"자코비 형식"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) (section '관련논문' updated) |
Pythagoras0 (토론 | 기여) |
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1번째 줄: | 1번째 줄: | ||
==정의== | ==정의== | ||
* 2변수 함수 | * 2변수 함수 | ||
− | * | + | * <math>(\tau,z)\in \mathcal{H}\times \mathbb{C}</math>에 대하여, 다음과 같이 정의된다 (<math>q=e^{\pi i \tau}</math>) |
\begin{align*} | \begin{align*} | ||
% \theta_1(z;\tau) | % \theta_1(z;\tau) | ||
119번째 줄: | 119번째 줄: | ||
==자코비 형식== | ==자코비 형식== | ||
− | * | + | * <math>q=e^{2\pi i \tau}</math>, <math>y=e^{2\pi i z}</math> |
− | * 다음을 만족시키는 함수 | + | * 다음을 만족시키는 함수 <math>\phi: \mathcal{H}\times \mathbb{C}\to \mathbb{C}</math> 를 자코비 형식(k : weight, m : index)이라 한다 |
− | + | :<math> | |
\phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right) = (c\tau+d)^ke^{\frac{2\pi i mcz^2}{c\tau+d}}\phi(\tau,z)=(c\tau+d)^k y^{mc \frac{z}{c\tau+d}}\phi(\tau,z) | \phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right) = (c\tau+d)^ke^{\frac{2\pi i mcz^2}{c\tau+d}}\phi(\tau,z)=(c\tau+d)^k y^{mc \frac{z}{c\tau+d}}\phi(\tau,z) | ||
− | + | </math> 여기서 <math>{a\ b\choose c\ d}\in SL_2(\mathbb{Z})</math> | |
− | + | <math>\lambda, \mu\in \mathbb{Z}</math>에 대하여 | |
− | + | :<math> | |
\phi(\tau,z+\lambda\tau+\mu) = e^{-2\pi i m(\lambda^2\tau+2\lambda z)}\phi(\tau,z)=q^{-m \lambda^2} y^{-2m\lambda}\phi(\tau,z) | \phi(\tau,z+\lambda\tau+\mu) = e^{-2\pi i m(\lambda^2\tau+2\lambda z)}\phi(\tau,z)=q^{-m \lambda^2} y^{-2m\lambda}\phi(\tau,z) | ||
− | + | </math> | |
* 푸리에 전개 | * 푸리에 전개 | ||
− | + | :<math> | |
\phi(\tau,z) = \sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)e^{2\pi i (n\tau+rz)}=\sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)q^nz^r | \phi(\tau,z) = \sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)e^{2\pi i (n\tau+rz)}=\sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)q^nz^r | ||
− | + | </math> | |
==메모== | ==메모== | ||
159번째 줄: | 159번째 줄: | ||
==관련논문== | ==관련논문== | ||
* Nathan C. Ryan, Nicolás Sirolli, Nils-Peter Skoruppa, Gonzalo Tornaría, Computing Jacobi Forms, arXiv:1602.07021 [math.NT], February 23 2016, http://arxiv.org/abs/1602.07021 | * Nathan C. Ryan, Nicolás Sirolli, Nils-Peter Skoruppa, Gonzalo Tornaría, Computing Jacobi Forms, arXiv:1602.07021 [math.NT], February 23 2016, http://arxiv.org/abs/1602.07021 | ||
− | * Labrande, Hugo. “Computing Jacobi’s | + | * Labrande, Hugo. “Computing Jacobi’s <math>\theta</math> in Quasi-Linear Time.” arXiv:1511.04248 [math], November 13, 2015. http://arxiv.org/abs/1511.04248. |
* Bringmann, Kathrin, Larry Rolen, and Sander Zwegers. “On the Fourier Coefficients of Negative Index Meromorphic Jacobi Forms.” arXiv:1501.04476 [hep-Th], January 19, 2015. http://arxiv.org/abs/1501.04476. | * Bringmann, Kathrin, Larry Rolen, and Sander Zwegers. “On the Fourier Coefficients of Negative Index Meromorphic Jacobi Forms.” arXiv:1501.04476 [hep-Th], January 19, 2015. http://arxiv.org/abs/1501.04476. |
2020년 11월 13일 (금) 02:52 판
정의
- 2변수 함수
- \((\tau,z)\in \mathcal{H}\times \mathbb{C}\)에 대하여, 다음과 같이 정의된다 (\(q=e^{\pi i \tau}\))
\begin{align*} % \theta_1(z;\tau)
\theta_{11}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{ \left( n+ \frac{1}{2} \right)^2} \, \E^{2 \pi i \left(n+\frac{1}{2} \right) \, \left( z+\frac{1}{2} \right) } = - i \, \theta_1(z; \tau)
\\ & = -2 q^{1/4} \sin (\pi z)+2 q^{9/4} \sin (3 \pi z)-2 q^{25/4} \sin (5 \pi z)+2 q^{49/4} \sin (7 \pi z)-2 q^{81/4} \sin (9 \pi z) +\cdots, % \\ % & = % i \, q^{\frac{1}{8}} \, \E^{\pi \I z} \, % \left( % q, \E^{-2 \pi \I z}, \E^{2 \pi \I z} \, q % \right)_\infty
\\[2mm] % % \theta_{2}(z;\tau) \theta_{10}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\left( n + \frac{1}{2} \right)^2} \, \E^{2 \pi i \left( n+\frac{1}{2} \right) z} = \theta_2(z;\tau) \\ & = 2 q^{1/4} \cos (\pi z)+2 q^{9/4} \cos (3 \pi z)+2 q^{25/4} \cos (5 \pi z)+2 q^{49/4} \cos (7 \pi z)+2 q^{81/4} \cos (9 \pi z)+\cdots , \\[2mm] % % \theta_3 (z;\tau) \theta_{00} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{n^2} \, \E^{2 \pi i n z} = \theta_3 (z;\tau) \\ & =1+2 q \cos (2 \pi z)+2 q^4 \cos (4 \pi z)+2 q^9 \cos (6 \pi z)+2 q^{16} \cos (8 \pi z)+2 q^{25} \cos (10 \pi z)+\cdots, \\[2mm] % % \theta_0 (z;\tau) \theta_{01} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{ n^2} \, \E^{2 \pi i n \left( z+\frac{1}{2} \right) } = \theta_4(z;\tau) \\ & =1-2 q \cos (2 \pi z)+2 q^4 \cos (4 \pi z)-2 q^9 \cos (6 \pi z)+2 q^{16} \cos (8 \pi z)+2 q^{25} \cos (5 \pi (2 z+1))+\cdots \end{align*} =='"`UNIQ--h-1--QINU`"'모듈라 성질== * 다음과 같은 모듈라 변환 성질을 갖는다 \begin{equation} \begin{pmatrix} \theta_{11}\left( z; \tau+1 \right) \\ \theta_{10}\left( z; \tau+1 \right) \\ \theta_{00}\left( z; \tau+1 \right) \\ \theta_{01}\left( z; \tau+1 \right) \end{pmatrix} = \sqrt{ \frac{1}{i} } \, \begin{pmatrix} i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & e^{\frac{i \pi }{4}} \\ 0 & 0 & e^{\frac{i \pi }{4}} & 0 \\ \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}= \begin{pmatrix} e^{\frac{i \pi }{4}} & 0 & 0 & 0 \\ 0 & e^{\frac{i \pi }{4}} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix} \end{equation} \begin{equation} \begin{pmatrix} \theta_{11}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{10}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{00}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{01}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \end{pmatrix} = \sqrt{ \frac{\tau}{i} } \, \E^{ \pi i \frac{z^2}{\tau}} \, \begin{pmatrix} -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix} \end{equation} * weight k=1/2이고, index m=1/2인 벡터 자코비 형식의 예이다 =='"`UNIQ--h-2--QINU`"'자코비 형식== * \(q=e^{2\pi i \tau}\), \(y=e^{2\pi i z}\) * 다음을 만족시키는 함수 \(\phi: \mathcal{H}\times \mathbb{C}\to \mathbb{C}\) 를 자코비 형식(k : weight, m : index)이라 한다 \[ \phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right) = (c\tau+d)^ke^{\frac{2\pi i mcz^2}{c\tau+d}}\phi(\tau,z)=(c\tau+d)^k y^{mc \frac{z}{c\tau+d}}\phi(\tau,z) \] 여기서 \({a\ b\choose c\ d}\in SL_2(\mathbb{Z})\)
\(\lambda, \mu\in \mathbb{Z}\)에 대하여 \[ \phi(\tau,z+\lambda\tau+\mu) = e^{-2\pi i m(\lambda^2\tau+2\lambda z)}\phi(\tau,z)=q^{-m \lambda^2} y^{-2m\lambda}\phi(\tau,z) \]
- 푸리에 전개
\[ \phi(\tau,z) = \sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)e^{2\pi i (n\tau+rz)}=\sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)q^nz^r \]
메모
- http://www.math.mcgill.ca/goren/Montreal-Toronto/Victoria.pdf
- http://www.cams.aub.edu.lb/events/confs/modular2012/files/choie_1.pdf
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 참고자료
에세이, 리뷰, 강의노트
- Kharchev, S., and A. Zabrodin. ‘Theta Vocabulary I’. arXiv:1502.04603 [math], 25 November 2014. http://arxiv.org/abs/1502.04603.
관련논문
- Nathan C. Ryan, Nicolás Sirolli, Nils-Peter Skoruppa, Gonzalo Tornaría, Computing Jacobi Forms, arXiv:1602.07021 [math.NT], February 23 2016, http://arxiv.org/abs/1602.07021
- Labrande, Hugo. “Computing Jacobi’s \(\theta\) in Quasi-Linear Time.” arXiv:1511.04248 [math], November 13, 2015. http://arxiv.org/abs/1511.04248.
- Bringmann, Kathrin, Larry Rolen, and Sander Zwegers. “On the Fourier Coefficients of Negative Index Meromorphic Jacobi Forms.” arXiv:1501.04476 [hep-Th], January 19, 2015. http://arxiv.org/abs/1501.04476.