Theta function of a quadratic form - 편집 역사

2021년 2월 17일 (수) 10:02에 Pythagoras0님의 편집

2021-02-17T10:02:25Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T08:10:18Z

메타데이터: 새 문단

2020년 11월 16일 (월) 12:27에 Pythagoras0님의 편집

2020-11-16T12:27:56Z

2020년 11월 13일 (금) 10:17에 imported>Pythagoras0님의 편집

2020-11-13T10:17:03Z

2018년 11월 14일 (수) 10:10에 imported>Pythagoras0님의 편집

2018-11-14T10:10:59Z

2018년 11월 14일 (수) 08:35에 imported>Pythagoras0님의 편집

2018-11-14T08:35:13Z

imported>Pythagoras0: /* introduction */

2018-11-14T03:02:03Z

introduction

imported>Pythagoras0: /* introduction */

2018-11-14T01:46:21Z

introduction

2018년 11월 13일 (화) 14:07에 imported>Pythagoras0님의 편집

2018-11-13T14:07:21Z

imported>Pythagoras0: 새 문서: ==introduction== * {{수학노트|url=리만_세타_함수}} ==computational resource== * https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view

2018-11-13T13:58:40Z

새 문서: ==introduction== * {{수학노트|url=리만_세타_함수}} ==computational resource== * https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view

새 문서

==introduction==
* {{수학노트|url=리만_세타_함수}}

==computational resource==
* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view

@@ 36번째 줄: / 36번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q7783550 Q7783550]

← 이전 판		2020년 12월 28일 (월) 08:10 판
35번째 줄:		35번째 줄:
	* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view		* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q7783550 Q7783550]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Let $Q$ be a positive definite integral quadratic form in $n$ variables, i.e. $Q(X) = X^t A_{Q} X$ for some positive definite half-integral symmetric square matrix $A_{Q}$
+* Let <math>Q</math> be a positive definite integral quadratic form in <math>n</math> variables, i.e. <math>Q(X) = X^t A_{Q} X</math> for some positive definite half-integral symmetric square matrix <math>A_{Q}</math>
-* $r(Q, m)$ : number of $X\in \Z^n$ such that $Q(X) = m$
+* <math>r(Q, m)</math> : number of <math>X\in \Z^n</math> such that <math>Q(X) = m</math>
-* theta function of $Q$
+* theta function of <math>Q</math>
-$$
+:<math>
 \theta_Q(\tau)=\sum_{m=0}^\infty r(Q, m)q^{m} = \sum_{{\mathbf{n}\in{\mathbb Z}^n}}e^{2\pi i\mathbf{n}^{t}A_{Q}\mathbf{n}}
-$$
+</math>
 * we can use the Riemann theta function to evaluate the above
-$$
+:<math>
 \Theta(\mathbf{z},\Omega):=\sum_{{\mathbf{n}\in{\mathbb Z}^n}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}^t\boldsymbol{\Omega}\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}}
-$$
+</math>
-$$
+:<math>
 \theta_Q(\tau)=\Theta(0,2A_{Q}\tau)
-$$
+</math>
 ;thm
-* set $\det Q := \det (2A_Q)$
+* set <math>\det Q := \det (2A_Q)</math>
-* level $N$ of $Q$ : smallest integer $N$ such that $N(2A_Q)^{-1}$ is twice of a half-integral matrix
+* level <math>N</math> of <math>Q</math> : smallest integer <math>N</math> such that <math>N(2A_Q)^{-1}</math> is twice of a half-integral matrix
-Assume that $n$ is even. For $\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)$ with $c\equiv 0 \pmod N$,
+Assume that <math>n</math> is even. For <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)</math> with <math>c\equiv 0 \pmod N</math>,
-$$
+:<math>
 \theta_Q\left(\frac{a\tau+b}{c\tau+d}\right) = \left(\frac{(-1)^{n/2}\det(Q)}{d}\right)(c\tau+d)^{n/2}\theta_Q(\tau)
-$$
+</math>
-i.e., $\theta_Q$ is a modular form of weight $n/2$ with a Dirichlet character w.r.t. $\Gamma_0(N)$
+i.e., <math>\theta_Q</math> is a modular form of weight <math>n/2</math> with a Dirichlet character w.r.t. <math>\Gamma_0(N)</math>

← 이전 판		2020년 11월 13일 (금) 10:17 판
34번째 줄:		34번째 줄:
	==computational resource==		==computational resource==
	* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view		* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view
		+	[[분류:migrate]]

@@ 22번째 줄: / 22번째 줄: @@
 $$
 i.e., $\theta_Q$ is a modular form of weight $n/2$ with a Dirichlet character w.r.t. $\Gamma_0(N)$
-* '''what is $N$?'''
 ==related items==

@@ 26번째 줄: / 26번째 줄: @@
 ==related items==
 * {{수학노트|url=리만_세타_함수}}
 ==computational resource==
 * https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view