EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

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1 EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS KATHRIN BRINGMANN AND OLAV K. RICHTER Abstract. In previous work, we introduced harmonic Maass-Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass-Jacobi-Poincaré series, as well as Zwegers real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass-Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass-Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.. Introduction and statement of results In 985, Eichler and Zagier [5] systematically developed a theory of Jacobi forms. That theory has since grown enormously, establishing deep connections to many other areas of mathematics and physics, such as the theory of Heegner points see Gross, Kohnen, and Zagier [6], the theory of elliptic genera see Zagier [0], string theory for example, see Cardy [4], and more recently, mock theta functions see Zwegers []. In [], we initiated a theory of harmonic Maass-Jacobi forms, which includes the holomorphic Jacobi forms of [5], the real-analytic Jacobi forms in [], and certain Maass-Jacobi-Poincaré series as explicit examples. Of particular interest is Ĵcusp, a distinguished subspace of harmonic Maass-Jacobi forms, which we will briefly review in Section. If φ Ĵcusp, then φ = φ+ + φ, where φ + τ, z = c + n, rq n ζ r n,r Z n,r Z D is the holomorphic part of φ, while the non-holomorphic part of φ is given by φ τ, z = 3 c πdy n, rγ k, q n ζ r. m Here and throughout the paper, τ = x + iy H the usual complex upper half plane, z = u + iv C, q := e πiτ, ζ := e πiz, D := r 4nm, and Γs, x := x e t t s dt is the incomplete Gamma-function. In particular, if φ Ĵcusp is holomorphic, then φ = φ+ is a weak Jacobi form. If in addition, the Fourier series in is only over D 0, then φ is a Jacobi form as in [5] and if the Fourier series in is only over D < 0, then φ is a Jacobi 000 Mathematics Subject Classification. Primary F50; Secondary F30, F37. The first author was partially supported by NSF grant DMS and by the Alfried Krupp prize. The paper was written while the second author was in residence at RWTH Aachen University and at the Max Planck Institute for Mathematics in Bonn. He is grateful for the hospitality of each institution and he thanks Aloys Krieg in particular for providing a stimulating research environment at RWTH Aachen University.

2 KATHRIN BRINGMANN AND OLAV K. RICHTER cusp form. Note that our notion of weak Jacobi form is slightly more general then the one in [5], where weak Jacobi forms have an expansion as in with the additional condition that n 0. If φ + is as in, then we call 3 P φ +τ, z := c + n, rq n ζ r the principal part of φ +. n,r Z In this paper, we extend ideas of [] to obtain exact formulas for the coefficients of φ + in. It turns out that the principal part P φ + in 3 dictates the coefficients of φ + for D < 0, as explained in the following theorem, which is the main result of this paper. Theorem. Let φ Ĵcusp D := r n < 0, then c + n, r = with holomorphic part φ+ as in. If k < 0 is even and Γ 5 k where Γ is the Gamma-function and r mod m c + n, rc k n,rn, r, 4 c k n,rn, r := b k n,rn, r + k b k n,rn, r with b k n,rn, r := πi k m D k D 3 4 c>0 c 3 Kc n, r, n, r I k 3 π D D. mc Here I s x is the usual I-Bessel function of order s and K c n, r, n, r is the Kloosterman sum K c n, r, n, r := e mc rr e c dmλ + n d r λ + dn + drλ, d mod c λ mod c where e c x := e πix c and the sum over d runs through the primitive residue classes modulo c and d is the inverse of d modulo c. In Section, we follow Bruinier and Funke [3] to define a pairing of harmonic Maass- Jacobi forms and skew-holomorphic Jacobi forms. We find that this pairing is determined by the principal part of the holomorphic part of a harmonic Maass-Jacobi form. In Section 3, we recall the Maass-Jacobi-Poincaré series from []. Given a harmonic Maass-Jacobi form, linear combinations of such Poincaré series allow one to construct a new harmonic Maass- Jacobi form whose holomorphic part has the same principal part as the given one. This is the key idea in our proof of Theorem. Finally, we give an explicit application of our results. The ring of Jacobi forms of even weights is generated over the ring of modular forms by certain weak Jacobi forms of index and weights and 0, which we denote by φ, and φ 0,, respectively see 8 and 9 of [5]. We show that φ, can be expressed as Jacobi-Poincaré series. It is likely that φ 0, can also be realized as a Jacobi-Poincaré series,

3 EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS 3 but this requires the existence of a Maass-Jacobi Poincaré series of weight 0, which we have not constructed yet.. The pairing In this section, we introduce a pairing between skew-holomorphic Jacobi forms and harmonic Maass-Jacobi forms, which is vital to our proof of Theorem in Section 3. Let J sk denote the space of skew-holomorphic Jacobi forms of weight k and index m and let J sk,cusp be the subspace of cusp forms for details, see Skoruppa [8, 9]. If φ, ψ J sk,cusp, then the Petersson scalar product of φ and ψ is defined by φ, ψ := Γ J \H C φτ, zψτ, ze 4πmv y y k dv, where Γ J := SL Z Z is the Jacobi group and dv := dudvdxdy is the Γ J -invariant volume y 3 element on H C for details, see Skoruppa [7, 9]. Note that skew-holomorphic Jacobi forms have a theta decomposition, i.e., if φτ, z = n,r Z cn, re πdy m q n ζ r J sk, then 5 φτ, z = where with 6 c µ N := and { µ mod m h µ τ := N 0 D 0 h µ τθ m,µ τ, z, c µ Nq N c r N, r for r Z, r µ mod m, and N µ mod 0 if N µ mod 7 θ m,µ τ, z := r Z r µ mod m q r ζ r. If φτ, z = h µ τθ m,µ τ, z J sk,cusp and ψτ, z = g µ τθ m,µ τ, z J sk,cusp, then analogous to Theorem 5.3 of [5], one finds that 8 φ, ψ = h µ τg µ τy k 5/ dxdy, F µ mod m where F is the standard fundamental domain for the action of SL Z on H. We wish to review the definition of harmonic Maass-Jacobi forms of []. Therefore, we need to recall the slash action for Jacobi forms and a certain differential operator which is invariant under that action. If φ : H C C, then 9 φ A τ, z := φ aτ + b cτ + d, z + λτ + µ cτ + d cτ + d k e πim cz+λτ+µ +λ cτ+d τ+λz «

4 4 KATHRIN BRINGMANN AND OLAV K. RICHTER for all A = [ ] a b c d, λ, µ Γ J. Furthermore, if w := w for a variable w, then C := τ τ τ τ ττ k τ τ τ + 4πim τzz kτ τ + 4πim τ τz z zz + zzz τ τz z τz + kz z z 4πim τ τ z z + 4πim kτ τ τ τz z τzz + + zz + zzz 4πim 4πim which up to the constant k k is the Casimir operator with respect to the action in 9. Definition. A function φ : H C C is a harmonic Maass-Jacobi form of weight k and index m if φ is real-analytic in τ H and z C and satisfies the following conditions: For all A Γ J, φ A = φ. We have that C φ = 0. 3 We have that φτ, z = O e ay e πmv /y as y for some a > 0. Of particular interest are harmonic Maass-Jacobi forms, which are holomorphic in z; the space of such forms is denoted by Ĵ. The differential operator ξ := y k 3/ iy τ iv z + y 4πm zz plays an important role in this context. It maps harmonic Maass-Jacobi forms that are holomorphic in the Jacobi variable z to skew-holomorphic Jacobi forms. Moreover, with Ĵ cusp Ĵ we denote the space of harmonic Maass-Jacobi forms, which appear in the pre-image of skew-holomorphic Jacobi cusp forms under ξ. The next proposition was proved in []. Proposition. [] The map is surjective. ξ : Ĵcusp J sk,cusp 3 If φ Ĵcusp sk,cusp and ψ J, then we define the non-degenerate pairing 3 0 {φ, ψ} := ξ φ, ψ. The following result extends Proposition 3.5 of [3] to Jacobi forms and shows that the pairing in 0 is determined by the principal part P φ + of φ +.

5 EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS 5 Proposition. Let φ = φ + + φ Ĵcusp ψτ, z = n,r Z dn, re πdy m q n ζ r J sk,cusp 3 {φ, ψ} = r mod m with φ+ and φ as in and and let. If k is even, then c + n, rdn, r. with Proof: Let ψτ, z = µ mod m g µ τ = g µ τθ m,µ τ, z d µ Nq N be the theta decomposition of ψ. It is easy to see that elements in Ĵcusp and hence also their holomorphic and non-holomorphic parts have theta decompositions. More precisely, we have φτ, z = h µ τθ m,µ τ, z, where h µ = h + µ + h µ, µ mod m h + µ τ = N c + µ Nq N, h µ τ = 3 c πny µ NΓ k, q N, m and c ± µ N is as in 6. We now follow the proof of Proposition 3.5 in [3]. First, if k is even, then one verifies that h µ g µ dτ is an SL Z-invariant -form on H. µ mod m Set L k := iy τ and ξ k := y k L k. From 8 we have {φ, ψ} = lim t j=0 F t,j µ mod m ξ k h µ τg µ τy k dxdy, where each { F t,j := τ H τ j, 0 x j, y t, or τ j, } x j, y t is a translation by j of the truncated fundamental domain F t := {τ F y t}, and where, in addition, the left half is cut and pasted to the right. Let dω := dxdy denote the usual y

6 6 KATHRIN BRINGMANN AND OLAV K. RICHTER invariant volume form on H. With the help of Stokes Theorem, we find that ξ k h µ τg µ τy k dxdy j=0 = = Stokes = = F t,j µ mod m j=0 j=0 0 µ mod m F t,j µ mod m F t,j µ mod m µ mod m c + µ Nd µ N + L k h µ g µ dω h µ g µ dτ h µ x + itg µ x + it dx µ mod m 3 c πnt µ Nd µ NΓ k,, m where the last equation follows from inserting the Fourier expansions of h µ and g µ. Finally, Γ 3 πnt k, m = O πnt k m e πnt m as t, which shows that {φ, ψ} = µ mod m c + µ Nd µ N = r mod m c + n, rdn, r. 3. Maass-Jacobi-Poincaré series and the proof of Theorem In [], we investigated Maass-Jacobi-Poincaré series, which we will now recall after introducing necessary notation. Let M ν,µ be the usual M-Whittaker function. Let D = r 4nm 0, and for s C, κ Z, and t R \ {0}, define and M s,κ t := t κ Msgnt κ,s t φ n,r,s τ, z := M s,k πdy e πirz πr y m +πinx. m Set Γ J := {[ ] } η 0, 0, n η, n Z. Then the Poincaré series P n,r,s τ, z := n,r φ,s k, m A τ, z A Γ J \ΓJ converges absolutely and uniformly for Res > 5 n,r 4 and P,s Ĵ if s = k 4, k > 3 or if s = 5 4 k n,r, k < 0. In [], we determined the Fourier expansion of P,s, which features a

7 certain theta series ϑ r EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS 7 3 ϑ r r τ, z = q. It is not difficult to see that θ m,r τ, z + k θ m, r τ, z For brevity, we will only recall the part of the Fourier expansion, which is needed for the purpose of this paper. Let P n,r + n,r,s denote the holomorphic part of P,s. If D > 0 and s = 5 4 k, k < 0, then Corollary of [] and 3 imply that 4 P n,r + 5,s = Γ k q D θ m,r τ, z + k θ m, r τ, z. + n,r Z D 0 where as before D = r 4n m. If D < 0, then c k n,rn, r is as in 4. c k n,rn, r q n ζ r, Proof of Theorem. Let φ Ĵcusp k < 0 with holomorphic part φ+ as in and principal part P φ +τ, z = c + µ Nq N θm,µ τ, z µ mod m with c + µ N as in 6. Equation 4 together with the identity c + µ N = k c + µ N shows that the holomorphic part of ϕ := Γ 5 k r mod m c + n, rp n,r, 5 4 k = Γ 5 k µ mod m c + µ NP «µ N,µ, 5 4 k has the same principal part as φ +. Consider ψ := φ ϕ Ĵcusp. Suppose that ψ has a non-trivial non-holomorphic part. Then ξ ψ 0 and hence {ψ, ξ ψ} = ξ ψ, ξ ψ 0. On the other hand, Proposition expresses {ψ, ξ ψ} in terms of the principal part of ψ, which is zero by construction. This gives a contradiction and thus ψ is necessarily holomorphic, i.e., ψ is a weak Jacobi form. Again, the principal part of ψ is zero and hence ψ is a holomorphic Jacobi form as in [5] of weight k < 0. We conclude that ψ = 0 and Theorem follows. We end with an explicit example. Let φ, τ, z = ζ + ζ +... be the weak Jacobi form of weight and index alluded to in the introduction. Note that q n ζ r = q r 4 ζ r = q 4 θ, τ, z n,r Z D= r mod appears as the principal part of φ, τ, z and also of Γ 9 P 0, that φ, = Γ 9 P 0,.,, 9 4,, 9 4 τ, z. Hence we find

8 8 KATHRIN BRINGMANN AND OLAV K. RICHTER References [] Bringmann, K., and Ono, K. Coefficients of harmonic weak Maass forms. To appear in the proceedings of the 008 University of Florida conference on partitions, q-series, and modular forms. [] Bringmann, K., and Richter, O. Zagier-type dualites and lifting maps for harmonic Maass-Jacobi forms. Preprint. [3] Bruinier, J., and Funke, J. On two geometric theta lifts. Duke Math. J. 5, no. 004, [4] Cardy, J. Operator content of two-dimensional conformally invariant theories. Nuclear Phys. B 70, no. 986, [5] Eichler, M., and Zagier, D. The theory of Jacobi forms. Birkhäuser, Boston, 985. [6] Gross, B., Kohnen, W., and Zagier, D. Heegner points and derivatives of L-series. II. Math. Ann. 78, , [7] Skoruppa, N.-P. Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms. J. Reine Angew. Math , [8] Skoruppa, N.-P. Developments in the theory of Jacobi forms. Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk 990, [9] Skoruppa, N.-P. Explicit formulas for the Fourier coeffcients of Jacobi and elliptic modular forms. Invent. Math. 0, no , [0] Zagier, D. Note on the Landweber-Stong elliptic genus, in: Elliptic curves and modular forms in algebraic topology Princeton, NJ, 986. Lecture Notes in Math. 36. Springer, 988, pp [] Zwegers, S. Mock theta functions, Ph.D. thesis, Universiteit Utrecht, The Netherlands, 00. Mathematisches Institut, Universität Köln, Weyertal 86-90, D-5093 Köln, Germany address: kbringma@math.uni-koeln.de Department of Mathematics, University of North Texas, Denton, TX 7603, USA address: richter@unt.edu