Differential operators on Jacobi forms and special values of certain Dirichlet series

Transcription

1 Differential operators on Jacobi forms and special values of certain Dirichlet series Abhash Kumar Jha and Brundaban Sahu Abstract We construct Jacobi cusp forms by computing the adjoint of a certain linear map constructed using bilinear holomorphic differential operators with respect to the Petersson scalar product The Fourier coefficients of the Jacobi cusp forms constructed involve special values of the shifted convolution of Dirichlet series of Rankin-Selberg type 1 Introduction The derivative of a modular form is not in general a modular form, but one can construct modular forms by using certain combinations of derivatives of modular forms Rankin [13, 14] gave a general description of differential operators which map modular forms to modular forms For every non-negative integer ν, Cohen [7] explicitly constructed certain bilinear operators from M k M l to M k+l+2ν, where M i denotes the space of holomorphic modular forms of weight i for the group SL 2 (Z) Zagier [16] studied algebraic properties of these bilinear operators and called them Rankin-Cohen brackets For example, the first bracket [, ] 1 satisfies the Jacobi identity [[f, g] 1, h] 1 + [[g, h] 1, f] 1 + [[h, f] 1, g] 1 = 0, (f M k, g M l, h M m ) giving M the structure of a graded Lie algebra Kohnen [12] constructed certain elliptic cusp forms whose Fourier coefficients involve special values of certain Dirichlet series of Rankin-Selberg type by computing the adjoint of the product map (ie the map f fg, for a fixed modular form g) with respect to the Petersson scalar product Recently, Herrero [10] generalized the work of Kohnen by computing the adjoint of certain linear maps constructed using Rankin-Cohen brackets with respect to the Petersson scalar product Herrero constructed cusp forms whose Fourier coefficients involve special values of Dirichlet series similar to those which appeared in the work of Kohnen, with additional factors arising due to the binomial coefficients appearing in the Rankin-Cohen brackets The work of Kohnen [12] has been generalized by Choie, Kim and Knopp [5] and Sakata [15] to the case of Jacobi forms Choie [2, 3] studied the Rankin- Cohen brackets for Jacobi forms by using the heat operator acting on Jacobi forms Recently in [11], we generalized the work of Herrero to the case of Jacobi forms We 2000 Mathematics Subject Classification Primary: 11F50; Secondary: 11F25, 11F66 Key words and phrases Jacobi Forms, Differential operators, Adjoint map 1

2 2 ABHASH KUMAR JHA AND BRUNDABAN SAHU explicitly computed the adjoint of certain linear maps constructed using Rankin- Cohen brackets with respect to the Petersson scalar product Böecherer [1] studied the general bilinear holomorphic differential operators on the Jacobi group by using the Maass operator and proved that the space of bilinear holomorphic differential operators raising the weight by ν is in general of dimension equal to ν Choie and Eholzer [4] explicitly constructed a family of dimension ν of Rankin- Cohen type operators defined on the space of Jacobi forms raising the weight by ν N In this article, we consider certain linear maps defined on the space of Jacobi forms which are constructed using Rankin-Cohen type operators (studied in [4]) We explicitly compute their adjoints with respect to the Petersson scalar product The Fourier coefficients of the image of a Jacobi cusp forms under the adjoint maps involve special values of shifted convolutions of Dirichlet series of Rankin-Selberg type 2 Preliminaries on Jacobi forms of scalar index Let C and H be the complex plane and the complex upper half-plane, respectively The Jacobi group Γ J := SL 2 (Z) (Z Z) acts on H C in the usual way by ( (a ) ) ( b aτ + b c d, (λ, µ) (τ, z) = cτ + d, z + λτ + µ ) cτ + d ( (a ) ) Let k and m be fixed positive integers If γ = b c d, (λ, µ) Γ J and φ is a complex-valued function defined on H C, then define φ γ := (cτ + d) k e c(z+λτ+µ)2 2πim( cτ+d +λ 2 τ+2λz) φ(γ (τ, z)) Let J be the space of Jacobi forms of weight k and index m on Γ J, ie the space of holomorphic functions φ : H C C satisfying φ γ = φ, for all γ Γ J and having a Fourier expansion of the form φ(τ, z) = c(n, r)q n ζ r (q = e 2πiτ, ζ = e 2πiz ) 4nm r 2 0 Furthermore, we say φ is a cusp form if and only if c(n, r) 0 = n > r 2 /4m We denote the space of all Jacobi cusp forms by J cusp We define the Petersson scalar product on J cusp as φ, ψ = φ(τ, z)ψ(τ, z)v k e 4πmy2 v dv J, Γ J \H C where τ = u + iv, z = x + iy and dv J = dudvdxdy v 3 is an invariant measure under the action of Γ J on H C The space (J cusp,, ) is a finite dimensional Hilbert space For more details on the theory of Jacobi forms, we refer the reader to [8] The following lemma describes the growth of the Fourier coefficients of a Jacobi form

3 Differential operators on Jacobi forms and special values of certain Dirichlet series 3 Lemma 21 If k > 3 and φ J has Fourier coefficients c(n, r), then and, moreover if φ is a cusp form, then For a proof, we refer to [6] c(n, r) r 2 4nm k 3 2, c(n, r) r 2 4nm k Poincaré series Let m, n and r be fixed integers such that r 2 < 4mn Let {( (1 ) ) } Γ J := t 0 1, (0, µ) : t, µ Z be the stabilizer of q n ζ r in Γ J Let (21) P ;(n,r) (τ, z) := γ Γ J \ΓJ e 2πi(nτ+rz) γ be the (n, r)-th Poincaŕe series of weight k and index m It is well-known that P ;(n,r) J cusp for k > 2 [9] This series has the following property: Then Lemma 22 Let φ J cusp has the Fourier expansion φ(τ, z) = c(n, r)q n ζ r 4nm r 2 >0 (22) φ, P ;(n,r) = mk 2 Γ(k 3 2 )(4mn r2 ) 3 2 k 2π k 3 2 where Γ( ) denotes the usual gamma function c(n, r), One can get explicit Fourier expansion of P ;(n,r), for details and a proof of the Lemma 22, we refer to [9] 22 Differential operators on Jacobi forms For an integer m, we define the heat operator L m := 1 (2πi) 2 (8πim τ 2 z), where τ and z are the derivative with respect to τ and z, respectively Note that L m (e (2πi(nτ+rz)) ) = (4mn r 2 )e (2πi(nτ+rz)) Let k 1, k 2, m 1 and m 2 be positive integers, and let φ and ψ be two complex-valued holomorphic functions defined on H C Then, for any complex number X and non-negative integer ν, define the 2ν-th and (2ν+1)-th Rankin-Cohen type brackets of φ and ψ as [φ, ψ] k1,k2,m1,m2 := α,β,γ N {0} C α,β,γ (k 1, k 2 )(1 + m 1 X) β (2 m 2 X) α L γ m 1+m 2 ( L α m1 (φ)l β m 2 (ψ) ), [φ, ψ] k1,k2,m1,m2 +1 := m 1 [φ, z ψ] k1,k2,m1,m2 m 2 [ z φ, ψ] k1,k2,m1,m2,

4 4 ABHASH KUMAR JHA AND BRUNDABAN SAHU where the coefficients C α,β,γ (k 1, k 2 ) are given by C α,β,γ (k 1, k 2 ) = (k 1 + ν 3/2) β+γ α! and (x) m = (k 2 + ν 3/2) α+γ β! 0 i m 1 (x i), ( (k 1 + k 2 + ν 3/2)) α+β, γ! with x! = Γ(x + 1) Using the action of the heat operator, one can verify that (23) [φ k1,m 1 γ, ψ k2,m 2 γ] k1,k2,m1,m2 This implies the following result, which was proved in [4] = [φ, ψ] k1,k2,m1,m2 k1+k 2+ν,m 1+m 2 γ, γ Γ J Theorem 23 Let φ J k1, m 1 and ψ J k2, m 2, where k 1, k 2, m 1 and m 2 are positive integers Then, for any X C and any non-negative integer ν, the function [φ, ψ] k1,k2,m1,m2 is a Jacobi form of weight k 1 + k 2 + ν and index m 1 + m 2 Moreover, [φ, ψ] k1,k2,m1,m2 is a Jacobi cusp form for ν > 1 In fact, [, ] k1,k2,m1,m2 is a bilinear map from J k1, m 1 J k2, m 2 to J k1+k 2+ν, m 1+m 2 ( ) ν/2 d Remark 21 For ν 2N, the operator [φ, ψ] k1,k2,m1,m2 is, up to dx a scalar multiple of the Rankin-Cohen bracket, studied in [2] If we take ν = 1 in Theorem 23, we obtain Theorem 95 [8] These are some applications of the Rankin-Cohen brackets 3 Statement of the Theorems Let ψ J cusp k 2, m 2 and ν be a non-negative integer For a complex number X, we define the map T ψ, : J cusp k 1, m 1 J cusp k 1+k 2+ν, m 1+m 2 by T ψ, (φ) = [φ, ψ] k1,k2,m1,m2 This is a C-linear map of finite dimensional Hilbert spaces and therefore there exists an adjoint map Tψ, : J cusp k 1+k 2+ν, m 1+m 2 J cusp k 1, m 1, such that φ, T ψ, (ω) = Tψ, X, ν(φ), ω, for all φ J cusp k 1+k 2+ν, m 1+m 2 and ω J cusp k 1, m 1 As an application one can obtain certain arithmetic information of Fourier coefficients of Jacobi forms using Tψ, (refer to [11]) and analogous results in the case modular forms (refer to [12, 10]) In the following theorems, we exhibit the Fourier coefficients of Tψ, cusp (φ) for φ Jk 1+k 2+ν, m 1+m 2 for ν even and odd These Fourier coefficients involve special values of certain shifted convolution of Dirichlet series of Rankin-Selberg type associated to φ and ψ Theorem 31 Let k 1, k 2, m 1, m 2 be positive integers, such that k 1 > 4, k 2 > 3 and ν be a non-negative integer Let ψ J cusp k 2, m 2 has Fourier expansion ψ(τ, z) = a(n 1, r 1 )q n1 ζ r1 n 1,r 1 Z,

5 Differential operators on Jacobi forms and special values of certain Dirichlet series 5 Then the image of any cusp form φ J cusp k 1+k 2+2ν, m 1+m 2 with Fourier expansion φ(τ, z) = b(n 2, r 2 )q n2 ζ r2 under T ψ,x,2ν where is given by T ψ,x,2ν(φ)(τ, z) = n 2,r 2 Z, 4(m 1+m 2)n 2 r 2 2 >0 4m 1n r 2 >0 c ν,x (n, r)q n ζ r, c ν,x (n, r) = (4m 1n r 2 ) k1 3/2 (m 1 + m 2 ) k1+k2+2ν 2 Γ(k 1 + k 2 + 2ν 3 2 ) π k2+2ν m k1 2 1 Γ(k ) C α,β,γ (k 1, k 2 )(1 + m 1 X) β (2 m 2 X) α (4m 1 n r 2 ) α n 1,r 1 Z 4(m 1+m 2)(n+n 1) (r+r 1) 2 >0 (4m 2 n 1 r 2 1) β a(n 1, r 1 ) b(n + n 1, r + r 1 ) (4(m 1 + m 2 )(n + n 1 ) (r + r 1 ) 2 ) k1+k2+2ν (γ+ 3 2 ) Theorem 32 Let k 1, k 2, m 1 and m 2 be positive integers such that k 1 > 4, k 2 > 3 and ν be a non-negative integer Let ψ J cusp k 2, m 2 has Fourier expansion ψ(τ, z) = a(n 1, r 1 )q n1 ζ r1 n 1,r 1 Z, Then the image of any cusp form φ J cusp k 1+k 2+2ν+1, m 1+m 2 with Fourier expansion φ(τ, z) = b(n 2, r 2 )q n2 ζ r2 under Tψ,X,2ν+1 is given by where T ψ,x,2ν+1(φ)(τ, z) = n 2,r 2 Z, 4(m 1+m 2)n 2 r 2 2 >0 4m 1n r 2 >0 c ν,x (n, r)q n ζ r, c ν,x (n, r) = 2i(4m 1n r 2 ) k1 3/2 (m 1 + m 2 ) k1+k2+2ν 1 Γ(k 1 + k 2 + 2ν 1 2 ) π k2+2ν m k1 2 1 Γ(k ) C α,β,γ (k 1, k 2 )(1 + m 1 X) β (2 m 2 X) α (4m 1 n r 2 ) α n 1,r 1 Z 4(m 1+m 2)(n+n 1) (r+r 1) 2 >0 (4m 2 n 1 r1) 2 β (m 1 r 1 m 2 r)a(n 1, r 1 ) b(n + n 1, r + r 1 ) (4(m 1 + m 2 )(n + n 1 ) (r + r 1 ) 2 ) k1+k2+2ν (γ+ 1 2 )

6 6 ABHASH KUMAR JHA AND BRUNDABAN SAHU 4 Proofs We follow the same exposition as in the proof of Theorem 31 in [11] The following lemma is used to prove the above theorems Lemma 41 Using the same notation as in Theorem 31, we have φ(τ, z) [ e 2πi(nτ+rz) k1,m 1 γ, ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m1+m2)(I(z)) 2 X,2ν γ Γ J \ΓJ Γ J \H C converges Proof The proof is analogous to that of Lemma 41 in [11] We now give a proof of Theorem 31 Write Tψ,X,2ν(φ)(τ, z) = c ν,x (n, r)q n ζ r 4m 1n r 2 >0 Consider the (n, r)-th Poincaŕe series of weight k 1 and index m 1, as given in (21) Using Lemma 22, we have T ψ, φ, P k1,m 1;(n,r) = mk1 2 1 Γ(k )(4m 1n r 2 ) 3 2 k1 2π k1 3 2 On the other hand, by the definition of the adjoint map, we have c ν,x (n, r) T ψ, φ, P k1,m 1;(n,r) = φ, T ψ, (P k1,m 1;(n,r)) = φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 X,2ν Hence, we obtain (41) c ν,x (n, r) = 2πk1 3 2 (4m 1 n r 2 ) k1 3 2 m k1 2 1 Γ(k ) φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 X,2ν By definition, φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 = Γ J \H C = Γ J \H C φ(τ, z) [ P k1,m 1;(n,r)(τ, z), ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m 1 +m 2 )(I(z)) 2 dv J dv J [ φ(τ, z) e 2πi(nτ+rz) k1,m 1 γ, ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m1+m2)(I(z)) 2 dv J γ Γ J \ΓJ By Lemma 41, we can interchange the order of summation and integration in the above equation Hence, φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 equals φ(τ, z) [ e 2πi(nτ+rz) k1,m 1 γ, ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m1+m2)(I(z)) 2 dv J γ Γ J \ΓJ Γ J \H C Using equation (23), the fact that φ(τ, z)ψ γ(τ, z) k e 4πmI(z) 2 = φ(γ(τ, z))ψ γ(τ, z) I(γτ) k e 4πmI(z)2 I(γτ) and the change of variable (τ, z) γ 1 (τ, z) and equation (23), the above equals φ(τ, z) [ e 2πi(nτ+rz), ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m1+m2)(I(z)) 2 dv J γ Γ J \ΓJ γ Γ J \H C

7 Differential operators on Jacobi forms and special values of certain Dirichlet series 7 Using Rankin s unfolding argument, φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 X,2ν equals [ φ(τ, z) e 2πi(nτ+rz), ψ(τ, z) ] k 1,k 2,m 1,m 2 k1+k2+2ν e 4π(m1+m2)(I(z)) Γ J \H C = C α,β,γ (k 1, k 2 ) (1 + m 1 X) β (2 m 2 X) α α,β,γ N {0} ) φ(τ, z)l γ m 1+m 2 (L α m 1 (e 2πi(nτ+rz) )L β m 2 (ψ(τ, z)) k1+k2+2ν e 4π(m 1 +m 2 )(I(z))2 dv J Γ J \H C Inserting the Fourier expansions of φ and ψ and using the iterated action of the heat operators L m1, L m2 : L α m 1 (e (2πi(nτ+rz)) ) = (4m 1 n r 2 ) α e (2πi(nτ+rz)), L β m 2 ψ(τ, z) = (4m 2 n 1 r1) 2 β a(n 1, r 1 )e (2πi(n1τ+r1z)), n 1,r 1 Z, and L m1+m 2 using (22), φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 equals C α,β,γ (k 1, k 2 ) (1 + m 1 X) β (2 m 2 X) α (4m 1 n r 2 ) α α,β,γ N {0} n 2,r 2 Z, 4(m 1+m 2)n 2 r 2 2 >0 b(n 2, r 2 ) n 1,r 1 Z 2 dv J (4m 2 n 1 r 2 1) β (4(m 1 + m 2 )(n + n 1 ) (r + r 1 ) 2 ) γ a(n 1, r 1 ) Γ J \H C e 2πi(n2τ+r2z) e 2πi((n+n1)τ+(r+r1)z) k1+k2+2ν e 4π(m 1 +m 2 )(I(z)) 2 dv J A fundamental domain for the action of Γ J on H C is given by the set {(u, v, x, y) : u [0, 1], v [0, ], x [0, 1], y R} Integrating in this region, φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 equals (m 1 + m 2 ) k1+k2+2ν 2 Γ(k 1 + k 2 + 2ν 3 2 ) 2π k1+k2+2ν 3 2 (4m 1 n r 2 ) α n 1,r 1 Z 4(m 1+m 2)(n+n 1) (r+r 1) 2 >0 α,β,γ N {0} C α,β,γ (k 1, k 2 )(1 + m 1 X) β (2 m 2 X) α (4m 2 n 1 r 2 1) β a(n 1, r 1 ) b(n + n 1, r + r 1 ) (4(m 1 + m 2 )(n + n 1 ) (r + r 1 ) 2 ) k1+k2+2ν (γ+ 3 2 ) In the above, we use the orthogonality relations for the exponential function to compute the integrals in x and u, the Gaussian integral to compute the integral in y and the integral representation of the Gamma function to compute the integral in v Inserting the expression for φ, [P k1,m 1;(n,r), ψ] k1,k2,m1,m2 in (41), we obtain the required expression for c ν,x (n, r) given in Theorem 31 The proof of Theorem 32 is analogous to that of Theorem 31

8 8 ABHASH KUMAR JHA AND BRUNDABAN SAHU Acknowledgements We thank the organizers for giving us the opportunity to participate in Building Bridges: 3rd EU/US Summer School and Workshop on Automorphic Forms and Related Topics during July 11-22, 2016 at the University of Sarajevo, Bosnia and Herzegovina The first author would like to thank the National Board of Higher Mathematics (NBHM), India for providing travel support to attend the workshop The first author is financially supported by the Council of Scientific and Industrial Research (CSIR), India The second author is partially funded by the SERB grant SR/FTP/MS-053/2012 Finally, the authors thank the referee for careful reading of the paper and many helpful suggestions References [1] S Böecherer, Bilinear holomorphic differential operators for the Jacobi group, Comment Math Univ St Paul, 47 (1998), no 2, [2] Y Choie, Jacobi forms and the heat operator, Math Z, 1 (1997), [3] Y Choie, Jacobi forms and the heat operator II, Illinois J Math, 42 (1998), [4] Y Choie and W Eholzer, Rankin-Cohen operators for Jacobi and Sigel forms, J Number Theory, 68 (1998), no2, [5] Y Choie, H Kim and M Knopp, Construction of Jacobi forms, Math Z, 219 (1995) [6] Y Choie and W Kohnen, Rankin s method and Jacobi forms, Abh Math Sem Univ Hamburg, 67 (1997) [7] H Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math Ann, 217 (1977), [8] M Eichler and D Zagier, The Theory of Jacobi Forms, Progr in Math vol 55, Birkhäuser, Boston, (1985) [9] B Gross, W Kohnen, and D Zagier, Heegner points and derivatives of L-series II, Math Ann 278, (1987), [10] S D Herrero, The adjoint of some linear maps constructed with the Rankin-Cohen brackets, Ramanujan J, 36, (2015), no 3, [11] A K Jha and B Sahu, Rankin-Cohen brackets on Jacobi forms and the adjoint of some linear maps, Ramanujan J, 39 (2016), no 3, [12] W Kohnen, Cusp forms and special value of certain Dirichlet Series, Math Z, 207 (1991), [13] R A Rankin, The construction of automorphic forms from the derivatives of a given form, J Indian Math Soc, 20 (1956), [14] R A Rankin, The construction of automorphic forms from the derivatives of given forms, Michigan Math J, 4 (1957), [15] H Sakata, Construction of Jacobi cusp forms, Proc Japan Acad Ser A, Math Sc, 74 (1998), [16] D Zagier, Modular forms and differential operators, Proc Indian Acad Sci Math Sci, 104 (1994), Department of Mathematics, Indian Institute of Science, Bangalore , India address: abhashkumarjha@gmailcom, abhashjha@iiscacin School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, PO: Jatni, Khurdha, Odisha , India address: brundabansahu@niseracin