# 자코비 형식

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• 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

$$$\begin{pmatrix} \theta_{11}\left( z; \tau+1 \right) \\ \theta_{10}\left( z; \tau+1 \right) \\ \theta_{00}\left( z; \tau+1 \right) \\ \theta_{01}\left( z; \tau+1 \right) \end{pmatrix} = \sqrt{ \frac{1}{i} } \, \begin{pmatrix} i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & e^{\frac{i \pi }{4}} \\ 0 & 0 & e^{\frac{i \pi }{4}} & 0 \\ \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}= \begin{pmatrix} e^{\frac{i \pi }{4}} & 0 & 0 & 0 \\ 0 & e^{\frac{i \pi }{4}} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}$$$

$$$\begin{pmatrix} \theta_{11}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{10}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{00}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\ \theta_{01}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \end{pmatrix} = \sqrt{ \frac{\tau}{i} } \, \E^{ \pi i \frac{z^2}{\tau}} \, \begin{pmatrix} -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \, \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}$$$

• weight k=1/2이고, index m=1/2인 벡터 자코비 형식의 예이다

## 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.[1]
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.[2]
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.[2]
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.[3]
5. The Transformation Law Jacobi forms are complex functions on H C which are invariant under an action of the Jacobi group.[3]
6. Notation Let Jk,m() denote the vector space of all Jacobi forms with weight k and index m on a congruence subgroup .[3]
7. The Jacobi-Eisenstein series Ek,m is a Jacobi form on SL2(Z).[3]
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.[4]
9. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).[5]
10. Building Jacobi forms.[6]
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).[7]
12. In Theorem 1.1, we prove that V + is a quantum Jacobi form with respect to the congruence subgroup 0(4) SL2(Z).[7]
13. R ei t22zt cosh(t) dt, and , are the familiar holomorphic modular and Jacobi forms dened in (2.3).[7]
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 .[7]
15. Jacobi forms of lattice index have applications in the theory of reective modular forms and that of vertex operator algebras, among other areas.[8]
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.[8]
17. This thesis concerns a certain generalization of elliptic modular forms called Jacobi forms of lattice index.[8]
18. Interest in Jacobi forms has increased in recent years due to their numerous applications to number theory, algebraic geometry and string theory.[8]

## 메타데이터

### Spacy 패턴 목록

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