Jacobi's theta function from a representation theoretic viewpoint
abstract
- title: Jacobi's theta function from a representation theoretic viewpoint
- Jacobi introduced his theta functions to develop the theory of elliptic functions. Weil's approach to theta functions opened up the way to study them from a representation theoretic point of view. This involves the Heisenberg group, the Stone-von Neumann theorem and the Weil representation of the metaplectic group. I will give an introduction to this topic focusing on the classical transformation properties of theta functions.
Jacobi theta function
- $\tau\in \mathcal{H}_1$, $z\in \mathbb{C}$
$$ \theta (z,\tau)= \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n z},\, q=e^{2\pi i \tau} $$
- for $\gamma=\left(
\begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\in SL_2(\mathbb{Z})$ and $ac,bd$ even, we have $$ \theta\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right) = \zeta_{\gamma}(c\tau+d)^{1/2}\exp(\frac{\pi i cz^2}{c\tau+d})\theta(z,\tau) $$
overview
- $g\in \mathbb{Z}$, $g\geq 1$
- Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
- Schrodinger representation of $H(2g, \mathbb{R})$ on $\mathcal{H}=L^2(\mathbb{R}^g)$
- Stone-von Neumann theorem induces an action of $Sp(2g,\mathbb{R})$ on $\mathcal{H}$
- but this is only a projective representation
- we can turn it into a genuine representation of the metaplectic group and we call it the Weil representation
- a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwartz space
- a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$, where $\mathcal{H}_{-\infty}$ is the space conjugate linear continuous maps from $\mathcal{H}_{\infty}$ to $\mathbb{C}$
- let $\mathbf{x}=(x_1,x_2)$ and $\underline{\mathbf{x}}=\Omega x_1+x_2$
- $\theta(\underline{\mathbf{x}},\Omega)$ appears as pairing
$$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \mathbf{x})\theta(\underline{\mathbf{x}},\Omega) $$
- modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$
Heisenberg group
- Let $V$ be a $2g$-dimensional symplectic space
- we can set $V=(\mathbb{R}^{2g},A)$, where $A$ is the form $A(x,y)=^tx_1y_2-^tx_2y_1$
- Heisenberg group $H(2g, \mathbb{R})$ : central extension of $V$ by $S^1$
- note that $\psi(x,y)=\exp(\pi i A(x,y))$ is a 2-cocycle
반주기성(quasi-periodicity)
- $\Omega\in \mathcal{H}_g$ 대하여 격자 $\Lambda_{\Omega}=\mathbb{Z}^g+\Omega \mathbb{Z}^g\subset \mathbb{C}^g$를 정의할 수 있다
- $\Theta(\mathbf{z},\Omega)$는 $\Lambda_{\Omega}$에 대하여 반주기성을 갖는다
- 정리
$\mathbf{a},\mathbf{b}\in \mathbb{Z}^g,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathcal{H}_g$라 하자. 다음이 성립한다. $$ \Theta (\mathbf{z}+\Omega \mathbf{a}+\mathbf{b},\Omega)=\exp(-\pi i\cdot \mathbf{a}^t \Omega a-2\pi i \mathbf{a}^t\mathbf{z})\Theta(\mathbf{z},\Omega) $$
모듈라 성질
지겔 모듈라 군
- 지겔 모듈라 군 $\Gamma_g:=\operatorname{Sp}(2g,\R)\cap \operatorname{GL}(2g,\mathbb{Z})$
- 행렬 $\gamma=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g$는 다음의 조건을 만족해야 한다
$$ \begin{align} A^tC=C^tA \\ B^tD=D^tB \\ A^tD-C^tB= I_g \end{align} $$
- 지겔 상반 공간 $\mathcal{H}_g$
$$ \mathcal{H}_g=\left\{\Omega \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \Im \Omega>0 \right\} $$
- 사교군 $\Gamma_g$ 은 $\mathcal{H}_g$에 다음과 같이 작용
$$ \Omega\mapsto \gamma(\Omega)=(A\Omega +B)(C\Omega + D)^{-1} $$
- $C\Omega + D$는 가역이고, $\Im{\gamma(\Omega)}>0 $임을 확인
이구사 부분군과 모듈라 성질
- 이구사 부분군 $\Gamma_{1,2}:=\{\gamma\in \Gamma_g|Q(\gamma \mathbf{x})=Q(\mathbf{x}) \pmod 2\}$, 여기서 $\mathbf{x}=(\mathbf(x_1),\mathbf(x_2))\in \mathbb{Z}^g\times \mathbb{Z}^g=\mathbb{Z}^{2g}$, $Q(\mathbf{x})=\mathbf(x_1)^t \cdot\mathbf(x_2)$
- $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$는 $A^tC, B^tD$의 대각성분이 짝수라는 사실과 동치
- 정리
이구사 부분군의 원소 $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$에 대하여 다음이 성립한다 $$ \Theta \left(((C\Omega + D)^{-1})^t \mathbf{z}, (A\Omega+B)(C\Omega + D)^{-1}\right)=\zeta_{\gamma}\det(C\Omega+D)^{1/2}\exp(\pi i\cdot ^t\mathbf{z}(C\Omega+D)^{-1}C\mathbf{z})\Theta(\mathbf{z},\Omega),\,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathcal{H}_g $$ 여기서 $\zeta_\gamma$는 $\gamma$에 의존하는 적당한 8-th root of unity