Singular moduli의 대각합 (traces of singular moduli)

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개요

• singular moduli의 대각합 $$\mathbf{t}(d)$$

$\mathbf{t}(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}J(\alpha_Q)$

• 생성함수는 weight 3/2인 모듈라 형식이 된다

singular moduli의 대각합

정의

$$d\in \mathbb{Z}_{>0}$$에 대하여, $\mathbf{t}(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}J(\alpha_Q)$ 여기서

$w_{Q} = \begin{cases} 2\mbox{ if } Q\sim [a,0,a] \\ 3\mbox{ if } Q\sim [a,a,a] \\ 1\mbox{ otherwise}. \end{cases}$

생성함수

• weight이 3/2인 모듈라 형식 $$g_1$$을 생각하자

$g_1(z)=\theta_1(\tau)\frac{E_4(4\tau)}{\eta(4\tau)^6}=q^{-1}-2+248q^3-492q^4+4119q^7-7256q^8+\cdots$ 이 때, $\theta_{1}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2},$ $E_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n},$ $\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n}),$

• $$B(d)$$는 $$g_1(z)$$의 푸리에 계수

$g_1(z)=\sum_{d \geq -1} B(d)q^d$

정리 (Zagier)

다음이 성립한다 $\mathbf{t}(d)=-B(d)$

테이블

\begin{array}{c|c} d & \mathbf{t}(d) \\ \hline -1 & 1 \\ 0 & 2 \\ 3 & -248 \\ 4 & 492 \\ 7 & -4119 \\ 8 & 7256 \\ 11 & -33512 \\ 12 & 53008 \\ 15 & -192513 \\ 16 & 287244 \\ 19 & -885480 \\ 20 & 1262512 \\ 23 & -3493982 \\ 24 & 4833456 \\ 27 & -12288992 \\ 28 & 16576512 \\ 31 & -39493539 \\ 32 & 52255768 \\ 35 & -117966288 \\ 36 & 153541020 \\ 39 & -331534572 \\ 40 & 425691312 \\ 43 & -884736744 \\ 44 & 1122626864 \\ 47 & -2257837845 \\ 48 & 2835861520 \\ \end{array}

메모

• Kaneko, M., The Fourier coefficients and the singular moduli of the elliptic modular function j(τ), Mem. Fac.Eng. Design, Kyoto Inst. Tech. 19 (1996), 1–5.
• Kaneko, M., Traces of singular moduli and the Fourier coefficients of the elliptic modular function j(τ), CRM Proceedings and Lecture Notes 19 (1999), 173–176.
• Goro Shimura established in his series of works the general principle that, the “arithmeticity” of modular forms (in far general setting) induced from the algebraicity of Fourier coefficients, and the one induced from the algebraicity of values at CM (complex multiplication) points, are equivalent.