모듈라 형식(modular forms) - 편집 역사

2020년 12월 28일 (월) 10:21에 Pythagoras0님의 편집

2020-12-28T10:21:33Z

2020년 11월 14일 (토) 05:24에 Pythagoras0님의 편집

2020-11-14T05:24:13Z

Pythagoras0: section '관련논문' updated

2016-05-20T01:01:26Z

section '관련논문' updated

Pythagoras0: section '관련논문' updated

2016-05-20T00:45:32Z

section '관련논문' updated

Pythagoras0: /* 관련논문 */

2016-02-07T10:15:44Z

Pythagoras0: /* 관련논문 */

2015-08-11T04:39:28Z

Pythagoras0: /* 리뷰논문, 에세이, 강의노트 */

2015-05-25T11:57:13Z

리뷰논문, 에세이, 강의노트

Pythagoras0: /* 구조 정리 */

2014-07-21T22:25:55Z

구조 정리

2014년 7월 21일 (월) 22:24에 Pythagoras0님의 편집

2014-07-21T22:24:24Z

Pythagoras0: /* 리뷰논문, 에세이, 강의노트 */

2013-08-29T12:30:53Z

리뷰논문, 에세이, 강의노트

@@ 26번째 줄: / 26번째 줄: @@
 </math>
-−
 ==예==
-* [[격자의 세타함수|even unimodular 격자의 세타함수]]
+* [[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>
@@ 49번째 줄: / 49번째 줄: @@
-−
 ==구조 정리==
@@ 59번째 줄: / 59번째 줄: @@
 \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>
+* 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==
@@ 70번째 줄: / 70번째 줄: @@
 </blockquote>
-−
 ==역사==
@@ 77번째 줄: / 77번째 줄: @@
-−
 ==관련된 항목들==
@@ 85번째 줄: / 85번째 줄: @@
 * [[격자의 세타함수]]
 * [[헤케 연산자(Hecke operator)]]
-−
 ==매스매티카 파일 및 계산 리소스==
@@ 95번째 줄: / 95번째 줄: @@
-−
-==사전 형태의 자료==
+==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]
+* 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]

@@ 7번째 줄: / 7번째 줄: @@
 ===기호===
-* $\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}$
+* <math>\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}</math>
-* [[모듈라 군(modular group)]] $\Gamma=SL(2, \mathbb Z) = \left \{ \left. \left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\right| a, b, c, d \in \mathbb Z,\ ad-bc = 1 \right \}$
+* [[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>
-* $\operatorname{PSL}(2,\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})/\{\pm I\}$ acts on $\mathbb{H}$ by
+* <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 $\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})$
+for <math>\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})</math>
-* $SL(2, \mathbb Z)$ is generated by $S$ and $T$
+* <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>
@@ 19번째 줄: / 19번째 줄: @@
 ==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
+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>
-# $f$ is "holomorphic at the cusp", i.e. it has a Fourier expansion of the following form
+# <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>
@@ 36번째 줄: / 36번째 줄: @@
 ===아이젠슈타인 급수===
 * [[아이젠슈타인 급수(Eisenstein series)]]
-* for an integer $k\geq 2$, define
+* 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 $G_{2k}$
+* 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 $\zeta$ denotes the Riemann zeta function and $\sigma_r(n)=\sum_{d|n}d^r$
+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 $2k$
+* 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>
@@ 53번째 줄: / 53번째 줄: @@
 ==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
+<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 $M_{12}$의 기저가 된다. 여기서 $\Delta=E_4^3-E_6^2$
+* 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==
@@ 102번째 줄: / 102번째 줄: @@
 ==리뷰논문, 에세이, 강의노트==
-* Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on $SL_2(\mathbb{Z})$]
+* 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]
 ==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, 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.

@@ 107번째 줄: / 107번째 줄: @@
 ==관련논문==
 * 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.

@@ 107번째 줄: / 107번째 줄: @@
 ==관련논문==
 * 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.

@@ 107번째 줄: / 107번째 줄: @@
 ==관련논문==
 * 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.

← 이전 판		2015년 8월 11일 (화) 04:39 판
107번째 줄:		107번째 줄:

	==관련논문==		==관련논문==
		+	* 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.

← 이전 판		2015년 5월 25일 (월) 11:57 판
104번째 줄:		104번째 줄:
	* Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on $SL_2(\mathbb{Z})$]		* Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on $SL_2(\mathbb{Z})$]
	* 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]		* 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==
		+	* 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.

@@ 3번째 줄: / 3번째 줄: @@
 * [[푸앵카레 상반평면 모델|푸앵카레 상반평면]]에서 정의된 해석함수
 * 모듈라 성질과 cusp에서의 푸리에전개를 가짐
-* 별다른 언급이 없을 경우 <math>q=e^{2\pi i\tau}</math> 를 의미함
+* 정수론에서 많은 중요한 역할
+===기호===
-* weight 2k 인 모듈라 형식
+==모듈라 형식==
-* [[모듈라 군(modular group)]]의 원소에 대하여 다음 조건을 만족시킴:<math>f \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} f(\tau)</math><br>  <br>
+;def
+==예==
-*  cusp에서도 해석함수의 성질을 갖도록 해주기 위한 조건:<math>f(\tau) = \sum_{n=0}^\infty a_n e^{2i\pi n\tau}</math><br>
+===아이젠슈타인 급수===
 * [[아이젠슈타인 급수(Eisenstein series)]]
-* [[판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)]]
+* for an integer $k\geq 2$, define
-:<math>\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots</math>
+$$
@@ 37번째 줄: / 52번째 줄: @@
 ==구조 정리==
 ==메모==
 :<math>d(\frac{az+b}{cz+d})=\frac{(acz+ad-acz-bc)}{(cz+d)^2}dz=(cz+d)^{-2}dz</math>
@@ 57번째 줄: / 78번째 줄: @@
 * [[수학사 연표]]
@@ 68번째 줄: / 88번째 줄: @@
 * [[헤케 연산자(Hecke operator)]]
@@ 77번째 줄: / 100번째 줄: @@
 ==사전 형태의 자료==
 * http://ko.wikipedia.org/wiki/보형형식