"모듈라 형식(modular forms)"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) |
||
| (같은 사용자의 중간 판 4개는 보이지 않습니다) | |||
| 7번째 줄: | 7번째 줄: | ||
===기호=== | ===기호=== | ||
| − | * | + | * <math>\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}</math> |
| − | * [[모듈라 군(modular group)]] | + | * [[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> | :<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math> | ||
| − | for | + | 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=\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>S: \tau\mapsto -1/\tau,T: \tau\mapsto \tau+1</math> | ||
| 19번째 줄: | 19번째 줄: | ||
==모듈라 형식== | ==모듈라 형식== | ||
;def | ;def | ||
| − | A holomorphic function | + | 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 \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} | f(\tau)=\sum_{n=0}^{\infty}c(n)e^{2\pi i n \tau} | ||
| − | + | </math> | |
| − | + | ||
==예== | ==예== | ||
| − | * [[격자의 세타함수|even unimodular | + | * [[격자의 세타함수|even unimodular 격자의 세타함수]] |
* [[판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)]] | * [[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> | :<math>\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots</math> | ||
| 36번째 줄: | 36번째 줄: | ||
===아이젠슈타인 급수=== | ===아이젠슈타인 급수=== | ||
* [[아이젠슈타인 급수(Eisenstein series)]] | * [[아이젠슈타인 급수(Eisenstein series)]] | ||
| − | * for an integer | + | * 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}} | G_{2k}(\tau) : =\sum_{(m,n)\in \mathbb{Z}^2\backslash{(0,0)}}\frac{1}{(m+n\tau )^{2k}} | ||
| − | + | </math> | |
| − | * Eisenstein series : normalization of | + | * 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> | :<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 | + | 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 | + | * this is a modular form of weight <math>2k</math> |
* for example | * 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_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}=1 + 240 q + 2160 q^2 + \cdots </math> | ||
| 49번째 줄: | 49번째 줄: | ||
| − | + | ||
==구조 정리== | ==구조 정리== | ||
;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> | :<math>M=\mathbb{C}[E_4,E_6]</math> | ||
* 차원 생성 함수 | * 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 | \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> | + | * 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== | ==메모== | ||
| 70번째 줄: | 70번째 줄: | ||
</blockquote> | </blockquote> | ||
| − | + | ||
==역사== | ==역사== | ||
| 77번째 줄: | 77번째 줄: | ||
| − | + | ||
==관련된 항목들== | ==관련된 항목들== | ||
| 85번째 줄: | 85번째 줄: | ||
* [[격자의 세타함수]] | * [[격자의 세타함수]] | ||
* [[헤케 연산자(Hecke operator)]] | * [[헤케 연산자(Hecke operator)]] | ||
| − | + | ||
==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== | ||
| 95번째 줄: | 95번째 줄: | ||
| − | + | ||
| − | ==사전 | + | ==사전 형태의 자료== |
* http://ko.wikipedia.org/wiki/보형형식 | * http://ko.wikipedia.org/wiki/보형형식 | ||
| − | + | ||
==리뷰논문, 에세이, 강의노트== | ==리뷰논문, 에세이, 강의노트== | ||
| − | * Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on | + | * 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. | + | * 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] |
==관련논문== | ==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. | * 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. | * 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.