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

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10 W. DUKE AND Y. LI the continuous spectrum of − ˜∆ ′ 1. So for any f ∈ ˜D 1 (M, ν), the contribution of the Eisenstein series to the spectral expansion of f is 1 4π ∑ ι ∫ ∞ −∞ 〈f(·), E ι (·, 1 2 + ir)〉E ι(z, 1 2 + ir)dr. If f is smooth, then the integral in r converges absolutely and uniformly for z in any fixed compact subset of H [35, Satz 12.3]. Applying the completeness theorem [35, Satz 7.2 ], we have the spectral expansion for any smooth f ∈ ˜D 1 (M, ν) in the form ∞∑ (19) f(z) = e n 〉e n (z) + n=1〈f, ∑ ι 1 4π ∫ ∞ −∞ 〈f(·), E ι (·, 1 2 + ir)〉E ι(z, 1 2 + ir)dr, where the sum over n ∈ N and the integral in r both converge uniformly and absolutely for z in any compact set of H. By setting f(z) = y 1/2 Q m (z, s) and comparing P m (z, s) to Q m (z, s) as in [31], we can deduce the following proposition. Proposition 2.1. For any positive integer m, the Poincaré series Q m (z, s) and P m (z, s) have analytic continuation to Re(s) > 0. At s = 1/2, P m (z, s) has at most a simple pole. Furthermore, the residues of P m (z, s) at s = 1/2 generate S 1 (M, ν). Proof. First, we will prove the analytic continuation Q m (z, s). For any m > 0, the function ˜Q m (z, s) := y 1/2 Q m (z, s) is square integrable for Re(s) > 1, hence contained in ˜D 1 (M, ν). So we can write out its spectral expansion (20) ˜Q m (z, s) = D(z, s) + C(z, s), D(z, s) := ∞∑ 〈 ˜Q m (·, s), e n (·)〉e n (z), n=1 C(z, s) := 1 4π ∑ ι ∫ ∞ −∞ 〈 ˜Q m (·, s), E ι (·, 1 2 + ir)〉E ι(z, 1 2 + ir)dr, where D(z, s) and C(z, s) are the contributions from the discrete and continuous spectrum of − ˜∆ ′ 1 respectively. By Satz 5.2 and Satz 5.5 in [35], e n (z) has eigenvalue λ n ∈ [1/4, ∞) under − ˜∆ ′ 1. If λ n = 1/4, then y −1/2 e n (z) is in S 1 (M, ν). Since each eigenvalue has finite multiplicity, we can let R ≥ 0 such that {y −1/2 e n (z) : 1 ≤ n ≤ R} is an orthonormal basis of S 1 (M, ν). Note S 1 (M, ν) could be empty, in which case R = 0. For n ∈ N, let t n = √ λ n − 1/4, s n = 1/2 + it n . We can use equation (66) in [20], the bound M µ,ν (y) = O µ,ν (e y ) as y → ∞ and the vanishing property of cusp forms at all cusps to write e n (z) = ∞∑ u=−∞,u≠0 c n,u W sgn(u) ,s 1 (4π|u|y)e 2πiux , 2 n− 2 where W µ,ν (z) is the W -Whittaker function and c n,u are constants. Note if t n = 0, i.e. y −1/2 e n (z) is a holomorphic cusp form, then W sgn(u)/2,sn−1/2(4π|u|y) = (4π|u|y) 1/2 e −2π|u|y and
MOCK-MODULAR FORMS OF WEIGHT ONE 11 c n,u = 0 for u ≤ 0. Applying the Rankin-Selberg unfolding trick to 〈 ˜Q m (·, s), e n (·)〉 gives us ∫ ∞ 〈 ˜Q m (·, s), e n (·)〉 = (4π|m|) 1/2 e −2π|m|y (4π|m|y) s−1 c n,m W sgn(m) 0 = (4π|m|) 1/2 Γ ( s − 1 c − it ) ( 2 n Γ s − 1 + it ) 2 n (21) n,m ( ) Γ s − sgn(m) 2 2 ,s n− 1 2 (4π|m|y) dy y The last step uses the Mellin transform of the W -Whittaker function [2, eq. (8b)] and the substitution s n = 1/2 + it n . When Re(s) > 1, the sum defining D(z, s) is absolutely convergent [35, Satz 8.1 ], since y 1/2 Q m (z, s) ∈ ˜D 1 (M, ν) for Re(s) > 1. When 0 < Re(s) ≤ 1, we can write D(z, s) as D(z, s) = (4π|m|) ∑ 1/2 Γ ( s + 1 − 1 c − it ) ( 2 n Γ s + 1 − 1 + it ) 2 n s − sgn(m) n,m ( ) 2 e n (z) · n∈N Γ s + 1 − sgn(m) (s − 1 2 2 )2 + t 2 n Since t 2 n = λ n − 1/4 and ∑ n>M λ−2 n converges [35, Satz 8.1], we can apply Cauchy-Schwarz inequality to see that the sum on the right hand side above converges absolutely on compact subsets of {s ∈ C : Re(s) > 0, s ≠ 1/2}. At s = 1/2, the first R terms in the sum produce a simple pole since t n = 0 for all 1 ≤ n ≤ R. The rest of the sum still converges absolutely. So the right hand side above gives the analytic continuation of D(z, s) to Re(s) > 0. For the continuous spectrum, the contribution C(z, s) can be treated similarly. For any cusp ι, we can write the Fourier expansion of E ι (z, s) at infinity in the following form (for ι = ∞, see [20, eq. (76)’]) E ι (z, s) = y s + ψ ι (s)y 1−s + ∑ ψ ι,m (s)W sgn(m) ,s− 1 (4π|m|y)e 2πimx , 2 2 m≠0 where ψ ι (s) and ψ ι,m (s) are products of gamma factors and Selberg-Kloosterman zeta functions. It is well-known that The Eisenstein series can be analytically continued in s to the whole complex plane. When Re(s) > 1/2, the poles of E ι (z, s) are in the interval s ∈ (1/2, 1] [35, Satz 10.3 ]. On the line Re(s) = 1/2, E ι (z 0 , s) is holomorphic in s for any fixed z 0 ∈ H [35, Satz 10.4]. So both ψ ι (s) and ψ ι,m (s) admit analytic continuation to Re(s) > 0 and are holomorphic on Re(s) = 1/2. Using the same unfolding trick above, we can evaluate 〈 ˜Q m (·, s), E ι (·, 1 + ir)〉 = ψ ( 1 2 ι,m + ir) (4π|m|) 1/2 Γ ( s − 1 − ir) Γ ( s − 1 2 ( ) + ir) 2 . 2 Γ s − sgn(m) 2 Then C(z, s) can be written as C(z, s) = ∑ ι (22) = ∑ ι ∫ 1 ∞ 4π −∞ ∫ 1 ∞ 4π −∞ ( ψ 1 ι,m + ir) (4π|m|) 1/2 Γ ( s − 1 − ir) Γ ( s − 1 2 ( ) + ir) 2 E 2 ι (z, 1 + ir)dr Γ s − sgn(m) 2 2 ( ψ 1 ι,m + ir) (4π|m|) 1/2 Γ ( s + 1 − 1 − ir) Γ ( s + 1 − 1 ( 2 ) + ir) 2 2 Γ s + 1 − sgn(m) 2 (s − sgn(m) ) 2 · (s − 1 − ir)(s − 1 + ir) · E ι(z, 1 + ir)dr. 2 2 2
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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