Construction of Vector-Valued Modular Integrals and Vector-Valued Mock Modular Forms

Transcription

1 Noname manusript No. will be inserted by the editor Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms Jose Gimenez Tobias Mühlenbruh Wissam Raji Reeived: date / Aepted: date Abstrat It is known that, given a vetor valued modular form of negative weight, its Fourier oeffiients an be alulated based on the prinipal part of the form. In this paper we start with an arbitrary prinipal part and omplete the Fourier expansion using the alulation. We show that the so obtained funtion is a vetor-valued modular integral of negative weight on the full modular group. Next, we onstrut the supplementary funtion assoiated to a vetor-valued modular usp form of positive weight. The onstrutions are inspired by the onstrution of Eihler integrals by Knopp. We onlude with a omparison of these forms and their integrals to vetor-valued weak harmoni Maass forms. Keywords modular forms Fourier oeffiients modular integrals supplementary funtion Mathematis Subjet Classifiation 2000 F2 F30 Contents Introdution Preliminaries and Notations Jose Gimenez Department of Mathematis Temple University Philadelphia, PA, USA gimenez@temple.edu Tobias Mühlenbruh Department of Mathematis and Computer Siene FernUniversität in Hagen Hagen, Germany tobias.muehlenbruh@fernuni-hagen.de Wissam Raji Department of Mathematis Amerian University of Beirut Beirut, Lebanon wr07@aub.edu.lb

2 2 Jose Gimenez et al. 3 Constrution of Vetor-Valued Modular Integrals Proof of Theorem The Supplementary Funtion Vetor-Valued Harmoni Weak Maass Forms Introdution Consider a vetor valued funtion F F,...,F p : H C p from the upper half plane H to the p-dimensional omplex vetor spae C p. We all suh a funtion F a vetor-valued modular integral of integer weight k on the full modular group Γ SL2,Z if it essentially satisfies the following two onditions: a For given ompatible multiplier system ε and p-dimensional omplex representation ρ : Γ GLp, C the funtion F satisfies Fτ εγ τ + d k ργ aτ + b F P γ τ τ + d a b for all τ H and γ Γ where P d γ τ C k [τ] p is a olumn vetor of polynomials in τ of degree at most k. b Eah omponent funtion F j τ has a onvergent q-expansion meromorphi at infinity: F j τ q m j a ν jq ν ν µ j with 0 m j < a rational number, µ j an integer and q e 2πiτ. The prinipal part of the q-expansion is the part with µ j ν < 0, similar to Laurent expansions of meromorphi funtions. We all F a vetor-valued modular form of integer weight k if P γ 0 holds for all γ Γ. We all F a vetor-valued usp form of integer weight k if F is a vetor valued modular from of weight k with µ j > 0 for all j {,..., p}. The preise definitions of vetor valued modular forms and integrals are given in Setion 2.3. The first and third author show in [] that if we onsider a vetor-valued modular form F F,...,F p of large negative weight on the full modular group, then its Fourier oeffiients a m j for m 0 are given by linear ombinations of the familiar Rademaher- Petersson series that are expressed in terms of Kloosterman sums and the modified Bessel funtion of the first kind. This means that the oeffiients a m j depend only on the prinipal part of F: a m j f m, j,{a ν l; ν < 0,0 l p} for all integers m 0 and j p. The funtion f is known expliitly. See Theorem in Setion 3 for the omplete statement. Our first result, see Theorem 2 in Setion 3 for the detailed version, is in some sense a onverse statement. It says that given an arbitrary prinipal part, i.e., an arbitrary set of oeffiients {b ν j; µ ν < 0,0 j p} of omplex numbers for some integer µ < 0, we an define a vetor valued funtion F F,...,F p on the omplex half-plane using the q-expansion of eah omponent: F j τ q m j µ ν<0 b ν jq ν + b m jq m m 0 with oeffiients b m j : f m, j,{a ν l; ν < 0,0 l p}

3 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 3 given by the same expliit funtion f as above. We prove that suh an F is in fat a modular integral. As an orollary, we also haraterize the ase that the modular integral F is indeed a modular form. Our seond main result starts with a vetor valued usp form g of positive integer weight k +2 large enough, multiplier system ε and representation ρ. It is known that we an write g as a linear ombination of vetor valued Poinaré series g s i b i Pρ,k + 2,ε, µ i,γ,r i ; for suitable s N, r i {,..., p} and nonnegative µ i Z 0 with Pρ,k + 2,ε, µ i,γ,r i ; denoting the Poinaré series of weight k + 2, multiplier ε and representation ρ. See Setion 5. for a preise introdution of Poinaré series. The supplementary series ĝ of g is the funtion given by with µ i : ĝτ s i b i Pρ,k + 2,ε, µ i,γ,r i ;τ { µ i if m j > 0, µ i if m j 0., ε γ : εγ and ρ γ : ργ. Our seond result desribes k + -fold integrals G of g and Ĝ of ĝ, the later is the supplementary funtion of g. We show in Theorem 3 that the funtions G and Ĝ in fat satisfy the transformation property of modular integrals of weight k. The assoiated polynomial terms are losely related. We refer to Theorem 3 in Setion 5 for the preise statement. We also introdue the auxiliary integral G 2 : H C p to the usp form g, given by i G 2 τ : gzz τ k dz τ τ H. This funtion also satisfies the transformation property of modular integrals of weight k and its assoiated polynomials are again losely related to the ones assoiated to G and Ĝ. The preise statement is given in Theorem 4 in Setion 5. As in the ase of lassial modular forms, vetor valued supplementary series and supplementary funtions play a role in the proof of Eihler isomorphism theorem for vetor valued modular forms on the full group in a reent paper of Gimenez [0]. The salar valued ase was disussed in [2]. The main idea is that the polynomials assoiated to the supplementary funtion are in fat period polynomials of the usp form g, see e.g. [2, Theorem ]. We onlude the paper by desribing our seond result in terms of the harmoni weak Maass wave forms. Vetor valued modular forms has been extensively studied in onnetion to metapleti over of SL2, R along with Weil representations [2]. In addition, a lot in their onnetion to harmoni weak Maass forms is known as well [4]. The two main results, Theorems 2 and 3, are based on Gimenez thesis work, see [9, Theorem 2. and 3.2]. The remaining results, Theorems 4 and 5, are new results. The paper is organized as follows: The next setion overs basi notations and definitions. Setion 3 ontains the first main theorem and its motivating question, along with two orollaries. The proof of this theorem is given in Setion 4. We ontinue with the onstrution of the supplementary funtion to a usp form in 5, our seond main result. We

4 4 Jose Gimenez et al. onlude our paper by sorting our result on the supplementary funtion into the modern setting of weak harmoni Maass wave forms in 6. 2 Preliminaries and Notations 2. The group, its ation and the slash-operator Let GLp,C the group of p p dimensional matries with omplex entries and nonvanishing determinant and let Γ SL2,Z denote the group of 2 2 matries with integers entries and determinant alled the full modular group. Γ is generated by These satisfy S 0 0 and T. 0 S 2 ST 3, 2 where Γ is the matrix with on the diagonal and 0 on the off-diagonal entries. Here denotes the identity matrix. The group Γ SL2, Z ats on the upper half-plane H {z C; Imz > 0} by frational linear transformations a b z : az + b d z + d. 3 The ation extends naturally to the boundary R {i} of H. The subgroup Γ of Γ generated by Γ :,T { } m m, ; m Z 0 0 denotes the stabilizer of the boundary point i. We also need the following j-funtion: j : SL2,R H H; j 4 a b,τ : τ + d. 5 d We all a funtion ε : SL2,Z C 0 multiplier or multiplier system ompatible with real weight k if ε satisfies and εγδ jγδ,z k εγεδ jγ,δ z k jδ,z k 6 for every γ,δ SL2,Z and z H. For k Z 6 simplifies to εγ 7 εγδ εγεδ. Let p N be a positive integer. A p-dimensional omplex representation ρ of Γ is a funtion ρ : Γ GLp,C

5 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 5 satisfying ργ δ ργρδ 8 for all γ,δ Γ. We denote by ρ j,m γ the omplex entry in the j th row and m th olumn of the matrix ργ. As notation, we use ργ j,m to denote j,mth entry of the matrix ργ. In [5], Knopp and Mason show that sometimes the representation ρ an be normalized suh that εt ρt e 2πim... 9 e 2πimp holds with 0 m j <, m j Q for j p and the m j s on the diagonal are given in the Fourier expansion 7. They argue that an assumption ρ is normalized is reasonable and does not lead to a signifiant loss of generality [5, above 0]. We also will always make this assumption throughout the rest of the paper. Remark In [5] Knopp and Mason assume 0 < m j. We prefer to hange the interval for onveniene. We assume the argument onvention π argz < π for all z C 0 throughout the paper. We introdue the slash notation. Let k Z a weight and ε be a multiplier. For f : H C and γ SL2,Z, we define f ε k γ τ : εγ jγ,τ k f γ τ 0 for every τ H. The slash notation extends naturally to vetor-valued funtions: tr f,..., f p ε k γ : f ε k γ,..., f p ε k γ tr. 2.2 The Eihler length and bounds on ρ To bound ργ, we introdue the Eihler length of γ [8] with respet to the generators S and T of Γ given in. Namely, we write V as a produt γ ±γ...γ L, where eah γ j is equal to either S or T n j for some integer n j, no two onseutive γ j are both equal to S or a power of T, and where L Lγ N is minimal. Eihler proved that where and n, n 2 are onstants independent of γ. η Lγ n logηγ + n 2, a b a 2 + b d 2, d Let ρ denote a p-dimensional omplex representation. As introdued above denote by ρ j,m γ the omplex entry in the j th row and m th olumn of the matrix ργ. Knopp and Mason [6] showed that ρ j,m γ p Lγ K Lγ for m, j p

6 6 Jose Gimenez et al. where K is a onstant that satisfies ρ j,m S K for all m, j p. Therefore if we use the usual norm ργ ρ j,m γ 2 m, j p we see that ργ K 2 ηγ α with α n log pk 2 and where the onstant K 2 is independent of γ. Note that sine the length of γ is the same as the length of γ, we an use the same bound for ργ ρ γ. The proof of the following lemma uses standard arguments and is omitted. Lemma For given Z 0 and d Z suh that gd,d define the matrix a b Γ satisfying > a 0 if > 0 and d γ,d 0 Γ if This matrix is uniquely defined. Let,d Z be two oprime integers, satisfying N > d 0 and > a 0. By [9, Lemma.2], we see that for suh a γ,d we have ργ,d j,m K 2 2δ with δ 2 logα log pk. 4 Here, K is a onstant satisfying ρs < K for all j,m p. Thus the estimate j,m ργ,d K N 2δ 5 j,m holds with onstants δ and K independent of γ. 2.3 Modular forms and modular integrals Now, we ontinue to define p-dimensional forms. Definition Let k be an integer weight, ε be a ompatible multiplier system and ρ : Γ GLp,C a p-dimensional omplex representation. Let H C p ; τ Fτ : F τ,...,f p τ tr be a p-tuple of meromorphi funtions in the omplex upper half-plane H. We all the pair F,ρ or simply F a vetor-valued meromorphi modular form of integer weight k on the modular group Γ if

7 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 7. We have F ε k γ ργf F τ,...,f p τ tr ε k γτ ργ F τ,...,f p τ tr 6 for all γ Γ and τ H. 2. Eah omponent funtion F j τ has a onvergent q-expansion meromorphi at infinity: F j τ q m j with 0 m j < a rational number, µ j an integer and q e 2πiτ. We denote the minimum of the µ j s by µ: ν µ j a ν jq ν 7 µ : minµ,..., µ p. 8 The spae of suh forms is denoted by F k,ε,ρ. We all a modular form F uspidal or usp form if eah µ j is positive. The spae of usp forms for given weight k, ompatible multiplier ε and representation ρ is denoted by S k,ε,ρ. The modular integral is inspired by the definition of the modular form above where we relax the transformation property 6. Definition 2 Let k be an integer weight, ε be a ompatible multiplier system and ρ : Γ GLp,C a p-dimensional omplex representation. Let H C p ; τ Fτ : F τ,...,f p τ tr be a p-tuple of meromorphi funtions in the omplex upper half-plane H. We all the pair F,ρ or simply F a vetor-valued modular integral of integer weight k on the modular group Γ if. The funtion F satisfies F ργ F ε k γ P γk,ε,ρ; Fτ εγ jγ,τ k ργ Fγτ Pγ k,ε,ρ;τ 9 for all τ H and γ Γ where P γ k,ε,ρ;τ C k [τ] p is a olumn vetor of polynomials in τ of degree at most k. 2. Eah omponent funtion F j τ has a onvergent q-expansion meromorphi at infinity: F j τ q m j with 0 m j < a rational number, µ j an integer and q e 2πiτ. We denote the minimum of the µ j s by µ: Obviously, every modular form is a modular integral. ν µ j a ν jq ν 20 µ : minµ,..., µ p. 2

8 8 Jose Gimenez et al. 3 Constrution of Vetor-Valued Modular Integrals Let us introdue our first question about modular integrals by stating Theorem below. It tells us the following: Start with a vetor-valued modular form F whih has a pole at infinity and onsider its Fourier expansions in 7. Then the Fourier oeffiients a m j for m Z 0 and j {,..., p} of the vetor-valued modular form F, see 7, are determined by the finite number of oeffiients a ν j with µ j ν < 0 and j {,..., p}. Theorem [] Let F be a vetor-valued modular form of weight k with k > 2δ > 0 with δ given in the estimate 5. Let ε denote the multiplier system and ρ the p-dimensional omplex representation assoiated to F, as used in Definition. Pik any j {,..., p} and onsider the Fourier expansion of F as given in 7. For m Z 0 the Fourier oeffiients a m j in 7 are given by the formula with a m j 2π The terms A,ν,m, j,l A,ν,m j,l i k A,ν,m, j,l p l µ ν<0 are given by d 0 d< gd,d a ν la,ν,m, j,l B,ν,m, j,l. 22 2πi dm+aν Ω,d,l, j e, 23 Ω,d,l, j Ω,d j,l εγ,d ργ,d j,l e2πi m l a+m jd. The oeffiient a in 23 is given by the matrix γ,d The terms B,ν,m, j,l are given by ν ml B,ν,m, j,l m + m j k+ 2 a b d in 3 I k+ 4π ν m l 2 m + m j for m + m j > 0 and B,ν,0, j,l k +! 2π ν ml k+ 26 for m m j 0. The I k+ z in 25 is the usual I k+ modified Bessel funtion of the first kind given by 0+ I k+ z 2 z k+ 2πi z2 2n+k+ n0 t k t+ z2 e 4t dt n + k +!, z R, 27 see e.g. [7, ].

9 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 9 In the theorem above, we started with a modular form for negative integer weight. Then, the prinipal part of its Fourier expansion determines the rest of its expansion. Is the onverse also true? This means: If we start with an arbitrary prinipal part and onsider the funtion defined by the prinipal part added to an expansion defined by Formula 22 stated above, is this funtion a modular form? The answer is No ; suh a funtion is just a modular integral. Theorem 2 For ν Z <0 a given negative integer, let b,...,b ν be a set of olumn vetors suh that b ν b ν,...,b ν p tr C p, b ν 0. Moreover, let ρ : Γ GLp,C be a p-dimensional omplex representation, ε be a multiplier system ompatible with integral weight k suh that k satisfies k > 2δ where δ is the same as the one in the estimate 4. Define Fτ µ ν<0 µ ν<0 b ν e 2πim +ντ +. b ν pe 2πim p+ντ + m0 m0 b m e 2πim+m τ, 28 b m pe 2πim+m pτ where we define the oeffiients b m b m,...,b m p tr, m Z 0, as the olumn vetors with omponents given by b m j 2π i k p l µ ν<0 with A,ν,m, j,l given by 23 and B,ν,m, j,l by 25. The funtion F satisfies the following properties:. F is holomorphi on H. 2. F is a modular integral: It satisfies b ν la,ν,m, j,l B,ν,m, j,l. 29 F ργ F ε k γ Q γk,ε,ρ; Fτ εγ jγ,τ k ργ Fγτ Qγ k,ε,ρ;τ, 30 for all γ Γ and τ H. The funtion Q γ k,ε,ρ;τ C k [τ] p is a olumn vetor of polynomials in τ of degree at most k. We prove this theorem in 4. Remark 2 The formula in 29 for alulating the Fourier oeffiients b m j, m Z 0 is the same as 22 in Theorem. In some sense, Theorem an be viewed as a speial ase of Theorem 2 where F in 28 is a priori known to be a vetor-valued modular form. Remark 3 The term A,ν,m, j,l 23 is a generalization of Kloosterman sums in the following sense: Consider the setting of the trivial -dimensional representation: p and ργ for all γ Γ. If we assume that the fator Ω,d,l, j in 24 is trivial, i.e., if we assume Ω,d,l, j, then the term A,ν,m, j,l 23 is just a usual Kloosterman sum. We have A,ν,m, j,l d 0 d< gd,d 2πi dm+aν e Km,ν,.

10 0 Jose Gimenez et al. where Km,ν, is given by Km,ν, : d e 2πi md+aν. mod Here d runs through the prime residue lasses modulo, and a is defined by the ongruene ad mod. Remark 4 Theorem 2 generalizes [7, Theorem 3.3]. Two simple orollaries of Theorem 2 are given below. Corollary Assume the setting of Theorem 2. The following two statements are equivalent:. The term Q S k,ε,ρ;τ in 30 vanishes for all τ H. 2. F defined in 28 is a vetor-valued modular form of weight k, multiplier ε, and representation ρ for Γ. Proof Assume F to be a vetor-valued modular form of weight k, multiplier ε and representation ρ for Γ. Then Theorem 2, in partiular 30, implies that Q S τ,k,ε, vanishes sine F satisfies 6. For the onverse diretion let b,...,b µ be a set of olumn vetors in C p with b µ 0, ρ : Γ GLp,C a p-dimensional omplex representation, ε a multiplier system for weight k with k > 2δ, k, µ N. Let τ Fτ be defined as in 28. Hene F has a Fourier expansion of the form speified in 7. The assumption Q S k,ε,ρ; vanishes everywhere implies that F satisfies the transformation property F ρs F ε k S. The Fourier expansion 28 together with the normalization 9 implies F ρt F ε k T. Hene F satisfies 7 for all generators of Γ. The oyle ondition 6 and indution in the Eihler length of eah γ Γ shows 7 for all elements of Γ. Corollary 2 The v.-v. polynomial Q g k,ε,ρ; in Theorem 2 satisfies the oyle relation Q gh k,ε,ρ;τ for all g,h Γ and τ H. εh ρh jh,τ k Q g k,ε,ρ;hτ + Q h k,ε,ρ;τ Proof This an be easily seen using 30 together with 6 and 8: Q gh k,ε,ρ;τ 30 Fτ εgh ρgh jgh,τ k Fghτ 6,8 εh ρh jh,τk Fhτ εg ρg jg,hτ k Fghτ 3 for all g,h Γ and τ H. + Fτ εh ρh jh,τ k Fhτ 30 εh ρh jh,τ k Q g k,ε,ρ;hτ + Q h k,ε,ρ;τ

11 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 4 Proof of Theorem 2 This setion ontains the proof of Theorem 2 whih is strutured into several steps. These steps are given in the following subsetions: Step : We define and disuss the sets D and D in 4.. They are losely related to the map,d γ,d in Lemma. These sets are needed as index sets for the matries γ,d throughout the proof. Step 2: In 4.2 we introdue the matrix-valued funtion R µ τ related to the Fourier expansion of F. This leads to the proof of F being holomorphi on H. Auxiliary Step I: We prove in 4.3 the Lemmas 4 and 5 whih we used in the step above. Step 3: We reall two variants of the the Lipshitz summation formula. They are used in the following steps. Step 4: In 4.5 we introdue and disuss matrix-valued funtions T ν τ, K µ and W µ τ. They are used to rewrite parts of the Fourier expansion of F suh that we an understand the ation of Γ on F in the steps below. Auxiliary Step II: We prove Lemma 6 and modified versions of [7, Lemma 2.5 and 2.3], the Lemmas 7 and 8, in 4.6. They are used in the step above. Step 5: We define and disuss the matrix-valued funtion S ν,k and its image under S: M S,ν,K in 4.7. Both funtions are ompared in Lemma 9. This forms the basis for the behavior of F under the ation of the group element S Γ. Step 6: We disuss matrix-valued funtions U S,ν, Q S,ν and Y S,ν in see 4.8. This explains basially how the Fourier expansion of F behave under S Γ. Step 7: In 4.9 we summarize the results in the step above. This shows expliitly the transformation behavior of F under S. Step 8: Extend the transformation property of F under S to all elements g Γ in 4.0. This onludes the proof. 4. On sets of oprime pairs of integers Next, we introdue and disuss some sets whih are needed in Setion 3. Let N be a positive integer for the moment. We define the set D : { d Z; 0 d < and gd,d } 32 where gd,d denotes the greatest ommon divisor of the integers and d. The next step is to define the set D for integer Z. We do this by defining three ases > 0, < 0 and 0. Consider first the ase of positive N. We define the set D + as the one ontaining all integers in the -equivalene lasses of all d D : D + : { d Z; d D suh that d d mod }. 33 The set D + is isomorphi to d + Z. d D Next we onsider negative s. We define the set We have D D +. D : { d Z; d D suh that d d mod }. 34

12 2 Jose Gimenez et al. As a final extension step, we define the set for 0: D 0 0 : { 0,,0, }. 35 Summarizing all ases above, we define for integer Z D + if > 0, D : D if < 0 and, D 0 0 if The following lemma follows immediately from the inversion on Γ \SL2,Z: Lemma 2 The map,d d, is an automorphism of the set {,d ; Z,d D }. 4.2 R µ τ and the holomorphiity of F Let R : H C p p ; τ R ν τ R ν j,l τ be the matrix-valued funtion defined by its entries R j,l ν τ m0 2π j,l p i k A,ν,m, j,l B,ν,m, j,l e 2πim+m jτ. 37 Lemmas 4 and 5 show that R ν j,l τ onverges absolutely in m and in. Therefore we an hange the order of summation in 28 and rewrite the j th omponent of Fτ in the form F j τ µ ν<0 + 2π b ν je 2πim j+ντ m0 i k p l µ ν<0 b ν je 2πim j+ντ µ ν<0 + p µ ν<0 l b ν lr ν j,l τ. b ν la,ν,m, j,l B,ν,m, j,l e 2πim+m jτ 38 Sine the series onverges uniformly on ompat sets of H with respet to the hyperboli metri by Lemma 5, Fτ is holomorphi on H. 4.3 Auxiliary lemmas I Lemma 3 For k Z 0 and z > 0 we have the estimate I k+ z z k sinhz. 39

13 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 3 Proof Using the power series expansion for e z we get z k sinhz ez e z z k z 2 m m! zm m! m0 2z k m0 m! 2 n0 2n +! n0 m odd z m z2 2n+k+ n +!n + k +! Comparing the last expression term wise to 27, we see that 39 follows if I k+ z n0 z 2n+k+ z2 2n+k+ n + k +!, 2 2n+k+2 n + k +! n + 22n + 2 2n+k+2 n + k +! n + 22n +. holds. Sine the left hand term is monotone in k, it suffie to prove the ase k 0: 2 2n+2 2n n+2 n +! 2n + 2 n + 22n + 2n+2 n+ whih follows diretly from the upper bound on the entral binomial oeffiient 2n < 22n 2 2n, n πn see e.g. the upper bound in [23, Corollary 2.3]. Lemma 4 For k > 2δ, the series onverges absolutely. Defining we have additionally as m. 2π i k A,ν,m, j,l B,ν,m, j,l 40 κ m minm,...,m p and κ M maxm,...,m p, 4 2π O i k A,ν,m, j,l B,ν,m, j,l 42 m + κ m k e 4πν κ m 2 m+κ M 2

14 4 Jose Gimenez et al. Proof The strategy is the same as the arguments leading to [5, Theorem 3.2]. First we will show that 2π i k 2 A,ν,m, j,l B,ν,m, j,l Cm + m j 2 e 2πm+m j 2 ν m l Then we show that as m the summation on is dominated by the term for. In order to bound A,ν,m, j,l in 23, we use 4 to get A,ν,m, j,l d 0 d< gd,d d 0 d< gd,d d 0 d< gd,d 7 d 0 d< gd,d Ω,d,l, j e 2πi dm+aν εγ,d ργ,d m l a+m jd j,l e2πi e εγ,d ργ,d ργ,d j,l m l a+m jd j,l e2πi e 2πi dm+aν 2πi dm+aν 4 O 2δ+. 44 On the other hand, from the power series definition 27 of I k+ z we have 39. Also, we have sinhz z sinhb, for 0 z B, 45 B sine sinh : R 0 R 0, restrited to the positive real line, is a onvex funtion. Now by 44, 39, 45 above, the definition of B,ν,m, j,l in 25 and the assumption k > 2δ, we find the following two ases: m + m j > 0 and m + m j 0. For m + m j > 0 and for suitable positive onstants C, C 2 we have 2π i k 2 A,ν,m, j,l B,ν,m, j,l 2π A,ν,m, j,l B,ν,m, j,l 44 2π 2 2 C 2δ+ B,ν,m, j,l 25,39 k+ ν 2π C 2δ ml 2 4π 2 m + m 2 j k

15 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 5 4π sinh ν m l 2 m + m j 2 45 k+ ν 2 4πk+ C 2δ k ml m + m 2 j sinh 2π ν m l 2 m + m j 2 4π k+ C ν ml k+ 2 m + m j 2 sinh 2π ν m l 2 m + m j 2 2δ k 2 C 2 m + m j 2 e 2π ν m l 2 m+m j 2. We also used the trivial estimate sinhz e z in the last line. We also remark that the term ν m l is positive sine ν Z <0 and 0 m l < by the assumptions around 7. For m + m j 0 and for suitable onstants C, C 3 we have 2π i k 2 A,ν,0, j,l B,ν,0, j,l 2π 44 2π 2 2 A,ν,0, j,l B,ν,0, j,l C 2δ+ B,ν,0, j,l 26 2π C 2δ 2 k +! 2π C 2π ν ml k+ k +! 2 2π C 3 2π ν ml k+. k +! Note that the onstants C, C 2 and C 3 depend on k. 2π ν ml k+ 2δ k Now, we disuss the remaining term of the series in 43. The defining equations 23 and 24 with imply A,ν,m, j,l 23 Ω,0,l, j e 2πiaν ε 0,,,0 ρ 0,,,0 24 j,l e2πim la e 2πiaν sine > d 0 and d 0 and 0 a < implies a d 0. Hene γ,0 S holds. However, we will keep writing a in our formulae instead of simplifying the terms by omitting vanishing terms. We have 2π i k A,ν,m, j,l B,ν,m, j,l 2π i k εs ρs j,l e2πim la e 2πiaν 46

16 6 Jose Gimenez et al. k+ ν ml 2 m+m j k+! I k+ 4π ν m l 2 m + m j 2 2π ν ml k+ We onsider the ases m + m j > 0 and m + m j 0 separately. if m + m j > 0 and if m + m j 0. Assume m + m j > 0. Using I k+ z ez 2πz, 47 see [24] or alternatively [7, ], we have 2π i k A,ν,m, j,l B,ν,m, j,l 46 2π i k εs ρs j,l e2πimla e 2πiaν 47 O k+ ν ml 2 m + m j ρs k j,l ν ml k m + m j e I k+ 4π ν m l 2 m + m j 2 4 m k O + m π ν m j e l 2 m+m j 2, 4π ν m l 2 m+m j 2 where the impliit onstant may depend on k, ν and m l. Assume m + m j 0. In this ase we get 2π i k A,ν,m, j,l B,ν,m, j,l 46 2π i k εs ρs j,l e2πimla e 2πiaν 2π ν ml k+ k +! ρs k+ ν ml O O Summarizing, we see that the series j,l ν ml k+. 2π in 43 onverges absolutely. We have 2π i k A,ν,m, j,l B,ν,m, j,l i k A,ν,m, j,l B,ν,m, j,l O m + κ m k e 4π ν κ m 2 m+κ M 2. This onludes the proof of Lemma 4. A diret onsequene of the lemma above is the following

17 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 7 Lemma 5 Let w > 0 be a given positive onstant. The series 2π i k A,ν,m, j,l B,ν,m, j,l e 2πim+m jτ m0 onverges absolutely for τ H. The onvergene is uniform for τ I w {τ H; Imτ > w}. Proof We only have to prove that the series in 48 onverges for m N and therefore omitting the term m 0. For m > 0 we have i k A,ν,m, j,l B,ν,m, j,l e 2πim+m jτ 2π 42 O m + κ m k e 4π ν κ m 2 m+κ M 2 e 2πim+m jτ Imτ > w O m + κ m k e 4π ν κ m 2 m+κ M 2 e 2πm+m jimτ m j 0 O m 34 k2 e 4πm+ 2 e 2πImτm We have used that ν, m j, κ m and κ M are onstants regarding the m-dependeny. The last estimate implies that the series in 48 onverges absolutely, sine the term e 2πmImτ dominates. The trivial estimate e 2πmImτ e 2πmw for τ I w implies that the onvergene is uniform on I w Variants of the Lipshitz summation formula The sum-star notation is defined as the symmetri limit : lim N N q N Reall the following variants of the Lipshitz summation formula.. 49 Variant : Let τ H and n Z 0 if m j Z respetively n N if m j Z be a non-negative respetively positive integer. The first variant of the Lipshitz summation formula is e 2πi n+ 2πim jq m + m j n e 2πim+m jτ τ + q n+ if n N and m+m j >0 e 2πim 50 jq if n 0. τ + q where the summation index m runs either through m N if m j Z or through m Z 0 if m j R Z, see e.g. [3]. Variant 2: We have 2πi m e 2πimτ πi + τ + q for n 0 and m j 0, see [9, 2.29] and [7, 2.9]. 5

18 { d; γ,d given in 3 } 55 8 Jose Gimenez et al. 4.5 T ν τ, K µ and W µ τ As 38 indiates we may rewrite Fτ as Fτ µ ν<0 where T ν τ is the matrix-valued funtion T : H C p p ; τ T ν τ T ν j,l τ given by b ν T ν τ., 52 b ν p j,l p T ν j,l τ δ j,l e 2πim j+ντ + R ν j,l τ, 53 where δ j,l denotes the Kroneker delta. For given oprime,d Z satisfying 0 d < we denote by γ,d the unique matrix given in 3: a b γ,d Γ satisfying 0 d < and 0 a <. d Reall D in 32. We have obviously D { d Z; 0 d < and gd,d } 54 { } a b d; Γ ; 0 d <,0 a < d whih is a partiular set of representatives of d mod with gd,d and 0 d <. From now on, we assume that τ lies on the upper imaginary ray: τ ir >0 H. Later on we extend obtained results by analyti ontinuation to all τ H. Using 37, 23 and the absolute onvergene of the double series R ν j,l in m and we find R j,l ν τ m0 m0 2π 2π 2π i k A,ν,m, j,l B,ν,m, j,l e 2πim+m jτ i k εγ,d ργ,d j,l d 0 d< gd,d e 2πi m l a+m j d d 0 d< gd,d 2πi dm+aν e B,ν,m, j,l e 2πim+m jτ εγ,d ργ,d j,l 56 e 2πia m l +ν i k B,ν,m, j,l e 2πim+m jτ+ d. m0

19 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 9 56 and Lemma 6 imply that where R ν j,l τ K j,l ν + d D εγ,d ργ,d e 2πim j q τ + d q k K ν j,l nk+ πi k A,ν,0, j,lb,ν,0, j,l m j,l e2πia l +ν 0 otherwise Lemma 4 shows that the series in 58 onverges absolutely. Now, for q Z define d d q and γ,d : γ,d T q 2πi ν ml n, 57 τ + d q if m j 0 and 58 a d. 59 As q runs through all integers and as d runs through the set D, d assumes exatly one eah value in the set D defined in 36. This leady to the set identity D 33 The definition of γ,d in 59 implies { d Z; d D suh that d d mod } { } 59 d a ; γ,d d Γ, 0 a <. εγ,d ργ,d 59 εγ,d T q ργ,d T q 6,8 εγ,d εt q ρt q ργ,d 9 εγ,d e 2πim q... ργ,d. e 2πimpq 60 Therefore we find the identity εγ,d ργ,d j,l εγ,d ργ,d j,l e 2πim jq 6 for the j,l th entry. Now, define the matrix-valued funtion W ν τ W ν j,l τ W j,l ν and write W j,l ν τ using 57: j,l p by its entries τ : δ j,l e 2πim j+ντ + R ν j,l τ K ν j,l T j,l τ K j,l ν ν, W ν j,l τ δ j,l e 2πim j+ντ 63

20 20 Jose Gimenez et al ,6 d D εγ,d ργ,d e 2πim j q τ + d q k lim N d D d N nk+ m j,l e2πia l +ν nk+ 2πi ν ml n τ + d q εγ,d rhoγ,d m j,l e2πia l +ν τ + d k 2πi ν ml τ + d n. To ontinue, reall that we assumed τ iy with y > 0. We will show that the series N lim N d D d N d D d N εγ,d ργ,d m j,l e2πia l +ν k+ iy + d onverges. Note that 64 is in fat the term with n k + in 63. To do so we write 64 as εγ,d ργ,d m j,l lim e2πia l +ν k+ iy + d d D εγ,d ργ,d m j,l e2πia l +ν e 2πim jq d D εγ,d ργ,d k+2 iy + d q m j,l e2πia l +ν 64 e 2πim jq k+2 iy + d q, 65 where we used 59, 6 and the relation between the sets D and D, whih is mentioned in 33 and 36. Next, we apply the Lipshitz summation formula 50 respetively 5 to the ases m j > 0 and m j 0. Assume m j > 0. Applying 50 for n 0 and τ iy + d to 65 we get d D εγ,d ργ,d 2πi d D m j,l e2πia l +ν k+2 εγ,d ργ,d k+2 iy + d q m j,l e2πia l +ν e 2πim jq e 2πim+m jiy+ d. m0 In the ase of m j 0 we use the variant 5 of Lipshitz summation formula. Starting again from 65, we get 65 d D εγ,d ργ,d m j,l e2πia l +ν e 2πim jq k+2 iy + d q

21 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms m j,l e2πia l +ν εγ,d ργ,d d D πi 2πi e 2πimiy+ d d D m k+2 πi εγ,d k+2 ργ,d m j,l e2πia l +ν 2πi εγ,d d D k+2 ργ,d m j,l e2πia l +ν Next, we will show that the following series lim N nk+2 τ + d k εγ,d ργ,d d D d N + e 2πimiy+ d. m0 m j,l e2πia l +ν 66 2πi ν ml n τ + d is an absolutely onvergent triple sum. Note that 66 is in fat the terms with n > k + in 63. To do so, we rewrite 66 as nk+2 2πi ν ml n d D εγ,d ργ,d m j,l e2πia l +ν 2n k e 2πim jq τ + d q n k. Lemma 7 implies that the above representation onverges absolutely. Hene 66 is indeed an absolutely onvergent triple series. Combining the absolute onvergene of the expressions in 64 and 66 we may rewrite the expression of W j,l ν τ δ j,l e 2πim j+ντ in 63 as : W ν j,l τ δ jl e 2πim j+ντ 2πi ν ml k+ k +! + lim N d D d N lim N d D d N nk+2 εγ,d ργ,d m j,l e2πia l +ν k+ iy + d + εγ,d ργ,d m j,l e2πia l +ν τ + d k n 2πi ν m l τ + d. 67

22 22 Jose Gimenez et al. Now, using Lemma 8 with t, we rewrite the expression in 67 as W j,l ν τ δ j,l e 2πim j+ντ K Z d D 0< K d K 7 lim + lim 2πi ν ml k+ k +! K Z d D 0< K d K nk+2 We ollet the summands and get W j,l ν τ δ jl e 2πim j+ντ K Z d D 0< K d K lim lim nk+ K Z d D 0< K d K e εγ,d ργ,d m j,l e2πia l +ν k+ τ + d εγ,d rhoγ,d j,l e2πia m l +ν τ + d k 2πi ν ml τ + d n. εγ,d ργ,d j,l e2πia m l +ν τ + d k 2πi ν ml n τ + d εγ,d ργ,d m j,l e2πia l +ν 2πi ν m l τ+d k n0 2πi ν ml n τ + d, where we used the power series expansion of the exponential funtion e x k x k n0 nk+ x n. τ + d k 4.6 Auxiliary lemmas II The following lemma gives a value for the summation of the speial funtions defined in Theorem. It was used in the previous part. Lemma 6 Depending on whether 0 m j < vanishes or not the term an be expressed as follows: ik m0 2π B,ν,m, j,l e 2πim+m jτ+ d

23 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 23 If 0 < m j < holds we have ik m0 2π If we assume m j 0 we have 2π B,ν,m, j,l e 2πim+m jτ+ d e 2πim jq τ + d q k ik m0 B,ν,m, j,l e 2πim+m jτ+ d πik B,ν,0, j,l + + e 2πiqm j τ + d q k The used sum-star notation is defined in 49. nk+ nk+ 2πi ν ml n. τ + d q 2πi ν ml n. τ + d q Proof The proof is a simple appliation of the Lipshitz summation formula given in 50 and in 5. Furthermore, we need the definition of B,ν,m, j,l in 25 and the power series expansion of I k+ z in 27. We onsider the ases m j > 0 and m j 0 separately.. We assume m j > 0 whih implies that m + m j > 0 for all m Z 0. Hene we have ik m0 2π 25,27 2π ik i k n0 50 i k n0 n0 B,ν,m, j,l e 2πim+m jτ+ d 2π m0 n0 2π 2n+k+ ν m l n+k+ n + k +! m + m j n e 2πim+m jτ+ d 2n+k+2 ν m l n+k+ n + k +! 2πi n+ 2πin+ 2π 2πi n n k m0 m + m j n e 2πim+m jτ+ d 2n+k+2 ν m l n+k+ n + k +! 2πi n+ τ + d + q n+ nk+ e 2πim jq n+k+ ν m l n+k+ e n + k +! 2πim jq τ + d + q n+ 2πi n ν m l n e 2πim jq τ + d + q n k q q τ + d q k e 2πim jq nk+ 2πi ν ml τ + d q n.

24 24 Jose Gimenez et al. 2. Assume m j 0. However, we still write the symbol m j to be able to ompare the formulae below to the ones derived in the ase above. Doing this allows us to indiate the influene of the onstant m j to the two ases. By the same arguments as above, we have ik m 2π 25,27 i k n0 B,ν,m, j,l e 2πim+m jτ+ d 2π 2n+k+2 ν m l n+k+ n + k +! 2πi n+ 2πin+ m + m j n e 2πim+m jτ+ d m 2π k+2 i k ν m l k+ k +! 2πi + i k n 2πi 2π m m + m j 0 e 2πim+m jτ+ d 2n+k+2 ν m l n+k+ n + k +! 2πi n+ 2πin+ m m + m j n e 2πim+m jτ+ d 50,5 2πik+ ν m l k+ e k+2 πi + k +! 2πim jq τ + d + q 2πi n+k+ ν m l + n+k+ e n + k +! 2πim jq τ + d + q n+ n 26 πik+ + n 2πi e 2πim j q B,ν,0, j,l + 2πi n+k+ ν m l n+k+ n + k +! k+ ν m l k+ e k +! 2πim jq τ + d + q e 2πim jq e 2πim jq τ + d + q n+. Using τ+d+q n+ we get furthermore τ+d+q n+ ik m 2π B,ν,m, j,l e 2πim+m jτ+ d πik+ B,ν,0, j,l 2πi n+k+ ν m l + n+k+ e n0 n + k +! 2πim jq τ + d + q n+ n + k + n πik+ + nk+ 2πi B,ν,0, j,l n ν m l n e 2πim jq τ + d + q n k

25 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 25 q q πik+ + Therefore we have B,ν,0, j,l τ + d q k e 2πim jq 2π ik B,ν,m, j,l e 2πim+m jτ+ d m0 πik B,ν,0, j,l+ + e 2πiqm j τ + d q k nk+ nk+ 2πi ν ml n. τ + d q 2πi ν ml n. τ + d q This onludes the proof. We now prove a modified version of Lemma 2.5 in [7] in order to show that a ertain sum in proof of Theorem 2 is onvergent. Lemma 7 Let k be a weight satisfying k > 2δ > 0 with δ as in 4. The series εγ,d ργ,d m j,l e2πia l +ν k 2 70 d D onverges. Here, the inner sum runs through the elements of the set D given in 54, and the matrix element γ,d is given in 3 and determines a. Proof The definition of the set D in 32 implies D. Hene we see that d D k 2 k onverges for all k > 0. Combine this with k > 2δ and 4 proves the lemma. To ontinue we need an adapted version of Lemma 2.3 in [7] and its proof in [8], with some modifiations to be appliable in the vetor-valued ase. Lemma 8 Let τ iy, with y > 0, k > 2δ, ν a negative integer and t a positive integer. Then lim N lim d D d N K Z d D 0< tk d K εγ,d ργ,d am j,l e2πi l +ν k+ τ + d εγ,d ργ,d am j,l e2πi l +ν k+ τ + d. 7 Here, the inner sums run through the entries of the set D given in 60.

26 26 Jose Gimenez et al. Proof Following Rademaher s proof [20], we will show lim lim K N Z 0< tk Reall that 6 implies d D K< d N εγ,d ργ,d am j,l e2πi l +ν k+ τ + d εγ,d q ργ,d q j,l εγ,d ργ,d j,l e2πiqm j with γ,d D and d d q suh that γ,d D as defined in 59 and 60. First, we will show the result for m l > 0. Rewrite d as d d q, see 59, and rewrite the inner sum of 6 as k+ d D εγ,d ργ,d l +ν [ ] j,l e2πi S + S 2, 73 K< d N with and As in [8, p. 343] we have S : S 2 : q< K+d K+d <q e 2πiqm j τ + d q e 2πiqm j τ + d q. with s given by t + t s S + S 2 < 2 sinπm j s K s, 74 { s : min, k 2δ } > 0 2 and t a positive integer appearing in 7 and given a priori as stated in the assumption of Lemma 8. Following [8, page 343] we obtain the estimate lim N d D K< d N εγ,d ργ,d am j,l e2πi l +ν k+ τ + d < C K s d D 2+s, where C is a suitable positive onstant taking are of the fator 2t+t s onstant K 2 in 4. The rest of the argument is the same as in [8]. sinπm j and the impliit

27 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 27 Next, onsider the ase m l 0. Similar to the previous ase, we rewrite the inner limit of 72 as lim N d D K< d N k+ εγ,d ργ,d am j,l e2πi l +ν k+ τ + d εγ,d ργ,d am l +ν j,l e2πi d D 75 [ S + S 2 + S 3 ], with and S : lim N S 2 : lim N S 3 : K+d K d N+d q N+d q< N d <q K d τ + d q, τ + d q τ + d q. Now apply the same argument as in [8] using the estimate 4 and the fat that k > 2δ. 4.7 S ν,k τ as approximation of W ν τ and M S,ν,K Put S j,l ν,k τ : δ jle 2πim j+ντ + 76 εγ,d ργ,d j,l e2πia l +ν τ + d 2πi ν m k l e τ+d. d D 0< K d K + Z In other words, we have Using γ,d W j,l ν τ 77 lim S j,l K ν,k τ εγ,d ργ,d j,l e 2πia m l +ν Z 0< K τ + d k d D d K k n0 a b d SL2,Z, see 59, we have 2πi ν ml n τ + d. a + τ + d aτ ad + ad b τ + d aτ + b τ + d γ,d τ and therefore e 2πia m l +ν 2πi ν m l e τ+d e 2πiν+m lγ,d τ.

28 28 Jose Gimenez et al. This implies that we an write S j,l ν,k τ as S j,l ν,k τ δ j,l e 2πim l+ντ + d D 0< K d K + Z εγ,d ργ,d j,l τ + d k e 2πiν+m l γ,d τ. 78 The next step inludes the first term in the summation. The trivial identity ερ p p, where p p denotes the p p unit matrix, implies δ j,l e 2πim l+ντ ε ργ0, j,l e2πim l+ντ. This allows us to inlude the,d 0, into the expression of S j,l ν,k τ: S j,l ν,k τ Z 0 K εγ,d ργ,d j,l τ + d k e 2πiν+m l γ,d τ. 79 d D d K,d 0, We an write eah γ,d τ as γ,d τ γ,d τ + γ 2, d τ with γ, d : γ,d. This implies e 2πim l+νγ,d τ e 2πim l+ν γ, τ, and also implies together with 6 εγ,d ργ,d τ + d k ε γ,d ρ γ,d τ d k. This shows that, if we inlude the transformations γ,d γ, d in the summation in 79, every term of S j,l ν,k τ ours twie: S j,l ν,k τ 2 Z d D K d K εγ,d ργ,d j,l τ + d k e 2πiν+m l γ,d τ. 80 We use this representation of the omponent funtion τ S j,l ν,k τ to define the the matrixvalued funtion S ν,k : H C p p ; τ S j,l ν,k τ. j,l p This allows us to let Γ at on the argument of S ν,k. In partiular, we define M S,ν,K : ρs Sν,K ε k S M S,ν,K τ : εs ρs τ k S ν,k Sτ The omponents of M S,ν,K are given by M j,l S,ν,K τ τ H. 8

29 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 29 d D K d K 2 Z 6 2 Z p s d D K d K εs εγ,d ρs ργ,d j,s s,l τk Sτ + d k e 2πim l +νγ,d Sτ We used the oyle relation 6 for the last identity. εγ,d S ργ,d S j,l d τ k e 2πim l+νγ,d Sτ. 82 Lemma 9 The limit funtions of M j,l j,l S,ν,K τ and S ν,k τ for K oinides: for all τ H. lim M j,l K S,ν,K τ lim In other words and using 8, Lemma 9 shows Proof of Lemma 9 S j,l K ν,k lim S j,l K ν,k lim ρs Sν,K ε K k. τ 83. Let CP : {,d Z 2 ; gd,d } be the set of all oprime pairs of integers. Then CP {,d ; K, d K } lim K Z d D K d K {,d } holds and,d d, is an automorphism on CP, see Lemma The oyle ondition 6 and the multipliity of ρ in 8 implies εt n γ,d ρt n γ,d jt n γ,d,τ k εt n εγ,d ρt n ργ,d jt n,γ,d,τ jγ,d,τ k εt ρt n εγ,d ργ,d jγ,d,τ k using jt n,γ,d,τ and εt n εt n. Sine we have the identity jt n γ,d,τ τ + d jγ,d,τ ε T n γ,d ρt n γ,d ε γ,d ργ,d εt ρt n 3. We have by 9 εt ρt n diag e 2πim,...,e 2πim p n diag e 2πinm,...,e 2πinm p.

30 30 Jose Gimenez et al. 4. If γ,d for suitable n Z. a b d we an write γ,d S Taking the arguments together we get lim M j,l K S,ν,K τ 82 2,d CP We use the substitution,d d,. b a d T n γ d, εγ,d S ργ,d S j,l d τ k e 2πim l+νγ,d Sτ 2 εt n γ,d ρt n γ,d j,l τ + d k e 2πim l+νt n γ,d τ,d CP 2 εγ,d ργ,d j,l e 2πinm l τ + d k e 2πim l+νt n γ,d τ,d CP Reall T γ,d τ γ,d τ +. 2 εγ,d ργ,d j,l e 2πinm l,d CP τ + d k e 2πim l+νγ,d τ e 2πinm l+ν 2 εγ,d ργ,d j,l τ + d k e 2πim l+νγ,d τ e 2πinν,d CP Reall e 2πinν for all n Z sine ν Z by assumption of Theorem lim K S j,l S,ν,K τ. Hene, we have proven lim M j,l K S,ν,K τ lim S j,l K ν,k τ. 4.8 The matrix-valued funtions U S,ν, Q S,ν and Y S,ν Now, define the matrix-valued funtion U S,ν : H C p p by U S,ν τ : εs ρs τ k W ν Sτ 84

31 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 3 for all τ H where W ν τ is given in 62. Using the alternate expression in 77 for W ν τ and the definition of M j,l S,ν,K τ in 8, we find U j,l S,ν τ lim K { M j,l S,ν,K τ Z 0< K Sτ + d k e 2πi am l +ν d D d K k n0 εγ,d S ργ,d S j,l τk using Lemma 9 and the respresentation of S ν,k in 78 δ j,l e 2πiml+ντ + { + lim K Z 0< K Z 0< K d D d K d D d K e 2πi am l +ν 2πi ν ml n } Sτ + d εγ,d ργ,d j,l τ + d k e 2πim l +νγ,d τ εγ,d S ργ,d S j,l τk Sτ + d k k n0 2πi ν ml n } Sτ + d. On the other hand, the alternate expression of W ν in 77 together with the representation of S ν,k in 78 gives analogously W ν j,l τ δ j,l e 2πim l+ντ { + lim K Z 0< K Z 0< K d D d K k n0 d D d K εγ,d ργ,d j,l τ + d k e 2πim l +νγ,d τ εγ,d ργ,d j,l 2πi ν ml n } τ + d. Comparing these two expressions above gives W j,l ν lim K Z 0< K d D d K τ + d k e 2πi am l +ν τ U j,l S,ν τ 85 { εγ,d S ργ,d S τ k Sτ + d k j,l e 2πi am l +ν k n0 2πi ν ml n Sτ + d

32 32 Jose Gimenez et al. lim Z 0< K d D d K e 2πi am l +ν εγ,d ργ,d j,l τ + d k k n0 e 2πi am l +ν K Z d D 0< K d K k n0 k n0 2πi ν ml n } τ + d { εγ,d S ργ,d S j,l 86 2πi ν ml n d τ k n τ n εγ,d ργ,d j,l 2πi ν ml n τ + d k n} where we used the trivial identity Sτ + d k n τ k d τ k n τ n. We easily see that the term inside the limit is a polynomial of degree at most k. This implies that W ν j,l τ U j,l S,ν τ C k[τ] is a polynomial of degree at most k, sine the limit of a sequene of polynomials of degree at most k onverging at k + points is a polynomial of degree at most k. Next, we hek the onstant term K ν in 58 and its image under S. Define Y S,ν : K ν ρs Kν ε k S Y S,ν τ : K ν τ εs τ k ρs Kν τ for all τ H. Clearly Y j,l j,l S,ν τ is a polynomial in τ of degree at most k sine K ν is a onstant, see 58. Adding the two polynomials in 85 and 87 shows that Q S,ν : W ν + K ν ρs ε W ν + K ν k S 88 W ν U S,ν + Y S,ν is a matrix-valued funtion whose entries are polynomials of degree at most k. Using the defining identities 87, 84 and 62 we find Q S,ν τ 88 W j,l τ U j,l τ + Y S,ντ ν S,ν 87 W ν τ + K ν ρs Wν + K ν ε k Sτ 62 T ν τ ρs Tν ε k S τ is also a matrix-valued funtion whose entries are polynomials of degree at most k. Eah entry is given by Q j,l S,ν j,l τ W τ U j,l j,l τ + Y τ ν S,ν T ν j,l τ εs τ k S,ν p s ρs j,s T ν s,l Sτ. 87

33 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms The transformation of F under S The defining property 52 of T ν and 92 implies that Fτ ρs F ε k S τ 89 b ν Q S,ν τ. µ ν<0 b ν p T ν τ ρs Tν ε k S µ ν<0 b ν + Y S,ν τ. µ ν<0 b ν p is a olumn vetor of polynomials of degree at most k. b ν. b ν p 4.0 The transformation of F under g Γ We just showed that F ρs F ε S is a olumn vetor of polynomials in the previous k step. Next, we extend this result to all elements of Γ to prove Theorem 2. We need a bit of notation for this. For g Γ define U g,ν : ρg Wν ε g, 90 k Y g,ν : K ν ρg Kν ε g and 9 k Q g,ν : W ν U g,ν + Y g,ν 92 T ν ρg Tν ε k g. Obviously, Y g,ν is a polynomial of degree at most k sine K ν is onstant. We want to show that Q g,ν is a matrix of polynomials of degree at most k. We prove this by indution on Lg, the Eihler length of g viewed as word in the generators S and T. To do so, we disuss the ases g S, g T and the indution steps g gs and g gt : Assume g S. We know it is true for g S by 88. Assume g T. The normalization 9 of the representation ρt implies that the Fouriertype series of T ν 53 is ompatible with the ation of T Γ : T ν ρt Tν ε T. 93 k Now, onsider the indution step g gs. We assume that Q g,ν τ is a matrix of polynomials of degree at most k. Sine Q S,ν τ is also a matries of polynomials of degree k, we have that T ν τ εgs ρgs jgs,τ k T ν gsτ

34 34 Jose Gimenez et al. 6 T ν τ εs ρs js,τk εg ρg jg,sτ k T ν gsτ T ν τ εs ρs js,τ k T ν Sτ Q g,ν Sτ Q S,ν τ + εs ρs js,τ k Q g,ν Sτ. The first term is a matrix of polynomials. The seond term also ontains polynomials sine the fator τ k ombines with Sτ k n, 0 n k. Next, onsider the indution step g gt. Similarly, using the fat that Q T,ν τ 0, we have T ν τ εgt ρgt jgt,τ k T ν gt τ 94 T ν τ εt ρt εg ρg jg,t τ k T ν gt τ T ν τ εt ρt T ν T τ Q g,ν T τ εt ρt Qg,ν T τ whih is a matrix of polynomials of degree at most k. Therefore we have shown that if Q M,ν τ is matrix of polynomials in τ of degree at most k, so are Q MS,ν τ and Q MT,ν τ. Thus 52 and 92 imply that Fτ εg ρg jg,τ k Fgτ is a vetor of polynomials of degree at most k. This onludes the proof of Theorem 2. 5 The Supplementary Funtion Let k +2 be a weight with k > max { 2δ,2α } with δ as in estimate 5 and α given in 2. Moreover, let ε be a multiplier and ρ be omplex p-dimensional normal representation ρ. The normal representation will be defined in Definition 5 in 5.. Let g S k + 2,ε,ρ be a usp form of weight k + 2, multiplier ε and representation ρ. It is known that g an be written as a linear ombination of vetor-valued Poinaré series Pρ,k,ε, µ,γ,r; for suitable parameters µ Z and r {,..., p}, see 5. for more details on Poinaré series. Suppose that we have gτ s i b i Pρ,k + 2,ε, µ i,γ,r i ;τ τ H 95 for suitable s N, r i {,..., p} and nonnegative µ i Z 0. Remark 5 Sine g is a usp form, the Fourier expansion of linear ombination of Poinaré series in 95 has no onstant term. We use this fat later to neglet the possible onstant term in the Fourier expansion of the Poinaré series sine we are only interested in the nonexistent linear ombinations of them. More details are given below.

35 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 35 Before we ontinue, let us introdue new notations. Reall that by 9 we have rational phase fators 0 m,...,m p <. We define the primed fators m j [0,, j,..., p and µ i Z, i,...,s, by { m m j if m i 0, j : m j if m i 0 and { 96 µ i µ i if m j > 0, : µ i if m j 0. We obviously have Furthermore, we define m i + µ i m i + µ i. 97 ε γ : εγ and ρ γ : ργ 98 for all γ Γ. Reall that ε is a multiplier system for Γ for weight k + 2 and for k sine k is an integer and ε is already a multiplier system for weight k + 2 and Γ. Also, ρ being a representation for Γ implies that ρ is a representation of Γ. With these notations we define the supplementary series ĝ : H C p to g by for all τ H. ĝτ s i b i Pρ,k + 2,ε, µ i,γ,r i ;τ 99 Remark 6. Note that ĝ is not a usp form. This omes from the fat that the fators µ i + m j are non-positive, see 96 and use that the µ i + m j s in 95 are non-negative. 2. To our knowledge, the term supplementary series was introdued by Knopp in in [4]. 3. The map indued by g ĝ orresponds to taking the flipping operator of g in [,.]. We now have the neessary notation to state the theorem. Our result generalizes results obtained by Husseini and Knoop in [7] and [2]. Theorem 3 Let g S k + 2,ε,ρ be a usp form of weight k + 2, and k > max { 2δ,2α } with δ as in 5 and α in 2, multiplier ε and omplex p-dimensional normal representation ρ given by the linear ombination 95 of Poinaré series and the supplementary series ĝ given by 99.. The k + -fold integral G of the usp form g given by the k + times termwise integrated Fourier expansion of g is a holomorphi vetor-valued funtion on H and satisfies the transformation property G ργ G ε k γ : p γ Gτ εγ jγ,τ k ργ Gγτ : pγ τ with p γ a p-dimensional olumn vetor of polynomials of degree at most k. Moreover, G an be written as the Eihler integral Gτ k+ k! i τ 00 gzz τ k dz τ H. 0

36 36 Jose Gimenez et al. 2. The k + -fold integral Ĝ of ĝ given by the k + times termwise integrated Fourier expansion of ĝ is a holomorphi vetor-valued funtion on H and satisfies the transformation property Ĝ ρ γ Ĝ ε k γ : p γ Ĝτ ε γ jγ,τ k ρ γ Ĝγτ : p γ τ 02 with p γ a p-dimensional olumn vetor of polynomials of degree at most k. 3. We have p γ τ p γ τ 03 for all γ Γ and τ H. Definition 3 The funtion Ĝ defined as the k + -fold integral of ĝ and given by the k + times termwise integrated Fourier expansion of ĝ, see the seond part of Theorem 3 above, is alled the supplementary funtion to g. Following Husseini-Knopp [2] we have the following Corollary 3 Let k > max { 2δ,2α }, gτ S k + 2,ρ,ε and Ĝτ the supplementary funtion to g. The following statements are equivalent:. g 0 and 2. Ĝ F k,ε,ρ. We prove it exatly in the same way as [2, Theorem 3]. Proof of Corollary 3 If gτ 0, then its k+-fold integral Gτ is also identially zero. Therefore p γ τ 0 for all γ Γ. Thus, by 38, we have p γ τ 0, implying ε γ ρ γ jγ,τ k Ĝγτ Ĝτ. Sine ĝτ is a vetor-valued modular form of weight k +2, representation ρ and multiplier ε for Γ we have that ĝτ is holomorphi on H and meromorphi at i. Therefore Ĝτ F Γ, k,ρ,ε. On the other hand, if Ĝτ F Γ, k,ρ,ε, then p γ τ 0 for all γ Γ. Using 38 we find p γ τ 0 for all γ Γ. This implies Gτ F Γ, k,ρ,ε. Sine gτ is a usp form, we know that g is holomorphi on H and also in the usp at i. Hene its k + -fold integral Gτ is holomorphi on H and at the usp. Then [6, Lemma 4.] implies Gτ 0 and hene gτ 0. Remark 7 Let g S k + 2,ε,ρ satisfy the assumptions of Theorem 3 and let G and Ĝ be the funtions given there. One might think in view of 03 to onsider a vetor-valued differene funtion H C p, τ Ĝτ Gτ. Negleting the issue of how to define Gτ for the moment, one an formally dedue. ε Ĝ G γ ρ γ k Ĝτ Gτ 2. τ 0, and Ĝ G for all γ Γ,

37 Constrution of Vetor-Valued Modular Integrals and Vetor-Valued Mok Modular Forms 37 The first statement follows from the formal alulation ρ γ ε Ĝ G τ k ρ γ Ĝ ε k τ ργ G ε k τ 00,02 Ĝτ p γ τ Gτ p γ τ 03 Ĝτ Gτ for all γ Γ and τ H. The seond part an be dedued from the Fourier expansions of G and Ĝ in 33 and 34, respetively 29 and 30 and the fat that e 2πiaτ e 2πiaτ holds for every real a R. However, we annot define Gτ even if we just take the Fourier expansion of G in 33 and replae τ with τ whih is then just a formal series and does not onverge. One of the most important results of Theorem 3 is, in our opinion, that the k + -fold integrals G and Ĝ have essentially the same transformation property stated in 00 and 02. In partiular, the orretion polynomial p γ respetively p γ satisfy the identity 03. There is also another way to onstrut an funtion G 2 replaing the k + -fold integral G whih has a similar transformation property. Definition 4 Let g S k + 2, ε, ρ be a usp form of weight k + 2 > 2 and p-dimensional omplex representation ρ. The auxiliary integral G 2 : H C p of the usp form g is given by i G 2 τ : gzz τ k dz τ H. 04 τ Remark 8 Sometimes, G 2 as defined above is alled a Niebur integral, see e.g. [6, p.5]. Some properties of the auxiliary integral are given in the following Theorem 4 Let g S k + 2,ε,ρ be a usp form of weight k + 2, and k > 2δ a positive integer with δ given in 5, multiplier ε and omplex p-dimensional normal representation ρ given by the linear ombination 95 of Poinaré series and the supplementary series ĝ given by 99.. The auxiliary integral G 2 of the usp form g satisfies the transformation property G 2 ρ γ G2 ε k γ : p 2,γ G 2 τ ε γ jγ,τ k ρ γ G2 γτ : p 2,γ τ 05 suh that p 2,γ is a p-dimensional olumn vetor of polynomials of degree at most k. 2. The vetor-valued polynomial p 2,γ in 05 satisfies k+ Γ k + p 2,γτ p γ τ C[τ] 06 for all τ H where p γ is given in 00 of Theorem 3.