Singular moduli의 대각합 (traces of singular moduli)

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

• $\mathcal{Q}_d$는 $-d=b^2-4ac$를 만족하는 정수계수 이변수 이차형식(binary integral quadratic forms) $Q=[a,b,c]=ax^2+bxy+cy^2$의 집합
• 모듈라군 $\Gamma=PSL(2,\mathbb{Z})$은 $\mathcal{Q}_d$에 작용
• 각각의 $Q$에 대하여, 자기동형군 $\Gamma_{Q}$을 생각, $w_{Q}=|\Gamma_{Q}|$,

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

• $J(z):=j(z)-744$, $j$는 타원 모듈라 j-함수 (elliptic modular function, j-invariant)
• $\alpha_Q\in \mathbb{C}$는 허수부가 양수인 $ax^2+bx+c=0$의 해

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.

관련된 항목들

• 후르비츠-크로네커 유수
• 타원 모듈라 j-함수의 singular moduli

매스매티카 파일 및 계산 리소스

• https://drive.google.com/file/d/0B8XXo8Tve1cxVjJJbElVcm9VT2M/edit

리뷰, 에세이, 강의노트

• Kaneko, Traces of singular moduli and the Fourier coefficients of the elliptic modular function j(τ)
• Duke and Jenkins, Notes on singular moduli and modular forms

관련논문

• Beneish, Lea, and Claire Frechette. “$p$-Adic Properties of Certain Half-Integral Weight Modular Forms.” arXiv:1509.00383 [math], September 1, 2015. http://arxiv.org/abs/1509.00383.
• Bruinier, Jan Hendrik, and Yingkun Li. “Heegner Divisors in Generalized Jacobians and Traces of Singular Moduli.” arXiv:1508.07112 [math], August 28, 2015. http://arxiv.org/abs/1508.07112.
• “Traces of CM Values of Modular Functions : Journal Fur Die Reine Und Angewandte Mathematik (Crelles Journal).”http://www.degruyter.com/view/j/crll.2006.2006.issue-594/crelle.2006.034/crelle.2006.034.xml.
• http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0965-11.pdf
• Zagier, Don. Traces of singular Moduli in Motives, Polyogarithms and Hodge Theory (Part II: Hodge Theory) http://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf