Mock-modular forms of weight one - UCLA Department of Mathematics

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8 W. DUKE AND Y. LI 2. Existence of Harmonic Maass Forms of Weight One Given a cusp form of weight one, we will show the existence of a weight one mock-modular form with it as shadow in this section. There are several different approaches to proving this result, such as a geometric approach in [9]. For completeness and since it might have some independent interest, in this section we prove the existence by analytically continuing weight one Poincaré series via the spectral expansion. We begin with some basic definitions. Let k ∈ Z. For any function f : H → C and γ ∈ GL + 2 (R), define the weight k slash operator | k γ by (f| k γ)(z) := (det(γ))k/2 f(γz), (cz+d) k where γz is the linear fractional transformation of z under γ. We will write f|γ for f| k γ when the weight of f is understood. For M ∈ Z + let Γ 0 (M) denote the usual congruence subgroup of SL 2 (Z) of level M, namely Γ 0 (M) = {( a c d b ) ∈ SL 2(Z) ( a c d b ) ≡ ( ∗ 0 ∗ ∗ ) mod M}. Let ν : Z/MZ → C × be a Dirichlet character such that ν(−1) = (−1) k and ν(γ) := ν(d) for γ = ( a c d b ) ∈ Γ 0(M). Let F k (M, ν) be the space of smooth functions f : H → C such that for all γ ∈ Γ 0 (M) (f | k γ)(z) = ν(γ)f(z). Recall from (1) and (2) the differential operator ξ k and the weight k hyperbolic Laplacian ∆ k . Let z = x + iy. Then ∆ k can be written as −∆ k = −y 2 ( ∂ 2 + ∂2 ∂x 2 ) ( + iky ∂y 2 ∂ + i ∂ ∂x ∂y Then f(z) ∈ F k (M, ν) is a weight k harmonic weak Maass form of level M and character ν (or more briefly, a weakly harmonic form) if it satisfies the following properties. (i) f(z) is real-analytic. (ii) ∆ k (f) = 0. (iii) The function f(z) has at most linear exponential growth at all cusps of Γ 0 (M). From now on we fix k = 1 and write | for the slash operator | 1 . Let H ! 1(M, ν) be the space of weakly harmonic modular forms of weight one, level M and character ν. Denote by M ! 1(M, ν) , M 1 (M, ν) and S 1 (M, ν) the usual subspaces of weakly holomorphic modular forms, holomorphic modular forms and cusp forms, respectively. Let M ! 1(M, ν) be the space of mock-modular forms with shadows in M 1 (M, ν) and M 1 (M, ν) ⊂ M ! 1(M, ν) the subspace of mock-modular forms whose shadows are in S 1 (M, ν). From (2), it is clear that the image of H ! 1(M, ν) under ξ 1 lies in M ! 1(M, ν). Let H 1 (M, ν) be the preimage of S 1 (M, ν) under ξ 1 . Then H 1 (M, ν) is canonically isomorphic to M 1 (M, ν). We want to show that the map ξ 1 is in fact a surjection onto S 1 (M, ν). To proceed, we will construct two families of Poincaré series P m (z, s), Q m (z, s), where the first family is similar to the one used in [6]. They are a priori defined for Re(s) > 1 and will be analytically continued to Re(s) > 0 through their spectral expansions. Unlike the cases k ≥ 2, this will only be a statement about existence, and not a formula that could be used to calculate the holomorphic coefficients of the preimage explicitly. To prove the analytic continuation, we will refer to results in [31] and [35]. Given any positive integer m, define (16) (17) ) . φ ∗ m(z, s) := e 2πimx (4π|m|y) −1/2 M sgn(m) ,s− 1 (4π|m|y), 2 2 ϕ ∗ m(z, s) := e 2πimx (4π|m|y) s−1/2 e −2π|m|y .
MOCK-MODULAR FORMS OF WEIGHT ONE 9 Here M µ,ν (y) is the M-Whittaker function defined by (18) M µ,ν (z) := z ν+1/2 e z/2 1F 1 ( 1 2 + µ + ν; 1 + 2ν; −z) , with 1 F 1 (α; β; z) being the generalized hypergeometric function. Averaging them over the coset representatives of Γ ∞ \Γ 0 (M), we can define the following Poincaré series ∑ P m (z, s, ν) := ν(γ)(φ ∗ m | γ)(z, s), Q m (z, s, ν) := γ∈Γ ∞\Γ 0 (M) ∑ γ∈Γ ∞\Γ 0 (M) ν(γ)(ϕ ∗ m | γ)(z, s). For convenience, we shall omit ν and write P m (z, s) and Q m (z, s) instead. When Re(s) > 1, both families have Fourier expansions in z that converges absolutely and uniformly on compact subset of H (see [31, Satz 6.5] for the weight zero case). In that region, both families are absolutely convergent and define a holomorphic function in s. Also, P m (z, s) is an eigenfunction of −∆ 1 with eigenvalue (s − 1/2)((1 − s) − 1/2) and belongs to F 1 (M, ν). Let ∆ ′ k be the differential operator defined by ( ) −∆ ′ k := −y 2 ∂ 2 + ∂2 + iky ∂ . ∂x 2 ∂y 2 ∂x It is related to ∆ k by the following equation Define the space D 1 (M, ν) by ∆ ′ k + k 2 ( 1 − k 2) = y k/2 ∆ k y −k/2 . D 1 (M, ν) := {y 1/2 f(z) : f(z) ∈ F 1 (M, ν) is smooth with compact support on H} Let ˜D 1 (M, ν) be the completion of D 1 (M, ν) with respect to the Petersson norm ∫ ||g|| 2 = 〈g, g〉 := |g(z)| 2 dxdy . y 2 Γ 0 (M)\H Satz 3.2 in [35] implies that −∆ ′ k has a self-adjoint extension − ˜∆ ′ k from D 1(M, ν) to ˜D 1 (M, ν). Also, − ˜∆ ′ k has a countable system of Maass cusp forms {e n(z)} n∈N ⊂ ˜D 1 (M, ν) forming the discrete spectrum. Each Maass cusp form e n (z) has eigenvalue λ n and each eigenvalue has finite multiplicity. For any f ∈ ˜D 1 (M, ν), the discrete spectrum contributes the following sum to its spectral expansion ∞∑ 〈f, e n 〉e n (z), n=1 which converges absolutely and uniformly for all z ∈ H [35, Satz 8.1]. Let ι be a cusp of Γ 0 (M), σ ι ∈ GL 2 (R) the scaling matrix sending ∞ to ι, and Γ ι ⊂ Γ 0 (M) the subgroup fixing ι. One can define the weight one real analytic Eisenstein series E ι (z, s) by E ι (z, s) := y1/2 2 ∑ γ∈Γ ι\Γ 0 (M) ν(γ) cz+d (Im(σ−1 ι γz)) s−1/2 . Selberg showed that the Eisenstein series has meromorphic continuation to the whole complex plane in s. For ι ranging over the cusps of Γ 0 (M), the Eisenstein series {E ι (z, s)} ι make up
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
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Page 31 and 32: Similar to the proof of Theorem 5.1
Page 39 and 40: where T := (T kk ′) 0≤k,k ′

Mock-modular forms of weight one - UCLA Department of Mathematics

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