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

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18 W. DUKE AND Y. LI Proof. The necessity part follows from Proposition 3.4. Since h(z) is a cusp form, the summation in (44) is only over n < 0. When ˜g(z) ∈ M −ɛ 1 (p) is modular, the sufficiency follows from an application of Serre duality [5, §3] (also see [8, Theorem 6]). When ˜g(z) is mock-modular, let ∑ n
MOCK-MODULAR FORMS OF WEIGHT ONE 19 For convenience, we write E(τ, s) for E 1 (τ, s). Kronecker’s first limit formula states that (49) ζ(2s)E(τ, s) = π s − 1 + 2π ( γ − log(2) − log( √ y|η(τ)| 2 ) ) + O(s − 1), where γ is the Euler constant. Using (49) and Rankin-Selberg unfolding trick, we can relate the inner product between dihedral newforms to logarithm of u A as follows. Proposition 4.1. Let ψ, ψ ′ be characters of Cl(F ), with ψ non-trivial. When ψ ′ = ψ or ψ, we have (50) 〈g ψ , g ψ ′〉 = 1 ∑ ψ 2 (A) log |u A |. 12 Otherwise, 〈g ψ , g ψ ′〉 = 0. A∈Cl(F ) Proof. Since p is prime, the Eisenstein series E p (z, s) has a simple pole at s = 1 with residue , which is independent of z. So we have the relationship 3 π(p+1) 3 π(p + 1) · 〈g ψ, g ψ ′〉 = Res s=1 g ψ (z)g ψ ′(z)E p (z, s)y ∫Γ dxdy . 0 (p)\H y 2 Now, we can use the Rankin-Selberg method to unfold the right hand side and obtain ∫ g ψ (z)g ψ ′(z)E p (z, s)y dxdy = Γ(s) ∑ r ψ (n)r ψ ′(n) y 2 (4π) s n s Γ 0 (p)\H Let ρ ψ : Gal(Q/Q) → GL 2 (C) be the representation induced from ψ. Then it is also the one attached to g ψ (z) via Deligne-Serre’s theorem. Up to Euler factors at p, the right hand side is L(s, ρ ψ ⊗ ρ ψ ′), the L-function of the tensor product of the representations ρ ψ and ρ ψ ′. From the character table of the dihedral group D 2H(p) , we see that ρ ψ ⊗ ρ ψ ′ = ρ ψψ ′ ⊕ ρ ψψ ′. So when ψ ′ ≠ ψ or ψ ′ , the L-function L(s, ρ ψ ⊗ρ ψ ′) is holomorphic at s = 1 and 〈g ψ , g ψ ′〉 = 0. Otherwise, we have ∑ r ψ (n)r ψ (n) = ζ(s)L(s, χ p)L(s, ρ ψ 2) , n s ζ(2s)(1 + p −s ) n≥1 Putting these together, we obtain (51) 〈g ψ , g ψ 〉 = p 2π L(1, χ p)L(1, ρ 2 ψ 2). From the theory of quadratic forms, we have √ ∑ pL(s, ρψ 2) = ζ(2s) ψ 2 (A)E(τ A , s). A∈Cl(F ) Since ψ is non-trivial and H(p) is odd, ψ 2 is non-trivial and (49) implies that L(1, ρ ψ 2) = −√ 2π ∑ ψ 2 (A) log( √ y A |η(τ A )| 2 ), p A∈Cl(F ) Along with the class number formula for p > 3 L(1, χ p ) = 2πH(p) w p √ p n≥1 = πH(p) √ 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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