자코비 형식

정의

• 2변수 함수로서의 정의

\begin{align*} % \theta_1(z;\tau) \theta_{11}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} \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) , % \\ % & = % 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^{\frac{1}{2} \left( n + \frac{1}{2} \right)^2} \, \E^{2 \pi i \left( n+\frac{1}{2} \right) z} = \theta_2(z;\tau) , \\[2mm] % % \theta_3 (z;\tau) \theta_{00} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n z} = \theta_3 (z;\tau) , \\[2mm] % % \theta_0 (z;\tau) \theta_{01} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n \left( z+\frac{1}{2} \right) } = \theta_4(z;\tau) , \end{align*}

모듈라 성질

\begin{equation} \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}

 =
 \sqrt{ \frac{i}{\tau} } \, \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}\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} .

\end{equation}

자코비 형식

• 다음을 만족시키는 함수 $\phi: \mathbb{H}\times \mathbb{C}\to \mathbb{C}$ 를 자코비 형식(k : weight, m : index)이라 한다

$$ \phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right) = (c\tau+d)^ke^{\frac{2\pi i mcz^2}{c\tau+d}}\phi(\tau,z) $$ 여기서 ${a\ b\choose c\ d}\in SL_2(\mathbb{Z})$ $\lambda, \mu\in \mathbb{Z}$에 대하여 $$ \phi(\tau,z+\lambda\tau+\mu) = e^{-2\pi i m(\lambda^2\tau+2\lambda z)}\phi(\tau,z) $$

• 푸리에 전개

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

메모

• http://www.math.mcgill.ca/goren/Montreal-Toronto/Victoria.pdf

관련된 항목들

• 자코비 세타함수
• 하이젠베르크 군과 대수

사전 형태의 참고자료

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