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

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22 W. DUKE AND Y. LI for any class A, B, C ∈ Cl(F ). Since M ′ P is a non-singular matrix with integer entries, its inverse, say (α i,j ) 1≤i,j≤d+ , has rational entries. Thus, r + ψ (−n i) can be written as δ(n i )r + ψ (−n i) = ∑ A (H(p)−1)/2 ∑ ψ 2 (A) α i,j log |u A (A j )| j=1 with α i,j ∈ Q for all 1 ≤ i, j ≤ d + . From this, we can choose β ∈ Q such that u(n i , A) = (H(p)−1)/2 ∏ j=1 u A (A j ) −α i,j/δ(n i )β ∈ O × H . α i,j 2H(p)β ∈ Z and Then r + ψ (−n i) satisfies conditions (ii) and (iii) in the proposition. Furthermore, the unit u(n i , A) is an H(p) th power in O × H . Finally, apply Propositions 4.1 and 3.4 to the Eisenstein series E 1 (z) and the cusp form g ψ (z), we get ∑ δ(n)R p (n)r + ψ (−n) = 0. n≥0 So we can write r + ψ (0) = −β log |u(0, A)| with u(0, A) = d + ∏ i=1 which satisfies condition (iii) as well. u(n i , A) −δ(n i)R p(n i )/H(p) ∈ O × H , □ 5. Rankin’s method and values of Green’s function I From the proof of Proposition 4.2, we saw that information about r + ψ (n), n ≤ 0 can be retrieved from their various linear combinations coming from inner products. In this section, we will consider a different linear combination of the coefficients r + ψ (n). They can be viewed as the regularized inner product between g ψ (z) and certain weakly holomorphic forms in M !,+ 1 (p) lifted from modular functions of level one. Following the same procedure as in [24], we will show in Theorem 5.1 that these linear combination of r + ψ (n)’s can be expressed as the twisted sum of logarithms of the modular polynomial evaluated at certain CM points. This result is more or less a twisted version of the Gross-Zagier formula at level one, where there is no weight two cusp form or L-function. Also, the information provided by the theorem, as formulated in Corollary 5.2, will be a stepping stone to the proof of Theorem 1.1. As a modular form on the degenerated Hilbert modular surface, the modular polynomial can be expressed in terms of a Borcherds lift of modular functions of level one. Thus, it is natural to phrase Theorem 5.1 in terms of a regularized inner product and this Borcherds lift. This prepares the way for §6, where we treat the level N case. For m ≥ 0, let j m (z) = q −m + O(q) be the unique modular function of level one with a pole of order m at infinity, e.g. j 0 (z) = 1 and j 1 (z) = j(z) − 744. For each B ∈ Cl(F ), choose an associated CM point τ B = x B + iy B ∈ H and define j lift,B m (z) ∈ M !,+ 1 (p) by jm lift,B (z) := j m (pz)ϑ B (z).
MOCK-MODULAR FORMS OF WEIGHT ONE 23 Let M 2 (Z) be the space of 2 × 2 matrices with integer coefficients. For m ≥ 1, (τ 1 , τ 2 ) ∈ (PSL 2 (Z)\H) 2 , define the function Ψ m (τ 1 , τ 2 ) by ∏ Ψ m (τ 1 , τ 2 ) := (j(τ 1 ) − j(γτ 2 )) . γ∈SL 2 (Z)\M 2 (Z) det(γ)=m It is the value of the modular polynomial ϕ m (X, Y ) at X = j(τ 1 ), Y = j(τ 2 ), and also the Borcherds lift of j m (z) to a function on the degenerated Hilbert modular surface. The main result of this section is as follows. Theorem 5.1. Let ψ be a non-trivial character of Cl(F ) and m ≥ 1. Then ∑ ( log |Ψm (τ (53) 〈jm lift,B , g ψ 〉 reg = −2 lim ψ 2 (A ′ A ) ′ B ′−1 + ɛ, τ A ′ B ′)| + ) ɛ→0 r B (m) log |y A ′ B ′−1| , A ′ ∈Cl(F ) where B ′ ∈ Cl(F ) is the unique class such that B ′2 = B. It will be clear in the proof that equation (53) is independent of the choice of τ B . Before stating the proof, we need a crucial result in [23]. For two distinct points τ j = x j + iy j ∈ H, the invariant hyperbolic distance d(τ 1 , τ 2 ) between them is defined by (54) cosh d(τ 1 , τ 2 ) = (x 1 − x 2 ) 2 + y1 2 + y2 2 . 2y 1 y 2 Notice for d(τ 1 , τ 2 ) = d(γτ 1 , γτ 2 ) for all γ ∈ SL 2 (R). The Legendre function of the second kind Q s−1 (t) is defined by ∫ ∞ Q s−1 (t) = (t + √ t 2 − 1 cosh u) −s du, Re(s) > 1, t > 1, (55) 0 Q 0 (t) = 1 log ( ) 1 + 2 2 t−1 For two distinct points τ 1 , τ 2 ∈ PSL 2 (Z)\H, the following convergent series defines the automorphic Green’s function (56) G s (τ 1 , τ 2 ) := ∑ g s (τ 1 , γτ 2 ), Re(s) > 1, where γ∈PSL 2 (Z) (57) g s (τ 1 , τ 2 ) := −2Q s−1 (cosh d(τ 1 , τ 2 )). Recall that E(τ, s) is defined in (48) and ϕ 1 (s) is the coefficient of y 1−s in the Fourier expansion of E(τ, s). Proposition 5.1 in [23] tells us that for distinct τ 1 , τ 2 ∈ PSL 2 (Z)\H, the values of the j-function is related to the values of the automorphic Green’s function by (58) log |j(τ 1 ) − j(τ 2 )| 2 = lim s→1 (G s (τ 1 , τ 2 ) + 4πE(τ 1 , s) + 4πE(τ 2 , s) − 4πϕ 1 (s)) − 24. Apply the m th Hecke operator to τ 2 on both sides gives us [23, eq.(5.2)] ( G m ) (59) log |Ψ m (τ 1 , τ 2 )| 2 s (τ 1 , τ 2 ) + 4πσ 1 (m)E(τ 1 , s)+ = lim s→1 4πm s σ 1−2s (m)E(τ 2 , s) − 4πσ 1 (m)ϕ 1 (s) where σ ν (m) = ∑ d|m dν and G m s (τ 1 , τ 2 ) = ∑ γ∈SL 2 (Z)\M 2 (Z) det(γ)=m g s (τ 1 , γτ 2 ). − 24σ 1 (m),
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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