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

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$Math 33B/1 S00 K. Solutions to Midterm Exam G1, Version B If you ...$

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26 W. DUKE AND Y. LI By (49) and the substitution A = A ′2 , we have lim s→1 4π ∑ A ψ(A)(E(τ √ AB −1 , s)+E(τ √ AB , s)) = −24(ψ(B)+ψ(B)) ∑ A ′ ψ 2 (A ′ ) log | √ y A ′η 2 (τ A ′)|. From (52) and H(p)ϑ B (z) = ∑ ψ ′ ψ ′ (B)g ψ ′(z), it is easy to see that 〈ϑ B , g ψ 〉 = −(ψ(B) + ψ(B)) ∑ A ′ ψ 2 (A ′ ) log | √ y A ′η 2 (τ A ′)|. After substituting these into equation (64) and changing A, B to A ′2 , B ′2 respectively, we obtain equation (53). □ For any m ≥ 1, r ≥ 0 and τ, τ ′ ∈ H, define the level one modular function Ψ ∗ m(z; τ, τ ′ , r) by Ψ ∗ m(z; τ, τ ′ Ψ m (τ, z) , r) := (j(τ ′ ) − j(z)) r In terms of Ψ ∗ m(z; τ, τ ′ , r), we can re-write the right hand side of (53) as −2 ∑ ψ 2 (A ′ ) (log |Ψ ∗ m(τ A ′ B ′; τ A ′ B ′−1, τ A ′ B ′, r B(m))| + r B (m) log |y A ′ B ′j′ (τ A ′ B ′)|) A ′ ∈Cl(F ) The quantity Ψ ∗ −1 m(τ A ′ B ′; τA ′ B , τ ′ A ′ B ′, r B(m)) is well-defined and non-zero, since the order of Ψ m (τ A ′ B ′−1, z) at z = τ A ′ B ′ is exactly r B(m) from the proof above. By equation (47) and the fact that (j ′ (z)) 6 = j(z) 4 (j(z) − 1728) 3 ∆(z), we have σ C ⎛ ⎝ ∏ C ′ ∈Cl(F ) y 6 A ′ B ′j′ (τ A ′ B ′)6 y 6 C ′ ∆(τ C ′) ⎞ ⎛ ⎠ = ⎝ ∏ C ′ ∈Cl(F ) y 6 C −1 A ′ B ′ j ′ (τ C −1 A ′ B ′)6 y 6 C ′ ∆(τ C ′) ⎞ ⎠ ∈ H. Similarly, the modular function Ψ ∗ m(z; τ A ′ B ′−1, τ A ′ B ′, r B(m)), which is defined over H, is sent to Ψ ∗ m(z; τ C −1 A ′ B ′−1, τ C −1 A ′ B ′, r B(m)) under σ C . So if we let ⎛ u m,B (A ′ ) := Ψ ∗ m(τ A ′ B ′; τ A ′ B ′−1, τ A ′ B ′, r B(m)) 6H(p) · ⎝ ⎞ ∏ r B (m) yA 6 ′ B ′j′ (τ A ′ B ′)6 ⎠ yC 6 ∆(τ ′ C ′) , then we can write with u m,B (A ′ ) ∈ H satisfying 〈j lift,B m , g ψ 〉 reg = − 1 3H(p) ∑ A ′ ∈Cl(F ) C ′ ∈Cl(F ) (65) σ C (u m,B (A ′ )) = u m,B (A ′ C −1 ). ψ 2 (A ′ ) log |u m,B (A ′ )| When m = 0, we have j lift,B 0 (z) = ϑ B (z) and ∑ 〈ϑ B , g ψ 〉 = − 1 ψ 2 (A ′ ) log |u 12H(p) A ′ B ′u A ′ B ′−1|, A ′
MOCK-MODULAR FORMS OF WEIGHT ONE 27 where u C ′ is defined in (46) for any C ′ ∈ Cl(F ). So we can let u m,B (A ′ ) = u A ′ B ′u A ′ B ′−1 when m = 0 and equation (65) still holds by (47). Since every modular function of weight one is a linear combination of j m (z), m ≥ 0, we can deduce the following corollary concerning the algebraic property of r + ψ (n) from (60) and Proposition 3.4. Corollary 5.2. Let ˜g ψ (z) = ∑ n∈Z r+ ψ (n)qn ∈ M − 1 (p) be a mock-modular form satisfying condition (i) in Proposition 4.2. For any modular function f(z) = ∑ n∈Z c(f, n)qn of level one with rational Fourier coefficients, there exists β f,B ∈ Q and u f,B (A) ∈ H × independent of the character ψ such that (66) 〈f lift,B , g ψ 〉 reg = ∑ c(f, n) ∑ δ(k)r B (pn − k)r + ψ (k) n∈Z k∈Z ∑ = −β f,B ψ 2 (A) log |u f,B (A)| A∈Cl(F ) and σ C (u f,B (A)) = u f,B (AC −1 ) for any C ∈ Cl(F ). If f has integral Fourier coefficients, then 12H(p)β f,B ∈ Z. 6. Rankin’s method and values of Green’s Function II In this section, we will work out the generalization of Corollary 5.2 to modular functions of prime level N. From Corollary 5.2, we see that linear combinations of r + ψ (n) have the desired algebraic property. To prove Theorem 1.1, we need different linear combinations arising from regularized inner product between g ψ and other f ∈ M !,+ 1 (p). By considering analogues of j lift,B for modular functions of level N, we can obtain these linear combinations, which will be sufficient to solve each r + ψ (n) (see §7). Other than being a step toward the proof of Theorem 1.1, this section is interesting on its own as the analogue of the Gross-Zagier formula with trivial Heegner point divisor in the Jacobian. Also, the interpretation of the regularized inner product in terms of Borcherds lifts continues to hold for modular functions of level N. Let N be a prime number different from p. For a modular function f(z) of level N and B ∈ Cl(F ), one can define f lift,N,B (z) ∈ M 1(p) ! by (67) f lift,N,B (z) := Tr Np p ((f|W N )(pz)ϑ B (Nz)), −1 where W N = ( N ) is the Fricke involution and ϑ B (z) is the weight one theta series corresponding to the class B ∈ Cl(F ). Notice that the form f lift,N,B (z) is the same as f lift,B (z) from §5 when N = 1. The main result of this section is the following generalization of Corollary 5.2. Proposition 6.1. Let f(z) is a modular function of prime level N ≠ p, with rational Fourier coefficients. There exist β f,B ∈ Q and u f,B (A) ∈ H independent of the character ψ such that ∑ (68) 〈f lift,N,B , g ψ 〉 reg = −β f,B ψ 2 (A) log |u f,B (A)|. A∈Cl(F ) For any class C ∈ Cl(F ), the algebraic integer u f,B (A) satisfies σ C (u f,B (A)) = u f,B (AC −1 ) where σ C ∈ Gal(H/F ) is associated to C ∈ Cl(F ) via Artin’s isomorphism. Furthermore, if f has integral principal part Fourier coefficients and satisfies equation (69) below, then the denominator of β f,B only depends on N and p.
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
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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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