From the transfer operator for geodesic flows on ... - TU Clausthal

From the transfer operator for geodesic flows on modular surfaces to the Hecke operators on period functions of Γ0 (n) D. Mayer and T. M¨ uhlenbruch Abstract. Recently it has been shown how from certain special eigenfunctions of the transfer operators for the congruence subgroups Γ0 (nm) a family of Hecke like operators can be constructed acting on the eigenfunctions of this operator for Γ0 (n). In the case n = 1 these operators are closely related to the Hecke operators acting on the space of period functions for the full modular group. In a different approach to Hecke operators on period functions M¨ uhlenbruch extended an integral transformation used by Lewis and Zagier relating Maass forms of the full modular group to their period functions to the congruence subgroups Γ0 (n). The well known Hecke operators on the Maass forms thereby get transformed into operators acting on vector valued period functions. In the present paper we analyze the exact relation between the two kinds of operators and show that for gcd(n, m) = 1 they are indeed identical. Our discussion leads also to a better understanding of the construction of these eigenfunctions of the transfer operators for Γ0 (nm) in terms of the so called old Maass forms for these groups.

Introduction There is a close connection between certain eigenfunctions of the transfer operator for the geodesic flow on the modular surface and the automorphic functions for the modular group SL(2, Z). Both the period polynomials respectively the rational period functions of the holomorphic modular forms and also the period functions of Lewis and Zagier [LZ01] of the real analytic Maass forms for SL(2, Z) give rise to eigenfunctions with eigenvalue λ = 1 of the transfer operator for the geodesic flow on the modular surface H\SL(2, Z). Of special interest is the connection of Maass forms with the period functions since the former are eigenfunctions of the hyperbolic Laplace-Beltrami operator and hence can be interpreted as the eigenstates or scattering states of a quantized particle moving freely on this surface (see [Ma03]). On the other hand, the period functions are eigenfunctions of the transfer 1991 Mathematics Subject Classification. Primary 11F60, 37A45; Secondary 11F67, 37C30 . Key words and phrases. Transfer operator, Geodesic flow, Modular groups, Period functions, Hecke operators. The first author was supported in part by the Deutsche Forschungsgemeinschaft through the DFG Research group “Zetafunktionen und lokalsymmetrische R¨ aume”. The second author was supported in part by the Deutsche Forschungsgemeinschaft through the DFG Research group “Zetafunktionen und lokalsymmetrische R¨ aume”. 1

2

¨ D. MAYER AND T. MUHLENBRUCH

operator which is constructed basically from datas describing the classical geodesic flow, namely the Poincar´e map and the classical length spectrum. In this case the classical system determines completely the properties of the quantum system especially its eigenstates and the scattering states. At least for this system the classical transfer operator offers hence a possible new approach to the study of quantum chaos. Different aspects of this theory are understood in the mean time also for general subgroups of SL(2, Z) with finite index, the so called modular groups. Typical examples of such modular groups are the Hecke congruence subgroups Γ0 (n), which play an important role in quite different areas of number theory. Their transfer operators have been investigated in [CM00], [CM01] and the relation between their eigenfunctions with eigenvalue λ = 1 and the Maass forms has been clarified quite recently by Deitmar and Hilgert [DH04]. Here we address another aspect of the theory of period functions for these groups. The Hecke operators play an important role in the theory of automorphic functions and L-functions for arithmetic groups [Sa90]. Their action on the Maass forms has a nice physical interpretation. Since these operators commute with the Laplace-Beltrami operator and hence the Hamiltonian for a quantized free particle on the corresponding surface they describe internal symmetries of this system. In the case SL(2, Z) one knows explicit expressions for the Hecke operators when acting on the period polynomials respectively rational period functions of the holomorphic modular forms [M73], [Z90], [Z93]. These results have been extended to the period functions of the Maass forms for the same group by M¨ uhlenbruch [M¨ u04a]. All these authors use the Eichler-Shimura respectively Lewis-Zagier correspondence relating the modular forms by an explicit integral transformation to the period polynomials respectively period functions. The well known Hecke operators on the modular forms then induce operators on the period functions via this correspondence. The form of these operators can be calculated explicitly. A completely different approach to the Hecke operators on period functions for SL(2, Z) has been given in [HMM03]. These authors construct to any eigenfunction of the transfer operator for SL(2, Z) with eigenvalue λ = 1 a nontrivial vector valued eigenfunction of this operator for any of the groups Γ0 (m). It then turns out that the sum of the components of these solutions are exactly the Hecke operators for m prime, and are closely related to the Hecke operators for arbitrary m. The same approach can be applied in the case of the congruence subgroups Γ0 (n) [HMM03]. The transfer operator for these groups involves the representation of SL(2, Z) induced from the trivial representation of Γ0 (n) and hence its eigenfunctions with eigenvalue λ = 1, the so called vector valued period functions, belong to this representation space. As eigenfunctions they fulfill a three-term functional equation as in the case of the full modular group SL(2, Z). The authors of [HMM03] then succeed in constructing to any eigenfunction of the transfer operator for Γ0 (n) a nontrivial eigenfunction of the corresponding operator for Γ0 (nm), m = 2, 3, . . .. Sums of certain components of these eigenfunctions again determine a new eigenfunction of the transfer operator for Γ0 (n) depending linearly on the original eigenfunction. In [HMM03] these linear operators are called “Hecke like” operators.

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Recently in [M¨ u04b] M¨ uhlenbruch worked out the classical approach via an integral transformation to vector valued period functions for the group Γ0 (n). Indeed, most of his results are valid for any modular group Γ ⊂ SL(2, Z) of finite index. Thereby he generalizes the integral transformation of Lewis and Zagier to vector valued Maass forms transforming under an induced representation of SL(2, Z) which leads to vector valued period functions fulfilling also a three-term functional equation closely related to the one derived from the transfer operator for the eigenfunctions. This allows him also in this case to determine explicit expressions for the Hecke operators acting on these vector valued period functions. In the present paper we analyze the exact relation between M¨ uhlenbruch’s Hecke operators and the Hecke like operators in [HMM03] for the group Γ0 (n). Even if the form of these operators seems to be rather different on a first glance, it turns out that for m prime and gcd(n, m) = 1 the operators are indeed conjugate to each other. On the way to this result we find a new interpretation of the eigenfunctions F~ for Γ0 (nm) constructed in [HMM03] for any eigenfunction f~ for Γ0 (n). They are the vector valued period functions corresponding to the old Maass form u(z) = v(mz) for Γ0 (nm) if the eigenfunction f~ corresponds to the Maass form v(z) for Γ0 (n). This shows that the eigenfunctions F~ for Γ0 (nm) are indeed “old eigenfunctions” in the sense of Atkin-Lehner. In detail the paper is organized as follows: In the first Chapter we recall the main steps in the derivation in [HMM03] of the Hecke like operators for Γ0 (n) from the eigenfunctions of the transfer operator for the groups Γ0 (nm). In the second Chapter we recall the Hecke operators acting on the vector valued period functions of Γ0 (n) induced by their action on vector valued Maass forms and determined by M¨ uhlenbruch via an integral transformation in [M¨ u04b]. The main part of this paper is the third Chapter. There we show that the eigenfunctions of the transfer operators for the groups Γ0 (nm) determined in Chapter 1 are indeed the vector valued period functions arizing from the old Maass forms v(mz) for v(z) a Maass form for Γ0 (n). We discuss the exact relation between the Hecke like operators in [HMM03] and the Hecke operators in [M¨ u04b]. We prove that for prime m with gcd(n, m) = 1 the operators are unitarily equivalent.

1. The “Hecke like” operators of Hilgert, Mayer and Movasati Let us fix some notations which we will use throughout this paper. We denote by H = {x + iy : y > 0} the hyperbolic plane with the hyperbolic metric ds2 = dx2 +dy 2 . A group Γ ⊂ SL(2, Z) of finite index µ = [SL(2, Z) : Γ] is called a y2 modular subgroup of the full modular group SL(2, Z). The surfaces MΓ = Γ\H with Γ a modular group are accordingly called modular surfaces. Obviously they are covering surfaces of M = SL(2, Z)\H. The geodesic flow on these modular surfaces is denoted by ΦΓt : SMΓ → SMΓ , where SMΓ is the unit tangent bundle of MΓ . This flow can be described symbolically as the special flow over a Poincar´e map PΓ : ΣΓ → ΣΓ , where ΣΓ is a Poincar´e section which can be chosen as follows [Se85], [CM00]: (1.1)

ΣΓ = I × I × Z2 × Γ\SL(2, Z)

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¨ D. MAYER AND T. MUHLENBRUCH

where I denotes the unit interval I = [0, 1] and Z2 = {+1, −1}. With respect to this Poincar´e section the Poincar´e map PΓ has the following explicit form [CM00]: µ ¶ 1 1 nε mod 1, PΓ (x1 , x2 , ε, [g]) = , −ε, [gT S] x1 x2 + n h i with n = x11 and T respectively S denoting the generators ¡ ¢ ¡ ¢ (1.2) T = 10 11 , S = 01 −1 0 of SL(2, Z). Later on we also need the matrices ¡ ¢ ¡ ¢ (1.3) M = 01 10 , T 0 = 11 01 It is then a standard procedure to write down the Ruelle transfer operator L¯Γβ for this map PΓ [CM00] ¶2β ∞ µ ³ ´ X ¢ ¡ 1 1 Γ~ ¯ χΓ (ST nε ) f~ , −ε (1.4) Lβ f (x1 , ε) = x1 + n x1 + n n=1 with f~ : I × Z2 → Cµ some vector ¡ ¢valued observable on the unstable directions of PΓ and χΓ : SL(2, Z) → GL Cµ the right regular representation of SL(2, Z) induced from the trivial representation of Γ. The operator L¯Γβ is well defined on the following Banach space B(D) of holomorphic functions: n o ¯ B(D) = f~ : D → Cµ , f holomorphic in D and continuous on D with D = {z : |z − 1| < 32 }. On this B-space the operator L¯Γβ has rather nice properties (see for instance [Ma91]): • L¯Γβ is nuclear for Re(β) > 12 . • L¯Γβ has a meromorphic extension as a nuclear operator to the entire complex β-plane with possible poles at βk = 1−k 2 , k = 0, 1, 2, . . .. • Z Γ (β) = det(1 − L¯Γβ ) with Z Γ (β) denoting the Selberg zeta function for the modular group Γ, that means ∞ Y³ ´ Y (1.5) Z Γ (β) = 1 − e−(β+k)l(γ) k=0 γ

where the second product is over the closed orbits γ of ΦΓt with prime period l(γ). In [MM02] Manin and Marcolli introduced another transfer operator LΓβ which leads to a factorization of the Selberg function for Γ well known for the full modular group. Assume that the modular group Γ has the property ³ ´ ³ ´ 1 0 1 0 0 −1 Γ 0 −1 = Γ. ³ ´ e := Γ ∪ Γ −1 0 and its extension Γ := For such Γ’s consider then the group Γ 0 −1 ³ ´ e∪Γ e 1 0 in GL(2, Z). Then obviously Γ\GL(2, Z) ∼ Γ = Γ\SL(2, Z). In [MM02] 0 −1 the following transfer operator LΓβ has been introduced ¶2β ¶ µ ∞ µ ³ ´ X ¡ −n 1 ¢ 1 1 Γ~ ~ χΓ 1 0 f (1.6) Lβ f (z) = z+n z+n n=1

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acting on functions f~ : D → Cµ holomorphic in D, where χΓ is the right regular representation of GL(2, Z) induced from the trivial representation of Γ. It turns out that the Selberg zeta function for the group Γ can be expressed also in terms of the operators LΓβ , indeed [MM02] one finds (1.7)

Z Γ (β) = det(1 − LΓβ ) det(1 + LΓβ )

and hence the zeros of this function are determined by those values of β for which LΓβ has the eigenvalues +1 or −1. ³

Remark 1.1. In the case Γ = SL(2, Z) the transfer operator LΓβ has the form ´ P∞ ³ 1 ´2β ³ 1 ´ LΓβ f (z) = f z+n and hence coincides with the well known n=1 z+n

transfer operator of the Gauss map TG x =

1 x

mod 1.

~ : C \ (−∞, 0] → Cµ with φ(z) ~ We call the function φ = f~(z − 1) and f~ = f~(z) Γ an eigenfunction of Lβ with eigenvalue λ = +1 or λ = −1 a vector valued period function for Γ. Obviously these vector valued period functions have the following properties: ~ is holomorphic in the complex z plane cut along the line (−∞, 0]. • φ ~ • φ fulfills the three-term functional equation µ ¶ z+1 −1 ~ −2β −1 ~ ~ (1.8) φ(z) − χΓ (T ) φ(z + 1) − λ z χΓ (T M ) φ =0 z the so called generalized Lewis equation. ~ fulfill certain growth properties on the real axis • The components φi of φ for z → 0 and z → ∞ depending on β. Remark 1.2. In the special case of the full modular group Γ = SL(2, Z) the period functions fulfill the original Lewis equation µ ¶ z+1 (1.9) φ(z) − φ(z + 1) − λ z φ = 0. z ~ = φ(z) ~ The vector valued period function φ can obviously be interpreted also as a ¡ ¢ ~ function on C \ (−∞, 0] × Γ\GL(2, Z) with φ(z, Γgi ) := φ(z) where i is just an i abbreviation for the rest class Γgi with gi , i = 1, . . . , µ representatives of the rest classes in Γ\GL(2, Z). There is a generalized slash action of the group GL(2, Z) on these functions given formally by ¯ (1.10) φ¯β h(z, Γg) := (cz + d)−2β φ(hz, ΓgT −1 hT ) ³ ´ for h = ac db ∈ GL(2, Z) and hz = az+b cz+d . For details see [HMM03]. The generalized Lewis equation (1.8) can then be written in the simple form ¯ ¯ (1.11) φ − φ¯β T − λφ¯β T M = 0. We do not know how to solve this equation in general, but it is possible to describe special solutions. For this, consider the ring R = Z[Mat∗ (2, Z)] of finite linear combinations of nonsingular 2 × 2 integer matrices and the right R-ideal J λ with J λ := (1 − T − λT M )R and 1 the unit matrix. The slash action in (1.10) extends

¨ D. MAYER AND T. MUHLENBRUCH

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obviously linearly to an action of R. Consider next the following set of equations in the ring R: ψi − ψiT −1 T − λψiT −1 M T M ≡ 0 mod J λ

(1.12)

with i denoting again the rest class Γgi in Γ\GL(2, Z) on which the group GL(2, Z) acts in a natural way. Then one has the obvious Lemma 1.3. Given any solution φ = φ(z) of the Lewis equation (1.9) for SL(2, Z) the functions ¯ φi = φi (z) := φ¯β ψi (z), i ∈ Γ\GL(2, Z) solve the Lewis equation (1.8) for the modular group Γ if the matrices ψi ∈ R solve equation (1.12). There is a trivial solution of equation (1.12) namely ψi = 1, i ∈ Γ\GL(2, Z). This leads to the special but trivial solution φi (z) = φ(z), i ∈ Γ\GL(2, Z), of equation (1.8). That such a solution exists is not surprising since we know in the case of the full modular group from the work of Lewis and Zagier [LZ01] and for general modular groups from the work of Deitmar and Hilgert [DH04] that there is a 1-1 correspondence between the period functions and the Maass forms for these groups. But every Maass form for SL(2, Z) is trivially a Maass form for any modular subgroup Γ and hence also a period function for SL(2, Z) should give rise to such a function for every of its modular subgroups. Consistently with the terminology for automorphic forms we should hence call the above solution of the Lewis equation for the modular subgroup Γ an “old solution”. It is not difficult to show that if ~ = φ(z) ~ φ = φ(z) is a period function for SL(2, Z) then φ with φi (z) = φ(z) for all i ∈ Γ\GL(2, Z) is indeed a vector valued period function for Γ with the correct asymptotic behavior for z → 0 and z → ∞. In the following we will restrict our discussion to the construction of other nontrivial solutions of equations (1.8) respectively (1.12) for the Hecke congruence subgroups Γ0 (n). For this we need a certain special characterization of the index set Γ0 (n)\GL(2, Z) as given in [HMM03]. Consider on Z × Z namely the following equivalence relation ∼n defined as (x, y)

∼n

(x0 , y 0 ) iff

∃k ∈ Z, gcd(k, n) = 1 such that x0 ≡ kx mod n, y 0 ≡ ky mod n

and the natural right action of GL(2, Z) ³ ´ (1.13) (x, y) ac db = (xa + yc, xb + yd) which is obviously compatible with ∼n . Hence GL(2, Z) acts also on (Z × Z)n := (Z × Z)/ ∼n . Denote the elements of (Z × Z)n by [x : y]n . It is easy to see that the stabilizer in GL(2, Z) of the element [0 : 1]n ∈ (Z × Z)n is just the subgroup Γ0 (n). Hence the following map π n : Γ0 (n)\GL(2, Z) → (Z × Z)n is well defined and injective: ³ ³ ´´ ³ ´ π n Γ0 (n) ac db := [0 : 1]n ac db = [c : d]n . (1.14) Denote by In the image of π n , that is ³ ´ (1.15) In := π n Γ0 (n)\GL(2, Z) .

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It is then not very difficult to show [HMM03] that (1.16)

In = {[x : y]n ∈ (Z × Z)n : gcd(x, y, n) = 1} .

Consider next the subset Pn ⊂ Z × Z defined as n o n n (1.17) Pn = (c, b) ∈ Z × Z : c ≥ 1, c|n, 0 ≤ b ≤ − 1, gcd(c, b, ) = 1 . c c Then there is a bijection between In and Pn given by the map [HMM03] (1.18)

Pn 3 (c, b) 7−→ [c : dn (c, b)]n ∈ In

with (1.19)

dn (c, b) =

o n n n c + b + k : gcd(c, b + k ) = 1 . 0≤k≤c−1 c c min

For simplicity we denote in the following the elements of In also by i. Relation (1.18) allows us to identify every element i ∈ In uniquely with a matrix Ai ∈ Matn (2, Z) with ³ ´ (1.20) Ai = 0c nb . c

where Matn (2, Z) denotes the 2 × 2 matrices with integer entries and determinant n. For the following discussion we need some sets of 2×2 matrices with nonnegative integer entries: n³ ´ o a b (1.21) Sn = c d ∈ Matn (2, Z) : a > c ≥ 0, d > b ≥ 0 , n³ ´ o n³ n ´ o c b c 0 Xn = (1.22) ∈ S , Y = ∈ S and n n n n 0 c b c n³ ´ o n c b (1.23) Xn? = ∈ Xn , gcd(c, b, ) = 1 . 0 n c c Obviously the matrix Ai in (1.20) belongs to Xn? for all i ∈ In . Consider next the map (1.24)

K : Sn \ Yn −→ Sn \ Xn

defined as (1.25)

³³ K

a b c d

´´

³ :=

−c+d db ea −d+d db eb a b

´

with dre ∈ Z determined for r ∈ R by dre − 1 < r ≤ dre. Obviously the map K is well defined and there exists for every A ∈ Sn \ Yn an integer kA > 0 such that K j A ∈ / Yn for 0 ≤ j < kA and K kA A ∈ Yn . For A ∈ Yn we set kA = 0 so that kA is well defined for all A ∈ Sn . The following Theorem has been proven in [HMM03]. Pki Theorem 1.4. The matrices ψi := j=0 K j (Ai ), i ∈ In , with ki = kAi and Ai as in (1.20) solve the Lewis equation (1.12) for the group Γ0 (n). As an immediate Corollary one then gets Corollary 1.5. For φ = φ(z) any ¯ solution of the Lewis equation (1.9) for SL(2, Z) the functions φi = φi (z) := φ¯β ψi , i ∈ In solve the Lewis equation (1.8) for the group Γ0 (n) with identical parameter β.

¨ D. MAYER AND T. MUHLENBRUCH

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This result can be generalized in the following way. Since SL(2, Z) = Γ0 (1) ⊃ Γ0 (n) ⊃ Γ0 (nm) for fixed n and all m = 1, 2, . . . one has a natural projection (1.26)

σm,n : Inm −→ In ³ ´ ³ ´ induced from the map Γ0 (nm) ac db 7→ Γ0 (n) ac db . In [HMM03] we proved Lemma 1.6. For any i ∈ Inm and 0 ≤ j ≤ kσn,m (i) there exists a unique index li,j ∈ In such that ¡ ¢ Ali,j K j (Aσn,m (i) ) A−1 ∈ SL(2, Z). i This on the other hand allows one to show [HMM03] ¡ ¢ ¡ ¢ ~ = ψi ~ = φi (z) Theorem 1.7. For ψ respectively φ any solution of i∈In i∈In the Lewis equations (1.12) respectively (1.8) with parameter β for the group Γ0 (n) ¡ ¢ ¡ ¢ ~ = Ψj ~ = Φj (z) the matrices Ψ respectively the functions Φ with j∈Inm j∈Inm kσn,m (j)

(1.27)

Ψj :=

X

ψlj,s K s (Aσn,m (j) )

s=0

respectively kσn,m (j)

(1.28)

Φj (z) :=

X

¯ φlj,s ¯β K s (Aσn,m (j) )(z)

s=0

solve the corresponding Lewis equations for the group Γ0 (nm) with the same parameter β. An immediate Corollary is ¡ ¢ ~ = φi (z) Corollary 1.8. For φ a vector valued period function for Γ0 (n) i∈In ¡ ¢ ~ the function Φ = Φj (z) j∈I in (1.28) is a vector valued period function for nm Γ0 (nm) with the same parameter β. ¡ ¢ Remark 1.9. The function F~ = Fj (z) j∈Inm with Fj (z) = Φj (z + 1) is an Γ (nm)

eigenfunction of the transfer operator Lβ 0 with eigenvalue λ = ±1 if the func¡ ¢ ~ tion f = fi (z) i∈In with fi (z) = φi (z + 1) is an eigenfunction of the operator Γ (n)

Lβ 0

with the same eigenvalue λ = ±1.

To derive from these special vector valued period functions operators similar to the Hecke operators on the automorphic forms we need another result from [HMM03]: ¡ ¢ ~ = Ψi Proposition 1.10. If the matrices Ψ solve the Lewis equation i∈Inm ¡ ¢ P ~ −1 (1.12) for Γ0 (nm) then the matrices ψ = ψj j∈In with ψj := i∈σm,n (j) Ψi solve ¡ ¢ ~ this equation for Γ0 (n). If on the other hand ψ = ψj j∈In solve equation (1.12) ¡ ¢ ~ = Ψi for Γ0 (n) then Ψ with Ψi := ψσ (i) solve this equation for Γ0 (nm). i∈Inm

m,n

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Remark 1.11. The second part of this Proposition shows that any period function for Γ0 (n) determines a “trivial” old period function for Γ0 (nm) whose components are just given by the components of the former one. An immediate consequence in the case n = 1, that is for the full modular group SL(2, Z), is ¡ ¢ ~ = Ψi Corollary 1.12. If Ψ solve equation (1.12) for Γ0 (m) then the i∈Im (m) matrix ψ with X Ψi (1.29) ψ (m) := i∈Im

solves this equation for the group SL(2, Z) for the same parameter β. A straightforward calculation shows that for any m prime one has indeed X (1.30) ψ (m) := A. A∈Sm

Corollary 1.13. For any period function φ = φ(z) for SL(2, Z) and for any ¯ ˜ m ∈ N the function φ˜ = φ(z) := φ¯β ψ (m) (z) is again a period function for the group with identical parameter β. In complete analogy one derives from Theorem 1.7 and Proposition 1.10 ¡ ¢ ~ = φi Theorem 1.14. For any vector valued period function φ for Γ0 (n) i∈In ~˜ ˜ ~ with and any m ∈ N the function φ = Tn,m φ ³ (1.31)

´ ~ (z) = ˜ n,m φ T i

X −1 (i) s∈σm,n

kσn,m (s)

X

¯ φls,j ¯β K j (Aσn,m (s) )(z)

j=0

is again a period function for Γ0 (n) with identical parameter β. ˜ n,m mapping the space Hence we have constructed this way linear operators T of vector valued period functions for Γ0 (n) with parameter β into itself. In the case ˜ 1,m reduces to the form n = 1 and m prime the operator T ³ ´ X ¯ ˜ 1,m φ (z) = (1.32) T φ¯β A(z) A∈Sm

˜ m in the form derived and hence coincides exactly with the mth Hecke operator H by M¨ uhlenbruch in [M¨ u04a] for period functions of Maass forms for SL(2, Z). For modular symbols this form of the Hecke operator was also derived earlier by Merel in [Me94]. ˜ 1,m are related to the Indeed in [HMM03] it was shown how the operators T ˜ Hecke operators Hm for arbitrary m ∈ N. ˜ n,m and the Hecke operators To understand the relation between the operators T ˜ Hn,m for an arbitrary subgroup Γ0 (n) we have to recall briefly in the next chapter the recent approach by M¨ uhlenbruch to the operators Hn,m on vector valued Maass forms.

¨ D. MAYER AND T. MUHLENBRUCH

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2. The Hecke operators on vector valued period functions for Γ0 (n) The approach by M¨ uhlenbruch to the vector valued period functions for the group Γ0 (n) works indeed for any modular subgroup Γ ∈ SL(2, Z) of finite index and also for the vector valued period polynomials for these groups. For simplicity we restrict however our discussion to the group Γ0 (n) and their Maass wave forms. Denote by S(n, β) the space of Maass forms u for Γ0 (n) with spectral parameter β where • • • •

u : H → C is real analytic, u(gz) = u(z) for all g ∈ Γ0 (n), ¡ ¢ ∆u = β(1 − β)u with ∆ = −y 2 ∂x2 + ∂y2 and u is of rapid decay at the cusps of Γ0 (n).

On the space S(n, β) one has an action of the Hecke operators Tp which for p prime are defined as follows [AL70]: if gcd(p, n) = 1 set ´ X ³ a b (2.1) T (p) = 0 d , ad=p 0≤b 0 Z (2.27) P~u(ζ) = η(~u, Rζβ ) L0,∞

Z

Z

= LT −1 0,T −1 ∞

=

ρΓ0 (n) (T

−1

η(~u, Rζβ ) +

LT 0−1 0,T 0 −1 ∞

) P~u(T z) + ρΓ0 (n) (T 0

−1

η(~u, Rζβ )

) (ζ + 1)−2s P~u(T 0 z).

Together with the growth condition in Lemma 2.7 this and a bootstrap argument similar as the one in [LZ01] for extending the function to the cut z-plane shows that P~u is a period function for Γ0 (n). ¤

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Remark 2.9. One can show that the map P : Sind (n, β) → FE(n, β) is injective and hence also surjective, since Sind (n, β) and FE(n, β) can be mapped bijectively onto each other [DH04]. To determine finally the form of the Hecke operators acting on the vector valued period functions ϕ ~ ∈ FE(n, β) of Γ0 (n) we follow again [M¨ u04b]. In a first step one defines for any rational q ∈ [0, 1) an element M (q) ∈ R1 = Z[SL(2, Z)] as follows: for q there exists a unique sequence (y0 , y1 , . . . , yL(q) ) of 0 rational numbers yl = abll , gcd(al , bl ) = 1, bl ≥ 0 with y0 = −1 0 = −∞, y1 = 1 = 0, yL(q) ³ = q, 0´ < yl < 1 for l = 2, . . . , L(q) such that y0 < y1 < . . . < yL(q) and det

al−1 al bl−1 bl

= −1 for l = 1, . . . , L(q). The element M (q) ∈ R1 is then defined as L(q)

(2.28)

M (q) = ³

with ml =

bl

−al −al−1

X

ml

l=1

´

for l = 1, . . . , L(q). Obviously one has ml ∈ SL(2, Z) for ¡ ¢ all l = 1, . . . , L(q) and m1 = 10 01 . An easy calculation [M¨ u04b] shows that the action of the operator K as defined in (1.25) when acting on any A ∈ Xm can be expressed through M (q) where q is PL(q) given by q = A0: if M (q) = l=1 ml then bl−1

K l (A) = ml+1 A

(2.29)

for l = 0, 1, . . . , L(q) − 1.

We have in particular that kA = L(A0) − 1. Consider first the case of period functions ϕ ∈ FE(1, β) for SL(2, Z). Then one has [M¨ u04b]: Lemma 2.10. For u ∈ S(1, β) a Maass form for SL(2, Z) and ϕ = Pu its period function the following identity holds for any A ∈ Xm : Z ¯ ¡ ¡ ¯ ¢ ¢ (2.30) η u¯0 A, Rζβ = ϕ¯β M (A0)A (ζ). L0,∞

Using relation (2.29) one can rewrite identity (2.30) as follows Z (2.31) L0,∞ th

Since the m

¡ ¯ ¢ η u¯0 A, Rζβ =

L(A0)−1

X

¡ ¯ l ¢ ϕ¯β K (A) (ζ).

l=0

Hecke operator Tm : S(1, β) → S(1, β) is given as X ¯ Tm u = u¯0 A A∈Xm

a straightforward application of relation (2.31) then gives for the Hecke operator ˜ m : FE(1, β) → FE(1, β) induced from Tm : S(1, β) → S(1, β): H ³ (2.32)

Z ´ ˜ Hm ϕ (ζ) = L0,∞

η(Hm u, Rζβ )

=

X A∈Xm

L(A0)−1 ³

X l=0

´ ¯ ϕ¯β K l (A) (ζ).

¨ D. MAYER AND T. MUHLENBRUCH

16

But the set of matrices {K l (A) : A ∈ Xm , l = 0, . . . , L(A0) − 1} coincides with the ˜ m can be written also as set Sm in (1.21) and hence the operator H X ¡ ¯ ¢ ˜ m ϕ(ζ) = (2.33) H ϕ¯β A (ζ). A∈Sm

˜ 1,m in [HMM03] and H ˜ m in [M¨ This shows that for m prime the operators T u04b] coincide. The main step in [M¨ u04b] for deriving the Hecke operators acting on the vector valued period functions ϕ ~ ∈ FE(n, β) of the groups Γ0 (n) is the following Lemma: Lemma 2.11. Let g1 , . . . , gµ be representatives of the right cosets of Γ0 (n) in SL(2, Z) and let P~u be the vector valued period function of ~u ∈ Sind (n, β). For A ∈ Xm set LA,j ¡ ¢ X (A,j) M σgj (A)0 = ml ∈ R1 for all j = 1, . . . , µ

(2.34) ¡

l=1

¢

with LA,j = L σgj (A)0 as in (2.28). The following identity holds for 1 ≤ j ≤ µ: Z ¯ ¡ ¢ η uΦA (j) ¯0 σgj (A), Rζβ (2.35) L0,∞

i Xh (A,j) −1 ρΓ0 (n) ((ml ) ) P ~u

¯ ³ ´ ¯ (A,j) σgj (A) (ζ). ¯ ml

LA,j

=

l=1

ΦA (j) β

Since any ~u ∈ Sind (n, β) is given by ~u = Π(u) with u ∈ S(n, β) in (2.5) and, according to relation (2.10), A gi ∈ Γ0 (n) gΦA (j) σgj (A) for all A ∈ Xm one has for u ∈ S(n, β) ¯ ¯ (2.36) u¯0 Agj = u¯0 gΦA (j) σgj (A) and hence identity (2.30) can be rewritten as Z ³ ¡ ¯ ¢ ´ η Π u¯0 A , Rζβ L0,∞

Xh

i

LA,j

=

(A,j) −1

ρΓ0 (n) ((ml

)

l=1

) PΠ(u)

¯ ¡ (A,j) ¢ ¯ m σgj (A) (ζ). l β

ΦA (j)

(A,j)

where LA,j and ml depend on A according to (2.34). From this an explicit ˜ n,p on FE(n, β) can be derived. For p representation for the Hecke operator H ˜ n,p has the form as given by prime H Proposition 2.12. For p prime, g1 , . . . , gµ representatives of the right cosets ˜ n,p has the form of Γ0 (n) in SL(2, Z) and ϕ ~ ∈ FE(n, β) the Hecke operator H ³ (2.37)

A,j h ´ i ´ ³¡ X LX (A,j) ¢−1 ˜ ϕ ~ Hn,p ϕ ~ = ρΓ0 (n) ml

j

A∈Ap l=1

¡ ¢ PLA,j (A,j) with M σgj (A)0 = l=1 ml ∈ R1 .

ΦA (j)

¯ (A,j) ¯ m σgj (A) β l

FROM TRANSFER OPERATORS TO HECKE OPERATORS FOR Γ0 (n)

17

˜ n,m and H ˜ n,m 3. The operators T ˜ n,m in (2.37) and Before discussing the exact relation between the operators H ˜ the operators Tn,m in (1.31) we want to give another derivation of the special solutions in Theorem 1.7. For this consider any Maass form u ∈ S(n, β) of Γ0 (n). Since Γ0 (n) ⊃ Γ0 (nm) for all m ∈ N one obviously has u ∈ S(mn, β). This u determines a vector valued form ~u = Πnm (u) ∈ Sind (nm, β) given by ¯ j = 1, . . . , µnm uj = (Πnm u) = u¯ gj , j

0

if Γ0 (nm)\SL(2, Z) = {Γ0 (nm)gj , j = 1, . . . , µnm }. Here µnm denotes the index of [SL(2, Z) : Γ0 (nm)]. The vector valued period function ϕ ~ corresponding to this ”old form ” Πnm u ∈ Sind (nm, β) then has the form Z ¡ ¢ ¡ ¯ ¢ ¡ ¢ ϕ ~ j= η u¯0 gj , Rζβ = Pnm ~u j L0,∞

for all j = 1, . . . , µnm . If hi , 1 ≤ i ≤ µn , are representatives of the cosets of ˜ k , k = 1, . . . , µnm , are representatives of the right cosets Γ0 (n) in SL(2, Z) and h µn ˜ k hi , 1 ≤ i ≤ µn , 1 ≤ k ≤ µnm are obviously in Γ0 (nm)\Γ0 (n) then gk,i := h µn

representatives of the rest classes in Γ0 (nm)\SL(2, Z). Identifying the index j with the tuple (k, i) we therefore get (3.1) Z Z ¡ ¢ ¡ ¢ ¡ ¯ ¢ ¡ ¯ ¢ ¡ ¢ ˜ k hi , R β = ~˜ (ζ) ϕ ~ j (ζ) = ϕ ~ (k,i) (ζ) = η u¯0 h η u¯0 hi , Rζβ = ϕ ζ i L0,∞

L0,∞

~˜ is the vector valued period function of ~u where ϕ ˜ = Πn u ∈ Sind (n, β). But i = σm,n (j) where we identify the index sets Iν and {1, . . . , µν } of the right cosets in Γ0 (ν)\SL(2, Z) for all ν ∈ N. Hence we have ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ~˜ (ζ). (3.2) ϕ ~ j (ζ) = Pnm ~u j (ζ) = Pn (~u ˜) σm,n (j) (ζ) = ϕ i This result coincides obviously with the second part of Proposition 1.10. On the other hand consider a Maass form u ∈ S(nm, β) with u(z) = v(mz) and v ∈ S(n, β) ⊂ S(nm, β). Such an u is called an old Maass form for Γ0 (nm). Then one finds for the corresponding vector valued Maass form ~u ∈ Sind (nm, β) ¯ ¡ ¢ ¡ ¢ ~u j = Πnm u j = v ¯0 Am gj , j = 1, . . . , µnm ¡ 0¢ where {g1 , . . . , gµnm } are representatives of Γ0 (nm)\SL(2, Z) and Am = m 0 1 . To determine the corresponding vector valued period function one has to apply the integral transform P = Pnm defined in (2.24): Z ¡ ¯ ¢ ¡ ¢ η v ¯0 Am gj , Rζβ , j = 1, . . . , µnm . Pnm ~u j (ζ) = L0,∞

¯ ¯ But v ¯0 Am gj = v ¯0 gΦAm (j) σgj (Am ) and hence Z ¢ ¡ ¢ ¡ ¯ (3.3) Pnm ~u j (ζ) = η v ¯0 gΦAm (j) σgj (Am ), Rζβ ,

j = 1, . . . , µnm .

L0,∞

¡ ¢ Therefore we get for Pnm ~u j Z ¡ ¢ (3.4) Pnm ~u j (ζ) =

L0,∞

¡ ¯ ¢ η v ¯0 gl(j) σgj (Am ), Rζβ ,

j = 1, . . . , µnm ,

¨ D. MAYER AND T. MUHLENBRUCH

18

where l(j) := φAm (j). Applying now identity (2.35) of Lemma 2.11 we get (3.5) LAm ,j h i ¯ ³ ´ X ¡ ¢ m ,j) −1 m ,j) ¯ m(A Pnm ~u j (ζ) = ρΓ0 (nm) ((m(A ) ) Pnm~v σ (A ) (ζ), g m r r j β l(j)

r=1

PLAm ,j (Am ,j) j = 1, . . . , µnm , where M (σgj (Am )0) = r=1 mr . But h i ¡ ¢ m ,j) −1 ρΓ0 (nm) (m(A ) Pnm~v = Pnm~v l(j,r) r l(j)

for a certain l depending on j and r and m ,j) m(A σgj (Am ) = K r−1 σgj (Am ) r

and hence ¡ ¢ Pnm ~u j (ζ)

(3.6)

LAm ,j

=

X ¡

Pnm~v

¯ r−1 ¯ K σgj (Am )

¢

l(j,r) β

r=1 LAm ,j −1

=

X

¯ ¡ ¢ Pnm~v l(j,r+1) ¯β K r σgj (Am ).

r=0

But according to relation (3.2) this can be written as (3.7)

¡

¢ Pnm ~u j (ζ) =

LAm ,j −1

X r=0

¡

¢ Pn~v˜

¡ σm,n l(j,r+1)

¯ ¢ ¯ K r σg (Am ). j β

¡ ¢ ~ = Φj (z) , This calculation indicates that the vector valued period function Φ j ∈ Inm , of Theorem 1.7 is exactly the vector valued period function corresponding to the vector valued Maass form ~u ∈ Sind (nm, β) with ~u = Πnm u and u ∈ S(nm, β) the old Maass form for Γ0 (nm) given by some v ∈ S(n, β) such that ¯ u(z) = v(mz) = v ¯ Am . 0

One can prove this statement using Lemma 3.3 and Lemma 3.12 to show that l(j, r) and lj,r coincide up to a certain conjugation and a modified version of the proof of Proposition 3.10 for the diagonal matrix Am . ~ = (Φj (z)), j ∈ Inm , of the Lewis The construction of the special solution Φ equation (1.8) for Γ0 (nm) relies strongly on the special parameterization of the rest classes Γ0 (n)\GL(2, Z) given by the set In in (1.15). For the following an explicit identification of In with Γ0 (n)\SL(2, Z) ≡ Γ0 (n)\GL(2, Z) is necessary. We take matrices αi ∈ SL(2, Z), i ∈ In satisfying (3.8)

[0 : 1]n αi = i.

Indeed one finds Lemma 3.1. The set Rn = {αi : i ∈ In } of matrices satisfying (3.8) for all i ∈ In is a set of representatives of the rest classes of Γ0 (n) in SL(2, Z). Proof. The matrices αi are defined up to an element γ ∈ Γ0 (n) since we have [0 : 1]n γαi = [0 : 1]n αi and hence Γ0 (n)αi 6= Γ0 (n)αj for i 6= j. On the other hand take any i ∈ In . Since i = [c : d]n with gcd(c, d)³ = 1´there exists a, b ∈ Z with ad−bc = 1 and hence [0 : 1]n αi = [c : d]n = i if αi =

a b c d

.

¤

FROM TRANSFER OPERATORS TO HECKE OPERATORS FOR Γ0 (n)

19

We can therefore interpret the map πn : Γ0 (n)\SL(2, Z) → In similar to (1.14) also as a map πn : Rn → In . On the other hand In can be identified via the map ? ? ω ³n : I´n → Xn with the set Xn of upper triangular matrices with ωn (i) = Ai = c b as defined in (1.20) for i = [c : d]n . Combining this map ωn with the map 0 n c πn then induces a bijection Rn → Xn? with αi 7→ ωn (πn (αi )). There is yet another bijection κn : Rn → Xn? given by ´ ³ b (3.9) κn (α[c:d]n ) = 0c n/c . with b ≡ d mod

n c,

such that ´ ³ 0 1 −n 0

α[c:d]n ∈ SL(2, Z) κn (α[c:d]n ) ³ ´ ¡ ¢ k with k satisfying holds. Indeed, for α[c:d]n = xc yd the matrix g = −1nx xb−yc c n d = b + k c is integer valued and one finds ³ ´ ³ ´ ³ ´ b c d 0 1 (3.11) g 0c n/c = −nx −ny = −n 0 α[c:d]n . ³ ´ (3.10)

det

0 1 −n 0

det α[c:d]n

Since det g = = 1 we get g ∈ SL(2, Z). det κn (α[c:d]n ) Combining the maps πn , κn and ωn we arrive at a bijective map hn : In → In with (3.12)

hn := ωn−1 ◦ κn ◦ πn−1 .

Then one finds for all i ∈ In that (3.13)

κn (αi ) = Ahn (i) .

There is an important relation between the matrices αi ∈ SL(2, Z) and the matrices Ai ∈ Xn? in (1.20). Lemma 3.2. For i, j ∈ In and g ∈ SL(2, Z) we have αi g αj−1 ∈ Γ0 (n)

Ahn (i) g A−1 hn (j) ∈ SL(2, Z). ³ ´ 0 1 Proof. For αi gαj−1 ∈ Γ0 (n) conjugating by the matrix −n 0 leads by using relations (3.10) and (3.13) to ³ ´ ³ ´−1 ¡ ¢−1 0 1 0 1 −1 (3.14) = g 0 Ahn (i) g g 00 Ahn (j) −n 0 αi g αj −n 0 ³ ³ ´ ´−1 0 1 0 1 ∈ −n 0 Γ0 (n) −n 0 for some g 0 , g 00 ∈ SL(2, Z). But

iff

³

0 1 −n 0

´

0 Ahn (i) g A−1 hn (j) ∈ g

³ Γ0 (n)

−1

0 1 −n 0

´−1

= Γ0 (n) and hence

Γ0 (n) g 00 ⊂ SL(2, Z).

On the other hand assume that for i, j ∈ In one has Ahn (i) g A−1 hn (j) ∈ SL(2, Z). For i ∈ In there exists an index j 0 ∈ In such that αi g αj−1 ∈ Γ0 (n) 0

¨ D. MAYER AND T. MUHLENBRUCH

20

S since SL(2, Z) = ι∈In Γ0 (n)αι . But then Ahn (i) g A−1 hn (j 0 ) ∈ SL(2, Z). Lemma 2.3 implies that there exists exactly one upper triangular matrix σg (Ahn (i) ) such that h i−1 Ahn (i) g σg (Ahn (i) ) ∈ SL(2, Z) and hence hn (j) = hn (j 0 ).

¤

This allows us to introduce another representation ρ˜n of the group SL(2, Z) in analogy to the right regular representation of ρΓ0 (n) induced from the trivial representation of Γ0 (n) as given in (2.4): ¡ ¢ (3.15) ρ˜n (g) := δSL(2,Z) (Ai gA−1 j ) i,j∈I . n

0

0

0

Then ρ˜n (g g ) = ρ˜n (g) ρ˜n (g ) for all g, g ∈ SL(2, Z). Lemma 3.2 shows that δΓ0 (n) (αi g αj−1 ) = δSL(2,Z) (Ahn (i) g A−1 hn (j) ). Hence, if Hn denotes the µn × µn matrix with entries ½ ¡ ¢ 1 if hn (j) = i and (3.16) Hn i,j = 0 otherwise one immediately shows that (3.17)

ρΓ0 (n) (g) = Hn−1 ρ˜n (g) Hn

and hence ρΓ0 (n) and ρ˜n are unitarily equivalent and one ¡Lemma 3.3. ¢ The¡ representations ¢ has ρΓ0 (n) (g) i,j = ρ˜n (g) hn (i),hn (j) for i, j ∈ In . For the following discussion let us recall the definition of the maps ΦA : In → In and σgj : Xm → Xm in (2.9) and (2.10) where A ∈ Xm and the {gj } are the representatives of Γ0 (n)\SL(2, Z) such that (3.18)

Agj ∈ Γ0 (n) gΦA (j) σgj (A)

Combining the two maps leads to a map λn,m : Xm × In → In × Xm with ¡ ¢ (3.19) λn,m (A, j) := ΦA (j), σgj (A) . ? On the other hand Lemma 1.6 allows us to introduce the map Λn,m : Inm → In ×Xm through ¢ ¡ (3.20) Λn,m (i) := li,0 , Aσn,m (i)

where according to Lemma 1.6 li,0 ∈ In is the unique index such that (3.21)

Ali,0 Aσn,m (i) ∈ SL(2, Z)Ai

where the matrices Ali,0 , Aσn,m (i) and Ai are associated to the different indices as given in (1.20). Obviously the projection of Λn,m onto its second component is surjective. Then the following Lemma can be derived Lemma 3.4. The map Λn,m is injective. It is surjective iff gcd(n, m) = 1.

FROM TRANSFER OPERATORS TO HECKE OPERATORS FOR Γ0 (n)

21

Proof. For i, j ∈ Inm with Λn,m (i) = Λn,m (j) equation (3.21) implies SL(2, Z) Ai 3 Ali,0 Aσn,m (i) = Alj,0 Aσn,m (j) ∈ SL(2, Z) Aj . ³ ´ ³ 0 0´ ? Assume gcd(n, m) = 1. For a0 db ∈ Xn? and a0 db 0 ∈ Xm put ³ 0 0 0´ ³ ´³ 0 0 ´ +bd B := aa0 abdd = a0 db a0 db 0 ∈ Matnm (2, Z). 0 0 0 0 0 If we show³ gcd(aa ´ , ab + bd , dd ) = 1 which implies the existence of a k ∈ Z such that B = 10 k1 Ai for a certain i ∈ Inm then B ∈ SL(2, Z) Ai for some i ∈ Inm and Λn,m is surjective. Therefore we assume that

(3.22)

∃p prime with p | gcd(aa0 , ab0 + bd0 , dd0 ).

Then p2 | aa0 dd0 = nm. Since gcd(n, m) = 1 we find either p2 | n or p2 | m. Say p2 | n. This implies that p | a and p | d since p | aa0 and p | dd0 . But p 6| b since gcd(a, b, d) = 1. Hence p 6| bd0 and the assumption (3.22) does not hold. If p2 | m then a similar argument holds. It remains the case gcd(n, m) > 1. Then Λn,m is not surjective since µnm < µm · µn and so #Inm < #(In × Im ). ¤ Furthermore one has Lemma 3.5. For each n, m ∈ N with gcd(n, m) = 1 there exists for each matrix ? ? A ∈ Xm a unique A0 ∈ Xm such that ¡ ¢ ¡ ¢ (3.23) A n0 01 ∈ SL(2, Z) n0 01 A0 ? ? and hence the map Λ : Xm → Xm with Λ(A) = A0 is a bijection. ³ ´ ? Proof. Let A ∈ Xm be given by A = a0 db . Since gcd(n, m) = 1 and hence also gcd(n, d) = 1 due to m = ad there exists an integer 0 ≤ k < n such that n | (b + kd). Then ³ ´³ ´¡ ¢ ³ na b+kd ´ ¡ n 0 ¢³ a b+kd ´ 1 k a b n 0 = 0 1 0 nd 0 1 0 d 0 1 = 0 d ³ b+kd ´ shows that A0 = a0 nd fulfills (3.23). Furthermore, since gcd(a, b, d) = 1, also 0 ? gcd(a, b+kd n , d) = 1 and the matrix A belongs indeed to Xm because 0 ≤ b + kd < d(1 + k) ≤ nd. ¶ µ ³ b+kd ´ 0 b0 +k0 d0 a n n = . Then a0 = a To show that Λ is bijective, assume a0 0 0 d d

and d0 = d as well as b0 + k 0 d0 = b + kd and hence b0 = b + (k − k 0 )d. But 0 ≤ b0 < d and therefore k = k 0 and hence b = b0 . ¤ Remark 3.6. Lemma ¡3.5 does not hold for gcd(n, m) > 1, a counter example ¢ 1 1 being n = m = 2 and A = 0 2 ∈ X2? . Furthermore one needs Lemma 3.7. For n, m with gcd(n, m) = 1 there exists for i ∈ Inm a matrix ? A ∈ Xm such that ³ ´ 0 1 ¡ ¢ ∈ SL(2, Z) Ai . (3.24) −n 0 Aα −1 hn

σn,m (i)

¨ D. MAYER AND T. MUHLENBRUCH

22

Proof. For i ∈ Inm consider Λm,n (i) = (li,0 , Aσm,n (i) ) ∈ Im × Xn? as defined in (3.20) such that (3.25)

Ali,0 Aσm,n (i) ∈ SL(2, Z) Ai .

Applying relation (3.10) to A = Aσm,n (i) ∈ Xn? there exists g ∈ SL(2, Z) with ³ ´ ¡ ¢ ¡ ¢ −1 ¡ ¢ 0 1 −1 n 0 −1 (3.26) Aσm,n (i) = g −n 0 κn Aσm,n (i) = g S 0 1 κn Aσm,n (i) . Hence relation (3.25) can be written as (3.27)

Ali,0 gS −1

¡n 0¢ 0 1

¡ ¢ κ−1 n Aσm,n (i) ∈ SL(2, Z) Ai .

Relation (3.18) then gives Ali,0 gS −1 ∈ SL(2, Z) σgS −1 (Ali,0 ) and hence (3.27) can be written as ¡ ¢ ¡ ¢ (3.28) σgS −1 (Ali,0 ) n0 01 κ−1 n Aσm,n (i) ∈ SL(2, Z) Ai . ? Lemma 3.5 then implies the existence of an A ∈ Xm such that (3.28) can be written as ¡n 0¢ ¡ ¢ −1 (3.29) 0 1 A κn Aσm,n (i) ∈ SL(2, Z) Ai .

But according to equation (3.13) ¡ ¢ κ−1 n Aσm,n (i) = α

¡

¢

−1 σm,n (i) hn

and hence acting by S −1 from the left on (3.29) one finds ³ ´ 0 1 ¡ ¢ ∈ SL(2, Z) Ai . −n 0 A α −1 hn

σm,n (i)

¤ ? Lemma 3.8. If³gcd(n, ´ m) = 1 there exists for j ∈ In and A ∈ Xm a unique 0 1 ? 0 0 A0 ∈ Xm such that −n 0 A αj ∈ SL(2, Z) A Ahn (j) . The map A 7→ A is bijective.

Proof. For A Ahn (j) there exists g ∈ SL(2, Z) such that ³ ´ ¡ ¢ 0 1 −1 n 0 (3.30) A Ahn (j) = A g −n 0 αj = A g S 0 1 αj ³ ´ 0 1 since Ahn (j) = κn (αj ) = g −n 0 αj according to relations (3.13) and (3.10). But A gS −1 = g 0 σgS −1 (A) for some g 0 ∈ SL(2, Z) by the definition of the map σg in (2.8) and hence ¡ ¢ (3.31) A Ahn (j) = g 0 σgS −1 (A) n0 01 αj . Lemma 3.5 shows that A Ahn (j) = g 00

¡n 0¢ 0 1

A0 αj = g 00 S −1

³

0 1 −n 0

´

A0 αj

? for some matrix A0 ∈ Xm and for some g 00 ∈ SL(2, Z). This proves the first part of the Lemma. Since both the maps A 7→ σgS −1 (A) and σgS −1 (A) 7→ A0 are bijective also A 7→ A0 is bijective. ¤

FROM TRANSFER OPERATORS TO HECKE OPERATORS FOR Γ0 (n)

23

¡ ¢ −1 Lemma 3.9. If gcd(n, m) = 1 then for any j ∈ In and i ∈ σm,n hn (j) one has ¡ ¢ ¡ ¡ ¢ ¢ li,0 , Aσn,m (i) = hn ΦB (j) , σαj (B) ¡ ¢ ? ? −1 for some B ∈ Xm . The map σm,n hn (j) → Xm with i 7→ B is bijective. ¡ ¢ −1 ? Proof. Fix an i ∈ σm,n hn (j) ⊂ Inm . Then there exists an A ∈ Xm with A Ahn (j) ∈ SL(2, Z) Ai since σm,n (i) = hn (j), see also Lemma 5.12 and Definition 5.13 in [HMM03]. Using Lemma 3.8 we then find ´ ³ 0 1 ? for some B ∈ Xm . SL(2, Z) Ai = SL(2, Z) A Ahn (j) 3 −n 0 B αj But

³

0 1 −n 0

´

³ B αj

∈ = ⊂

0 1 −n 0

´

Γ0 (n) αΦB (j) σαj (B) ´ 0 1 Γ0 (n) −n 0 αΦB (j) σαj (B) ¢ σα (B) SL(2, Z) A ¡ ³

hn ΦB (j)

j

where relations (3.13) and (3.10) imply the last step. Hence ¢ σα (B) with g ∈ SL(2, Z) Ai = g A ¡ j hn ΦB (j)

and this decomposition is unique as shown in the proof of Lemma 3.7. Indeed it coincides with the decomposition Ai = g Ali,0 Aσn,m (i) ¡ ¢ given there and since Λn,m (i) = li,0 , Aσn,m (i) we find ¡ ¡ ¢ ¢ Λn,m (i) = hn ΦB (j) , σαj (B) . ¡ ¢ −1 ? The map σm,n hn (j) 3 i 7→ B ∈ Xm is bijective because the maps involved in the construction of B are bijective. ¤ This leads to Proposition 3.10. If gcd(n, m) = 1 then for j ∈ In the two sets © ¡ ¢ª −1 (li,0 , Aσn,m (i) ) : i ∈ σm,n hn (j) and

©¡

¡ ¢ ¢ ª ? hn ΦA (j) , σαj (A) : A ∈ Xm

are identical. From this follows Corollary 3.11. If gcd(m, n) = 1 then for each j ∈ In we have ¡ −1 ¡ ¢¢ ©¡ ¡ ¢ ¢ ª ? Λn,m σm,n hn (j) = hn ΦA (j) , σαj (A) : A ∈ Xm . ¡ ¢ Proof of Corollary. Since Λ−1 n,m li,0 , Aσn,m (i) = i the corollary follows from the proposition. ¤ Proof of Proposition 3.10. The first part of Lemma 3.9 shows that the first set is contained in the second set. The second part of this Lemma shows that the two sets are identical. ¤

¨ D. MAYER AND T. MUHLENBRUCH

24

To derive our main result on the relation between the “Hecke like” operators in [HMM03] and the Hecke operators in [M¨ u04b] let us reformulate Lemma 1.6 in which the existence of an index li,j ∈ In for any i ∈ Inm and any 0 ≤ j ≤ kσn,m (i) with ¡ ¢ ∈ SL(2, Z) (3.32) Ali,j K j (Aσn,m (i) ) A−1 i PL(Aσn,m (i) 0) (i) is shown. For this consider the matrix sum M (Aσn,m (i) 0) = mj . j=1 j But kσn,m (i) = L(Aσn,m (i) 0) − 1 and according to relation (2.29) K Aσn,m (i) = (i)

mj+1 Aσn,m (i) . and hence (3.32) can be rewritten as (i)

Ali,j mj+1 Aσn,m (i) A−1 ∈ SL(2, Z). i But for j = 0 relation (3.32) reads Aσn,m (i) A−1 ∈ A−1 i li,0 SL(2, Z) and hence (i)

Ali,j mj+1 A−1 li,0 ∈ SL(2, Z) respectively

¡ (i) ¢−1 −1 Ali,0 mj+1 Ali,j ∈ SL(2, Z)

Using the definition of the presentation ρ˜n in (3.15) this then shows : PLi (i) mr one has Lemma 3.12. For i ∈ Inm and M (Aσn,m (i) 0) = r=1 ½ h ¡ ¢−1 i 1 if r = j + 1 and = δ = ρ˜n (m(i) ) r,j+1 r 0 if r 6= j + 1. li,0 ,li,j Hence we get ˜ n,m and H ˜ n,m Proposition 3.13. For gcd(n, m) = 1, m prime the operators T −1 ˜ ˜ are conjugate with Hn Tn,m Hn = Hn,m . ~ a vector valued period function for Γ0 (n) as defined in §1 conProof. For φ ~ ˜ n,m φ as given in (1.31). Then using Lemma 3.12 and Equation (2.29) we sider T get ³ ´ ~ ˜ n,m φ (3.33) T hn (j)

=

LX i −1 ³

X ¡

¢

−1 i∈σm,n hn (j)

=

¢ r=1

−1 i∈σm,n hn (j)

¯ ¯ (i) ¯ mr Aσn,m (i) .

h−1 n (li,0 ) β

But according to Proposition 3.10 this can be rewritten as (3.34) ´ ³ ´ X L(A0) X ³ −1 −1 ~ ~ ˜ n,m φ ρΓ0 (n) (m(A,j) ) H φ T = r n hn (j)

with M (σαj (A)0) =

? r=1 A∈Xm

PL(A0) r=1

¯ ¯ (i) ¯ mr+1 Aσn,m (i)

li,0 β

r=0 Li ³ X ¡ ¢ −1 ´ −1 ~ ρΓ0 (n) (m(i) Hn φ r )

X ¡

´ ¢ ¡ (i) ~ Hn ρΓ0 (n) (mr+1 )−1 Hn−1 φ

(A,j)

mr

¯ ¯ (A,j) σαj (A) ¯ mr

ΦA (j) β

˜ n,m = H ˜ n,m Hn−1 . and hence Hn−1 T

¤

FROM TRANSFER OPERATORS TO HECKE OPERATORS FOR Γ0 (n)

25

References [AL70] A. O. L. Atkin, J. Lehner, Hecke operators on Γ0 (m), Math. Ann. 185 (1970), 134 – 160. [CM00] C. Chang, D. Mayer, Thermodynamic formalism and Selberg’s zeta function for modular groups, Regul. Chaotic Dyn. 5 (2000), 281–312. [CM01] C. Chang, D. Mayer, Eigenfunctions of the transfer operators and the period functions for modular groups, in “Dynamical, spectral and arithmetic zeta functions” (San Antonio, TX. 1999) Contemp. Math. 290, (2001), 1–40, AMS Providence, RI. [DH04] A. Deitmar and J. Hilgert, The Lewis Correspondence for submodular groups. e-arxiv, 2004. http://arXiv.org/abs/math/0404067. [HMM03] J. Hilgert, D. Mayer, H. Movasati, Transfer operators for Γ0 (n) and the Hecke operators for the period functions of PSL(2, Z), Math. Camb. Phil. Soc. 139 (2005), 81–116. [LZ01] J. Lewis, D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. 153 (2001), 191–258. [M73] Yu. Manin, Periods of cusp forms and p-adic Hecke series, Math. USSR-Sb. 92 (1973), 371–393. [MM02] Yu. Manin, M. Marcolli, Continued fractions, modular symbols and non commutative geometry, Selecta Math. (N.S.) 8 (2002), 475–520. [Mr01] F. Martin, Periodes des formes modulaires de poid 1, These, Universite Paris 7, 2001. [Ma91] D. Mayer, Continued fractions and related transformations, in “Ergodic theory, symbolic dynamics and hyperbolic spaces”, Eds.: T. Bedford et al., pp. 175–222, Oxford University Press, Oxford 1991. [Ma03] D. Mayer, Transfer operators, the Selberg zeta functions and the Lewis-Zagier theory of period functions, Lectures at the “International School on Mathematical Aspects of Quantum Chaos II”, G¨ unzburg 2003, to appear in the Proceedings. [Me94] L. Merel, Universal Fourier expansions of modular forms, in “On Artin’s Conjecture for Odd 2-dimensional Representations”, Springer-Verlag, Lecture Notes in Mathematics, 1585 (1994) [Mi89] T. Miyake, Modular Forms, Springer-Verlag, Berlin 1989 [M¨ u04a] T. M¨ uhlenbruch, Hecke operators on period functions for the full modular group, IMRN 77 (2004), 4127-4145 [M¨ u04b] T. M¨ uhlenbruch, Hecke operators on period functions for Γ0 (n), J. Number Theory, to appear [Sa90] P. Sarnak, Some Applications of Modular Forms, Cambridge University Press, Cambridge 1990. [Se85] C. Series, The modular surface and continued fractions, J. London Math. Soc. 31 (2) (1985), 69–80. [Z90] D. Zagier, Hecke operators and periods of modular forms, Israel Math. Conf. Proc. 3 (1990), 321–336. [Z93] D. Zagier, Periods of modular forms, traces of Hecke operators and multiple zeta values, in “Research into automorphic forms and L-functions”, Kyoto 1992, Surikaisekikenkyusko Kokyuroku 843 (1993), 162–170. ¨ r Theoretische Physik, Technische Univerista ¨ t Clausthal, ClausthalInstitut fu Zellerfeld, Germany E-mail address: [email protected] ¨ r Theoretische Physik, Technische Univerista ¨ t Clausthal, ClausthalInstitut fu Zellerfeld, Germany E-mail address: [email protected]