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.

questions

why consider conjugate linear functionals?
precise definition of metapletic group

theta function

Jacobi theta function

$\theta:\mathbb{C}\times \mathcal{H}\to \mathbb{C}$

$$ \theta (z,\tau)= \sum_{n \in \mathbb{Z}} e^{\pi i n^2 \tau} \, \E^{2 \pi i n z},\, \tau\in \mathcal{H}_1,z\in \mathbb{C} $$

$\theta (z+a\tau +b,\tau)=\exp(-\pi i a^2 \tau -2\pi i az)\theta(z,\tau)$
for $\gamma=\left(

\begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\in SL_2(\mathbb{Z})$ and $ac,bd$ even, we have $$ \theta\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right) = \zeta_{\gamma}(c\tau+d)^{1/2}\exp(\frac{\pi i cz^2}{c\tau+d})\theta(z,\tau) $$

Riemann theta function

지겔 상반 공간 $\mathcal{H}_g=\left\{\Omega \in M_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \operatorname{Im}(\Omega) \text{ positive definite} \right\}$
리만세타함수 $\Theta:\mathbb{C}^g\times \mathcal{H}_g\to \mathbb{C}$ 를 다음과 같이 정의

$$ \Theta(\mathbf{z},\Omega):=\sum_{{\mathbf{n}\in{\mathbb Z}^g}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{\Omega}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}} ,\, \Omega\in \mathcal{H}_g,\mathbb{z}\in \mathbb{C}^g $$

$\Omega\in \mathcal{H}_g$ 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 반주기성(quasi-periodicity)을 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. $$ \Theta (\mathbf{z}+\Omega \mathbf{a}+\mathbf{b},\Omega)=\exp(-\pi i\cdot \mathbf{a}^t \Omega a-2\pi i \mathbf{a}^t\mathbf{z})\Theta(\mathbf{z},\Omega) $$

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

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

\[ 1 \rightarrow S^1~\rightarrow~H(2g, \mathbb{R})~\rightarrow~V \rightarrow 0\]

metaplectic group

coverting of the symplectic group

\[ 1 \rightarrow S^1~\rightarrow~\widetilde{Mp}(2g,\mathbb{R})~\overset{\rho}{\rightarrow}~Sp(2g,\mathbb{R}) \rightarrow 1\]

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