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

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24 W. DUKE AND Y. LI Using this result and appropriate Rankin-Selberg unfolding trick as in [24], we can prove Theorem 5.1. Proof of Theorem 5.1. Let ˜g ψ (z) ∈ M − 1 (p) be a mock-modular form as in Proposition 4.2 with Fourier expansion ˜g ψ (z) = ∑ r + ψ (n)qn n∈Z and ĝ ψ (z) the associated harmonic Maass form. For a class B ∈ Cl(F ), define Φ B (ĝ ψ , z) := Tr p 1(−i(ĝ ψ |W p )(z)ϑ B (z)) = (ĝ ψ |U p )(z)ϑ B (z) + (ĝ ψ (z)ϑ B (z))|U p . This should be compared to ˜Φ s (z) in Proposition (1.2) of [24, §IV]. When N = 1, the derivative of ˜Φ s (z) at s = 1 is more or less our Φ B (ĝ ψ , z) for ψ trivial. Here, Φ B (ĝ ψ , z) transforms like a level one, weight two modular form and has the following Fourier expansion at infinity Φ B (ĝ ψ , z) = ∑ n∈Z(b B (ĝ ψ , n, y) + a B (ĝ ψ , n))q n , b B (ĝ ψ , n, y) = − ∑ k∈Z a B (ĝ ψ , n) = ∑ k∈Z δ(k)r ψ (k)β 1 (k, y p δ(k)r + ψ (k) r B(pn − k). ) r B (pn + k), By the choice of ˜g ψ (z), we have ord ∞ (˜g ψ (z)) ≥ − p+1 > −p. So the sum a 24 B(ĝ ψ , n) is zero for all n < 0. It is easy to see from the definition above that jm lift,B ∈ M !,+ 1 (p). So we can apply Proposition 3.4 to obtain (60) 〈j lift,B m , g ψ 〉 reg = a B (ĝ ψ , m). In particular when m = 0, we have j lift,B 0 = 1 lift,B = ϑ B (z) and 〈ϑ B , g ψ 〉 = a B (ĝ, 0). Now, let Ẽ2(z) the non-holomorphic Eisenstein series of level one, weight two defined by Ẽ 2 (z) = 1 − 24 ∑ n≥1 σ 1 (n)q n − 3 πy . Then we can use Poincaré series to apply holomorphic projection to the function Φ ∗ B(ĝ ψ , z) := Φ B (ĝ ψ , z) − a B (ĝ ψ , 0)Ẽ2(z), since it satisfies the growth condition Φ ∗ B (ĝ ψ, z) = O(1/y) at the cusp infinity. For m > 0, let P m,1 (z, s) be the Poincaré series of level one, weight two defined by ∑ P m,1 (z, s) := (y s e 2πimz )| 2 γ. γ∈Γ ∞\PSL 2 (Z) The limit as s goes to zero of the inner product between P m,1 (z, s) and Φ ∗ B (ĝ ψ, z) is the m th Fourier coefficient of a cusp form of weight two, level one. Since the space of such forms is trivial, we have (61) lim s→0 〈Φ ∗ B(ĝ ψ , z), P m,1 (z, s)〉 = 0.
MOCK-MODULAR FORMS OF WEIGHT ONE 25 Using the Fourier expansion of Φ B (ĝ ψ , z) and Rankin-Selberg unfolding, we can rewrite equation (61) as ( ) k (62) a B (ĝ ψ , m) + 24σ 1 (m)a B (ĝ ψ , 0) = lim δ(k)r B (pm + k)r ψ (k)φ s , where φ s (µ) := ∫ ∞ 1 s→0 ∑ k≥1 du (1 + µu) 1+s u . Comparing Q s−1 (t) and φ s−1 (µ), we see that φ 0 (µ) = 2Q 0 (1 + 2µ) and φ s−1 (µ) − 2Γ(2s) Q sΓ(s) 2 s−1 (1 + 2µ) = O(µ −1−s ). ( ) ( ) 2 That means we can substitute Q sΓ(s) 2 s−1 1 + 2k k for φ pm s−1 and in the limit as s approaches 1. Together with (8) and (60), equation (61) pm becomes (63) 〈j lift,B m , g ψ 〉 reg +24σ 1 (m)〈ϑ B , g ψ 〉 = − lim s→1 ∑ A ψ(A) ∑ k≥1 pm ( )) δ(k)r B (pm + k)r A (k) (−2Q s−1 1 + 2k . pm Now the counting argument in Proposition (3.11) in [23] tells us that δ(k)r B (pm + k)r A (k) = ρ m (A, B, k), where ρ m (A, B, k) counts the number of γ ∈ M 2 (Z)/{±1} such that det(γ) = m and cosh d(τ √ AB −1 , γτ √ AB ) = 1 + 2k pm . In particular when k = 0, the number of γ ∈ M 2 (Z)/{±1} such that det(γ) = m and γτ √ AB = τ√ AB is exactly r −1 B (pm) = r B (m). Since k ≥ 1 in the summation, equation (63) becomes (64) 〈jm lift,B ,g ψ 〉 reg + 24σ 1 (m)〈ϑ B , g ψ 〉 = − lim = − lim ɛ→0 lim s→1 ∑ A s→1 ∑ A ψ(A) ∑ ( g s τ √ AB −1, γτ √ AB γ∈M 2 (Z)/{±1} det(γ)=m, γτ √ AB≠τ√ AB −1 ψ(A) ( G m s (τ √ AB −1 + ɛ, τ √ AB ) − r B(m)g s (τ √ AB −1 + ɛ, τ √ AB −1 ) ) where g s is defined in (57). From this expression, it is clear that the choice of these CM points do not matter. ) Since g 1 (τ + ɛ, τ) = − log (1 + 4Im(τ)2 and ψ is non-trivial, we see that ɛ 2 lim lim ɛ→0 s→1 ∑ A ψ(A)g s (τ √ AB −1 + ɛ, τ √ AB −1 ) = − ∑ A Also by equation (59), we have ∑ lim ψ(A)G m s (τ √ s→1 AB + ɛ,τ √ −1 AB ) = ∑ A A 4πσ 1 (m) ∑ A ψ(A) log(y 2 √ AB −1). ψ(A) log |Ψ m (τ √ AB −1 + ɛ, τ √ AB )|2 − ψ(A) lim s→1 (E(τ √ AB −1 + ɛ, s) + E(τ √ AB , s)). )
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
Page 23: MOCK-MODULAR FORMS OF WEIGHT ONE 23
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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