모듈라 형식(modular forms)

개요

• 푸앵카레 상반평면에서 정의된 해석함수
• 모듈라 성질과 cusp에서의 푸리에전개를 가짐
• 정수론에서 많은 중요한 역할

기호

• \(\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}\)
• 모듈라 군(modular group) \(\Gamma=SL(2, \mathbb Z) = \left \{ \left. \left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\right| a, b, c, d \in \mathbb Z,\ ad-bc = 1 \right \}\)
• \(\operatorname{PSL}(2,\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})/\{\pm I\}\) acts on \(\mathbb{H}\) by

\[\tau\mapsto\frac{a\tau+b}{c\tau+d}\] for \(\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})\)

• \(SL(2, \mathbb Z)\) is generated by \(S\) and \(T\)

\[S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \] \[S: \tau\mapsto -1/\tau,T: \tau\mapsto \tau+1\]

모듈라 형식

def

A holomorphic function \(f:\mathbb{H}\to \mathbb{C}\) is a modular form of weight \(k\) (w.r.t. \(SL(2, \mathbb Z)\)) if

1. \(f \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{k} f(\tau)\)
2. \(f\) is "holomorphic at the cusp", i.e. it has a Fourier expansion of the following form

\[ f(\tau)=\sum_{n=0}^{\infty}c(n)e^{2\pi i n \tau} \]

예

• even unimodular 격자의 세타함수
• 판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)

\[\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots\]

아이젠슈타인 급수

• 아이젠슈타인 급수(Eisenstein series)
• for an integer \(k\geq 2\), define

\[ G_{2k}(\tau) : =\sum_{(m,n)\in \mathbb{Z}^2\backslash{(0,0)}}\frac{1}{(m+n\tau )^{2k}} \]

• Eisenstein series : normalization of \(G_{2k}\)

\[E_{2k}(\tau):=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)\] where \(\zeta\) denotes the Riemann zeta function and \(\sigma_r(n)=\sum_{d|n}d^r\)

• this is a modular form of weight \(2k\)
• for example

\[E_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}=1 + 240 q + 2160 q^2 + \cdots \] \[E_6(\tau)=1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n}=1 - 504 q - 16632 q^2 - \cdots \]

구조 정리

정리

\(M_k\) be the space of modular forms of weight \(k\) and \(M:=\bigoplus_{k\in \mathbb{Z}_{\geq 0}} M_k\). We have \[M=\mathbb{C}[E_4,E_6]\]

• 차원 생성 함수

\[ \sum_{k=0}^{\infty}\dim M_k x^k=\frac{1}{\left(1-x^4\right)\left(1-x^{6}\right)}=1+x^4+x^6+x^8+x^{10}+2 x^{12}+x^{14}+2 x^{16}+2 x^{18}+2 x^{20}+\cdots \]

• 가령 \(\{E_6^2, \Delta\}\)는 \(M_{12}\)의 기저가 된다. 여기서 \(\Delta=E_4^3-E_6^2\)

메모

\[d(\frac{az+b}{cz+d})=\frac{(acz+ad-acz-bc)}{(cz+d)^2}dz=(cz+d)^{-2}dz\]

• 마르틴 아티클러 (Martin Eichler)는 다음과 같은 말을 남김

There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms.

다섯개의 기본적인 산술적 연산이 있다 : 더하기, 빼기, 곱하기, 나누기, 그리고 ... 모듈라 형식.

역사

• 수학사 연표

관련된 항목들

• 아이젠슈타인 급수(Eisenstein series)
• 판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)
• 자코비 세타함수
• 격자의 세타함수
• 헤케 연산자(Hecke operator)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxNGxwUTkwdV9fakE/edit

수학용어번역

• modular - 대한수학회 수학용어집

사전 형태의 자료

• http://ko.wikipedia.org/wiki/보형형식

리뷰논문, 에세이, 강의노트

• Finch, Modular Forms on \(SL_2(\mathbb{Z})\)
• Vaughan, modular forms I, modular forms II

관련논문

• Kevin Buzzard, Computing weight one modular forms over \(\C\) and \(\Fpbar\), arXiv:1205.5077 [math.NT], May 23 2012, http://arxiv.org/abs/1205.5077
• Kevin Buzzard, Alan Lauder, A computation of modular forms of weight one and small level, arXiv:1605.05346 [math.NT], May 17 2016, http://arxiv.org/abs/1605.05346
• Schulze-Pillot, Rainer, and Abdullah Yenirce. “Petersson Products of Bases of Spaces of Cusp Forms and Estimates for Fourier Coefficients.” arXiv:1602.01803 [math], February 4, 2016. http://arxiv.org/abs/1602.01803.
• Bellaiche, Joel, and Kannan Soundararajan. “The Number of Non-Zero Coefficients of Modular Forms (mod P).” arXiv:1508.02095 [math], August 9, 2015. http://arxiv.org/abs/1508.02095.
• Jorgenson, Jay, Lejla Smajlovic, and Holger Then. “Certain Aspects of Holomorphic Function Theory on Some Genus Zero Arithmetic Groups.” arXiv:1505.06042 [math], May 22, 2015. http://arxiv.org/abs/1505.06042.