and (p,p

) is conjugate in PSL(2,R)2 to

(p,p)

N(p)1/2 00 N(p)1/2

,

cos sin sin cos

.

N(p) > 1, (0, ) and / Q. N such that K(1) N, (K(s) :Dedekind zeta function of K)

and

1

j N

(1 j N) {1, 2, . . . , N} : the orders of primitive ellipticelements in K.Problem... .

.

....1 Analytic properties of ZK(s; m)...2 Functional equation of ZK(s; m)


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Analytic properties ofZK(s;m)

.Theorem 1... .

..

For an even integer m 4, ZK(s; m) a priori defined for Re(s) 0 has ameromorphic extension over the complex plane C.

.Theorem 2.

.. .

.

.

ZK(s, m) has the following essential zeros and poles ats = 12 ij j = 0, 1, 2, : zeros

s = 12 ik k = 0, 1, 2, : poles

Here,

{14 + 2j |j = 0, 1, 2, } = Spec(

(1)0 |Ker((2)m )

)

{14 + 2k | k = 0, 1, 2, } = Spec(

(1)0 |Ker((2)m2)

)

are the sets of eigenvalues of the Laplacian (1)0 acting on Hilbert-Maass forms

of weight (0, m) or (0, m 2) and (2)m ,

(2)m2 are Maass operators.


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Functional equation ofZK(s;m)

ZK(s, m) has another series of zeros and poles coming from the idenity, elliptic,type 2 hyperbolic conjugacy classes of K and scattering terms..Theorem 3.

.. .

.

.ZK(s, m) satisfies the following functional equation

ZK(s; m) = ZK(1 s; m).

Here the completed zeta function ZK(s, m) is given by

ZK(s; m) := ZK(s; m)

Zid(s) Zell(s)Zsct/hyp2(s)

with


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Gamma and local factors

.Gamma and local factors ofZK(s;m)

.

.. .

.

.

Zid(s) :=

2(s)2(s + 1)2K(1)

Zell(s) :=

N

j=1

j1

l=0

( s+lj)j1l(m,j)

j

Zsct/hyp2(s) := (s m2 1)(s

m2 2)

1

{1, 2, . . . , N} : the orders of primitive elliptic elements in K l(m, j) {0, 1, , 2j 2}

(s) := (1 2s)1 : the fundamental unit of KThe zeros and poles of Zid(s), Zell(s) and Zsct/hyp2(s) are easily calculated. All zeros and poles of ZK(s; m) are determined ! These analytic properties and functional equation of ZK(s; m) are obtained bythe differences of the Selberg trace formula for Hilbert modular groups.


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What is the differences of the

Selberg trace formula ?(short sketch)


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Maass forms of weight m (m 2Z0)

SL(2,R) : discrete subgroup, m := y2(

2

x2 +2

y2 ) + im yx.

.

. .

.

.

L2(\H ; m) :=

f: H C, C

f(z) = cz + d|cz + d|

mf(z)

m f(z) = f(z) ||f||2 =

\H

f(z)f(z)dxdy

y2< .

.STF for L2(\H ; m), h: test function, G: Fourier trans. ofh... .

.

.

Spec(m)

h() =

[]Conj()

G().

By considering the differences of the above STF, we have... .

..

m(min) h(min) =[]S

G(). S Conj().


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Hilbert Maass forms of weight (m1,m2) (mj 2Z0)

SL(2,R)2 : discrete subgroup, mj := y2j (

2

x2j

+ 2

y2j

) + imj yjxj

.

.. .

.

.

L2(\H2 ; (m1, m2)) :=

f: H2 C, C

f is weight (m1, m2) w.r.t m1 f =

(1)f m2 f = (2)f ||f||2 < .

.STF for L2(\H ; (m1,m2)), h: test function... .

.

.

((1),(2))Spec(m1 ,m2 )

h((1), (2)) =

[]Conj()

G().

By considering the differences of the above STF, we have... .

..

((1),

(2)min)Spec(m1 ,m2 )

h((1), (2)min) =

[]S

G(). S Conj().


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Differences of the Selberg trace

formula for compactRiemann surfaces


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Notation

First of all, we recall the differences of the Selberg trace formula for compactRiemann surfaces.

G := SL(2,R) =

g =

a bc d

det g = 1.

G = N AK : the Iwasawa decomposition, M := the centralizer of A in K

N (R, +), A R>0, K = SO(2), M = {I2}

G/K H := {z C | Im z > 0} : the upper half plane

G acts on H by g.z := az+bcz+d H

G : discrete subgroup

.

.. .

.

.

...1 is hyperbolic |tr()| > 2 Fix() = {, 1} R {}...2 is elliptic |tr()| < 2 Fix() = {, }, H...3 is parabolic |tr()| = 2 Fix() = {} R {}Yasuro GON (Kyushu Univ.) Selberg type zeta functions April 23, 2011 16 / 32

G di b

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G : co-compact discrete subgroup

\H is compact. has no parabolic ellements..Assumption on ... .

.

. G : co-compact discrete subgroup X := \G/K is a compact Riemann surface

is hyperbolic is conjugate in G to

N()1/2 00 N()1/2

with N() > 1.

is elliptic is conjugate in G to

cos sin sin cos

SO(2)


S lb f l f Ri f

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Selberg trace formula for compact Riemann surfaces

Fix m 2Z0 : weight

j(z) :=cz+d

|cz+d| for m := y

2( 2

x2 +2

y2 ) + im yx : the Laplacian acting on

.

.. .

.

.

L2(\H ; m) :=

f: H C, C f(z) = j(z)mf(z)

m f(z) = f(z) ||f||2 =

\H

f(z)f(z)dxdy

y2< .

Let {n

= 1/4 + r2n

} is the eigenvalues of the Laplacian m

acting onL2(\H ; m) enumerated as 0 1 2 n

h(r) = h(r): test function, analytic on | Im(r)| < max{m12 ,12} +

( > 0) and |h(r)| A[1 + |r|]2

g(u) := 12

h(r)eiru dr


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.Selberg trace formula for L2(\H ; m) ( : co-compact, m 2Z0).

.. .

.

.

n=0 h(r

n) =

vol(\H)

4

rh(r) tanh(r) dr

+

m/21k=0

(m 1 2k) h( i(m 1 2k)

2

)

+

hyp

log N(0)

N()1/2 N()1/2 g(log N())

+Rell

1

R sin R

14

cosh(( 2)r)

cosh rh(r) dr

+

m/21k=0

iei(m12k)2

h(

i(m 1 2k)2

)

hyp (resp. ell) : hyperbolic (resp. elliptic) -conjugacy classes mR : the order of the elliptic element R, 0 < R <


M t

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Maass operators

.Definition (Maass operators (m 2Z)).

.. .

.

.

Km := iy

x+ y

y+

m

2: L2(\H ; m) L2(\H ; m + 2)

m := iy

x y

y+

m

2: L2(\H ; m) L2(\H ; m 2)

m+2Km = Kmm and m2m = mm Let L2(\H ; , m) be a eigen-subspace with the eigenvalue .

.Proposition.

.. .

.

.

m[L2(\H ; , m)] = L2(\H ; , m 2) whenever = m

2

(1 m

2

)

Km[L2(\H ; , m)] = L2(\H ; , m + 2) whenever = m2 (1 +

m2 )

m[L2(\H ; , m)] = 0 when = m2 (1

m2 )

Km[L2(\H ; , m)] = 0 when = m2 (1 +

m2 )

{j(m)} = {m

2 (1 m

2 )}d

k=1 {j(m2) | j(m2) =m

2 (1 m

2 )}Yasuro GON (Kyushu Univ.) Selberg type zeta functions April 23, 2011 20 / 32

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m2 (1 m2 ) =

14 +

( i(m1)22

)2Let m 2 be an even integer..Differnce of STF for L2(\H ; m) L2(\H ; m 2).

.. .

.

.

d(m) 2,m

h( i(m 1)

2

)=

vol(\H)

4(m 1) h

( i(m 1)2

)

+ Rell

iei(m1)

2R sin R

h(i(m 1)

2)

d(m) 2,m : the multiplicity of the eigenvalue =m2 (1

m2 ) of m on

L2(\H ; m)

.Dimension formula for the holomorphic modular forms of weight m... .

.

.d(m) = 2,m +

vol(\H)

4(m 1) +

Rell

iei(m1)

2R sin R


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Differences of the Selberg trace

formula for the Hilbert modulargroup


Preliminaries

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Preliminaries

G := PSL(2,R)2 = SL(2,R)/{I}2

G acts on H2 by (g1, g2).(z1, z2) := (a1z1+b1c1z1+d1

, a2z2+b2c2z2+d2 ) H2

G : irreducible discrete subgroup i.e, not comensurable with any directproduct 1 2 of two discrete subgroups of PSL(2,R)

.Classification of the elements of irreducible .

.. .

.

.

...1 = (I, I) is the identity...2 = (1, 2) is hyperbolic |tr(1)| > 2 and |tr(2)| > 2...3 = (1, 2) is elliptic |tr(1)| < 2 and |tr(2)| < 2

...4 = (1, 2) is hyperbolic-elliptic |tr(1)| > 2 and |tr(2)| < 2...5 = (1, 2) is elliptic-hyperbolic |tr(1)| < 2 and |tr(2)| > 2...6 = (1, 2) is parabolic |tr(1)| = |tr(2)| = 2 There are no other types in . (parabolic-elliptic etc.) (Cf. Shimizu 63)


Hilbert modular group of a real quadratic field

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Hilbert modular group of a real quadratic field

K : real quadratic field of the class number 1

D : the discriminiant of K

OK K : the ring of integers

: the fundamental unit

a = (a), is the nontrivial element of Gal(K/Q)

N(a) := aa

.Hilbert modular group.

.. .

.

.K := (,

) = a b

c d

, a b

c d a b

c d

PSL(2, OK)

.

K is an irreducible discrete subgroup of G = PSL(2,R)2 with the only one

cusp := (, ).


Selberg trace formula for Hilbert modular surfaces

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Selberg trace formula for Hilbert modular surfaces

Fix (m1, m2) (2Z0)2 : weight

j(zj) :=czj+d

|czj+d| for PSL(2,R) (j = 1, 2)

(j)mj := y

2j (

2

x2j

+ 2

y2j

) + imj yjxj

(j = 1, 2)

.

.. .

.

.

L2(K\H2 ; (m1, m2)) :=

f: H2 C, C

f((, )(z1, z2)) = j(z1)m1j(z2)

m2f(z1, z2) (, ) K

(1)m1 f(z1, z2) = (1)f(z1, z2),

(2)m2 f(z1, z2) =

(2)f(z1, z2)

((1), (2)) R2

||f||2 =K\H2

f(z)f(z) d(z) < .

d(z) = dx1dy1y21

dx2dy2y22

for z = (z1, z2) H2

Yasuro GON (Kyushu Univ.) Selberg type zeta functions April 23, 2011 25 / 32 .

P iti

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Proposition.

.. .

.

.

We have a direct sum decomposition:

L2(K\H2 ; (m1, m2)) = L

2dis(K\H

2 ; (m1, m2)) L2con(K\H

2 ; (m1, m2))

and there is an ortonormal basis {j}j=0 of L

2dis(K\H

2 ; (m1, m2)).

Let ((1)j ,

(2)j ) R

2 such that

(1)

m1j =

(1)

j j and (2)

m2j =

(2)

j j

Let Spec(m1, m2) :={

(r(1)j , r

(2)j )

}j=0

R2. (discrete subset)

Here, we write (l)j =

14 + (r

(l)j )

2. (l = 1, 2)Now we can say about the Selberg trace formula:

h(r1, r2) = h(r1, r2): test function (satisfying certain analytic conditions)g(u1, u2) :=

142

h(r1, r2)ei(r1u1+r2u2) dr1dr2

is type 1 hyperbolic is hyperbolic and whose all fixed points are not fixedby parabolic elements. is type 2 hyperbolic is hyperbolic and not type 1 hyperbolic.


.2( \ 2 ( ))

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Selberg trace formula for L2(K\H2 ; (m1,m2))

((m1,m2) (2Z0)2) : Zograf 82, Efrat 87 for (m1,m2) = (0, 0).

.. .

.

.

j=0

h(r(1)j , r(2)j ) (Contribution from Eisenstein series)

=vol(K\H

2)

162

R2

2

u1u2g(u1, u2)

sinh(u1/2) sinh(u2/2)e

m12 u1e

m22 u2 du1du2

+

hyp1

vol(\G) g(log N(), log N())(N()1/2 N()1/2)(N()1/2 N()1/2)

+Rell

E(m1, m2; R) +

hyp-ell

HE(m1, m2; ) +

ell-hyp

EH(m1, m2; )

+ P(m1, m2) +

hyp2

H2(m1, m2; )

Hereafter, we assume that h(r1, r2) = h1(r1) h2(r2).

(2)m := iy2

x2

y2y2

+ m2 : L2(K\H

2 ; (0, m)) L2(K\H2 ; (0, m 2))


Let {14 + 2j}j=0 := Spec(

(1)0 |Ker((2)m )

) and recall that

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( m )

Ker((2)m ) = L2

(K\H

2 ;(

, m2 (1 m2 ))

, (0, m))

i.e. 2 =m2 (1

m2 )-eigenspace.

Differences of STF for L2(K\H2 ; (0,m)) L2(K\H

2 ; (0,m 2)).

.. .

.

.

j=0

h1(j) h2(i(m1)

2 ) m,2 h1(i2 )h2(

i2)

= (m 1)h2(i(m1)

2 )vol(K\H

2)

162

r1h1(r1) tanh(r1) dr1

+

R(1,2)ell

ie(m1)2

8R sin 1 sin 2h2(

i(m1)2 )

cosh(( 21)r1)

cosh r1h1(r1) dr1

+ (,)hyp-ell

log N(0)

N()1/2

N()1/2

g1(log N())iei(m1)

2sin

h2(i(m1)

2 )

log g1(0) h2(i(m1)

2 ) 2log h2(i(m1)

2 )

k=1

g1(2k log )k(m1)

We write the above formula as L(m) L(m 2) for m 2.


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Next we consider (for m 4) :(L(m) L(m 2)

)h2(

i(m1)2 )

1 (

L(m 2) L(m 4))

h2(i(m3)

2 )1

.Theorem (Double differences of STF for L

2

(K\H2

; (0,m))).

.. .

.

.

Let m 2N and m 4. We have

j=0

h1(j) k=0

h1(k) + m,4h1(i2

) =vol(K\H

2)

82

rh1(r) tanh(r) dr

R(1,2)ell

ei(m2)2

4R sin 1

cosh(( 21)r)

cosh rh1(r) dr

(,)hyp-ell

log N(0)

N()1/2 N()1/2g1(log N()) e

i(m2)

2log k=1

g1(2k log ) (k(m1) k(m3)).


Test function h(r1, r2) = h1(r1)h2(r2)

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Test function h(r1, r2) h1(r1)h2(r2)

Here, {j}, {k} are given by.

.. .

.

.

{14 + 2j}j=0 = Spec((1)0 |Ker((2)m )) with

(2)m : L2

(K\H

2 ; (0, m))

L2(

K\H2 ; (0, m 2)

). Note that

Ker((2)m ) = L2

(K\H

2 ;(

, m2 (1 m2 ))

, (0, m))

i.e. 2 =m2 (1

m2 )-eigenspace

{14 + 2k}k=0 = Spec(

(1)0 |Ker((2)m2)

) with

(2)m2 : L2(

K\H2 ; (0, m 2))

L2(

K\H2 ; (0, m 4))

. Note that

Ker((2)m2) = L

2(

K\H2 ;(

, m22 (2 m2 ))

, (0, m))

i.e.2 =

m22 (2

m2 )-eigenspace

Let us consider the following test function h(r1, r2) = h1(r1)h2(r2) :

h1(r) = 1r2 + (s 12)

2 1

r2 + 2 g1(u) = 1

2s 1e(s

12 )|u| 1

2e |u|

(or(

12s1

dds

)nh1(r) for n 0 )

h2(r) such that h2(i(m1)

2 ) = 0 and h2(i(m3)

2 ) = 0We consider DD-STF for the above h(r1, r2)


Analytic continuation of ZK(s;m)

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Analytic continuation ofZK(s;m)

.DD-STF for the above test function h1 and h2.

.. .

.

.

j=0

[ 12j + (s

12

)2

1

2j + 2

]

k=0

[ 12k + (s

12

)2

1

2k + 2

]

= K(1)

k=0

[ 1s + k

1

+ 12 + k]

+1

2s 1

ZK(s)

ZK(s)

1

2

ZK(12 + )

ZK(12

+ )+

2s 1

Zell(s)

Zell(s)

2

Zell(12 + )

Zell(12

+ )

+

2s 1

d

dslog(1

(2s+m4))

(1 (2s+m2))

2(s 1

2

) in the left. vol(K\H

2)82 = K(1) N.

Analytic continuation and functional equation of dds log ZK(s; m). Analytic continuation and functional equation of ZK(s; m).


Remark

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We remark that the scattering and type 2 hyperbolic components of ZK(s; m)

are local Selberg zeta functions for PSL(2,Z) :.

.. .

.

.Zsct/hyp2(s) = (s +

m2 1)(s +

m2 2)

1

with (s) = (1 2s)1

: the fundamental unit of KLet = PSL(2,Z). The Selberg (Ruelle) zeta function for is given by

(s) := pPrim()

(1 N(p)s

)1

(s) = K

(1 (K)2s)h(K),

where, K run through all real quadratic fields over Q and (K) and h(K) arethe fundamental unit and the class number of K.


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