자코비 형식

개요

• 2변수 함수
• 보형 함수의 예

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

\[ \phi(\tau,z) = \sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)e^{2\pi i (n\tau+rz)}=\sum_{n\ge 0} \sum_{r^2\le 4mn} c(n,r)q^nz^r \]

자코비 세타함수

정의

• \((\tau,z)\in \mathcal{H}\times \mathbb{C}\)에 대하여, 다음과 같이 정의된다 (\(q=e^{\pi i \tau}\))

\[ \begin{align*} % \theta_1(z;\tau) \theta_{11}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{ \left( n+ \frac{1}{2} \right)^2} \, \E^{2 \pi i \left(n+\frac{1}{2} \right) \, \left( z+\frac{1}{2} \right) } = - i \, \theta_1(z; \tau) \\ & = -2 q^{1/4} \sin (\pi z)+2 q^{9/4} \sin (3 \pi z)-2 q^{25/4} \sin (5 \pi z)+2 q^{49/4} \sin (7 \pi z)-2 q^{81/4} \sin (9 \pi z) +\cdots, % \\ % & = % i \, q^{\frac{1}{8}} \, \E^{\pi \I z} \, % \left( % q, \E^{-2 \pi \I z}, \E^{2 \pi \I z} \, q % \right)_\infty \\[2mm] % % \theta_{2}(z;\tau) \theta_{10}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\left( n + \frac{1}{2} \right)^2} \, \E^{2 \pi i \left( n+\frac{1}{2} \right) z} = \theta_2(z;\tau) \\ & = 2 q^{1/4} \cos (\pi z)+2 q^{9/4} \cos (3 \pi z)+2 q^{25/4} \cos (5 \pi z)+2 q^{49/4} \cos (7 \pi z)+2 q^{81/4} \cos (9 \pi z)+\cdots , \\[2mm] % % \theta_3 (z;\tau) \theta_{00} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{n^2} \, \E^{2 \pi i n z} = \theta_3 (z;\tau) \\ & =1+2 q \cos (2 \pi z)+2 q^4 \cos (4 \pi z)+2 q^9 \cos (6 \pi z)+2 q^{16} \cos (8 \pi z)+2 q^{25} \cos (10 \pi z)+\cdots, \\[2mm] % % \theta_0 (z;\tau) \theta_{01} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{ n^2} \, \E^{2 \pi i n \left( z+\frac{1}{2} \right) } = \theta_4(z;\tau) \\ & =1-2 q \cos (2 \pi z)+2 q^4 \cos (4 \pi z)-2 q^9 \cos (6 \pi z)+2 q^{16} \cos (8 \pi z)+2 q^{25} \cos (5 \pi (2 z+1))+\cdots \end{align*} \]

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

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

• 자코비 세타함수
• 하이젠베르크 군과 대수

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxRk94Yk53YlRFdzQ/edit

사전 형태의 참고자료

• http://en.wikipedia.org/wiki/Jacobi_form

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.

노트

말뭉치

1. Meromorphic Jacobi forms appear in the theory of Mock modular forms.
2. IntroductionIn this note we give a geometrical approach to the theory of Jacobi forms, whichwas originated in an analytic way by Eichler and Zagier (cf.
3. To solve it, we determine the divisor of a suitablychosen (meromorphic) Jacobi form of weight k, index m with respect to r.We first recall two elementary lemmas.
4. Notation Let e(x) denote e2ix for x C. Let q = e( ) and = e(z) where H and z C. Jacobi forms are meant to be a natural generalization of Jacobi theta series.
5. The Transformation Law Jacobi forms are complex functions on H C which are invariant under an action of the Jacobi group.
6. Notation Let Jk,m() denote the vector space of all Jacobi forms with weight k and index m on a congruence subgroup .
7. The Jacobi-Eisenstein series Ek,m is a Jacobi form on SL2(Z).
8. We introduce a concept of Jacobi forms and try to explain a various connection with other fields, such as quasi-modular forms and Mock modular forms if time allows.
9. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).
10. Building Jacobi forms.
11. Here we prove that two additional combinatorial functions are quantum Jacobi forms, one of which is a normalized version of the function in (1.1).
12. In Theorem 1.1, we prove that V + is a quantum Jacobi form with respect to the congruence subgroup 0(4) SL2(Z).
13. R ei t22zt cosh(t) dt, and , are the familiar holomorphic modular and Jacobi forms dened in (2.3).
14. (1) Theorem 1.1 shows that the function V + is a quantum Jacobi form of weight 1 2 and (cid:8) (cid:7) 1 0 .
15. Jacobi forms of lattice index have applications in the theory of reective modular forms and that of vertex operator algebras, among other areas.
16. We dene Poincare series for Jacobi forms of lattice index and show that they reproduce Fourier coecients of cusp forms under the Petersson scalar product.
17. This thesis concerns a certain generalization of elliptic modular forms called Jacobi forms of lattice index.
18. Interest in Jacobi forms has increased in recent years due to their numerous applications to number theory, algebraic geometry and string theory.

메타데이터

위키데이터

• ID : Q6119638

Spacy 패턴 목록

• [{'LOWER': 'jacobi'}, {'LEMMA': 'form'}]