600 K. Bringmann, K. Ono Before we prove this result, we first recall the construction ( of) these forms. Suppose that λ is an integer, and that k := λ+ 1 αβ. For each A = Ɣ γ δ 0 (4),let j(a, z) := ( ) γ ɛδ 1 (γ z + δ) 1 δ be the usual factor of automorphy for half-integral weight modular forms. If f : H C is a function, then for A Ɣ 0 (4) we let (f k A) (z) := j(a, z) λ 1 f (Az). (.5) As usual, let z = x + iy be the standard variable on H. Fors C and y R {0}, welet M s (y) := y k M k sgn(y), s 1 ( y ), (.6) where M ν,μ (z) is the standard M-Whittaker function which is a solution to the differential equation ( ) u z + 1 4 + ν 1 z + 4 μ z u = 0. If m is a positive integer, then define ϕ m,s (z) by and consider the Poincaré series ϕ m,s (z) := M s ( 4πmy)e( mx), F λ ( m, s; z) := A Ɣ \Ɣ 0 (4) (ϕ m,s k A)(z). (.7) It is easy to verify that ϕ m,s (z) is an eigenfunction, with eigenvalue of the weight k hyperbolic Laplacian k := y ( s(1 s) + (k k)/4, (.8) ) x + y ( + iky x + i ). y ( ) Since ϕ m,s (z) = O y Re(s) k as y 0, it follows that F λ ( m, s; z) converges absolutely for Re(s) >1, is a Ɣ 0 (4)-invariant eigenfunction of the Laplacian, and is real analytic.
Coefficients of half-integral weight Poincaré series 601 These series provide examples of weak Maass forms of half-integral weight. Following Bruinier and Funke [5], we make the following definition. Definition. A weak Maass form of weight k for the group Ɣ 0 (4) is a smooth function f : H C satisfying the following: (1) For all A Ɣ 0 (4) we have (f k A)(z) = f (z). () We have k f = 0. (3) Thefunctionf(z) has at most linear exponential growth at all the cusps. Remark If a weak Maass form f (z) is holomorphic on H, then it is a weakly holomorphic modular form. In view of (.8), it follows that the special s-values at k/ and 1 k/ of F λ ( m, s; z) are weak Maass forms of weight k = λ + 1 when the defining series is absolutely convergent. If λ {0, 1} and m 1 is an integer for which ( 1) λ+1 m 0, 1 (mod 4), then we recall the definition ) 3 F λ ( m, k ; z pr λ if λ, F λ ( m; z) := ) (.9) 3 ( m,1 k ; z pr λ if λ 1. (1 k)ɣ(1 k) F λ By the discussion above, it follows that F λ ( m; z) is a weak Maass form of weight k = λ + 1 on Ɣ 0(4). Ifλ = 1 and m is a positive integer for which m 0, 1 (mod 4), then define F 1 ( m; z) by F 1 ( m; z) := 3 F 1 ( m, 34 ) ; z pr 1 + 4δ,m G(z). (.10) The function G(z) is given by the Fourier expansion G(z) := H(n)q n 1 + 16π y n=0 where H(0) = 1/1 and β(s) := 1 n= t 3 e st dt. β(4πn y)q n, Proposition 3.6 of [7] proves that each F 1 ( m; z) is in M! 3. These series form the basis given in (1.5).
60 K. Bringmann, K. Ono Remark Note that the integral β(s) is easily reformulated in terms of the incomplete Gamma-function. We make this observation since the non-holomorphic parts of the F λ ( m; z),forλ 7 and λ = 5, will be described in such terms. Remark We may define the series F 0 ( m; z) M! 1 using an argument analogous to Proposition 3.6 of [7]. Instead, we simply note that the existence of the basis (1.6) of M! 1, together with the duality of Theorem 4 [3] and an elementary property of Kloosterman sums (see Proposition 3.1), gives a direct realization of the Fourier expansions of F 0 ( m; z) in terms of the expansions of the F 1 ( n; z) described above. To compute the Fourier expansions of these weak Maass forms, we require some further preliminaries. For s C and y R {0},welet W s (y) := y k W k sgn(y), s 1 ( y ), (.11) where W ν,μ denotes the usual W-Whittaker function. For y > 0, we shall require the following relations: M k ( y) = e y, (.1) W 1 k (y) = W k (y) = e y, (.13) and W 1 k ( y) = W k ( y) = e y Ɣ (1 k, y), (.14) where Ɣ(a, x) := x e t t a dt t is the incomplete Gamma-function. For z C, the functions M ν,μ (z) and M ν, μ (z) are related by the identity W ν,μ (z) = Ɣ( μ) Ɣ(μ) Ɣ( 1 μ ν)m ν,μ(z) + Ɣ( 1 + μ ν)m ν, μ(z). From these facts, we easily find, for y > 0, that M 1 k ( y) = (k 1)e y Ɣ(1 k, y) + (1 k)ɣ(1 k)e y. (.15) Proof of Theorem.1 Although the conclusions of Theorem.1 (1) and (3) were obtained previously by Bruinier, Jenkins and the second author in [7], for completeness we consider the calculation for general λ {0, 1}. In particular,
1.3 Non-holomorphic Poincaré series of negative weight 7 lies in S κ,l. This shows that M κ,l = M Eis κ,l S κ,l and (1.0) is a formula for the codimension of S κ,l in M κ,l. In later applications we will mainly be interested in the Eisenstein series E 0 (τ) which we simply denote by E(τ). In the same way we write q(γ,n) for the Fourier coefficients q 0 (γ,n) of E(τ). Let us finally cite the following result of [BK]: Proposition 1.7. The coefficients q(γ,n) of E(τ) are rational numbers. 1.3 Non-holomorphic Poincaré series As in the previous section we assume that L is an even lattice of signature (b +,b ) = (,l), (1,l 1), or (0,l ) with l 3. Put k = 1 l/ and κ = 1 + l/. We now construct certain vector valued Maass-Poincaré series for Mp (Z) of weight k. Series of a similar type are well known and appear in many places in the literature (see for instance [He, Ni, Fa]). Let M ν, µ (z) and W ν, µ (z) be the usual Whittaker functions as defined in [AbSt] Chap. 13 p. 190 or [E1] Vol. I Chap. 6 p. 64. They are linearly independent solutions of the Whittaker differential equation d w dz + ( 1 4 + ν z µ 1/4 z ) w = 0. (1.1) The functions M ν, µ (z) and M ν, µ (z) are related by the identity W ν, µ (z) = Γ( µ) Γ(µ) Γ( 1 µ ν)m ν, µ(z) + Γ( 1 + µ ν)m ν, µ(z) (1.) ([AbSt] p. 190 (13.1.34)). This implies in particular W ν, µ (z) = W ν, µ (z). As z 0 one has the asymptotic behavior M ν, µ (z) z µ+1/ (µ / 1 N), (1.3) Γ(µ) W ν, µ (z) Γ(µ ν + 1/) z µ+1/ (µ 1/). (1.4) If y R and y one has M ν, µ (y) = Γ(1 + µ) Γ(µ ν + 1/) ey/ y ν (1 + O(y 1 )), (1.5) W ν, µ (y) = e y/ y ν (1 + O(y 1 )). (1.6) For convenience we put for s C and y R >0 :
8 1 Vector valued modular forms for the metaplectic group M s (y) = y k/ M k/, s 1/ (y). (1.7) In the same way we define for s C and y R {0}: W s (y) = y k/ W k/ sgn(y), s 1/ ( y ). (1.8) If y < 0 then equation (1.) implies that M s ( y ) = Γ(1 + k/ s) W s (y) Γ(1 s) Γ(1 + k/ s)γ(s 1) M 1 s ( y ). Γ(1 s)γ(s + k/) (1.9) The functions M s (y) and W s (y) are holomorphic in s. Later we will be interested in certain special s-values. For y > 0 we have M k/ (y) = y k/ M k/, k/ 1/ (y) = e y/, (1.30) W 1 k/ (y) = y k/ W k/, 1/ k/ (y) = e y/. (1.31) Using the standard integral representation Γ(1/ ν + µ)w ν, µ (z) = e z/ z µ+1/ 0 e tz t 1/ ν+µ (1 + t) 1/+ν+µ dt (R(µ ν) > 1/, R(z) > 0) of the W-Whittaker function ([E1] Vol. I p. 74 (18)), we find for y < 0: W 1 k/ (y) = y k/ W k/, 1/ k/ ( y ) = e y / y 1 k = e y / y 1 k 0 1 e t y (1 + t) k dt e t y t k dt. If we insert the definition of the incomplete Gamma function (cf. [AbSt] p. 81) Γ(a,x) = e t t a 1 dt, (1.3) x we obtain for y < 0 the identity W 1 k/ (y) = e y/ Γ(1 k, y ). (1.33) The usual Laplace operator of weight k (cf. [Ma1])
54 KEN ONO Suppose that k 1 + Z. We define a class of Poincaré series P k(s; z). For matrices ( a c d b ) Γ 0(), with c 0, define the character χ( ) by (( )) { ( ) a b e b if c = 0, 4 (11.7) χ := c d i 1/ ( 1) 1 (c+ad+1) e ( a+d a + ) 3dc 4c 4 8 ω 1 d,c if c > 0, where (11.8) ω d,c := e πis(d,c). Here s(d, c) denotes the classical Dedekind sum. Throughout, let z = x + iy, and for s C, k 1 + Z, and y R \ {0}, and let (11.9) M s (y) := y k M k sgn(y), s 1 ( y ), where M ν,µ (z) again is the M-Whittaker function. Furthermore, let ( ϕ s,k (z) := M s πy ) ( e x ). 6 4 Using this notation, define the Poincaré series P k (s; z) by (11.10) P k (s; z) := χ(m) 1 (cz + d) k ϕ s,k (Mz). π M Γ \Γ 0 () Here Γ again is the subgroup of translations in SL (Z). ( The defining series is absolutely convergent for P k 1 k ; z) for k < 1/, and is conditionally convergent when k = 1/. We are interested in P 3 ( 1 ; z), which we define by 4 analytically continuing the Fourier expansion. This argument is not straightforward (see Theorem ( 3. and Corollary 4. of [53]). Thanks to the properties of M ν,µ, we find that 3 ; 4z) is a Maass form of weight 1/ for Γ 4 0 (144) with Nebentypus χ 1. P 1 A long calculation gives the following Fourier expansion ( ) ( ( 3 (11.11) P 1 4 ; z = 1 π 1 1 Γ, πy )) 6 q 1 4 + 0 n= γ y (n)q n 1 4 + β(n)q n 1 4, where for positive integers n we have ) ( 1) k+1 (11.1) β(n) = π(4n 1) 1 A k (n k(1+( 1)k ) ( ) 4 π 4n 1 4 I 1. k 1k k=1 ( The Poincaré series P 3 1 ; z) was defined so that (11.1) coincides with the conjectured 4 expressions for the coefficients α(n). For convenience, we let (11.13) P (z) := P 1 ( ) 3 4 ; 4z. Canonically decompose P (z) into a non-holomorphic and a holomorphic part (11.14) P (z) = P (z) + P + (z). n=1
In particular, we have that HARMONIC MAASS FORMS AND NUMBER THEORY 55 P + (z) = q 1 + β(n)q 4n 1. Since P (z) and D ( 1 ; z) are Maass forms of weight 1/ for Γ 0 (144) with Nebentypus χ 1, (11.11) and (11.1) imply that the proof of the conjecture reduces to proving that these forms are equal. This conclusion is obtained after a lengthy and somewhat complicated argument. n=1 11.. Exact formulas for harmonic Maass forms with weight 1/. Generalizing the results of the previous section, Bringmann and the author have obtained exact formulas for the coefficients of the holomorphic parts of harmonic Maass forms with weight k 1/ [59]. Suppose that f is in H k (N, χ), the space of weight k harmonic Maass forms on Γ 0 (N) with Nebentypus character χ, where we assume that 3 k 1 Z. As usual, we denote its Fourier expansion by (11.15) f(z) = c f (n)γ(k 1, 4π n y)qn. n c + f (n)qn + n<0 It is our objective to determine exact formulas for the coefficients c + f (n) of the holomorphic part of f. We now define the functions which are required for these exact formulas. Throughout, we let k 1 Z, and we let χ be a Dirichlet character modulo N, where 4 N whenever k 1 Z \ Z. Using this character, for a matrix M = ( a b c d ) Γ 0(N), we let { χ(d) if k Z, (11.16) Ψ k (M) := χ(d) ( ) c d ɛ k d if k 1Z \ Z, where ɛ d is defined by (7.), and where ( c d) is the usual extended Legendre symbol. In addition, if T = ( a c d b ) SL (Z), then we let (11.17) µ(t ; z) := (cz + d) k. Moreover, for pairs of matrices S, T SL (Z), we then let (11.18) σ(t, S) := µ(t ; Sz)µ(S; z). µ(t S; z) Using this notation, we now define certain generic Kloosterman sums which are naturally associated with cusps of Γ 0 (N). Suppose that ρ = aρ c ρ = L 1, (L SL (Z)) is a cusp of Γ 0 (N) with c ρ N and gcd(a ρ, N) = 1. Let t ρ and κ ρ be the cusp width and parameter of ρ with respect to Γ 0 (N) (see 11.1). Suppose that c > 0 with c ρ c and N c ρ c. Then for integers n and m we