Series expansion of the period function and representations of Hecke operators

Journal of Number Theory 171 (2017) 301–340

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Journal of Number Theory www.elsevier.com/locate/jnt

Series expansion of the period function and representations of Hecke operators Dohoon Choi a , Subong Lim b,∗ , Tobias Mühlenbruch c , Wissam Raji d a

School of liberal arts and sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi 412-791, Republic of Korea b Department of Mathematics Education, Sungkyunkwan University, 25-2, Sungkyunkwan-Ro, Jongno-gu, Seoul 03063, Republic of Korea c Department of Mathematics and Computer Science, FernUniversität in Hagen, 58084 Hagen, Germany d Department of Mathematics, American University of Beirut, Beirut, Lebanon

a r t i c l e

i n f o

Article history: Received 25 February 2016 Received in revised form 2 July 2016 Accepted 3 July 2016 Available online 7 September 2016 Communicated by A. El-Guindy MSC: primary 11F37 secondary 20C08, 32N10, 33B20 Keywords: Period functions Eichler–Shimura cohomology Hecke operators

a b s t r a c t The period polynomial of a cusp form of an integral weight plays an important role in the number theory. In this paper, we study the period function of a cusp form of real weight. We obtain a series expansion of the period function of a cusp form of real weight for SL(2, Z) by using the binomial expansion. Furthermore, we study two kinds of Hecke operators acting on cusp forms and period functions, respectively. With these Hecke operators we show that there is a Hecke-equivariant isomorphism between the space of cusp forms and the space of period functions. As an application, we obtain a formula for a certain L-value of a Hecke eigenform by using the series expansion of its period function. © 2016 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (D. Choi), [email protected] (S. Lim), [email protected] (T. Mühlenbruch), [email protected] (W. Raji). http://dx.doi.org/10.1016/j.jnt.2016.07.020 0022-314X/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction For an even integer k ≥ 2, denote by Sk the space of cusp forms of weight k and the trivial multiplier system on SL(2, Z). For each cusp form f ∈ Sk , the period polynomial of f is defined by i∞ f (τ )(τ − z)k−2 dτ. 0

Then the Eichler–Shimura cohomology theory tells us that there is an isomorphism between the direct sum Sk ⊕ Sk and a subspace of the vector space      Wk = P ∈ Pk−2 ; P + P 2−k S = P + P 2−k U + P 2−k U 2 = 0 with codimension 1. Here, Pk−2 is the space of polynomials of degree at most k − 2, S, T and U denote the matrices S :=  in SL(2, Z) and 

 0 −1  1 0

,

T :=

1 1 01

and U :=

 1 −1  1 0

is the slash operator as in (2.2). We refer to [Lan] for more details on this isomorphism. The coefficients of the period polynomial are also called periods. These periods give two additional rational structures on Sk besides the usual rational structure given by the rationality of Fourier coefficients. Periods are also related with the critical values of L-functions and have been much studied by many researchers (for example, see [CZ,Kno2,KZ,Man,Za]). Another related objects are the Hecke operators acting on cusp forms. On Sk , the mth Hecke operator is represented with triangle matrices of the form a b of non-negative integer entries satisfying ad = m and 0 ≤ b < d which act on f 0 d by extending the slash operator accordingly. Their representation on period polynomials were also studied (for example, see [CZ,Man,PP]). In this paper, we consider cusp forms of real weight. In general the associated period function is not a polynomial any more even when the weight is half-integral. This is a big difference between the cases of integral and real weight. We discuss some aspects of such period functions in this paper. More precisely, we obtain a series expansion of the period function of a cusp form of real weight for SL(2, Z) by using the binomial expansion. The coefficients of this expansion can be expressed in terms of the incomplete gamma function. Furthermore, we study two kinds of Hecke operators acting on cusp forms and period functions, respectively. With these Hecke operators we show that there is a Hecke-equivariant isomorphism between the space of cusp forms and the space of period functions. As an application, we obtain an exact formula for a certain L-value of a Hecke eigenform by using the series expansion of its period function. Throughout this paper let k ∈ R be a real weight with compatible multiplier system χ. We denote by Sk,χ (N ) the space of cusp forms of weight k and multiplier system χ on Γ0 (N ). If f ∈ Sk,χ (1), then f has a Fourier expansion of the form 2−k

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340 ∞ 

303

a(m)e2πi(m+κ)z ,

(1.1)

i∞ P f (z) := f (τ ) (τ − z)k−2 dτ

(1.2)

f (z) =

m+κ>0

where κ ∈ [0, 1). Then the function

0

is the period function of the non-holomorphic Eichler integral associated with f . This period function has the following series expansion. Theorem 1.1. Let k ∈ R with k > 2. Let f ∈ Sk,χ (1) with a Fourier expansion as in (1.1). Then P f can be written as

k−1−n  ∞  a(m) k−2 i P f (z) = 2π (m + κ)k−1−n n n=0 m+κ>0

2π (m + κ), k − 1 − n z k−2−n + × χ(S)(−1)n Γ |z|

  n k−1 + (−1) Γ 2π|z|(m + κ), k − 1 − n z , where Γ(a, n) =

∞ a

e−t tn−1 dt denotes the incomplete gamma function.

Next, we construct a representation of Hecke operators acting on period functions for congruence subgroups of level N . For integral weights Choie and Zagier [CZ] defined the action of Hecke operators on period polynomials for SL(2, Z). Paşol and Popa [PP] generalized this approach to congruence subgroups. We construct a representation of Hecke operators acting on period functions for arbitrary weights and congruence subgroups Γ0 (N ). To do this, we consider the space of vector-valued cusp forms. The mth Hecke operator on Sk,χ (N ) has the form (for more details see §3.4)   HN,m f (z) :=





 f

k,ψ

A∈Xm

where j(A, z) = (cz + d) and Az = upper triangle matrices  Xm :=



A (z) =

ψ −1 (A) j(A, z)−k f (Az),

A∈Xm az+b cz+d

for

a b c d

and the sum runs through the set of

 a b ; a, b, d ∈ Z≥0 , ad = m, 0 ≤ b < d . 0 d

(1.3)

Here, we consider a map ψ −1 : Xm → C, see §3.2, which encodes some additional scalar valued action of Xm similar to the multiplier system does for the group Γ0 (N ). Note that

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we allow the case ψ −1 (A) = 0 for some A ∈ Xm . We also extended the slash-notation slightly to accommodate ψ −1 . The exact definition and some explicit examples of Hecke operators HN,m can be found in §3.4. Let f ∈ Sk,χ (1) be a cusp form. If a function ψ −1 : Xm → C satisfies conditions in Definition 3.17, then we call this suitable. For m ∈ N and a suitable function ψ −1 , the ˜ 1,m acting on P f is given by mth Hecke operator H L        ˜ 1,m P f := H (P f )2−k,χ ml 2−k,ψ A

(1.4)

A∈Xm l=1

  L with M (A0) = l=1 ml ∈ Z SL(2, Z) a finite formal sum of matrices in SL(2, Z), see Definition 3.13. Here, A0 denotes the Mobius transformation defined by A evaluated at

a b the real number 0, i.e., A0 = db for A = . c d Remark 1.2. There are matrices ml appearing in (1.4) which seem to come from nowhere. To explain their role recall the definition of P f in (1.2). If we replace f (τ ) by f (Az), then we get 

 P f (A·) (z) =

i∞ i∞  k−2 d k−2 f (Aτ ) (τ − z) dτ = f (τ ) A−1 τ − z dτ a 0

A0

a b

 i∞ ∈ Xm . The role of the matrices ml is to reduce the above integral A0 to  i∞ (a sum of) integrals 0 which we can interpret again as (transformations of) a period function P f . More details on the matrices ml and its use in the integral representation of P f can be found in §3.4 and §5. for A =

0 d

A first result on Hecke operators for period functions is the following. ˜ 1,m satisfies Theorem 1.3. The operator H     ˜ 1,m P f = P H1,m f . H For N, m ∈ N, Theorem 1.3 can be extended to Γ0 (N ) by taking a vector-valued   approach. Let μ := SL(2, Z) : Γ0 (N ) and fix representatives α1 , . . . , αμ of Γ0 (N ) \   SL(2, Z). To shorten notation we borrow the multi-index notation α ¯ = α1 , . . . , αμ to refer to the right coset representatives. A change of right coset representatives are new   representatives α¯ = α1 , . . . , αμ satisfying αi ∈ Γ0 (N )αi for all i ∈ {1, . . . , μ}. For f ∈ Sk,χ (N ), define the associated period function Pα¯ Πα¯ f by  Pα¯ Πα¯ f :=

 Pα¯ Πα¯ f 1

  , · · · , Pα¯ Πα¯ f , μ

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where 



i∞ Pα¯ Πα¯ f (z) := f (αi τ ) (αi τ − αi z)k−2 d(αi τ ). i

0

For the representatives α ¯ = (α1 , · · · , αμ ) of Γ0 (N ) \ SL(2, Z), we define the companion weight matrix ωα¯ = (ωi,j )i,j by (2.9). For vector-valued function f : H → Cμ , z ∈ H and γ ∈ SL(2, Z), we define the matrix slash action ωα¯ by (f ωα¯ γ)(z) := ωα¯ (γ, z)−1 f (γz). ˜ N,m For a suitable function ψ −1 and f ∈ Sk,χ (N ), we define the mth Hecke operator H acting on vector-valued period function by    ˜ N,m Pα¯ Πα¯ f (z) := Υ−1 (A, z) H A∈Xm

L

  Pα¯ Πα¯ f ω ¯ ml (Az), l=1

α

where α¯ denotes the change of representatives given by (5.13) and Υ−1 : Xm × H → Mat(μ, C) is the matrix-valued map defined by (3.8). Then, we have the following result. ˜ N,m satisfies Theorem 1.4. The operator H     ˜ N,m Pα¯ Πα¯ f = Pα¯ Πα¯ HN,m f . H

(1.5)

Next we consider an isomorphism between the space of cusp forms and the space of period functions. Let P be the space of holomorphic functions g in H which satisfy the growth condition   r −σ |g(z)| < K |z| + Im (z)

(1.6)

for some positive constants K, r and σ. This is the space which was introduced by Knopp in [Kno] to establish the Eichler–Shimura cohomology theory for real weights. We define  V2−k,χ¯ (P) :=

   h ∈ P; h + h2−k,χ¯ S = 0, h + h2−k,χ¯ U + h2−k,χ¯ U 2 = 0



and       U2−k,χ¯ (P) := h − h 2−k,χ¯ S; h ∈ P such that h − h 2−k,χ¯ T = 0 . Knopp and his collaborators proved that there is an isomorphism (see [Kno,KM])

306

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340 1 Sk,χ (1) ∼ = H2−k, χ ¯ (P),

1 where H2−k, χ ¯ (P) is the Eichler–Shimura cohomology group associated with the module P (for the precise definition, see §5). This induces an isomorphism

 Sk,χ (1) ∼ = V2−k,χ¯ (P) U2−k,χ¯ (P)

(1.7)

which is given by the period function f → P f . In the following theorem, we prove that this isomorphism is Hecke-equivariant. Theorem 1.5. The isomorphism in (1.7) is Hecke-equivariant. For a cusp form f ∈ Sk,χ (1) with the Fourier expansion as in (1.1) its L-function L(f, s) is given by the analytic continuation of the series 

a(m) . (m + κ)s m+κ>0 ˜ 1,m and the series expansion of the period As an application of the Hecke operators H function in Theorem 1.1, we obtain a formula for the special L-value L(f, k − 1) of a Hecke eigenform f in Sk,χ (1). This formula provides the relation between the mth Hecke eigenvalue λm of f and the special L-value L(f, k − 1) of f . Theorem 1.6. Let f ∈ Sk,χ (1) be a Hecke eigenform such that H1,m f = λm f for m ∈ N. If k > 2, then λm L(f, k − 1) = i1−k

L (2π)k−1   χ(ml )ψ(A)j(ml , Az)k−2 j(A, z)k−2 Sf (ml A0), Γ(k − 1) A∈Xm l=1

where Sf (x) is given by

k−1−n  ∞  k−2 −i n=0

a(m) χ(S)(−1)n k−1−n 2π (m + κ) n m+κ>0



  n 2π k−2−n k−1 (m + κ), k − 1 − n x ×Γ + (−1) Γ 2π|x|(m + κ), k − 1 − n x . |x|

This paper is organized as follows. In Sections 2 and 3, we recall basic definitions from the theory of modular forms, specifically Hecke operators on cusp forms and lifts to a representation of Hecke operators on vector-valued cusp forms. Section 4 discusses properties of period functions, the scalar valued cases as well as the vector-valued one. Section 5 contains the proofs to our main results: Theorems 1.1, 1.3, 1.4, 1.5 and 1.6.

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2. Modular forms First we recall the definition of scalar valued modular forms. Consider a congruence subgroup Γ0 (N ) ⊂ SL(2, Z). Let k ∈ R denote a real weight. We call a function χ : Γ0 (N ) → C a compatible multiplier system χ to weight k if χ satisfies (1) |χ(γ)| = 1 for all γ ∈ Γ0 (N ), (2) the compatibility condition χ(γδ) j(γδ, z)k = χ(γ) j(γ, δz)k χ(δ) j(δ, z)k

(2.1)

for all γ, δ ∈ Γ0 (N ) and z ∈ H,   (3) the non-triviality condition χ(−1) = e−πik with 1 := 10 01 . We also introduce the trivial multiplier system 1 given by 1 : Γ0 (N ) → C;

γ → 1(γ) := 1.

We later need the extension to 1 : Xm → C=0 , 1(A) := 1. The slash notation now comes in two flavors, one with the multiplier system included and one without it:    f k,χ γ (z) : = χ(γ)−1 j(γ, z)−k f (γz) and    k f k M (z) : = (det M ) 2 j(M, z)−k f (M z).

(2.2)

The first one is defined for all γ ∈ Γ0 (N ) and the last one makes sense for all 2 × 2 matrices M ∈ Mat(2, Z) with positive determinant as long as j(M, z) does not vanish. The following is the precise definition of modular forms and cusp forms. Definition 2.1. A holomorphic function f : H → C is called a modular form of weight k and multiplier system χ on Γ0 (N ) if it satisfies (1) for all γ ∈ Γ0 (N )  f k,χ γ = f

(2.3)

 (2) for each i, f k αi admits a right sided Fourier expansion of the form    f k αi (z) =



ai (n) e2πi(n+κi )z/λi

n+κi ≥0

with some λi ∈ N (the width of the cusp αi ∞) and κi ∈ [0, 1).

(2.4)

308

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A cusp form is a modular form such that, for each 1 ≤ i ≤ μ, the coefficient ai (n) vanishes when n + κi ≤ 0. The space of cusp forms is denoted by Sk,χ (N ). Now we introduce the notion of vector-valued cusp forms. 2.1. Weight matrices   Recall that μ ∈ N denotes the index of Γ0 (N ) in SL(2, Z) and α ¯ = α1 , . . . , αμ denotes a vector of right representatives of the coset decomposition of Γ0 (N ) in SL(2, Z). By following [MR1, Definition 5.1] we define the weight matrix. Definition 2.2. The weight matrix w : SL(2, Z) × H → GL(μ, C) is a map satisfying the cocycle condition w(γδ, z) = w(γ, δz) w(δ, z)

(2.5)

for all γ, δ ∈ SL(2, Z) and z ∈ H.  Similar to the slash action k,χ γ in (2.2) we define the matrix slash action. Let f : H → Cμ be avector-valued function and let w be a weight matrix. We define the matrix slash action w by    f w γ (z) := w(γ, z)−1 f (γz)

(2.6)

for all z ∈ H and γ ∈ SL(2, Z). As an example to a weight matrix with trivial multiplier system we consider the induced representation ρ0 : SL(2, Z) → GL(μ, Z) [MR1, (5.1)], which is given by ρ0 (γ)i,j :=



  δΓ0 (N ) αi γαj−1

for all γ ∈ SL(2, Z) and i, j ∈ {1, . . . , μ}. Here, ρ0 (γ)i,j is the (i, j)th entry of the matrix ρ0 (γ) ∈ GL(μ, Z), and δΓ0 (N ) (γ) is 1 if γ ∈ Γ0 (N ), and 0 otherwise. The associated weight matrix w0 (γ, z) = j(γ, z)k ρ0 (γ) for even k ∈ 2Z is given by multiplying the scalar automorphic factor with the induced representation. For a multiplier system χ of Γ0 (N ) compatible to weight k, we define the matrix-valued   function wα¯ (γ, z) = wi,j (γ, z) 1≤i,j≤μ by      k wi,j (γ, z) = δΓ0 (N ) αi γαj−1 χ αi γαj−1 j αi γαj−1 , αj z

(2.7)

for γ ∈ SL(2, Z) and z ∈ H. Lemma 2.3. The map wα¯ : SL(2, Z) × H → GL(μ, C) defined by (2.7) is a weight matrix.

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Proof. The proof is basically the same as the one of [MR1, Lemma 5.3] just replacing  k the factor e−ikarg(j(γ,z)) in [MR1] with j αi γαj−1 , z . 2 We may write wα¯ = w hiding the dependency on the representatives of the right cosets. Remark 2.4. The weight matrix w in Lemma 2.3 resembles a permutation matrix in the following sense: (a) In each line and column there is exactly one non-zero entry and all other entries vanish. (b) The matrix is invertible. The inverse matrix of w is given by 

w(γ, z)−1

 i,j

−k    −1  = δΓ0 (N ) αj γαi−1 χ αj γαi−1 j αj γαi−1 , αi z

(2.8)

for all z ∈ H and γ ∈ SL(2, Z). We conclude this part defining a companion matrix to the weight matrix w in (2.7).   Definition 2.5. We define the companion weight matrix ωα¯ = ωi,j 1≤i,j≤μ to the weight matrix wα¯ by      2−k ωi,j (γ, z) : = δΓ0 (N ) αi γαj−1 χ αi γαj−1 j αi γαj−1 , αj z

(2.9)

for all γ ∈ SL(2, Z) and z ∈ H. We may write ωα¯ = ω hiding the dependency on the representatives of the right cosets. Example 2.6. For level N = 1 the 1 × 1-matrices w−1 (A) and ω −1 (A) defined in (2.7) and (2.9) are scalar functions given by w−1 (γ, z) = χ(γ)−1 j(γ, z)−k

and ω −1 (γ, z) = χ(γ) j(γ, z)k−2

for all γ ∈ SL(2, Z) and z ∈ H. 2.2. Vector-valued cusp forms In analogy to [MR1, §5.2] we introduce vector-valued cusp forms. Definition 2.7. Let w be the weight matrix given in (2.7). We call a function f = (f1 , . . . , fμ )tr : H → Cμ a vector-valued cusp form for weight matrix w if f satisfies:

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(1) each component function fi : H → C is holomorphic on H, (2) the function f satisfies the transformation law f (gz) = w(g, z) f (z)

(2.10)

for all g ∈ SL(2, Z) and z ∈ H, (3) each component function fi vanishes at the cusps. vec (N ). The space of vector-valued cusp forms is denoted by Sw vec We have maps between two spaces Sk,χ (N ) and Sw (N ). vec Lemma 2.8. Maps between Sk,χ (N ) and Sw (N ) are given by

    f → f 0 α1 , . . . , f 0 αμ    f = f1 , . . . , fμ → f1 0 α1−1 .

vec Πα¯ : Sk,χ (N ) → Sw (N ); vec πα¯ : Sw (N ) → Sk,χ (N );

(2.11)

We may write Πα¯ = Π and πα¯ = π hiding the dependency on the representatives. Proof. We follow the proof of [MR1, Lemma 5.6] adapted to the weight matrix given in (2.7). Let f ∈ Sk,χ (N ) and g ∈ SL(2, Z). Then 

   Π(f )w g (z)

 i

   μ   = w(g, z)−1 Π(f )(gz) = w(g, z)−1 i,j f (αj g z) i

=

μ 

j=1

   −1  −k δΓ0 (N ) αj gαi−1 χ αj gαi−1 j αj gαi−1 , αi z f (αj gαi−1 αi z)

j=1

 −1  −k = χ αj0 gαi−1 j αj0 gαi−1 , αi z f (αj0 g z), where the index 1 ≤ j0 ≤ μ is uniquely determined by αj0 gαi−1 ∈ Γ0 (N ). Using the modular identity (2.3) for the cusp form f we get  



  Π(f )w g (z)

 −1  −k = χ αj0 gαi−1 j αj0 gαi−1 , αi z f (αj0 gαi−1 αi z) i

  = f (αi z) = Π(f ) . i

Hence Π(f ) satisfies (2.10).

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vec On the other hand, if we consider f ∈ Sw (N ) and γ ∈ Γ0 (N ), then

        −1 −1 −1 −1

  π f w γ (z) = f w γ (α1 z) = w(γ, α1 z) f (γα1 z) 1

=

μ 

1

  −k    −1  δΓ0 (N ) αj γα1−1 χ αj γα1−1 j αj γα1−1 , α1 α1−1 z f (γα1−1 z) j

j=1



−1  −k   = χ α1 γα1−1 j α1 γα1−1 , z f (γα1−1 z) 1  −k   −1  = χ α1 γα1−1 j α1 γα1−1 , z π f (α1 γα1−1 z)    since γ ∈ Γ0 (N ) and α1 ∈ Γ0 (N ). Consider π f w γ again. By using (2.10) we get      π f w γ (z) = π f (z). Hence the equation    −k   −1  π f (z) = χ α1 γα1−1 j α1 γα1−1 , z π f (α1 γα1−1 z) holds for all z ∈ H and γ ∈ Γ0 (N ). Since α1 γα1−1 runs through all elements of Γ0 (N ) if and only if γ does it, we just showed that π(f ) satisfies (2.3). 2 Let α1 , . . . , αμ ∈ SL(2, Z) denote another set of right coset representatives of Γ0 (N ) in SL(2, Z). We make sure that the enumeration is consistent, i.e., αi and αi denote the same right coset: Γ0 (N )αi = Γ0 (N )αi for all i ∈ {1, . . . , μ}. Then there exist γi ∈ Γ0 (N ) such that γi αi = αi for all i. Next, vec we define an operation on Sw (N ) which corresponds to the change of representatives under the map Π. Lemma 2.9. Let γ1 , . . . , γμ ∈ Γ0 (N ) denote the matrices as introduced above. The operation     f (z) → Dγ f (z) := diag χ(γ1 ) j(γ1 , α1 z)k , . . . , χ(γμ ) j(γμ , αμ z)k f (z)

(2.12)

corresponds to a change of right representatives αi → γi αi =: αi in the definition of Π in (2.11). The inverse map γi αi → αi corresponds to applying the diagonal matrix   diag χ(γ1−1 ) j(γ1−1 , γ1 α1 z)k , · · · , χ(γμ−1 ) j(γμ−1 , γμ αμ z)k   = diag χ(γ1 )−1 j(γ1 , α1 z)−k , · · · , χ(γμ )−1 j(γμ , αμ z)−k .

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Furthermore, we have the relation

   diag χ(γ1 ) j(γ1 , α1 ·)k , . . . , χ(γμ ) j(γμ , αμ ·)k Πα¯ (f ) w g 

= diag χ(γ1 ) j(γ1 , α1 ·) , . . . , χ(γμ ) j(γμ , αμ ·) k

k





  Πα¯ f w g



(2.13)

for all f ∈ Sk,χ (N ) and g ∈ SL(2, Z), where w (resp. w ) is the weight matrix given by (2.7) with respect to the representatives αi (resp. αi = γi αi ). Proof. Note that     Πα¯ f (z) = f (αi z) = f (γi αi z) = χ(γi )j(γi , αi z)k f (αi z) i

k

= χ(γi )j(γi , αi z)

  f 0 αi (z)

      k k = diag χ(γ1 ) j(γ1 , α1 z) , . . . , χ(γμ ) j(γμ , αμ z) Πα¯ f (z) i



   = Πα¯ diag χ(γ1 ) j(γ1 , ·)k , . . . , χ(γμ ) j(γμ , ·)k f (z) i

for all z ∈ H and indices i ∈ {1, . . . , μ}. This proves (2.12). The inverse mapping with   the operator diag χ(γ1−1 ) j(γ1−1 , γ1 α1 z)k , · · · , χ(γμ−1 ) j(γμ−1 , γμ αμ z)k follows from that χ(γ −1 ) = χ(γ)−1 and the compatibility condition (2.1). On the other hand, if we use identities (2.8), (2.11) and (2.12), then 

=

      diag χ(γ1 ) j(γ1 , α1 ·)k , . . . , χ(γμ ) j(γμ , αμ ·)k Π f w g(z)

μ 

i

 −k −1    −1 −1   −1 δΓ0 (N ) αj gαi χ αj gαi j αj gαi , αi z f (αj gz)

j=1

  −k    −1 −1  −1 −1 w(g, z) Π f (gz) j γi , γi αi z =χ γi i



     k k  = diag χ(γ1 ) j(γ1 , α1 z) , . . . , χ(γμ ) j(γμ , αμ z) Π f w g (z) . i

We just showed (2.13).

2

Remark 2.10. Let γ := (γ1 , . . . , γμ ) ∈ Γ0 (N )μ denote the vector of matrices used in the change of right representatives αi → γi αi =: αi as discussed in Lemma 2.9 by the operation defined in (2.12). Lemma 2.9 shows that the diagram

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

Sk,χ (N )

−→

Sk,χ (N ) Πα¯ ↓

↓ Πα¯ vec (N ) Sw

313

Dγ

−→

vec Sw  (N )

commutes. 3. Hecke operators on modular forms In this section we introduce Hecke operators on modular forms. 3.1. Some technical results For m ∈ N we consider the set Xm of triangle matrices defined in (1.3). We recall a few necessary technical results. Lemma 3.1 ([Mue, Lemma 3.5]). For each m ∈ N and g ∈ SL(2, Z) there exists a bijective map σg : Xm → Xm ;

A → σg (A)

satisfying  −1 Ag σg (A) ∈ SL(2, Z)

(3.1)

for all A ∈ Xm . Remark 3.2. The inverse map σg−1 : Xm → Xm of σg , which is defined in Lemma 3.1, is given by σg−1 (A) := σg−1 (A) for all A ∈ Xm and g ∈ SL(2, Z). This can be seen by  −1 looking at the inverse of Ag −1 σg−1 (A) :   −1 −1 = σg−1 (A) g A−1 ∈ SL(2, Z). Ag −1 σg−1 (A) Together with the injectivity of σg , which is already shown in Lemma 3.1, we conclude that σg−1 is indeed the inverse map of σg . In fact, Lemma 3.1 implies a bit more. Recall that we denote the right coset representatives of Γ0 (N ) in SL(2, Z) by α1 , . . . , αμ . For given A ∈ Xm and i ∈ {1, . . . , μ}, we can write Aαi as Aαi = γA,i αj B

(3.2)

with uniquely determined γA,i ∈ Γ0 (N ), j ∈ {1, . . . , μ} and B ∈ Xm . The reverse statement also holds: For given B ∈ Xm and j ∈ {1, . . . , μ}, we can write αj B as

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γ˜j,B Aαi = αj B with uniquely determined γ˜j,B ∈ Γ0 (N ), i ∈ {1, . . . , μ} and A ∈ Xm . The above reasoning leads to the following. Definition 3.3. For each A ∈ Xm and index i ∈ {1, . . . , μ} consider σαi (A) introduced in Lemma 3.1. Its defining relation by (3.1) allows us to define the maps φ :Xm × {1, . . . , μ} → Γ0 (N ) × {1, . . . , μ} × Xm ;

  (A, i) → φ1 (A, i), φ2 (A, i), φ3 (A, i)

and φ˜ : {1, . . . , μ} × Xm → Γ0 (N ) × Xm × {1, . . . , μ};   (j, B) → φ˜1 (j, B), φ˜2 (j, B), φ˜3 (j, B) such that the equations A αi = φ1 (A, i) αφ2 (A,i) φ3 (A, i)

and

αj B = φ˜1 (j, B) φ˜2 (j, B) αφ˜3 (j,B)

(3.3)

hold for all A, B ∈ Xm and i, j ∈ {1, . . . , μ}. The uniqueness of the decomposition in (3.2) also implies the following. Lemma 3.4. The reduced maps Xm × {1, . . . , μ} → {1, . . . , μ} × Xm ;

  (A, i) → φ2 (A, i), φ3 (A, i)

{1, . . . , μ} × Xm → Xm × {1, . . . , μ};

  (j, B) → φ˜2 (j, B), φ˜3 (j, B)

and

are inverse to each other. Furthermore, the first components of the maps φ and φ˜ satisfy

  −1 φ˜1 φ2 (A, i), φ3 (A, i)

φ1 (A, i) = and φ˜1 (j, B) =





φ1 φ˜2 (j, B), φ˜3 (j, B)

−1

for all A, B ∈ Xm and i, j ∈ {1, . . . , μ}. Proof. The lemma follows from the uniqueness of the decomposition in (3.2): Given (A, i) we get unique (B, j) and vice versa such that A αi = γ B αj for suitable unique γ ∈ Γ0 (N ). 2

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3.2. Triangle matrix representations and matrix actions Next we define an action from element A ∈ Xm on functions f : H → C=0 . We consider a map ψ −1 : Xm → C such that A → ψ −1 (A) induces a scalar action of Xm . The trivial map 1−1 is obviously defined by 1−1 : Xm → C with 1−1 (A) = 1 for all A ∈ Xm . Remark 3.5. (a) The map ψ −1 may also depend on the level N of the group Γ0 (N ) and on the weight k. This implicit dependency is hidden in the notation. In slight abuse of notation, its action on a scalar valued cusp form f ∈ Sk,χ (N ) is given by the slash action    k f (z) → f k,ψ A (z) = det(A) 2 ψ −1 (A)j(A, z)−k f (Az).

(3.4)

The above formula also indicates why we define ψ −1 (A). This allows us to use the  slash action f k,ψ A, where this factor appears. (b) In general, the map ψ −1 is not an inverse of another map, i.e., we allow for the term ψ −1 (A) to vanish for some A. We use this notation to follow the traditional way to define the slash operator as in (3.4). Later, we will use this slash operator to write Hecke operators. An example of such a Hecke operator is given by Atkin and Lehner in [AL] (see also Example 3.18). Now we consider how to define the action of a triangle matrix A ∈ Xm on a cusp form f ∈ Sk,χ (N ). Note that      Π f k,ψ A (z)   = f

i

    αi (z)= det(A) k2 ψ −1 (A)j(A, αi z)−k f A αi z A k,ψ 0

  k = det(A) 2 ψ −1 (A)j(A, αi z)−k f φ1 (A, i) αφ2 (A,i) φ3 (A, i) z    k k = det(A) 2 ψ −1 (A)j(A, αi z)−k χ φ1 (A, i) j φ1 (A, i), αφ2 (A,i) φ3 (A, i)z   × f αφ2 (A,i) φ3 (A, i) z for all A ∈ Xm and z ∈ H. Since the map Xm × {1, . . . , μ} → {1, . . . , μ} × Xm ;

  (A, i) → φ2 (A, i), φ3 (A, i)

  is bijective by Lemma 3.4, we can replace the image φ2 (A, i), φ3 (A, i) with (j, B) and get

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     Π f k,ψ φ˜2 (j, B)

˜3 (j,B) φ

 −1 k (z) = det(φ˜2 (j, B)) 2 ψ φ˜2 (j, B) j(φ˜2 (j, B), αφ˜3 (j,B) z)−k    k   × χ φ˜1 (j, B)−1 j φ˜1 (j, B)−1 , αj Bz f αj B z  −1 k = det(φ˜2 (j, B)) 2 ψ φ˜2 (j, B) j(φ˜2 (j, B), αφ˜3 (j,B) z)−k ⎤ ⎡       k ⎥ ⎢ × ⎣ χ φ˜1 (j, B)−1 j φ˜1 (j, B)−1 , · f (·) 0 αj ⎦ (Bz).    Change of right coset representatives

We just calculated the identity      Π f k,ψ φ˜2 (j, B) (z)

(3.5)

˜3 (j,B) φ

 −1 k = det(φ˜2 (j, B)) 2 ψ φ˜2 (j, B) j(φ˜2 (j, B), αφ˜3 (j,B) z)−k     k × diag χ φ˜1 (1, B)−1 j φ˜1 (1, B)−1 , α1 · ,

      k . . . , χ φ˜1 (μ, B)−1 j φ˜1 (μ, B)−1 , αμ · Π f (·) (Bz) j

k 2



= det(φ˜2 (j, B)) ψ φ˜2 (j, B)

−1

j(φ˜2 (j, B), αφ˜3 (j,B) z)−k

    → D−−−−− Π f (·) (Bz) −1 α α

j

for all j ∈ {1, . . . , μ}, B ∈ Xm and z ∈ H. The form of the right hand side indicates the three components of the above formula. The first is some kind of twisted permutation matrix which changes the φ˜3 (j, B)th component to the jth component, since the map j → φ˜3 (j, B) for fixed B ∈ Xm is bijective. The second is the corresponding diagonal entry of the operation changing the representatives of the right cosets from αi to αi := → = D− −−−−−−− → for this diagonal operator was φ˜1 (i, B)−1 αi ; the short notation D−−−−− −1 −1 ˜ α α

φ1 (·,B)

introduced in (2.12). The last component of the formula is the action of the matrix B ∈ Xm on the vector-valued form Π(f ). The above calculation motivates the definition of the matrix Ψ−1 (B, z) which is similar to the weight matrix w(γ, z) but is based on ψ instead on the multiplier system χ. To do so, we need one more index map. Lemma 3.6. For each B ∈ Xm the map JB : {1, . . . , μ} → {1, . . . , μ};

j → JB (j) := φ˜3 (j, B)

(3.6)

is bijective. Proof. Let B ∈ Xm be fixed. For j ∈ {1, . . . , μ} and i := φ˜3 (j, B) there exists an A ∈ Xm such that (3.3) holds:

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A αi ∈ Γ0 (N ) αj B.   Lemma 3.4 shows that for given (j, B) the pair φ˜2 (j, B), φ˜3 (j, B) satisfying φ˜2 (j, B) αφ˜3 (j,B) ∈ Γ0 (N ) αj B is unique. This implies injectivity of j → φ˜3 (j, B). The map JB is also surjective since the set {1, · · · , μ} is finite. 2 Definition 3.7. We define the matrix-valued map Ψ−1 : Xm × H → Mat(μ, C) by its   entries Ψ−1 (A, z) 1≤i,j≤μ for all i, j ∈ {1, . . . , μ} and A ∈ Xm . These entries are given by

 −1 k ˜ Ψ−1 (A) : = δ (j, A) det(φ˜2 (j, A)) 2 ψ φ˜2 (j, A) j(φ˜2 (j, A), αφ˜3 (j,A) z)−k (3.7) φ i 3 i,j for all A ∈ Xm , z ∈ H and i, j ∈ {1, . . . , μ}. In analogy to ω in (2.9) we define its companion Υ−1 : Xm × H → Mat(μ, C) by

 −1 ˜3 (j, A) det(φ˜2 (j, A))1− k2 ψ φ˜2 (j, A) Υ−1 (A) : = δ j(φ˜2 (j, A), αφ˜3 (j,A) z)k−2 . φ i i,j (3.8) Here, the symbol δj (i) denotes the Kronecker-δ. Note that the inverse notation in Ψ−1 (A, z) and Υ−1 (A, z) is a formal notation. The individual matrices Ψ−1 (A, z) and Υ−1 (A, z) may not be invertible. Example 3.8. For level N = 1, the 1 × 1-matrices Ψ−1 (A, z) and Υ−1 (A, z) are scalars given by Ψ−1 (A, z) = (det A) 2 ψ −1 (A) j(A, z)−k k

Υ−1 (A, z) = (det A)

1− k 2

and

ψ −1 (A) j(A, z)k−2

for all A ∈ Xm and z ∈ H. Lemma 3.9. For A ∈ Xm the matrices Ψ−1 (A, z) and Υ−1 (A, z) defined in (3.7) and (3.8) are indeed independent of z ∈ H.   Proof. Assume that A ∈ Xm has the form A = a0 db . The automorphic factor then reads as j(A, z) = d which is independent of the argument z. This shows that appearing terms j(φ˜2 (j, A), αφ˜3 (j,A) z) are independent of the argument z. Hence Ψ−1 (A, z) and Υ−1 (A, z) are indeed matrices only depending on A and the choice of representatives αi . 2

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  Next, we define a formal matrix slash action Ψ in analogy to w defined in (2.6). Definition 3.10. Let Ψ−1 denote the matrix as defined in (3.7) and let f ∈ Sk,χ (N ). For any A ∈ Xm we define 

      k Πα¯ (f )Ψ A (z) : = Ψ−1 (A) diag χ φ˜1 (1, A)−1 j φ˜1 (1, A)−1 , α1 · , . . .

   k      −1 −1 ˜ ˜ Πα¯ f (·) A z . j φ1 (μ, A) , αμ · . . . , χ φ1 (μ, A) (3.9)

→ = D− −−−−−−− → for this diagonal operator Remark 3.11. With the short notation D−−−−− −1 −1 ˜ φ1 (·,B)

α α

introduced in (2.12) we can rewrite the right hand side of (3.9) as −1

Ψ

         − − − − − → (A) D  −1 Πα¯ f A z = Ψ−1 (A) Πα¯ f A z . α α

Lemma 3.12. Let f ∈ Sk,χ (N ) and Ψ−1 : Xm × H → C be the matrix defined in (3.7). Then

     ˜   Πα¯ f k,ψ φ2 (i, A) = Π(f ) Ψ A

(3.10)

−1 JA (i)

−1 for any component i = 1, . . . , μ and A ∈ Xm , where JA (·) is the inverse map of JA defined in (3.6) with the map Π given in (2.11). If we sum over all elements of Xm , then we get



      Π f k,ψ A = Π(f ) Ψ A.

A∈Xm

(3.11)

A∈Xm

Proof. The first equation of the lemma follows directly from the calculation in (3.5) together with the Definitions 3.7 and 3.10. For the second equation, consider (3.10). If we sum (3.10) over all A ∈ Xm , then we get    A∈Xm

 Π(f ) Ψ A

 = j

  A∈Xm

    ˜  Π f k,ψ φ2 (j, A)

i

for all j ∈ {1, . . . , μ}. By the bijectivity of maps in Lemma 3.4 we conclude that the map from Xm to Xm given by A → φ˜2 (j, A) is also bijective for each j. This means that the sum on the right hand side also runs through all elements of Xm . We may renumerate the summation from summing over the preimage of A → φ˜2 (j, A) to summing over the image of A → φ˜2 (j, A). We get

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           Π(f ) Ψ A = Π f k,ψ A . j

A∈Xm

j

A∈Xm

This proves (3.11) for each component j. 2 3.3. Farey sequences and the formal sum M (q) =

L l=1

ml , based on [Mue, §2]

We briefly recall the properties of M (q) based on the discussion in [Mue, §2]. Consider the following problem: By starting from a rational number q ∈ [0, 1) ∩ Q, we would like to find a finite sequence of qi with −∞ = q0 < q1 < q2 < . . . < qL−1 < qL = q

(3.12)

such that q1 , . . . qL ∈ [0, 1) ∩ Q are rational. Moreover, by writing the ql ’s in its normal form ql = abll with coprime integers 0 ≤ al < bl , l ∈ {1, . . . , L}, and writing −∞ = −1 0 which gives a0 = −1 and b0 = 0, we additionally require bl−1 < bl and that the matrices ml :=

−al−1 −bl−1

al bl

∈ SL(2, Z)

for all l ∈ {1, . . . , L}

(3.13)

have all determinant 1. The number L = L(q) in (3.12) depends on the rational number q. The existence of such matrices was shown in [Mue, §2] by an explicit construction based on Farey sequences. By collecting the results in [Mue, §2] we have the following definition. Definition 3.13 ([Mue, §2, Definition 2.8]). To each q ∈ [0, 1) ∩ Q we attach the formal   L sum M (q) = l=1 ml ∈ Z SL(2, Z) , with L = L(q) ∈ N, of the form M (q) =

−a0 −b0

a1 b1

−1

+ ... +

−al−1 −bl−1

al bl

−1

+ ... +

−aL−1 −bL−1

aL bL

−1 .

(3.14)

  Here, Z SL(2, Z) denotes the set of finite linear combinations of SL(2, Z) with coefficients in Z. Lemma 3.14 ([Mue, Lemmata 2.9–2.11]). Let A = L l=1 ml .

a b 0 d

∈ Xm and consider M

b d

=

∗ ∗ and one has cl ζ + dl > 0 for all ζ ≥ db and l ∈ {1, . . . , L}. l=1 d cl dl (2) The matrices ml A, l = 1, . . . , L, contain   only nonnegative integer entries. (1) M

b

=

L



(3) The entries of the matrix ml A =

a b c d

satisfy a > c ≥ 0 and d > b ≥ 0.

Remark 3.15. The reason for the construction of the matrices ml is that we would like to use Lemmas 5.4 and 5.5 later. One step in our construction of Hecke operators for

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∞ period functions involves a calculation of an integral of the form A0 . . . dτ with A ∈ Xm ∞ in terms of integrals of the form 0 . . . dτ (with slightly different integrants). Since A0 ∈ [0, 1) ∩ Q is a rational number, we may use the matrices ml to peace together the ∞ L  m−1 l ∞ integral path of A0 . . . dτ by l=1 m−1 . . . dτ and use the transformation τ → ml τ l 0 to put the ml ’s into the integrant. We refer to §5, in particular Lemmas 5.6 and 5.7 for the exact details. Remark 3.16. A construction similar to the one in [Mue, §2] leading to matrices m ˜ l was done by Manin in [Man]. His construction is based on the continued fraction expansion of the rational number q ∈ [0, 1) ∩ Q. One important difference of the two constructions is that our construction leads to Lemma 3.14. The properties listed there guarantee that     we can evaluate the map g(z) → g k ml (Az) for all z ∈ H. 3.4. Hecke operators for cusp forms We introduce Hecke operator for scalar valued cusp forms of real weight. Definition 3.17. We call a function ψ −1 : Xm → C suitable if ψ −1 satisfies: (1) ψ −1 induces a scalar action of Xm . (2) ψ −1 is non-trivial: there exists A ∈ Xm such that ψ −1 (A) = 0.  (3) For f ∈ Sk,χ (N ), a function A∈Xm f k,ψ A is also a cusp form in Sk,χ (N ). Here, the summation is taken for all A ∈ Xm such that ψ −1 (A) = 0. In this case we call the operator HN,m : Sk,χ (N ) → Sk,χ (N );

f →



 f k,ψ A

(3.15)

A∈Xm

the mth Hecke operator on Sk,χ (N ). Obviously, HN,m depends on the choice of ψ. We will give some examples below. Example 3.18. (1) Consider level N = 1, an even integer weight k ∈ 2N and m ∈ N. The mth Hecke operator H1,m on Sk,1 (1) is given by (for example see [Miy]) H1,m f =



 f k,1 A.

A∈Xm

(2) Consider the case N ∈ N, an even integer weight k ∈ 2N with trivial multiplier system 1 and a prime number m ∈ N. We define

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⎧ ⎪ ⎪ ⎨1 if p  N , ψ(A) :=

  1 if p | N and A = p0 10 ⎪ ⎪ ⎩0 if p | N and A =  p 0 . 0 1

The induced map HN,p : Sk,1 (N ) → Sk,1 (N );

f →



 f k,ψ A

A∈Xp

is usual pth Hecke operator (for example see [AL]). (3) Half-integral weight Hecke operators were introduced by Shimura in [Shi]. He gave a concrete example of the Hecke operator Tp2 on Γ0 (N ) with 4 | N and prime with p  N for half-integral weights. Let 4 | N be a level divisible by 4 and k ∈ 12 N be a half-integral weight. Let p  N be a prime. Shimura defined the set Ap2 := Xp2 



p 0 0 p



=: Aa ∪ Ab ∪ Ac

partitioned in the disjoint sets  Aa : =  Ab : =  Ac : =

 b 2 ; 0 ≤ b < p , p2

 h ; 0
 0 . 1

1 0 p 0 p2 0

He also defined

ψ −1 (A) =

⎧ ⎪ 1 ⎪ ⎪

k ⎪ ⎪ ⎪ ⎨ ε−1 −h ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ 0

p

p

if A ∈ Aa ,   if A = p0 hp ∈ Ab , if A ∈ Ac , if A ∈ / Ap2 ,

where εp -constant is ' εp :=

1 if p ≡ 1 mod 4, i

if p ≡ 3 mod 4.

The multiplier system χ is given accordingly. Then Shimura’s Hecke operator HN,p2 is defined by

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HN,p2 : Sk,χ (N ) → Sk,χ (N );



f → HN,p2 f =

 f k,ψ A.

A∈Xp2

3.5. Hecke operators for vector-valued cusp forms vec We will derive a formula for the Hecke operators acting on Sw (N ) with weight matrix     w given in (2.7). To do so we write the vector-valued cusp form Π f k,ψ A∈Xm A in terms of certain matrix sums acting on the vector-valued cusp form Π(f ).   Consider the ith component of the vector-valued cusp form Π HN,m f . Since the map Π is linear we can write

Π



 f

k,ψ

A

A∈Xm

=



   Π f k,ψ A i .

A∈Xm

i

Then, by Lemma 3.12, we can write each component with the Ψ -notation and have ( Π



 f

k,ψ

A∈Xm

) A i

     Π(f ) Ψ A . = i

A∈Xm

This calculation motivates the following definition of Hecke operators for vector-valued cusp forms. Definition 3.19. For m ∈ N, fix a Ψ−1 as in (3.7). We call Ψ−1 suitable if: (1) Ψ−1 is non-trivial: there exists A ∈ Xm such that Ψ−1 (A) is not the μ×μ zero-matrix.   vec (N ), a vector-valued function A∈Xm f Ψ A is also a vector-valued cusp (2) For f ∈ Sw vec form in Sw (N ). In this case we call the operator vec vec vec HN,m : Sw (N ) → Sw (N );

f →



 f Ψ A

(3.16)

A∈Xm vec the mth Hecke operator on Sw (N ). vec vec The mth Hecke operator HN,m on Sw (N ) corresponds to the mth Hecke operator HN,m on Sk,χ (N ) as the following lemma shows.

Lemma 3.20. For all f ∈ Sk,χ (N )     vec Π(f ) . Π HN,m f = HN,m

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Proof. By Lemma 3.12 and the linearity of Π defined in (2.11)       Π HN,m f = Π f k,ψ A A∈Xm

=



 Π(f )Ψ A

A∈Xm vec = HN,m Π(f ). 2

This immediately implies the following. Corollary 3.21. For f ∈ Sk,χ (N )       ˜ N,m Pα¯ Πα¯ f = Pα¯ Πα¯ HN,m f = Pα¯ H vec Πα¯ f . H N,m Remark 3.22. In [Mue] the third author discussed Hecke operators for Maass cusp forms with trivial multiplier system. The situation in [Mue] was a bit simpler compared to our present discussion. For example, Maass cusp forms are invariant under group action Γ0 (N ). This allows us to skip the discussion about change of representatives induced by the elements γA,i in (3.2). 4. Period functions In this section we review the basic notions of period functions. Recall the definition of period functions P f in (1.2) for cusp forms f ∈ Sk,χ (1). This definition can be extended to higher level cusp forms f ∈ Sk,χ (N ). The integral in (1.2) exists since the exponential decay of the cusp form f near the cuspidal points i∞ and 0 compensates for the growth of the part (τ − z)k−2 in the integrand. Remark 4.1. There are other related integrals which associate cusp forms to period functions or period polynomials. (a) Consider even weight k, trivial multiplier system 1 and level 1. For f ∈ Sk,1 (1) the integral i∞ f (τ ) (τ − z)k−2 dτ ∈ C[z] 0

gives the associated period polynomial of degree less than k − 2 (for example see [Za]). (b) The third author considered a similar integral transform with replaced 1-form or integration kernel and studied Hecke operators on period functions associated to Maass cusp forms of zero weight, trivial multiplier system and level 1 (for example

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see [Mue]). This form was extended to cover the case of Maass cusp forms of real weights (and level 1) in [MR2]. A recent discussion of period polynomials can be found in [PP]. vec (N ), the associated vector-valued For the weight matrix w given by (2.7) and f ∈ Sw vec

period function P f is defined as the integral transform P from the space Sw (N ) to the space of functions on H given by

 Pα¯ f :=

 Pα¯ f 1

  , · · · , Pα¯ f , μ

where   i∞  

Pα¯ f (z) := f i (τ ) (αi τ − αi z)k−2 d(αi τ ). i

(4.1)

0

Formally, we write (4.1) as 

 Pα¯ f (z) =

i∞ −−−−−−−−−→ f (τ ) (τ − z)k−2 dτ

(4.2)

0

−−−−−−−−−→ with f (τ ) (τ − z)k−2 dτ as a shortcut for   −−−−−−k−2 −−−→  

dτ = f i (τ ) (αi τ − αi z)k−2 d(αi τ ) f (τ ) (τ − z)

(4.3)

i

for all i. We may write Pα¯ f = P f hiding the dependency on (the choice of representa ¯ = α1 , . . . , αμ ). tives) α Remark 4.2. The definition of vector-valued period functions (of real weight) follows closely [Mue, Definition 4.2] (up to some complex conjugations). This allows us to use most of the machinery which was introduced and discussed there. The action of SL(2, Z) on H extends naturally to H := H ∪ Q ∪ {i∞}. By a simple path L connecting points z0 , z1 ∈ H we understand a piecewise smooth curve which lies inside H except possibly for the initial and end point z0 , z1 and is analytic in all points H  H. Two simple paths Lz0 ,z1 and Lz0 ,z1 are always homotopic. A path L connecting points z0 , z1 ∈ H is given by the union of finitely many simple paths Ln , n = 1, . . . , N connecting the points z0,n , z1,n ∈ H such that z0,1 = z0 , z1,n = z0,n+1 and z1,N = z1 . For distinct z0 , z1 ∈ H H the standard path Lz0 ,z1 is the geodesic connecting z0 and z1 . A standard path L is also a simple path (for more details see [Lan]).

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For cusp forms f ∈ Sk,χ (N ) it is known that the integral transformations P f satisfy (see the proof of Lemma 2.2 in [KM]) 

 −1  −1 k−2   χ γ −1 j γ ,z P f γ −1 z =

f (τ ) (τ − z)k−2 dτ

(4.4)

Lγ0,γ∞

for all γ ∈ Γ0 (N ) and z ∈ H. Similarly, for the vector-valued case, we find 

 −1   ω g −1 , z P f g −1 z =

−−−−−−−−−→ f (τ ) (τ − z)k−2 dτ

(4.5)

Lg0,g∞ vec for f ∈ Sw (N ), g ∈ SL(2, Z) and z ∈ H. Here, ω denotes the companion weight matrix to w defined in (2.9).

5. Proof of the main theorems First, we investigate a series expansion of the period function of a cusp form f ∈ Sk,χ (1). We define two functions F1 and F2 as follows. For z ∈ H we define i|z| F1 (z) : = f (τ ) (τ − z)k−2 dτ

(5.1)

0

and i∞ F2 (z) : = f (τ ) (τ − z)k−2 dτ .

(5.2)

i|z|

We can see that P f (z) = F1 (z) + F2 (z) holds. In the following lemma, we compute these two functions more explicitly. Lemma 5.1. Let f ∈ Sk,χ (1) be a cusp form. Then F1 and F2 can be written as

F1 (z) = lim ε↓0

and

∞  k−2 n=0

n

i|z|−iε 

z

k−2−n

k−2−n

f (τ ) τ n dτ

(−1)

0

(5.3)

326

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

F2 (z) = lim ε↓0

∞  k−2 n

n=0

i∞ n

(−1) z

n

f (τ ) τ k−2−n dτ .

(5.4)

i|z|+iε

  Proof. Note that if |τ | < |z|, then  τz  < 1. Then by binomial theorem we obtain

k−2 

∞ τ k − 2 k−2−n n (τ − z)k−2 = z k−2 (−1)k−2 1 − = z τ (−1)k−2−n , n z n=0   where k−2 . If we use the Lebesgue dominated convergence the= (k−2)(k−3)···(k−1−n) n! n orem, then we have

∞  k−2

i|z|−iε 

F1 (z) = lim ε↓0

0

= lim ε↓0

n=0

∞  k−2 n=0

n

n

z k−2−n (−1)k−2−n f (τ ) τ n dτ i|z|−iε 

z

k−2−n

k−2−n

f (τ ) τ n dτ .

(−1)

0

  On the other hand, if |τ | > |z|, then  τz  < 1. Then we obtain the following equality analogously to the case of F1

F2 (z) = lim ε↓0

∞  k−2 n=0

n

i∞ n

(−1) z

n

f (τ ) τ k−2−n dτ . i|z|+iε

This completes the proof. 2 Now we show that F1 and F2 can be written as a series by using the incomplete gamma functions. Lemma 5.2. Functions F1 and F2 can be written as 

k−1−n ∞  k−2 i n (−1) F1 (z) = lim χ(S) z k−2−n ε↓0 n 2π n=0   2π(m+κ)   a(m)Γ |z|−ε , k − 1 − n × (m + κ)k−1−n m+κ>0 and

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327

k−1−n

k−2 i zn (−1)k−1 ε↓0 2π n n=0    a(m)Γ 2π(m + κ)(|z| + ε), k − 1 − n  . × (m + κ)k−1−n m+κ>0

F2 (z) = lim

 ∞

Proof. To prove Lemma 5.2 we compute the integral i∞ Gk−2−n (ε) := f (τ ) τ k−2−n dτ iε

for ε > 0. By the Lebesgue dominated convergence theorem, we obtain ∞ k−2−n

Gk−2−n (ε) =

f (iv) (iv)

∞ 

k−1−n

idv = i

ε

ε

= ik−1−n



∞ a(m)

m+κ>0

=

a(m) e−2π(m+κ)v v k−2−n dv

m+κ>0

e−2π(m+κ)v v k−2−n dv

ε

 

k−1−n  a(m) Γ 2π(m + κ)ε, k − 1 − n i . 2π (m + κ)k−1−n m+κ>0

If we use Lemma 5.1 and the transformation property (2.3) applies to S, then

k−1

F1 (z) = lim χ(S) (−1) ε↓0

∞  k−2 n=0

i∞ z

n

k−2−n

f (τ ) τ k−2−n dτ . i |z|−ε

With the expression of Gk−2−n (ε) as above we find F1 (z) = lim χ(S) (−1)k−1 ε↓0

∞  k−2 n=0

n

z k−2−n Gk−2−n

1 |z| − ε



k−1−n ∞  k−2 i = lim χ(S) (−1)n z k−2−n ε↓0 n 2π n=0   2π(m+κ)  , k − 1 − n  a(m) Γ |z|−ε . × (m + κ)k−1−n m+κ>0 By the same way, we see that F2 (z) = lim ε↓0

∞  k−2 n=0

n

(−1)n z n Gk−2−n (|z| − +ε)



D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

328

k−1−n

k−2 i zn (−1)k−1 ε↓0 2π n n=0    a(m)Γ 2π(m + κ)(|z| − +ε), k − 1 − n  . × (m + κ)k−1−n m+κ>0

= lim

 ∞

Therefore, we get the desired result. 2 If k > 2, then we can remove the limit in the expressions of F1 and F2 as in Lemma 5.1. Lemma 5.3. Suppose that k > 2. Then

F1 (z) =

∞  k−2 n

n=0

z

k−2−n

k−2−n

(−1)

i|z| f (τ ) τ n dτ 0

and

F2 (z) =

∞  k−2 n

n=0

i∞ f (τ ) τ k−2−n dτ . (−1) z n

n

i|z|

Proof. We define the Gauß brackets [x] as the largest integer n such that x ≥ n. Define   a := [k − 1]. Then for n > a we see that k−2 is equal to n   (k − 2)(k − 3) · · · k − 2 − (n − 1) 1 · 2···n



k − 2 − (a − 1) k − 2 − (n − 1) k−2−a k−2 ··· × ··· . = n n − (a − 1) 1 n−a     , . . . ,  k−2−(n−1)  are less than 1 since k − 2 < a ≤ k − 1. Therefore Note that  k−2−a 1 n−a



k−2 n

≤

k−2 k − 2 − (a − 1) ··· . n n − (a − 1)

Furthermore, we see that  iε   ε      ε     εn+1 n n  f (τ ) τ dτ  ≤  f (it) (it) idt ≤ Mε , tn dt = Mε     n+1     0

0

0

where Mε is a positive number such that |f (it)| < Mε for all 0 ≤ t ≤ . Such a number exists since f is a cusp form.

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

329

By the above analysis, for F1 , we obtain     i|z|  k−2  k − 2

k − 2 − (a − 1) k−1 1   z k−2−n (−1)k−2−n ··· |z| f (τ ) τ n dτ  ≤ M|z|    n n n − (a − 1) n+1   0 =

(k − 2) · · · (k − 2 − (a − 1))|z|k−1 M|z| . (n + 1)n · · · (n − (a − 1))

Therefore, we see that

i|z| ∞  k − 2 k−2−n (−1)k−2−n f (τ ) τ n dτ z n n=0 0

converges absolutely. So, we may remove the limit in (5.3). For F2 we use Lemma 5.1 and the property of f that f k,χ S = f so that we have

F2 (z) = lim χ(S) ε↓0

∞  k−2 n

n=0

i |z|+ε



(−1)k−1 z n

f (τ )τ n dτ . 0

The same arguments as used in the case of F1 show that i

∞  k−2

(−1)k−1 z n

n

n=0

|z| f (τ ) τ n dτ 0

converges absolutely. This completes the proof. 2 We are ready to prove Theorem 1.1. Proof of Theorem 1.1. Assume 2 < k ∈ R. We see that k−1

F1 (z) = χ(S) (−1)

∞  k−2 n=0

n

z

k−2−n

i∞ f (τ ) τ k−2−n dτ i |z|

and

F2 (z) =

∞  k−2 n=0

n

n

(−1) z

n

i∞ f (τ ) τ k−2−n dτ i|z|

hold. Therefore, by the same computation as in Lemma 5.2 we see that

(5.5)

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330

Fig. 1. Diagram (a) illustrates the paths used in Lemma 5.4. Diagram (b) illustrates the paths Lq,∞ and * l Lm−1 0,m−1 ∞ used in Lemma 5.5. l

l

F1 (z) = χ(S)

∞  k−2 n=0

×

n

 a(m)Γ



(−1)n

2π |z| (m

i 2π

k−1−n z k−2−n

 + κ), k − 1 − n

(m + κ)k−1−n

m+κ>0

and F2 (z) =

∞  k−2 n=0

n

(−1)k−1

i 2π

k−1−n zn

 a(m)Γ(2π|z|(m + κ), k − 1 − n) . (m + κ)k−1−n m+κ>0

From this we get the desired result. 2 To prove Theorem 1.3 and Theorem 1.4 we recall a few results from [Mue]. Lemma 5.4 ([Mue, Lemma 5.3]). Let θ be a closed 1-form on H such that all simple paths L in H . Then 

 θ= Lq,∞

L

θ exists for

 θ+

Lq,q



θ

Lq ,∞

for all q, q  ∈ Q.  Lemma 5.5 ([Mue, Lemma 5.1]). For a rational number q ∈ [0, 1), put M (q) = l ml as * in Definition 3.13. Then, the two paths Lq,∞ and l Lm−1 0,m−1 ∞ have the same initial l l and end point. The diagrams in Fig. 1, taken from [Mue, Figure 1], illustrate the integration paths used in Lemmas 5.4 and 5.5.

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

331

First we consider the Hecke operator H1,m acting on f ∈ Sk,χ (1). To lift  this operator to an operator on period functions P f we need to understand how f k,ψ A can be translated to period functions for A ∈ Xm . Lemma 5.6. For A ∈ Xm and M (A0) =

L l=1

  ml ∈ Z SL(2, Z) , we have

L     P f 2−k,χ ml 2−k,ψ A(z) l=1 1− k 2

= (det A)



(5.6)

ψ −1 (A)j(A, z)k−2

f (τ ) (τ −

Az)k−2

dτ

LA0,A∞

for all z ∈ H. Proof. By (4.4) we obtain 

−1

f (τ ) (τ − Az)k−2 dτ = χ(ml )

  j(ml , Az)k−2 P f ml A z

Lm−1 0,m−1 ∞ l

l

for l ∈ {1, . . . , L}. With the help of the slash action in (2.2) and the extension introduced in (3.4), we have

    k P f 2−k,χ ml 2−k,ψ A(z) = (det A)1− 2 ψ −1 (A)j(A, z)k−2 P f 2−k,χ ml (Az) k

= (det A)1− 2 ψ −1 (A)j(A, z)k−2 × χ(ml )−1 j(ml , Az)k−2 P f (ml Az) k

= (det A)1− 2 ψ −1 (A)j(A, z)k−2  × f (τ ) (τ − Az)k−2 dτ Lm−1 0,m−1 ∞ l

l

for all l ∈ {1, . . . , L}. By Lemma 5.4 and the structure of the matrices ml in (3.14), we may sum the integrals on the right hand side and obtain L     P f 2−k,χ ml 2−k,ψ A(z) l=1 k

= (det A)1− 2 ψ −1 (A)j(A, z)k−2



L  l=1 L

f (τ ) (τ − Az)k−2 dτ −1 −1 0,m ∞ l l

m

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

332

1− k 2

= (det A)

 ψ −1 (A)j(A, z)k−2

f (τ ) (τ − Az)k−2 dτ

Lm−1 0,∞ L

1− k 2

= (det A)



ψ −1 (A)j(A, z)k−2

f (τ ) (τ − Az)k−2 dτ ,

LA0,A∞

2

where we used LA0,A∞ = Lm−1 0,∞ in the last step. L

The vector-valued case follows similarly. Lemma 5.7. Let w denote a weight matrix as constructed in (2.7) and ω its companion vec given in (2.9). Take a vector-valued cusp form f ∈ Sw (N ) and consider the associL

ated period function P f as defined in (4.1). For A ∈ Xm and M (A0) = l=1 ml ∈   Z SL(2, Z) , we have

L  

 

 P f ω ml (Az) =

l=1



−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ

(5.7)

LA0,A∞

for all z ∈ H. Proof. We basically copy the proof of Lemma 5.6. By (4.5) we obtain 

−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ

Lm−1 0,m−1 ∞ l

l

  = ω(ml , Az)−1 P f ml A z

for l ∈ {1, . . . , L}. With the help of the matrix slash notation in (2.6), we have 



  P f ω ml (Az) = ω(ml , Az)−1 P f (ml Az)=

−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ

Lm−1 0,m−1 ∞ l

l

for all l ∈ {1, . . . , L}. By Lemma 5.4 and the structure of the matrices ml in (3.14), we may sum the integrals on the right hand side and get 

L L      P f ω ml (Az) = l=1

l=1 L



−1 −1 0,m ∞ l l

m

= Lm−1 0,∞ L

−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ

−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340



333

−−−−−−−−−−→ f (τ ) (τ − Az)k−2 dτ ,

= LA0,A∞

where we used LA0,A∞ = Lm−1 0,∞ in the last step. L

2

The next two lemmas now deal with the integrals on the right hand side of (5.6) respectively (5.7). Lemma 5.8. For A ∈ Xm let M (σαj (A)0) = Then the following identity holds

L l=1

  ml ∈ Z SL(2, Z) as in Definition 3.13.

   k P f k,ψ A (z) = det(A)1− 2 ψ −1 (A)j(A, z)k−2

 f (τ ) (τ − Az)k−2 dτ .

(5.8)

LA0,A∞

Proof. By the substitution τ = At, one can see that the right hand side of (5.8) is the same as  1− k k−2 −1 2 det(A) ψ (A)j(A, z) f (At) (At − Az)k−2 dAt. (5.9) L0,∞

If we use the same argument as in the proof of Lemma 2.2 in [KM], then one can check that (5.9) is equal to 1− k 2

det(A)

 =



 ψ −1 (A) j(A, z)k−2

f (At)

t−z det(A) j(A, t) j(A, z)

k−2 det(A)

dt j(A, t)2

L0,∞

 k−2 k det(A) 2 ψ −1 (A) j(A, t)−k f (At) t − z dt.

(5.10)

L0,∞

By exploiting the extended slash notation in (3.4) we see that (5.10) equals     f k,ψ A (t) (t − z)k−2 dt. L0,∞

Then we get the desired result due to (1.2) 2 Lemma 5.9. Let α1 , . . . , αμ be the representatives of the right cosets of Γ0 (N ) in SL(2, Z). For A ∈ Xm we denote new representatives αi of the cosets which are defined by αφ 2 (A,i) := φ1 (A, i) αφ2 (A,i)

(5.11)

for all i (for the definition of φi see Definition 3.3). Then the following identity holds for f ∈ Sk,χ (N )

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

334



    k  P Π f ψ A (z) = det(A)1− 2 ψ −1 (A) j(A, αi z)k−2 i



  Πα¯ f φ

2 (A,i)

(τ )

Lφ3 (A,i)0,φ3 (A,i)∞

 k−2 × αφ 2 (A,i) τ − αφ 2 (A,i) φ3 (A, i)z d(αφ 2 (A,i) τ ). (5.12) Proof. One can check that α· are indeed a new set of representatives respecting the enumeration of the cosets since the map i → φ2 (A, i) is bijective on the index sets for given (fixed) A. By the same argument as in the proof of Lemma 2.2 in [KM], one can see that 

      P Π f k,ψ A (z) = i

      Π f k,ψ A (τ ) (αi τ − αi z)k−2 d(αi τ ) i

L0,∞

 =

   f k,ψ A (αi τ ) (αi τ − αi z)k−2 d(αi τ )

L0,∞ k

= det(A) 2  × ψ −1 (A) j(A, αi τ )−k f (Aαi τ ) (αi τ − αi z)k−2 d(αi τ ) L0,∞ k

= det(A)1− 2 ψ −1 (A) j(A, αi z)k−2   k−2 × f (Aαi τ ) Aαi τ − Aαi z d(Aαi τ ). L0,∞

Then by a direct computation we obtain 

     P Π f k,ψ A (z)

i



k

= det(A)1− 2 ψ −1 (A) j(A, αi z)k−2

f (φ1 (A, i) αφ2 (A,i) φ3 (A, i)τ )

L0,∞

 k−2 × φ1 (A, i) αφ2 (A,i) φ3 (A, i)τ − φ1 (A, i) αφ2 (A,i) φ3 (A, i)z × d(φ1 (A, i) αφ2 (A,i) φ3 (A, i)τ ) k

 f (αφ 2 (A,i) φ3 (A, i)τ )

= det(A)1− 2 ψ −1 (A) j(A, αi z)k−2 L0,∞

 k−2 × αφ 2 (A,i) φ3 (A, i)τ − αφ 2 (A,i) φ3 (A, i)z d(αφ 2 (A,i) φ3 (A, i)τ )

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

1− k 2

= det(A)

335

 f (αφ 2 (A,i) τ )

ψ −1 (A) j(A, αi z)k−2 Lφ3 (A,i)0,φ3 (A,i)∞

 k−2 d(αφ 2 (A,i) τ ) × αφ 2 (A,i) τ − αφ 2 (A,i) φ3 (A, i)z    k = det(A)1− 2 ψ −1 (A) j(A, αi z)k−2 Πα¯ f φ

2 (A,i)

(τ )

Lφ3 (A,i)0,φ3 (A,i)∞

 k−2 d(αφ 2 (A,i) τ ) × αφ 2 (A,i) τ − αφ 2 (A,i) φ3 (A, i)z for all z ∈ H and i ∈ {1, . . . , μ}. This shows the identity (5.12) of the lemma.

2

Lemma 5.10. Let B ∈ Xm be an upper triangle matrix. Denote new representatives αi of the cosets which are defined by αi := φ˜1 (i, B)−1 αi

(5.13)

for all i. Then the following identity holds for f ∈ Sk,χ (N ) (

(

)    ˜  P Π f ψ φ2 (j, B) (z) = Υ−1 (B) j



−−−→   − −−−−−−− k−2 Πα¯ f (τ ) τ − Bz dτ

LB0,B∞

) ,

(5.14)

j

where the vector  −−−−−−−−−−−→ k−2    k−2  dτ = f j αj τ − αj Bz d(αj τ ) f τ − Bz j

is taken with respect to the new representatives α· .   Proof. By Lemma 3.4 we have the tuple φ2 (A, i), φ3 (A, i) = (j, B). This allows us to   write the tuple (A, i) as (A, i) = φ˜2 (j, B), φ˜3 (j, B) . By using this substitution in (5.12) we find the identity 

    P Π f ψ φ˜2 (j, B) (z)

˜3 (j,B) φ

 1− k2  −1  k−2 = det φ˜2 (j, B) ψ φ˜2 (j, B) j φ˜2 (j, B), αφ˜3 (j,B) z     k−2 × Πα¯ f j (τ ) αj τ − αj Bz d(αj τ ). LB0,B∞

− −−−−−−−−−−→

) τ − Bz k−2 dτ in (4.3) and the defining relation of Υ−1 (B) in By the definition of f(τ (3.8) we conclude

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

336

(

(

)    P Π f ψ φ˜2 (j, B) (z) = Υ−1 (B) j



−−−→   − −−−−−−− k−2 Πα¯ f (τ ) τ − Bz dτ

) j

LB0,B∞

with the index permutation map JB defined in (3.6). This proves (5.14).

2

Lemma 5.11. For f ∈ Sk,χ (N ), we have PΠ



 f k,ψ A



 (z) = Υ(A)−1

A∈Xm

A∈Xm



−−−→ −−−−−−−− k−2 Πα¯ f (τ ) τ − Az dτ ,

(5.15)

LA0,A∞

where the notation  −−−−−−−−−−−→ k−2    k−2  dτ = f j αj τ − αj Az d(αj τ ) f τ − Az j

is taken with respect to the new representatives α· defined in (5.13). Proof. Similar to the proof of (3.11) we realize that the map A → φ˜2 (j, A)

Xm → Xm ;

is a bijection for fixed j ∈ {1, . . . , μ}. This allows us to renumerate the summation over all elements of Xm : For fixed i ∈ {1, . . . , μ} ( ( 

)

)      PΠ f k,ψ A (z) = P Π f k,ψ A (z) A∈Xm

A∈Xm

j

=

 A∈Xm

=

 A∈Xm

j

)    ˜  P Π f k,ψ φ2 (j, A) (z)

(

j

( −1

Υ

 (B)

) −−−→   −−−−−−−− k−2 Πα¯ f (τ ) τ − Bz dτ . j

LB0,B∞

We can repeat above calculation for every j which proves (5.15).

2

If we combine Lemmas 5.6 and 5.8, then we derive an explicit formula for the action of the Hecke operators on the period functions for SL(2, Z) induced from the action of these operators H1,m on Sk,χ (1) defined in (3.15) for this group. This is our Theorem 1.3. Proof of Theorem 1.3. By Lemma 5.6 and Lemma 5.8 we obtain L        ˜ 1,m P f (z) = (P f )2−k,χ ml 2−k,ψ A H A∈Xm l=1

D. Choi et al. / Journal of Number Theory 171 (2017) 301–340

=



1− k 2

(det A)

 ψ −1 (A)j(A, z)k−2

A∈Xm

=



337

f (τ ) (τ − Az)k−2 dτ

LA0,A∞

  P f

 A (z). k,ψ

A∈Xm

  ˜ 1,m P f (z) for all z ∈ H follows from the first statement in The well definiteness of H Lemma 3.14 since z ∈ H ensures j(ml , Az) = 0. 2 Remark 5.12. It was shown in [Mue] that the set of matrices ml A with M (A0) = satisfies ' ml A; A ∈ Xm , M (A0) =  =

L l=1

ml

+

L 

ml

l=1

 a b ; m ≥ a > c ≥ 0, m ≥ d > c ≥ 0, ad − bc = m . c d

Similarly we can combine Lemmas 5.7 and 5.11 and derive an explicit formula for the action of the Hecke operators on the period functions for Γ0 (N ) induced from the action of these operators H1,m on Sk,χ (1) defined in (3.15) for this group. This is our Theorem 1.4. Proof of Theorem 1.4. The mth Hecke operator Hn,m acts on u as   Pα¯ Πα¯ Hn,m f (z) =

PΠ

  f k,ψ A (z)

  A∈Xm

=



−1

Υ(A)

A∈Xm

=

 A∈Xm



−−−→ −−−−−−−− k−2 Πα¯ f (τ ) τ − Az dτ

LA0,A∞



−1

Υ(A)

L  



Pα¯ Πα¯ f ωα¯ ml

(Az).

l=1

  ˜ N,m P f (z) for all z ∈ H follows This shows formula (1.5). The well-definedness of H from the first statement in Lemma 3.14 since z ∈ H ensures j(ml , Az) = 0. 2 To prove Theorem 1.5 we review the Eichler–Shimura cohomology theory for real weight, which was established by Knopp [Kno2]. Let k ∈ R with a compatible multiplier system χ on SL(2, Z). Recall that P is the space of holomorphic functions with a certain 1 1 growth condition (1.6). Let Z2−k, χ ¯ (P) be the set of parabolic cocycles in P and B2−k,χ ¯ (P) 1 be the set of coboundaries in P. Let H2−k,χ¯ (P) be the quotient space  1 1 1 H2−k, χ ¯ (P) := Z2−k,χ ¯ (P) B2−k,χ ¯ (P).

(5.16)

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This is called the Eichler–Shimura cohomology group. We define another cohomology group associated with T . Let , 1 1 Z2−k, χ ¯ (P; T ) := ϕ ∈ Z2−k,χ ¯ (P); ϕ(T ) = 0} and 1 1 B2−k, χ ¯ (P; T ) := {ϕ ∈ B2−k,χ ¯ (P); ϕ(T ) = 0}. 1 Then let H2−k, χ ¯ (P; T ) be the quotient space

 1 1 1 H2−k, χ ¯ (P; T ) := Z2−k,χ ¯ (P; T ) B2−k,χ ¯ (P; T ).

(5.17)

By following the argument in [Lan], we obtain the following isomorphism. 1 1 Lemma 5.13. The natural map H2−k, χ ¯ (P; T ) → H2−k,χ ¯ (P) is an isomorphism. 1 1 1 Proof. Let ϕ ∈ Z2−k, χ ¯ (P) ∩ B2−k,χ ¯ (P; T ). By the definition of B2−k,χ ¯ (P) one can see 1 that ϕ ∈ B2−k,χ¯ (P) and the natural map is injective. We see that the map is surjective  1  by making a translation of cocycles. That is, if ϕ ∈ Z2−k, T χ ¯ (P), then ϕ(T ) = f −f 2−k,χ ¯ for some f ∈ P since ϕ is a parabolic cocycle. Let

 ϕ (g) := ϕ(g) − (f − f 2−k,χ¯ g) 1 for g ∈ SL(2, Z). Then ϕ ∈ Z2−k, χ ¯ (P) and 1  1 ϕ + B2−k, χ ¯ (P) = ϕ + B2−k,χ ¯ (P). 1 This implies that [ϕ] = [ϕ ] in H2−k, χ ¯ (P), where [ϕ] is the coset represented by ϕ ∈ 1   Z2−k,χ¯ (P). Moreover, ϕ (T ) = 0 so [ϕ ] is in the image. 2

Lemma 5.14. We have an isomorphism 1 1 ∼ 1 H2−k, χ ¯ (P; T ) = V2−k,χ ¯ (P)/U2−k,χ ¯ (P).

Proof. The map is given by [ϕ] → ϕ(S).  1  If ϕ ∈ B2−k, g for some f ∈ P and ϕ(T ) = 0. This implies χ ¯ (P; T ), then ϕ(g) = f −f 2−k,χ ¯  1 1 that f − f 2−k,χ¯ T = 0. Therefore, ϕ(S) ∈ U2−k,χ¯ (P). Conversely, if f ∈ U2−k, χ ¯ (P), then there is a cocycle ϕ which can be defined by ϕ(T ) = 0 and ϕ(S) = f . This is possible because S and T are generators of SL(2, Z). We conclude 1 ∼ 1 B2−k, χ ¯ (P; T ) = U2−k,χ ¯ (P).

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1 1 On the other hand, if ϕ ∈ Z2−k, χ ¯ (P; T ), then one can check that ϕ(S) ∈ V2−k,χ ¯ (P). 1 Conversely, if f ∈ V2−k,χ¯ (P), then it induces a cocycle ϕ such that ϕ(S) = f and 1 1 1 ϕ(T ) = 0, so ϕ ∈ Z2−k, χ ¯ (P; T ). Therefore, Z2−k,χ ¯ (P; T ) is isomorphic to V2−k,χ ¯ (P). 1 From this, we can deduce that the quotient space H2−k,χ¯ (P; T ) is isomorphic to the 1 1 quotient space V2−k, χ ¯ (P)/U2−k,χ ¯ (P). 2

Proof of Theorem 1.5. Knopp [Kno2] proved that there is an isomorphism 1 Sk,χ (1) ∼ = H2−k, χ ¯ (P).

Then by Lemma 5.13 and 5.14 1 1 ∼ 1 Sk,χ (1) ∼ = H2−k, χ ¯ (P; T ) = V2−k,χ ¯ (P)/U2−k,χ ¯ (P).

By Theorem 1.3 one can see that this map is Hecke-equivariant. 2 Finally we prove Theorem 1.6 Proof of Theorem 1.6. By Theorem 1.3 we have     ˜ 1,m P f (z) = P H1,m f (z). H Now we evaluate both sides at z = 0. Since f is an eigenform with an eigenvalue λm , we obtain   P H1,m f (0) = λm P f (0). Note that we have i∞ Γ(k − 1) P f (0) = f (τ )τ k−2 dτ = (−i)k−1 L(f, k − 1). (2π)k−1 0

˜ 1,m we obtain On the other hand, by the definition of the mth Hecke operator H L     P H1,m f (0) = χ(ml )ψ(A)j(ml , Az)k−2 j(A, z)k−2 P f (ml A0). A∈Xm l=1

If we use Theorem 1.1, then we see that P f (ml A0) = Sf (ml A0). Therefore, we have (−i)k−1

L   Γ(k − 1) λ L(f, k − 1) = χ(ml )ψ(A)j(ml , Az)k−2 j(A, z)k−2 Sf (ml A0). m (2π)k−1 A∈Xm l=1

This completes the proof. 2

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Acknowledgment The authors would like to thank the referee for helpful comments and suggestions which improved the exposition of this paper. References [AL] [CZ]

[Kno] [Kno2] [KM] [KZ] [Lan] [Man] [Miy] [Mue] [MR1] [MR2] [PP] [Shi] [Za]

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