리만 세타 함수

개요

• 아벨-야코비 정리에서 야코비 반전 (jacobi inversion) 문제를 해결하기 위해 도입
• $\mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^t=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\}$
• $\Omega\in \mathcal{H}_g$, $\mathbb{z}\in \mathbb{C}^g$
• 리만세타함수 $\Theta: \mathcal{H}_g\times \mathbb{C}^g\to \mathbb{C}$ 를 다음과 같이 정의 ($\mathbf{\nu _1}, \mathbf{\nu _2}\in \mathbb{C}^g$ : characteristic)

$$ \Theta \left[ \begin{array}{c} \mathbf{\nu _1} \\ \mathbf{\nu _2} \\ \end{array} \right] (\Omega ,\mathbf{z}) =\sum_{{\mathbf{n}\in{\mathbb Z}^g}} e^{2 \pi i \left(\frac{1}{2}\left(\mathbf{\nu _1}+ \mathbf{n} \right)\Omega \left(\mathbf{\nu _1}+ \mathbf{n} \right)+ \left(\mathbf{\nu _1}+ \mathbf{n}\right)\left(\mathbf{\nu _2}+\mathbf{z}\right)\right)} $$

• characteristic이 $\mathbf{\nu _1}=\mathbf{\nu _2}=0\in \mathbb{C}^g$인 경우

$$ \Theta \left[ \begin{array}{c} \mathbf{0} \\ \mathbf{0} \\ \end{array} \right] (\Omega ,\mathbf{z})=\sum_{{\mathbf{n}\in{\mathbb Z}^g}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{\Omega}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}} $$

모듈라 성질

사교군

• $\Gamma_g:=\operatorname{Sp}(2g,\Z)=\{M\in \operatorname{GL}(2g,\mathbb{Z})|M^t J_{g} M = J_{g}\}$

여기서 $$ J_{g} =\begin{pmatrix}0 & I_g \\-I_g & 0 \\\end{pmatrix} $$

• $2g\times 2g$ 행렬

$$M=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g,$$ $$ \begin{align} A^tC=C^tA \\ B^tD=D^tB \\ A^tD-C^tB= I_g \end{align} $$

• 지겔 상반 공간 $\mathcal{H}_g$

$$ \mathcal{H}_g=\left\{\tau \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \tau^t=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} $$

• 사교군 $\Gamma_g$ 은 $\mathcal{H}_g$에 다음과 같이 작용

$$ \tau\mapsto \gamma(\tau)=(A\tau +B)(C\tau + D)^{-1} $$

• $C\tau + D$는 가역이고, $\Im{\gamma(\tau)}>0 $임을 확인
• 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

모듈라 변환

정리

$\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$, $\mathbf{z}\in \mathbb{C}^g$, $\tau\in \mathcal{H}_g$에 대하여 다음이 성립한다 $$ \theta \left(((C\tau + D)^{-1})^t \mathbf{z}, (A\tau+B)(C\tau + D)^{-1}\right)=\zeta_{\gamma}\det(C\tau+D)^{1/2}\exp(\pi i\cdot ^t\mathbf{z}(C\tau+D)^{-1}C\mathbf{z})\theta(\mathbf{z},\tau) $$ 여기서 $\zeta_\gamma$는 $\gamma$에 의존하는 적당한 8-th root of unity

예:자코비 세타함수

• 자코비 세타함수와 자코비 형식
• $g=1$인 경우, $q=e^{2\pi i \tau}$

$$ \begin{align*} \theta_{11}(z;\tau) &:= \Theta \left[ \begin{array}{c} 1/2 \\ 1/2 \\ \end{array} \right](\tau ,z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} \left( n+ \frac{1}{2} \right)^2} \, \E^{2 \pi i \left(n+\frac{1}{2} \right) \, \left( z+\frac{1}{2} \right) } \\ \theta_{10}(z;\tau) &:= \Theta \left[ \begin{array}{c} 1/2 \\ 0 \\ \end{array} \right](\tau ,z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} \left( n + \frac{1}{2} \right)^2} \, \E^{2 \pi i \left( n+\frac{1}{2} \right) z} \\ \theta_{00} (z;\tau) &:= \Theta \left[ \begin{array}{c} 0 \\ 0 \\ \end{array} \right](\tau ,z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n z} \\ \theta_{01} (z;\tau) &:= \Theta \left[ \begin{array}{c} 0 \\ 1/2 \\ \end{array} \right](\tau ,z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n \left( z+\frac{1}{2} \right) } \end{align*} $$

역사

• 자코비 fundamenta nova
• 수학사 연표

메모

• Computing Riemann theta functions in Sage with applications
• http://mathoverflow.net/questions/64261/whats-the-difference-between-a-riemann-theta-and-a-siegel-theta-function
• Arizona Winter School 2009: Quadratic Forms
• 자코비의 네 제곱수 정리
• 'singular series'
  • Dickson
  • Mordell
  • Hardy
  • Bateman

관련된 항목들

• 리만 곡면의 주기 행렬과 겹선형 관계 (bilinear relation)
• 자코비 세타함수와 자코비 형식
• 격자의 세타함수
• 모듈라 형식(modular forms)
• 지겔 모듈라 형식

사전 형태의 자료

• https://en.wikipedia.org/wiki/Siegel_modular_form
• Chapter 21 Multidimensional Theta Functions

에세이

• Arnaud Beauville Theta functions, old and new

관련논문

• Beshaj, L., A. Elezi, and T. Shaska. “Theta Functions of Superelliptic Curves.” arXiv:1503.00297 [math], March 1, 2015. http://arxiv.org/abs/1503.00297.
• Matsuda, Kazuhide. “The Algebraic Structure of Certain Theta Constants.” arXiv:1410.4153 [math], October 14, 2014. http://arxiv.org/abs/1410.4153.
• Gelca, Razvan, and Alastair Hamilton. 2014. “The Topological Quantum Field Theory of Riemann’s Theta Functions.” arXiv:1406.4269 [math-Ph], June. http://arxiv.org/abs/1406.4269.