"모듈라 형식(modular forms)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 30개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
| − | + | * [[푸앵카레 상반평면 모델|푸앵카레 상반평면]]에서 정의된 해석함수 | |
| + | * 모듈라 성질과 cusp에서의 푸리에전개를 가짐 | ||
| + | * 정수론에서 많은 중요한 역할 | ||
| − | |||
| − | < | + | ===기호=== |
| + | * <math>\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}</math> | ||
| + | * [[모듈라 군(modular group)]] <math>\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 \}</math> | ||
| + | * <math>\operatorname{PSL}(2,\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})/\{\pm I\}</math> acts on <math>\mathbb{H}</math> by | ||
| + | :<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math> | ||
| + | for <math>\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})</math> | ||
| + | * <math>SL(2, \mathbb Z)</math> is generated by <math>S</math> and <math>T</math> | ||
| + | :<math>S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} </math> | ||
| + | :<math>S: \tau\mapsto -1/\tau,T: \tau\mapsto \tau+1</math> | ||
| − | |||
| − | |||
| − | |||
| − | <math>\ | + | ==모듈라 형식== |
| + | ;def | ||
| + | A holomorphic function <math>f:\mathbb{H}\to \mathbb{C}</math> is a modular form of weight <math>k</math> (w.r.t. <math>SL(2, \mathbb Z)</math>) if | ||
| + | # <math>f \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{k} f(\tau)</math> | ||
| + | # <math>f</math> is "holomorphic at the cusp", i.e. it has a Fourier expansion of the following form | ||
| + | :<math> | ||
| + | f(\tau)=\sum_{n=0}^{\infty}c(n)e^{2\pi i n \tau} | ||
| + | </math> | ||
| − | + | ||
| − | + | ==예== | |
| + | * [[격자의 세타함수|even unimodular 격자의 세타함수]] | ||
| + | * [[판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)]] | ||
| + | :<math>\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots</math> | ||
| − | |||
| − | ( | + | ===아이젠슈타인 급수=== |
| + | * [[아이젠슈타인 급수(Eisenstein series)]] | ||
| + | * for an integer <math>k\geq 2</math>, define | ||
| + | :<math> | ||
| + | G_{2k}(\tau) : =\sum_{(m,n)\in \mathbb{Z}^2\backslash{(0,0)}}\frac{1}{(m+n\tau )^{2k}} | ||
| + | </math> | ||
| + | * Eisenstein series : normalization of <math>G_{2k}</math> | ||
| + | :<math>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)</math> | ||
| + | where <math>\zeta</math> denotes the Riemann zeta function and <math>\sigma_r(n)=\sum_{d|n}d^r</math> | ||
| + | * this is a modular form of weight <math>2k</math> | ||
| + | * for example | ||
| + | :<math>E_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}=1 + 240 q + 2160 q^2 + \cdots </math> | ||
| + | :<math>E_6(\tau)=1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n}=1 - 504 q - 16632 q^2 - \cdots </math> | ||
| − | |||
| − | + | ||
| − | + | ==구조 정리== | |
| + | ;정리 | ||
| + | <math>M_k</math> be the space of modular forms of weight <math>k</math> and <math>M:=\bigoplus_{k\in \mathbb{Z}_{\geq 0}} M_k</math>. We have | ||
| + | :<math>M=\mathbb{C}[E_4,E_6]</math> | ||
| + | * 차원 생성 함수 | ||
| + | :<math> | ||
| + | \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 | ||
| + | </math> | ||
| + | * 가령 <math>\{E_6^2, \Delta\}</math>는 <math>M_{12}</math>의 기저가 된다. 여기서 <math>\Delta=E_4^3-E_6^2</math> | ||
| − | + | ==메모== | |
| + | :<math>d(\frac{az+b}{cz+d})=\frac{(acz+ad-acz-bc)}{(cz+d)^2}dz=(cz+d)^{-2}dz</math> | ||
| + | * 마르틴 아티클러 (Martin Eichler)는 다음과 같은 말을 남김 | ||
| + | <blockquote> | ||
| + | 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. | |
| + | </blockquote> | ||
| − | + | ||
| − | + | ==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== | |
| + | * {{학술용어집|url=modular}} | ||
| − | |||
| − | + | ||
| − | + | ==사전 형태의 자료== | |
| + | * http://ko.wikipedia.org/wiki/보형형식 | ||
| + | |||
| − | * [ | + | ==리뷰논문, 에세이, 강의노트== |
| + | * Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on <math>SL_2(\mathbb{Z})</math>] | ||
| + | * Vaughan, [http://www.personal.psu.edu/users/r/c/rcv4/567c09.pdf modular forms I], [http://www.personal.psu.edu/users/r/c/rcv4/567c10.pdf modular forms II] | ||
| − | |||
| − | * [http:// | + | ==관련논문== |
| + | * Kevin Buzzard, Computing weight one modular forms over <math>\C</math> and <math>\Fpbar</math>, 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. | ||
2020년 12월 28일 (월) 02:21 기준 최신판
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
- \(f \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{k} f(\tau)\)
- \(f\) 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
\[\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots\]
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
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
수학용어번역
- modular - 대한수학회 수학용어집
사전 형태의 자료
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.