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

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46 W. DUKE AND Y. LI be a mock-modular form with shadow g ψ (z). By Proposition 3.4, the regularized inner product 〈f lift,θ , g ψ 〉 reg can be expressed in terms of r + ψ (n) as (101) 〈f lift,θ , g ψ 〉 reg = ∑ ∑ ( ) −pm − k r + 2 ψ c(f, m)δ(k). 4 m∈Z k∈Z The following theorem relates this inner product, hence the linear combination of r + ψ (n), to the values of the Borcherds lift Ψ f (z). Theorem 9.1. Let f(z) ∈ M 1/2 ! be a weakly holomorphic modular form with integral Fourier coefficients c(f, n), and Ψ f (z) its Borcherds lift. Suppose ψ is a non-trivial character of Cl(F ). Then we have ∑ (102) 〈f lift,θ , g ψ 〉 reg = −2 lim ψ 2 (A) ( log |Ψ f (τ A + ɛ)| 2 + C f log |y A | ) , ɛ→0 A∈Cl(F ) where C f = ∑ k∈Z c(f, −pk2 ) is the constant term of f lift,θ (z), and τ A is a CM point associated to the class A. Remark. Recall that jm lift,B (z) = j m (pz)ϑ B (z) for B ∈ Cl(F ). When B = A 0 is the principal class in Cl(F ), we can write 2j lift,A 0 m (z) = j m (pz)U 4 (θ(pz)θ(z)) = (j m (4z)θ(z)) lift,θ = ∑ t∈Z f lift,θ 4m−t (z). 2 t 2 ≤4m Suppose r A0 (m) = 0, then m is not a perfect square and the right hand side of equation (53) becomes −2 ∑ ψ 2 (A) log |Ψ m (τ A , τ A )| . A∈Cl(F ) By Kronecker’s identity, this is the same as ⎛ −2 ∑ A∈Cl(F ) ∑ ψ 2 (A) ⎜ ⎝ log |Ψ f4m−t 2 (τ A )| ⎟ ⎠ . t∈Z t 2 ≤4m Thus, this case of Theorem 5.1 is a consequence of Theorem 9.1. The plan of the proof goes as follows. First, notice that both sides of equation (102) are additive with respect to f(z). So it suffices to prove the theorem when f(z) = f d (z) for all d ≥ 0 and d ≡ 3 or 0 modulo 4. For convenience, we denote (103) a(d, ψ) := 〈f lift,θ d , g ψ 〉 reg . Then, we define a function Φ(z) analogous to G D (z) in [29], where the inner product between G D (z) and a weight k+1/2 level 4 eigenform g(z) gives special values of L-function associated to the Shimura lift of g(z). For us, Φ(z) is non-holomorphic, transforms with weight 3/2 and level 4, and satisfies Kohnen’s plus space condition. Since there is no holomorphic cusp form of weight 3/2, level 4 satisfying the plus space condition, the inner product between the holomorphic projection of (almost) Φ(z) and the Poincaré series with an s-variable giving rise to such cusp forms is zero as s approaches zero. This gives us the identity relating finite ⎞
MOCK-MODULAR FORMS OF WEIGHT ONE 47 linear combinations of r + ψ (n) to the limit of an infinite sum. The finite linear combination will be exactly the right hand side of equation (101). Then, we will relate the limit of the infinite sum to Green’s function, hence Ψ fd (z), via the counting argument in §8. In some sense, the proof is in the same spirit as Zagier’s proof of Borcherds theorem in [44]. The following lemma is analogous to Lemma 6.6 and relates a(d, ψ) with the limit of an infinite sum involving r ψ (n). Lemma 9.2. Let d ≥ 1 be an integer congruent to 0 or 3 modulo 4. Then we have ( a(0, ψ)H(d) ∑ ( ) ( ) ) (104) a(d, ψ) − = lim δ(k)ϱ s−1 −1 + k2 −pd + k 2 r ψ , H(0) s→1 pd 4 k∈Z where the function ϱ s (µ) is defined by ϱ s (µ) := ∫ ∞ 1 du (µu + 1) 1+s 2 u , µ > 0. Proof. Let ˜g ψ (z) ∈ M − 1 (p) be a mock-modular form with shadow g ψ (z) and ord ∞˜g ψ (z) ≥ − p 4 . Forms satisfying these conditions exist by Proposition 4.2. The restriction on the order of ˜g ψ (z) at infinity simplifies the proof and is not important. But the requirement that ˜g ψ (z) be in the minus space is crucial for the validity of the statement. Denote ĝ ψ (z) ∈ H − 1 (p) the associated harmonic Maass form and define the function Φ(z) by Φ(z) = Tr 4p 4 ((ĝ ψ |U p )(4z)θ(pz)), where Tr 4p 4 is the trace down operator from level 4p to level 4. The function Φ(z) transforms like a weight 3/2 modular form of level 4 and should be compared to G D (z) in [29], which was defined by applying the trace down operator Tr 4D 4 to the product of the holomorphic weight k Eisenstein series and θ(|D|z). If one replaces the weight k holomorphic Eisenstein series by the weight one real-analytic Eisenstein series with an s variable, and consider the derivative with respect to s, then it is something quite similar to this Φ(z). Using the coset representatives Γ 0 (4p)\Γ 0 (4) = p−1 ⊔ µ=0 ( 1 4 1 ) ( 1 µ 1 ) ⊔ ( 1 1) , and the fact that (ĝ ψ |U p )(z) = −i(ĝ ψ |W p )(z), We can calculate the Fourier expansion of Φ(z) = Tr 4p 4 ((ĝ ψ |U p )(4z)θ(pz)) as Φ(z) = ∑ n∈Z(b(n, y) + a(n))q n , b(n, y) = − ∑ a(n) = ∑ k∈Z k 2 −4m=pn δ(k)r + ψ δ(k)r ψ (m)β 1 (m, 4y p ( pn − k 2 Since p ≡ 3 (mod 4), Φ(z) satisfies Kohnen’s plus space condition and n ≡ 0 or 3 (mod 4). Since ord ∞˜g ψ (z) ≥ −p/4, the right hand side of equation (101) equals to a(d) for f(z) = f d (z). 4 ) . ) ,
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
Page 31 and 32: Similar to the proof of Theorem 5.1
Page 39 and 40: where T := (T kk ′) 0≤k,k ′
Page 45: MOCK-MODULAR FORMS OF WEIGHT ONE 45

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

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