DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES. Yasuro Gon

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1 Publ. Mat. 6 08, DOI: /PUBLMAT6808 DETERMINANTS OF LAPLACIANS ON HILBERT MODULAR SURFACES Yasuro Gon Abstract: We study regularized determinants of Laplacians acting on the space of Hilbert Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in [5, 6]. 00 Mathematics Subject Classification: M36, F7, 58J5. Key words: Hilbert modular surface, Selberg zeta function, regularized determinant.. Introduction Determinants of the Laplacian acting on the space of Maass forms on a hyperbolic Riemann surface X are studied by many authors. See for example [5,,, 0]. It is known that the determinants of are described by the Selberg zeta function cf. [6] for X. On the other hand, two Laplacians, act on the space of Hilbert Maass forms on the Hilbert modular surface X K of a real quadratic field K. For this reason, it seems that there are no explicit formulas for Determinants of Laplacians on X K until now. In this article we consider regularized determinants of the first Laplacian acting on its certain subspaces V m, indexed by m N. We show that these determinants are described by Selberg type zeta functions for X K introduced in [5, 6]. Let K/Q be a real quadratic field with class number one and O K be the ring of integers of K. Let D be the discriminant of K and ε > be the fundamental unit of K. We denote the generator of GalK/Q by σ and put a := σa for a K. We also put γ = a b c d for γ = a b c d PSL, OK. Let Γ K = {γ, γ γ PSL, O K } be the Hilbert modular group of K. It is known that Γ K is a co-finite non-cocompact irreducible discrete subgroup of PSL, R PSL, R This work was partially supported by JSPS Grant-in-Aid for Scientific Research C no and no. 7K0578.

2 66 Y. Gon and Γ K acts on the product H of two copies of the upper half plane H by component-wise linear fractional transformation. Γ K has only one cusp,, i.e. Γ K -inequivalent parabolic fixed point. X K := Γ K \H is called the Hilbert modular surface. Let γ, γ Γ K be hyperbolic-elliptic, i.e., trγ > and trγ <. Then the centralizer of hyperbolic-elliptic γ, γ in Γ K is infinite cyclic. Definition. Selberg type zeta function for Γ K with the weight 0, m. For an even integer m, we define. Z m s:= e im ω Np n+s for Res>. p,p PΓ HE n=0 Here, p, p run through the set of primitive hyperbolic-elliptic Γ K - conjugacy classes of Γ K, and p, p is conjugate in PSL, R to p, p Np / 0 cos ω sin ω 0 Np /,. sin ω cos ω Here, Np >, ω 0, π, and ω / πq. The product is absolutely convergent for Res >. Analytic properties of Z m s are known. Theorem. [6, Theorems 5.3 and 6.5]. For an even integer m, Z m s a priori defined for Res > has a meromorphic extension over the whole complex plane. In this article, we also consider the square root of Z s. Definition.3 Z s.. Z s := p,p PΓ HE n=0 = exp p,p k= Np n+s / k Np ks Np k for Res >. By [6, Theorem 6.5] and the fact that the Euler characteristic of X K is even see Lemma., we see that d ds log Z s has even integral residues at any poles. Therefore, we find that Z s has a meromorphic continuation to the whole complex plane. Let us introduce the completed Selberg type zeta functions Ẑ s and Ẑms m 4, which are invariant under s s. See [6, Theorems 5.4 and 6.6].

3 Determinants of Laplacians on Hilbert Modular Surfaces 67 Definition.4 Completed Selberg zeta functions..3 Ẑ s := Z sz id sz ell s; Z par/sct s; Z hyp/sct s; with Z id s := Γ sγ s + ζ K, Z ell s; := Z par/sct s; := ε s, N j Z hyp/sct s; := ζ ε s. Γ s+l j l,.4 Ẑ m s := Z m s Z id s Z ell s; m Z hyp/sct s; m m 4 with Z ell s; m := Z id s := Γ sγ s + ζ K, N j Γ s+l j αlm,j αlm,j, Z hyp/sct s; m := ζ ε s + m ζ ε s + m. Here, Γ s is the double Gamma function for definition, we refer to [] or [7, Definition 4.0, p. 75], the natural numbers,,..., N are the orders of the elliptic fixed points in X K and the integers α l m, j, α l m, j {0,,..., } are defined in., ζ K s is the Dedekind zeta function of K, ζ ε s := ε s and ε is the fundamental unit of K. Let m N. We recall that two Laplacians.5 0 := y x + y, m := y x + y +im y x are acting on L dis Γ K\H ; 0, m, the space of Hilbert Maass forms for Γ K with weight 0, m. See Definition.6. We consider a certain subspace of L dis Γ K\H ; 0, m given by.6 V m = { fz, z L disγ K \H ; 0, m The set of eigenvalues of 0 V m m f = m are enumerated as 0 < λ 0 m λ m λ n m m } f. Let s be a fixed sufficiently large real number. We consider the spectral zeta function by using these eigenvalues..7 ζ m w, s = λ n m + ss w Rew 0. n=0 We can show that ζ m w, s is holomorphic at w =0. See Proposition 4.3.

4 68 Y. Gon Let us define the regularized determinants of the Laplacian 0 V m Definition.5 Determinants of restrictions of 0. Let m N. For s 0, define.8 Det 0 V m +ss :=exp w w=0 n=0 λ n m+ss w. We see later that Det 0 V + ss can be extended to an m entire function of s. See Corollary.7. Our main theorem is as follows. Theorem.6 Main Theorem. Let m := 0 V m for m N. We have the following determinant expressions of the completed Selberg type zeta functions. Ẑ s = e s ζ K +C Det + ss. ss Ẑ4s = e s ζ K +C 4 ss Det 4 + ss. Det + ss 3 For m 6, Ẑms = e s ζ K +C m Det m + ss Det m + ss. Here, the constants C m are given by C = log ε + N C m = N j log, j α 0m, j{ α 0 m, j} 6 log m 4, the natural numbers,,..., N are the orders of the elliptic fixed points in X K, and the integers α 0 m, j {0,,..., } are defined in.. We know the following Weyl s law: N + mt := #{j λ j m T } m ζ K T T. See [6, Theorem 6.]. Therefore, we may say that Z m s m 4 have more zeros than poles. We have several corollaries from Theorem.6 by direct calculation..

5 Determinants of Laplacians on Hilbert Modular Surfaces 69 Corollary.7. Let m = 0 V m for m N. For m N, we have Det + ss = ss e s ζ K C Ẑ s. Det m +ss =e m s ζ K C +C 4+ +C m Ẑ sẑ4s Ẑms for m 4. It follows from the above corollary that Det m + ss m N can be extended to entire functions of s. By putting s = in the above, we have Corollary.8. For m N, we have Det = e 4 ζ K C Res s= Ẑ s. Det 4 = e 3 4 ζ K C +C 4 Res s= Ẑ sẑ 4. 3 Det m = e m 4 ζ K C +C 4+ +C m Res s= Ẑ sẑ 4Ẑ6 Ẑm for m 6. V m Here, m = 0 for m N. Finally, we have a few comments on related works. Let Z Y s be the Selberg zeta function for a modular curve Y. As kindly pointed out by the referee, the special value Z Y is evaluated by an arithmetic Riemann Roch formula in the realm of Arakelov geometry in [3, 4]. Therefore, we might imagine that our results on regularized determinants and special Selberg type zeta values should play the role of the missing holomorphic analytic torsion of sheaves of higher weight Hilbert modular forms, in a conjectural arithmetic Riemann Roch formula à la Gillet Soulé. For this reason, to work out the case for congruence subgroups of Γ K of a quadratic field K with arbitrary class number would be interesting and make the range of application wider. We hope to treat this problem in a future paper since our results depend on explicit Selberg trace formulas for Γ K with class number one in [6].. 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 forms for the Hilbert modular group and review Differences of the Selberg trace formula, introduced in [6], which play a crucial role in this article... Hilbert modular group of a real quadratic field. Let K/Q be a real quadratic field with class number one and O K be the ring of integers of K. Put D be the discriminant of K and ε > be the fundamental

6 60 Y. Gon unit of K. We denote the generator of GalK/Q by σ and put a := σa for a K. We also put γ = a b c d for γ = a b c d PSL, OK. Let G be PSL, R = SL, R/{±I} and H be the direct product of two copies of the upper half plane H := {z C imz > 0}. The group G acts on H by a z + b g.z = g, g.z, z =, a z + b H c z + d c z + d for g = g, g = a b c d, a b c d and z = z, z H. A discrete subgroup Γ G is called irreducible if it is not commensurable with any direct product Γ Γ of two discrete subgroups of PSL, R. We have classification of the elements of irreducible Γ. Proposition. Classification of the elements. Let Γ be an irreducible discrete subgroup of G. Then any element of Γ is one of the following: γ = I, I is the identity. γ = γ, γ is hyperbolic trγ > and trγ >. 3 γ = γ, γ is elliptic trγ < and trγ <. 4 γ = γ, γ is hyperbolic-elliptic trγ > and trγ <. 5 γ = γ, γ is elliptic-hyperbolic trγ < and trγ >. 6 γ = γ, γ is parabolic trγ = trγ =. 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, Γ K := { γ, γ = a b, c d a b c d a b c d } PSL, O K. It is known that Γ K is an irreducible discrete subgroup of G = PSL, R with the only one cusp :=,, i.e. Γ K -inequivalent parabolic fixed point. X K = Γ K \H is called the Hilbert modular surface. We have a lemma about the Euler characteristic of the Hilbert modular surface X K. Lemma.. Let EX K be the Euler characteristic of the Hilbert modular surface X K = Γ K \H. Then we have EX K N. Proof: By noting the formula EX K = ζ K + N see, 4 in [9, pp ], EX K is a positive integer. Let Y K and Y K be the non-singular algebraic surfaces resolved singularities, in the canonical

7 Determinants of Laplacians on Hilbert Modular Surfaces 6 minimal way, of compactifications of Γ K \H and Γ K \H H respectively. Here H is the lower half plane. Let χy K and χy K be the arithmetic genera of Y K and Y K respectively. By formulas and 4 in [9, p. 48], we have We complete the proof. EX K = χy K + χy K. We fix the notation for elliptic conjugacy classes in Γ K. Let R, R,..., R N be a complete system of representatives of the Γ K -conjugacy classes of primitive elliptic elements of Γ K.,,..., N N, denote the orders of R, R,..., R N. We may assume that R j is conjugate in PSL, R to R j cos π sin π sin π cos π, cos t jπ sin tjπ sin tjπ cos tjπ, t j, =. For an even natural number m and l {0,,..., }, we define α l m, j, α l m, j {0,,..., } by. l + t jm α l m, j mod, l t jm α l m, j mod. We divide hyperbolic conjugacy classes of Γ K into two subclasses according to their types. Definition.3 Types of hyperbolic elements. For a hyperbolic element γ, we define that: γ is type hyperbolic whose all fixed points are not fixed by parabolic elements. γ is type hyperbolic not type hyperbolic. We denote by Γ H, Γ E, Γ HE, Γ EH, and Γ H, type hyperbolic Γ K -conjugacy classes, elliptic Γ K -conjugacy classes, hyperbolic-elliptic Γ K -conjugacy classes, elliptic-hyperbolic Γ K -conjugacy classes and type hyperbolic Γ K -conjugacy classes of Γ K respectively... The space of Hilbert Maass forms. Fix the weight m, m Z. Set the automorphic factor j γ z j = czj+d cz j+d for γ PSL, R j =,. Let j m j := yj + x j y + i j mj y j x j j =, be the Laplacians of weight m j for the variable z j.

8 6 Y. Gon Let us define the L -space of automorphic forms of weight m, m with respect to the Hilbert modular group Γ K. Definition.4 L -space of automorphic forms of weight m, m. L Γ K \H ; m, m is defined to be the set of functions f : H C, of class C satisfying: i fγ, γ z, z = j γ z m j γ z m fz, z, γ, γ Γ K ; ii λ, λ R such that m fz, z = λ fz, z, m fz, z = λ fz, z ; iii f = Γ K \H fzfz dµz <. Here, dµz = dxdy y dx dy y Then, it is known that: for z = z, z H. Proposition.5. Let L dis Γ K\H ; m, m be the subspace of the discrete spectrum of the Laplacians and L conγ K \H ; m, m be the subspace of the continuous spectrum.then, we have a direct sum decomposition: L Γ K \H ; m, m =L disγ K \H ; m, m L conγ K \H ; m, m and there is an orthonormal basis {φ j } j=0 of L dis Γ K\H ; m, m. Definition.6 Hilbert Maass forms of weight m,m. Let m,m Z. We call L disγ K \H ; m, m the space of Hilbert Maass forms for Γ K of weight m, m. Let {φ j } j=0 be an orthonormal basis of L dis Γ K\H ; m, m and j, λ j R such that λ We write λ l j. r l j := for l =,. m φ j = λ j φ j and m φ j = λ j φ j. = 4 + rl j and r i j i are defined by λ l j 4 if λ l 4 λl j j 4, if λ l j < 4,

9 Determinants of Laplacians on Hilbert Modular Surfaces Double differences of the Selberg trace formula. Let m be an even integer. We studied and derived the full Selberg trace formula for L Γ K \H ; 0, m in [6]. Let hr, r be an even test function which satisfy certain analytic conditions. Roughly speaking, [6, Theorem.] is as follows: hr j, r j = Ih + II a h + II b h + IIIh. j=0 Here, the right hand side is a sum of distributions of h contributed from several conjugacy classes of Γ K and Eisenstein series for Γ K. Assuming that the test function hr, r is a product of h r and h r, we derived differences of STF [6, Theorem 4.] and double differences of STF [6, Theorem 4.4]. We explain for this. Let us consider the subspace of L dis Γ K\H ; 0, m given by { V m = f L disγ K \H ; 0, m m f = m m } f. Let h r be an even function, analytic in imr < δ for some δ > 0, h r = O + r δ for some δ > 0 in this domain. Let g u := π h re iru dr. Then we have Proposition.7 Double differences of STF for L Γ K \H ; 0,. Let m =. We have i h ρ j h j=0 = volγ K\H 6π Rθ,θ Γ E ie iθ rh r tanhπr dr 8 R sin θ [ g ue u/ e u ] e iθ du cosh u cos θ log Nγ 0 g log Nγ Nγ / Nγ log εg 0 / γ,ω Γ HE log ε g k log εε k. k= Here, {λ j = /4 + ρ j } j=0 is the set of eigenvalues of the Laplacian 0 acting on V.

10 64 Y. Gon Proof: See [6, Corollary 6.3]. Proposition.8 Double differences of STF for L Γ K \H ; 0, m. Let m N and m 4. We have i h ρ j m h ρ j m + δ m,4 h j=0 = volγ K\H 8π Rθ,θ Γ E j=0 rh r tanhπr dr ie iθ eim θ 4 R sin θ [ g ue u/ e u ] e iθ du cosh u cos θ log Nγ 0 Nγ / Nγ g log Nγe im ω / γ,ω Γ HE log ε g k log εε km ε km 3. k= Here, {λ j q = /4 + ρ j q } j=0 is the set of eigenvalues of the Laplacian 0 acting on V q q = m, m. Proof: See [6, Theorem 4.4] and [6, 5.3]. 3. Asymptotic behavior of the completed Selberg zeta functions We have to know the asymptotic behavior of the completed Selberg zeta functions Ẑ s and Ẑms m 4 when s, to prove Main Theorem Theorem.6. We calculate their asymptotic behavior in this section. Lemma 3. Stirling s formula for Γ z. We have 3. log Γ z + = 3 z 4 z log z + o z, where Γ z := exp s s=0 m,n=0 m + n + z s denotes the double Gamma function. Proof: Let Gz be the Barnes G-function defined by Gz + = π z e z+z +γ k= { + z k k e z+ z k }.

11 Determinants of Laplacians on Hilbert Modular Surfaces 65 See [, p. 68]. Here, γ = Γ is the Euler constant. By using the relation Γ z = e ζ π z Gz see [4, Proposition 4.] and the asymptotic formula log Gz+= z logπ+ζ 3 z 4 z + log z+o see [, p. 69] we have the desired formula. Lemma 3. Asymptotics of the identity factors. We have { log Z 3 id s=ζ K s s s s+ } 3. log s +o z, s, { 3 log Z id s=ζ K s s s s+ } log s +o s. 3 Proof: By Definition.4, log Z id s = ζ K log Γ s + log Γ s + and Lemma 3., we have the desired 3.. We see that the relation log Z id s = log Z id s implies 3.3. It completes the proof. Lemma 3.3 Asymptotics of the elliptic factors. We have log Z ell s; = log Z ell s; m N j log s + o s, N j = α 0m, j{ α 0 m, j} log s +o s 6 for m N and m 4. Here α 0 m, j {0,,..., } are defined in.. Proof: We use Stirling s formula of Γz see [3, p. ]: log Γz = z log z z + logπ + o z.

12 66 Y. Gon By Definition.4, log Z ell s; m = N j α l m, j α l m, j s + l log Γ We see that {α l m, j 0 l } = {α l m, j 0 l } = {0,,,..., } for each j. Thus we have α l m, j α l m, j = 0, and find that j = = = = α l m, j α l m, j j j j j = α l m, j α l m, j { s + l log s + l log Γ s + l α l m, j α l m, j s + l + } logπ + o { s + l logs + l l log l } + o. { α l m, j α l m, j s log s+ l log s } +o α l m, j α l m, j log s j l log s + o α l m, j + α l m, j l log s + o s. By., we can check that { α 0 m, j + l 0 l α 0 m, j, α l m, j = α 0 m, j + l α 0 m, j l,

13 Determinants of Laplacians on Hilbert Modular Surfaces 67 hence we calculate further, j l α l m, j j = α 0m,j + j l= α 0m,j lα 0 m, j + l j lα 0 m, j + l j = 6 + α 0m, jα 0 m, j. By noting α 0 m, jα 0 m, j = α 0 m, jα 0 m, j, we have j α l m, j α l m, j = log s j s + l log Γ α l m, j + α l m, j l log s + o = log s { j + α } 0m, jα 0 m, j log s +o 6 = j α 0m, j{ α 0 m, j} 6 log s +o s. Thus we have 3.5. In addition, we note that log Z ell s; = log Z ells; m. m= Since α l, j = l, we see that α 0, j = 0 for any j. have 3.4. It completes the proof. Therefore we

14 68 Y. Gon Proposition 3.4 Asymptotics of the completed Selberg zeta functions. We have { log Ẑ 3 s = ζ K s s s s + } log s N j log s s log ε + o s, 3.7 { 3 log Ẑms=ζ K s s s s + } log s 3 N j α 0m, j{ α 0 m, j} log s +o 6 s, for m N and m 4. Here α 0 m, j {0,,..., } are defined in.. Proof: We note that log Z s, log Z m s = o s. By Definition.4 and Lemmas 3. and 3.3, we complete the proof. 4. Asymptotic behavior of the regularized determinants To investigate the analytic nature of the spectral zeta function ζ m w, s at w = 0, we introduce the theta function θ m t in this section. Since the regularized determinants of the Laplacians Det m +ss are defined by the derivative of ζ m w, s at w = 0, we need to know the asymptotics of w ζ mw, s w=0 when s. We calculate their asymptotics in this section. Definition 4.. For m N and t > 0, define 4. θ m t := e t λjm. j=0 We investigate the asymptotic behavior of θ m t as t +0 by using Propositions.7 and.8, which are called Double differences of the Selberg trace formula for Hilbert modular surfaces introduced and proved in [6].

15 Determinants of Laplacians on Hilbert Modular Surfaces 69 Proposition 4.. We have the following asymptotic formulas. θ t = ζ K t log ε π t ζ K + b o t +0, 4.3 Here, θ m t = m + b 0 = b 0 m = N N ζ K t log ε π t m ζ K +b 0 +b b 0 m 6 j 4, +o t +0, m N, m 4. j α 0m, j{ α 0 m, j} m 4. Proof: For t > 0, let us take the pair of test functions h r = e tr +/4 and g u = 4πt exp t 4 u 4t in Proposition.7, then we have 4.4 θ t = I t + E t + HE t + P S t + HS t. Here, I t = volγ K\H 6π E t = Rθ,θ Γ E ie iθ 8 R sin θ exp 4πt exptr + /4r tanhπr dr, [ t4 u e u/ 4t HE t = log Nγ 0 Nγ / Nγ / γ,ω Γ HE exp 4πt P S t = log ε exp t, 4πt 4 HS t = log ε exp 4πt k= e u ] e iθ du, cosh u cos θ t4 log Nγ 4t t4 k log ε ε k. 4t,

16 630 Y. Gon Firstly, we see that HE t and HS t are exponentially decreasing as t +0. Secondly, by changing the variable u to tu in E t, we see that there is a constant b 0 such that E t = b 0 + o t +0. Thirdly, P S t = log ε 4πt t/4 + ot t +0. Lastly, noting volγ K \H 8π = ζ K and integration by parts, we have I t = ζ K exp t r + t 4 = π 4 ζ K n t n n=0 n! r + 4 n cosh πr dr π cosh πr dr = a + a 0 + o t +0. t We calculate the coefficients a n n =, 0. a = π 4 ζ K = π 4 ζ K 4 π a 0 = π { 4 ζ K 0 dr cosh πr = π 4 ζ K 6π + 4 Here, we used the formula 0 x x + dx = ζ K, r cosh πr dr + 4 = π 6 ζ K. r cosh πr dr = π π π 6 = π dr cosh πr in [8, 3.57 no. 5]. Besides, we calculate the coefficient b 0 appearing in E t. [ b 0 = ie iθ exp u e iθ ] du 8 R sin θ 4π 4 cos θ = Rθ,θ Γ E N j k= 4 N cos = πk j 4. Summing up each terms appearing in the right hand side of 4.4, we have the desired formula 4.. }

17 Determinants of Laplacians on Hilbert Modular Surfaces 63 Let us prove 4.3 with m = 4. For t > 0, we also take the pair of test functions h r = e tr +/4 and g u = 4πt exp t 4 u 4t in Proposition.8 with m = 4, then we have 4.5 θ 4 t θ t + = I 4 t + E 4 t + HE 4 t + HS 4 t. Here, I 4 t = volγ K\H 8π E 4 t = Rθ,θ Γ E exp tr + /4r tanhπr dr, ie iθ eiθ 4 R sin θ exp 4πt k= [ t4 u e u/ 4t e u ] e iθ du, cosh u cos θ HE 4 t = log Nγ 0 Nγ / Nγ / γ,ω Γ HE t4 log Nγ exp 4πt 4t t4 k log ε HS 4 t = log ε exp 4πt 4t e iω, ε 3k ε k. Similarly, we see that HE 4 t and HS 4 t are exponentially decreasing as t +0, and there is a constant b 0 4 such that E 4 t = b o t +0, and I 4 t = ζ K /t /3 + o t +0. Summing up each terms appearing in the right hand side of 4.5 and using 4. in the left side, we have the desired formula 4.3 with m = 4. One can prove 4.3 for m 6 similarly. We complete the proof. Proposition 4.3. Let s be a fixed sufficiently large real number. For m N, let ζ m w, s := λ n m + ss w Rew 0 n=0 be the spectral zeta function for m.then ζ m w, s is holomorphic at w = 0. Proof: We follow [, p. 448]. For w C with Rew 0, we have 4.6 ζ m w, s = Γw 0 θ m te ss t t w dt t.

18 63 Y. Gon We consider the first three terms of θ m t in Proposition 4.. Let 4.7 η p w, s := Γw 0 t p e ss t t w dt = Γw ss p w Γw p with p = 0,,. Then we see that η pw, s p = 0,, are holomorphic at w = 0. The reminder term is 4.8 η f w, s := Γw 0 fte ss t t w dt t with ft = o t +0 and O t. Since Γw at w = 0, it completes the proof. vanishes Proposition 4.4. Let m be an even natural number. We have w ζ w, s = ζ K s s + log s + w=0 3 ζ K s 4.9 s log ε + b 0 + log s 4 ζ K + log ε + o s, and for m 4, w ζ mw, s = m ζ K s s+ log s w= m ζ K s s log ε + b b 0 m log s m ζ K + log ε+o 4 s.

19 Determinants of Laplacians on Hilbert Modular Surfaces 633 Besides, we have for m 4, 4. w ζ mw, s + w=0 w ζ m w, s = ζ K w=0 s s + log s 3 + ζ K s + b 0 m δ 4,m log s ζ K + o s. Proof: By the formulas 4.7 and 4.8, we find that w η 0w, s = logss = log s + o w=0 w η w, s w=0 = πss = π w η w, s = ss logss w=0 s +o s, s, w η f w, s = o w=0 = ss log s + s + o s, Therefore, by using 4., we have w ζ w, s = w=0 ζ K ss log s+ s s. s log ε + 6 ζ K +b 0 + log s+o s = ζ K s s + log s + 3 ζ K s s log ε + b 0 + log s 4 ζ K + log ε + o s.

20 634 Y. Gon For m 4, by using 4.3, we have w ζ mw, s = m ζ K s s + log s w=0 3 + m ζ K s s log ε + b b 0 m log s m ζ K + log ε + o 4 We complete the proof. s. 5. Proof of Main Theorem In this section we prove Theorem.6. We prove the following two propositions. The first proposition connect the completed Selberg zeta functions: Ẑ s, Ẑ4s,..., Ẑms with the regularized determinants of Laplacians: Det + ss, Det 4 + ss,..., Det m + ss. The second proposition determines the explicit relations among them. Theorem.6 is deduced from these two propositions. Proposition 5.. Let m := 0 V for m N. There exit polynomials P s,..., P m s such m that Ẑ s = e Ps Det + ss, ss Ẑ 4 s = e P4s ss Det 4 + ss, Det + ss Ẑ m s = e Pms Det m + ss Det m + ss m 6. Proof: Let k be a sufficiently large natural number. We note that k+ d ζ m w, s = ww + w + kζ m w + k +, s. s ds Taking of both sides, we have w w=0 5. k+ d k! log Det m +ss = s ds λ j m+ss k+. j=0

21 Determinants of Laplacians on Hilbert Modular Surfaces 635 Let m =, we use the following double differences of STF with the certain test function see [6, Theorem 6.4]: [ ] [ ] ρ j + s + c h s ρ j + βh ss + c h s βh 4 j=0 k=0 h= [ = ζ K s + k h= h= h= h= c h s β h c h s β h c h s β h c h s β h d h= ] c h s β h + + k + s dβ h Z + β h + Z + β s h N j l j ψ N + β h + l j h= d ds Z s Z s l s + l j ψ + s d logε βh+/ + { d dβ h s ds log { d log dβ h ε β h+ }. ε s d ds logε s Here, ψz is the digamma function, β β are constants and c s, c s are quadratic polynomials invariant under s s. Operating d s ds on both sides, we have 5. j=0 k! λ j + ss k+ = s k d d ds s ds logẑ s ss. By 5. and 5., we have k+ d log Det + ss s ds = k+ d logẑ s ss. s ds Therefore, we find that there exists a polynomial P s such that 5.3 log Det + ss + P s = logẑ s ss. } k

22 636 Y. Gon Thus we have Ẑ s = e Ps Det + ss. ss Let m 4 be an even integer. We use the following double differences of STF with the certain test function see [6, Theorem 5.]: [ ] ρ j m + s + c h s ρ j m + βh j=0 j=0 h= [ ] ρ j m + s + c h s ρ j m + βh [ + δ m,4 ss + k=0 h= c h s β h 4 h= ] h= [ ] = ζ K s + k + c h s β h + + k + Z ms s Z m s + + s + h= N c h s β h j N j h= c h s β h Z m + β h Z m + β h α l m, j α l m, j j ψ α l m, j α l m, j j ψ + { d ε s+m 4 } s ds log ε s+m + h= c h s β h d log dβ h { ε β h +m 3 }. ε β h+m s + l + β h + l Here, β β are constants and c s, c s are quadratic polynomials invariant under s s.

23 Determinants of Laplacians on Hilbert Modular Surfaces Operating j=0 = d s ds k on both sides, we have k! λ j m + ss k+ k! λ j m + ss k+ s j=0 k d d ds s ds log Ẑms δ m,4 logss. By 5. and 5.4, there exists a polynomial P m s such that 5.5 log Det m + ss log Det m + ss + P m s We complete the proof. = log Ẑms δ m,4 logss. Proposition 5.. We have P s= s ζ K N log ε + j log, j P m s= s N ζ K + j α 0m, j{ α 0 m, j} log 6 Proof: Substituting 3.6 and 4.9 in 5.3, we have P s=logẑ s ss log Det + ss { 3 =ζ K s s s s + } log s 3 m 4. N j log s s log ε + log s + ζ K s s + log s 3 ζ K s + s log ε b 0 + log s + 4 ζ K log ε + o = s ζ K N log ε + j log + o s. j Since P s is a polynomial, we have the desired formula for P s.

24 638 Y. Gon Let m 4. Substituting 3.7 and 4. in 5.5, we have P m s=log Ẑms δ m,4 logss log Det m + ss + log Det m + ss = s ζ K N + j α 0m, j{ α 0 m, j} 6 log +o s. Since P m s is a polynomial, we have the desired formula for P m s. It completes the proof. Acknowledgement The author would like to thank the referee for careful reading of the first version and many helpful suggestions for the revision of this paper. References [] E. W. Barnes, The theory of the G-function, Quart. J , [] I. Efrat, Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys , ; Erratum: Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys , 607. [3] G. Freixas i Montplet, An arithmetic Riemann Roch theorem for pointed stable curves, Ann. Sci. Éc. Norm. Supér , DOI: /asens.098. [4] G. Freixas i Montplet and A. von Pippich, Riemann Roch isometries in the non-compact orbifold setting, Preprint 06. arxiv: [5] Y. Gon, Selberg type zeta function for the Hilbert modular group of a real quadratic field, Proc. Japan Acad. Ser. A Math. Sci , DOI: 0.379/pjaa [6] Y. Gon, Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces, J. Number Theory 47 05, DOI: 0.06/j.jnt [7] Y. Gon and J. Park, The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps, Math. Ann , DOI: 0.007/s

25 Determinants of Laplacians on Hilbert Modular Surfaces 639 [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM Windows, Macintosh and UNIX, Seventh edition, Elsevier/Academic Press, Amsterdam, 007. [9] F. Hirzebruch and D. Zagier, Classification of Hilbert modular surfaces, in: Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 977, pp [0] S. Koyama, Determinant expression of Selberg zeta functions. I, Trans. Amer. Math. Soc , DOI: 0.307/ [] N. Kurokawa, Parabolic components of zeta functions, Proc. Japan Acad. Ser. A Math. Sci , 4. DOI: 0.379/ pjaa.64.. [] N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math , DOI: 0.55/form [3] N. N. Lebedev, Special Functions and their Applications, Revised edition, Translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication, Dover Publications, Inc., New York, 97. [4] K. Onodera, Weierstrass product representations of multiple gamma and sine functions, Kodai Math. J , DOI: 0.996/kmj/ [5] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys , 3 0. [6] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. N.S , Faculty of Mathematics Kyushu University 744 Motooka, Nishi-ku Fukuoka Japan address: ygon@math.kyushu-u.ac.jp Primera versió rebuda el 7 de gener de 07, darrera versió rebuda el 8 d abril de 07.