Selberg zeta functions and the Shimura correspondence for Maass wave forms
Tsuneo Arakawa (Rikkyo University)
Contents
0. Introduction
1. Resolvent Selberg trace formulas
2. Katok-Sarnak-Shimura correspondence for Maass wave forms 3. Comparison of the traces of the two corresponding spaces
0 Introduction
The aim of this article is to explain how we can observe the Shimura correspondence for Maass wave forms due to Katok-Sarnak via Selberg zeta functions (or resolvent trace formulas). We present a conjecture which will reduce the conjectural bijectivity of the correspondence to some simple relationship of the two different Selberg zeta functions concerned. This article is based on our previous one [Ar2] in which we have discussed some dimension formulas of small weights involving Selberg zeta functions in some details.
1 Resolvent Selberg trace formulas
We briefly recall resolvent Selberg trace formulas for SL2(R) due to Hejhal [He]
and Fischer [Fi].
The symbole(w) is used as an abbreviation for exp(2πiw) (w∈C). We choose the principal branch of holomorphic function ws := eslogw in C− {z = x | x ≤ 0} with
−π < argw ≤ π. Let V be a d-dimensional C-vector space equipped with a positive definite hermitian scalar product ⟨v, w⟩ (v, w ∈ V) and let U(V) denote the group of
unitary tansformations of V. Set d = dimV. Let Γ denote a congruence subgroup of SL2(Z) including −12.
Definition (Multiplier systems and spaces of automorphic forms on Γ).
A mapχ: Γ−→ U(V) is called a (unitary) multipier system of Γ of weight 2k(k ∈R), if it satisfies
(i) χ(−12) = e−2πikidV,
(ii) χ(AB) =σ2k(A, B)χ(A)χ(B) for allA, B ∈Γ, whereσ2k(A, B) is a cocycle given by
σ2k(A, B) = J(A, Bz)2kJ(B, z)2k J(AB, z)2k with J(A, z) := cz+d for A =
( a b c d
)
, the usual factor of automorphy for SL2(R).
LetHk,χ denote the space of V-valued measurable functions on H with the properties (i) f|[M, k] =χ(M)f (∀M ∈Γ),
where f|[M, k](z) :=jM(z)−1f(M z) with jM(z) = exp(2ikargJ(M, z)).
(ii) (f, f) :=
∫
Γ\H
⟨f(z), f(z)⟩dω(z)<∞,dω(z) being the invariant measure dxdy y2 . A typical example is the case ofk = 0, d= 1, and χ= 1; H0 :=H0,χ =L2(Γ\H).
We consider the differential operator
∆k:=y2( ∂2
∂x2 + ∂2
∂y2
)−2iky ∂
∂x
which is consistent with the action f|[A, k] of SL2(R). Namely, for any C2-class functionf onH,
−∆kf|[A, k] =−∆k(f|[A, k]) (∀A∈SL2(R)) (see [Ro] I, pp.305-306). We set
Dk ={f ∈ Hk | C2-class functions and (∆kf,∆kf)<∞}, which is dense in Hk. If f, f′ ∈ Dk, then
(−∆kf, f′) = (f,−∆kf′) ([Ro], I, pp.308-309).
This means that −∆k is symmetric on Dk. It is known by [Ro],I, Satz3.2 that there exists the unique self-adjoint extension −∆ek :Dek −→ Hk (whose domain of definition is denoted byDek) such that
(−∆ekf, g) = (f,−∆ekg) ∀f, g ∈Dek. Let
λn= 1
4+r2n (λ0 < λ1 <· · ·< λn<· · ·)
denote the eigen values of the differential operator −∆ek (the eigen values are real numbers because of the self-adjointness). We may choose rn so that rn ∈ i(0,∞)∪ [0,∞). Moreover let dn denote the multiplicity of λn. For s, a∈C we set
SΓ,χ(s, a) :=
∑∞ n=0
( dn
(s−1/2)2
+r2n − dn
(a−1/2)2
+rn2 )
(spectral side).
Take Re(a) sufficiently large. It is known that the infinite series is absolutely convergent forswiths̸= 12±irn. ThenSΓ,χ(s, a) indicates a meromorphic function ofswhose poles are located ats= 12 ±irn. They are simple poles except for s= 1/2 (rn= 0). Denote byHk,χ(s) the space ofC2-class functionsf ∈ Hk,χ satisfying−∆kf =s(1−s)f. Then
dn= dimHk,χ
(1 2 +irn
) .
The Selberg zeta function attached to the multiplier system (Γ, χ) is given by ZΓ,χ(s) := ∏
P∈P rm+(Γ)/Γ
∏∞ m=0
det(
idV −χ(P)N(P)−s−m) ,
where P rm+(Γ)/Γ denotes the set of Γ-conjugacy classes of primitive hyperbolic ele- ments of Γ with trP >2, N(P) is the norm of P, and the product on the right hand side converges absolutely for Re(s)>1. Set
ξEis(s) := 2s−1 4π
∫ ∞
−∞
( 1
(s−12)2
+t2 − 1
1 4 +t2
)φ′ φ
(1 2+it)
dt,
where φ(s) = det Φ(s) with Φ(s) being the scattering matrix of the Eisenstein series attached to (Γ, χ). Hejhal [He] and Fischer [Fi] obtained the resolvent trace formula explicitly forHk,χ using the Green kernel.
Theorem 1 (RTF) Assume Re(s), Re(a)>1. Then, SΓ,χ(s, a) = −dv(Γ)
4π
(ψ(s+k) +ψ(s−k))
+ 1
2s−1 (ZΓ,χ′
ZΓ,χ(s) +ξEis(s) +“something”
)
−{
the same expression with s replaced by a} ,
where ψ(s) = Γ′(s)/Γ(s) and “something” is contributions from elliptic and parabolic conjugacy classes of Γ. By this trace formula ZΓ,χ′
ZΓ,χ(s) (and also ZΓ,χ(s)) can be ana- lytically continued to a meromorphic function in the wholes plane satisfying a certain functional equation.
Remark. For the precise computation of the part “something” we refer to [Fi], [Ar2].
2 Katok-Sarnak-Shimura correspondence for Maass wave forms
From now on we assume that Γ = SL2(Z). Our special concern is the following theta multiplier U(M) for Γ:
( θ0(M(τ, z)) θ1(M(τ, z))
)
=e
( cz2 J(M, τ)
)
J(M, z)1/2U(M)
( θ0(τ, z) θ1(τ, z)
) ( M =
( a b c d
)
∈Γ )
, where
θi(τ, z) =∑
n∈Z
e (
(n+i/2)2τ + (2n+i)z )
(i= 0, 1).
For the convenience we consider the complex conjugateχ of U: χ(M) = U(M) (M ∈Γ).
LetH0 =L2(Γ\H) andHeven0 (resp. Hodd0 ) denote the subspace consisting of even (resp.
odd) functionsf ∈ H0; namely f(−z) = f(z) (resp. f(−z) = −f(z)). For any s ∈ C we denote by Heven0 (s) the space consisting of C2-class functions f ∈ Heven0 satisfying
−∆0f =s(1−s)f.
We are interested in the following two spaces Heven0 (s′) (⊂ H0) and H−1/4,χ(s), wheres′ and s are related with each other by s′ = 2s−1/2. It is known essentially by Kohnen-Skoruppa-Zagier that the following isomorphisms hold:
H−1/4,χ(1/4)∼=J1,1∗ ∼=M1/2+ (Γ0(4)), (1)
where Jk,1∗ (resp. Mk+−1/2(Γ0(4))) is the space of skew-holomorphic Jacobi forms of weightk (resp. the Kohnen plus space of weightk−1/2). This isomorphism can easily be extended to the situation of Maass wave forms. LetTs+ denote the space consisting of C2-class functions g :H−→C satisfying the following two conditions:
(i) g(M z) =g(z)j(M, z)|cz+d|−1/2 for allM ∈Γ0(4) and |g(z)|is square-integrable on Γ0(4)\H. Here j(M, z) is Shimura’s factor of automorphy.
(ii) g has a Fourier expansion of the form:
g(z) = ∑
n∈Z
B(n, y)e(nx),
where we impose the condition that ifn ≡2,3 mod 4, then necessarilyB(n, y) = 0. Moreover the Fourier coefficients B(n, y) for n̸= 0 are given by
B(n, y) =b(n)Wsignn/4, s−1/2(4πy|n|) (Wα,β : the usual Whittaker function).
Then a modified version of (1) above generalized to Maass wave forms is given by Proposition 2 There exists the following anti C-linear isomorphism
H−1/4,χ(s)∼=Ts+ (2)
given by H−1/4,χ(s)∋g = ( g0
g1 )
7→G(τ) =g0(4τ) +g1(4τ)∈Ts+.
Remark. We note that, ifsis real or of the forms= 12+irwithrreal, thenTs+ =Ts+, and moreover thatTs+ ={0}, otherwise.
Katok-Sarnak [KS] asserted that there exists a correspondence between the spaces Heven0 (s′) and Ts+ with s′ = 2s−1/2 which is consistent with Hecke actions. Kojima [Ko] also studied a similar correspondence in a more general setting.
Theorem 3 (Katok-Sarnak) Let s′ = 2s−1/2 and let f be an even Hecke eigen Maass wave form of Heven0 (s′). Then there exists g =∑
n∈ZB(n, y)e(nx) ∈Ts+ which satisfies the relation
b(−n) = n−3/4 ∑
T,det 2T=n
f(zT)|AutT|−1 (n ∈Z>0),
whereT runs through all theSL2(Z)-equivalence classes of positive definite half-integral symmetric matricesT with det 2T =n and |AutT| denotes the order of the unit group ofT. MoreoverzT is the point inHcorresponding toT; namely if we writeT =tg−1g−1 with g ∈GL+2(R), then zT =g(i).
Remark. In [KS] they include a possibility of g being zero. So the correspondence is neither injective nor surjective. But it is expected that for each Hecke eigen Maass wave formf there exists at least one non-zero g corresponding tof. Under this expectation
dimHeven0 (s′)≤dimTs+ with s′ = 2s−1/2 (?).
(3)
LetT :H0 −→ H0 be the operator given by (T f)(z) :=f(−z) (f ∈ H0). Obviously Heven0 ={f ∈ H0 |T f =f}. Since the differential operator −∆e0 commutes with T in De0, we set De0even ={f ∈De0 |T f =f} and Deodd0 ={f ∈De0 |T f =−f}. Let
λ+n = 1
4+ (r+n)2 (λ+0 < λ+1 <· · ·< λ+n <· · ·) (
resp.λ−n = 1
4 + (rn−)2 (λ−0 < λ−1 <· · ·< λ−n <· · ·) )
denote all the eigen values of the differential operator −∆e0 in the space De0even (resp.
De0odd). Moreover let d+n (resp. d−n) denote the multiplicity of λ+n (resp. λ−n) in De0even
(resp. De0odd). Set, fors, a∈C, SΓeven(s, a) : =
∑∞ n=0
( d+n (s−1/2)2
+ (rn+)2 − d+n (a−1/2)2
+ (rn+)2 )
=
∑∞ n=0
( d+n
λ+n −λ − d+n λ+n −µ
)
and
SΓodd(s, a) :=
∑∞ n=0
( d−n (s−1/2)2
+ (rn−)2 − d−n (a−1/2)2
+ (rn−)2 )
withλ=s(1−s) andµ=a(1−a). The infinite series on the right hand sides converge absolutely for Re(s), Re(a)>1.
Here if the equality holds for anys in (3), then it is expected that SΓ,χ(s, a) = 4SΓeven(s′, a′) (s′ = 2s−1/2, a′ = 2a−1/2), whereSΓ,χ(s, a) is the spectral side of the RTF for H−1/4,χ.
3 Comparison of the traces of the two correspond- ing spaces
We now calculate the geometric sides of the resolvent trace formulas of the both spaces ofHeven0 and H−1/4,χ in view of (2) and Theorem 3. In [Ar1] we have computed the resolvent trace formula for H−1/4,χ explicitly.
Theorem 4 (RTF for H−1/4,χ) Choose Re(s), Re(a) sufficiently large. Then SΓ,χ(s, a) = −1
6
(ψ(s+1
4) +ψ(s− 1 4))
+ 1
2s−1 Zχ′ Zχ(s)
+ 1
9(2s−1) {(
ψ
(s+94 3
)−ψ
(s−14 3
))−( ψ
(s+ 54 3
)−ψ
(s+34 3
))}
+ 1
2s−1 {
−3
2log 2 + 1 4
( ψ(
s−1 4
)−ψ( s+1
4
))+ψ( s+1
4
)−ψ(s)−ψ( s+ 1
2 )}
+ 2
(2s−1)2 + 1 4π
∫ ∞
−∞
( 1
(s− 12)2
+t2 − 1
1 4 +t2
)Φ′ Φ
(1 2+it)
dt
−[
the same expression with s replaced by a] , where Zχ =ZΓ,χ is characterized by
Zχ′
Zχ(s) = ∑
P∈P rm+(Γ)/Γ
∑∞ m=0
tr(χ(Pm)) logN(P)
1−N(P)−m N(P)−ms
and Φ(s) = 23/2−2sπΓ(2s−1)
Γ(s+ 1/4)Γ(s−1/4)· ζ(4s−2) ζ(4s−1).
To obtain RTF for the space Heven0 we have to introduce a little modified Green kernel:
Gevenλ (z, w) := 1 2 (
Gλ(τ, w) +Gλ(τ,−w) )
, where Gλ(τ, w) = ∑
M∈Γ Ks(σ(z, M w)) (λ = s(1−s)) is the ordinary Green kernel forH0 with
σ(z, w) = |z−w|2
4(Imz)(Imw) and Ks(σ) =σ−s Γ(s)2 4πΓ(2s)·F
(
s, s; 2s; 1 σ
)
(σ >1), F(α, β;γ;z) denoting the hyper geometric function.
The preliminary resolvent trace formula for Heven0 in this situation is formulated as follows:
SΓeven(s, a) = − 1 12
(ψ(s)−ψ(a)) +
∫
Γ\H
[ 1 4
∑
M∈Γ−{±12}
(
Ks(σ(z, M z))−Ka(σ(z, M z)) )
+1 4
∑
M∈Γ
(
Ks(σ(z∗, M z))−Ka(σ(z∗, M z)) )
− 1 4π
∫ ∞
−∞
( 1
1
4 +t2−λ − 1
1
4 +t2−µ ) E
( z,1
2+it) 2 dt
]
dω(z),
whereE(z, s) denotes the ordinary real analytic Eisenstein series for Γ =SL2(Z):
E(z, s) := ∑
M∈Γ∞\Γ
(ImM z)s (Γ∞ ={M ∈Γ|M∞=∞})
and we put λ = s(1−s), µ = a(1−a), and z∗ := −z. The infinite series and the integrals on the both hand sides are absolutely convergent.
Computing the integrals on the right hand side of the preliminary resolvent trace formula explicitly, we obtain the resolvent trace formula for the space Heven0 . For the convenience of the later comparison we use the parameters s′, a′ instead of s, a.
Theorem 5 (RTF for Heven0 ) Assume Re(s′), Re(a′)>1. Then
SΓeven(s′, a′) = − 1
12ψ(s′) + 1 2(2s′ −1)
Zeven′
Zeven(s′) + 1 4(2s′−1)
{1 2
( ψ
(s′ + 1 2
)−ψ (s′
2 ))
+4 9 (
ψ
(s′+ 2 3
)−ψ (s′
3 ))}
+ 1
4(2s′−1) {
2ψ(s′)−4ψ( s′+1
2
)−log 2 + 8 2s′−1
}
+ 1 4π
∫ ∞
−∞
( 1
(s′− 12)2
+t2 − 1
1 4 +t2
)φ′ φ
(1 2+it)
dt
−[
the same expression with s′ replaced by a′] , where φ(s′) = π1/2Γ(s′−1/2)ζ(2s′−1)
Γ(s′)ζ(2s′) and Zeven(s′) is given by Zeven′
Zeven(s′) = ∑
P∈P rm+(Γ)/Γ
∑∞ m=1
logN(P)
1−N(P)−mN(P)−ms′ (4)
+ ∑
P0∈P rm+(eΓ)/Γ
∑
n>0 odd
logN(P0)2
1 +N(P0)−nN(P0)−ns′,
where Γ =e {M ∈ GL2(Z) | detM = −1} and P rm+(Γ)/Γe denotes the set of Γ- conjugacy classes of primitive (hyperbolic) elements P0 of eΓ with trP0 > 0. Moreover for P0 ∈eΓ with trP0 >0, N(P0) denotes the square of the eigen value >1 of P0.
The Selberg zeta functionZodd(s′) can be introduced by exchanging the sign + into
−on the right hand side of the above identity (4); namely Zodd′
Zodd(s′) = ∑
P∈P rm+(Γ)/Γ
∑∞ m=1
logN(P)
1−N(P)−mN(P)−ms′
− ∑
P0∈P rm+(eΓ)/Γ
∑
n>0 odd
logN(P0)2
1 +N(P0)−nN(P0)−ns′.
Then the resolvent trace formula for the space Hodd0 of odd functions can be similarly established.
Theorem 6 (RTF for Hodd0 ) Assume Re(s′), Re(a′)>1. Then
SΓodd(s′, a′) = − 1
12ψ(s′) + 1 2(2s′−1)
Zodd′
Zodd(s′) + 1 4(2s′−1)
{1 2 (
ψ
(s′+ 1 2
)−ψ (s′
2 ))
+4 9 (
ψ
(s′+ 2 3
)−ψ (s′
3 ))}
+ 1
4(2s′−1)
{−2ψ(s′)−3 log 2}
−[
the same expression with s′ replaced by a′] .
If we writeZ(s′) for the ordinary Selberg zeta function for Γ =SL2(Z) characterized by
Z′
Z(s′) = ∑
P∈P rm+(Γ)/Γ
∑∞ m=1
logN(P)
1−N(P)−mN(P)−ms′, then
Z(s′)2 =Zeven(s′)Zodd(s′).
Comparing these two resolvent trace formulas, we conclude that the both geometric sides coincide except hyperbolic contributions up to the multiplication by 4. Namely, we have
Theorem 7 Let s′ = 2s−1/2 and a′ = 2a−1/2 with Re(s)>1, Re(a)>1. Then SΓ,χ(s, a)−( 1
2s−1 Zχ′
Zχ(s)− 1 2a−1
Zχ′ Zχ(a)
)
= 4 (
SΓeven(s′, a′)−( 1 2(2s′−1)
Zeven′
Zeven(s′)− 1 2(2a′−1)
Zeven′ Zeven(a′)
)) .
The proof of Theorem 7 is based on the duplication formula 2ψ(2z) = 2 log 2 +ψ(z) +ψ
( z+1
2 )
for the logarithmic derivative ψ(z) of the Gamma function and also on the following simple relation of Φ andφ:
Φ(s′) =φ(s) withs′ = 2s−1/2.
Therefore it will be reasonable to expect that the hyperbolic contributions of the both hand sides coincide. We present the following conjecture (hope).
Conjecture 8 Let s′ = 2s−1/2. We have Zχ′
Zχ(s) = Zeven′
Zeven(s′) or eqivalently, Zχ(s)2 =Zeven(s′).
As a consequence of this conjecture we may predict the following three assertions:
(i) If Conjecture 8 is true, then for any s,
dimHeven0 (s′) = dimTs+ withs′ = 2s−1/2.
(ii) If the Katok-Sarnak correspondence is surjective for any s and moreover Conjec- ture 8 is true, then the Katok-Sarnak correspondence is bijective.
(iii) The Selberg zeta function Zχ(s) will have no zeros in the intervals [1/2,3/4) and (3/4,1].
Remark. As for (iii) we have proved in [Ar1] that the order of zero at s= 3/4 of the zeta functionZχ(s) exactly coincides with dimJ1,1∗ = 1.
An explicit expression of the logarithmic derivative of the original Selberg zeta functionZ(s) is known to Sarnak [Sa] and also to Hejhal [He, p.518]. Following them we exhibit here that ofZeven(s), Zodd(s).
Let D range all positive discriminants, namely all positive integers with D ≡ 0, 1 mod 4 that are not perfect squares and let h(D) denote the number of SL2(Z)- equivalence classes of binary quadratic forms with discriminant D. Let ϵD = t+β
√D 2
denote the minimal solution of the Pell equationt2−β2D= 4 witht, β ∈Z>0. Moreover we denote by ϵ0D = t0+β0
√D
2 the minimal solution of the Pell equation t20 −β02D=−4 witht0, β0 ∈Z>0 if it exists. We have
Zeven′ Zeven
(s) = ∑
D>0
∑∞ m=1
2h(D) logϵD
1−(ϵD)−2m(ϵD)−2ms + ∑
D>0
♯∑
n>0 odd
2h(D) logϵD
1 + (ϵD)−n (ϵD)−ns, where the summation ∑♯
D>0 indicates that D runs over all positive discriminants for which ϵ0D with norm −1 exist. In that case ϵD = (ϵ0D)2. Similarly,
Zodd′
Zodd(s) = ∑
D>0
∑∞ m=1
2h(D) logϵD
1−(ϵD)−2m(ϵD)−2ms − ∑
D>0
♯∑
n>0 odd
2h(D) logϵD
1 + (ϵD)−n (ϵD)−ns.
References
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[Ar2] Arakawa, T.: Selberg trace formulasa forSL2(R) and dimension formulasa with some related topics. in Report of the third autumn workshop in number theory (in 2000) edited by Ibukiyama. pp.112-152.
[Fi] Fischer, J.: An approach to the Selberg trace formula via the Selberg zeta- function. Lecture Notes in Math. 1253, Springer, 1987.
[He] Hejhal, D.: The Selberg trace formula forP SL(2,R). Vol. 1, 2, Lecture Notes in Math. 548(1976) and 1001(1883), Springer.
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Math. 84(1984), 193-227.
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Tsuneo Arakawa
Department of Mathematics Rikkyo University
