"Theta function of a quadratic form"의 두 판 사이의 차이

2021년 2월 17일 (수) 02:02 기준 최신판

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 \(A_{Q}\)
\(r(Q, m)\) : number of \(X\in \Z^n\) such that \(Q(X) = m\)
theta function of \(Q\)

\[ \theta_Q(\tau)=\sum_{m=0}^\infty r(Q, m)q^{m} = \sum_{{\mathbf{n}\in{\mathbb Z}^n}}e^{2\pi i\mathbf{n}^{t}A_{Q}\mathbf{n}} \]

we can use the Riemann theta function to evaluate the above

\[ \Theta(\mathbf{z},\Omega):=\sum_{{\mathbf{n}\in{\mathbb Z}^n}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}^t\boldsymbol{\Omega}\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}} \] \[ \theta_Q(\tau)=\Theta(0,2A_{Q}\tau) \]

thm

set \(\det Q := \det (2A_Q)\)
level \(N\) of \(Q\) : smallest integer \(N\) such that \(N(2A_Q)^{-1}\) 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 \(c\equiv 0 \pmod N\), \[ \theta_Q\left(\frac{a\tau+b}{c\tau+d}\right) = \left(\frac{(-1)^{n/2}\det(Q)}{d}\right)(c\tau+d)^{n/2}\theta_Q(\tau) \] i.e., \(\theta_Q\) is a modular form of weight \(n/2\) with a Dirichlet character w.r.t. \(\Gamma_0(N)\)

related items

틀:수학노트

references

Iwaniec, Topics in classical automorphic forms 174p. Equation (10.39)

computational resource

https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view

메타데이터

위키데이터

ID : Q7783550

Spacy 패턴 목록

[{'LOWER': 'theta'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'lattice'}]

@@ 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
-Assume that $n$ is even. For $\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)$ with $c|N$,
+* set <math>\det Q := \det (2A_Q)</math>
-$$
+* level <math>N</math> of <math>Q</math> : smallest integer <math>N</math> such that <math>N(2A_Q)^{-1}</math> is twice of a half-integral matrix
+Assume that <math>n</math> is even. For <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)</math> with <math>c\equiv 0 \pmod N</math>,
+:<math>
 \theta_Q\left(\frac{a\tau+b}{c\tau+d}\right) = \left(\frac{(-1)^{n/2}\det(Q)}{d}\right)(c\tau+d)^{n/2}\theta_Q(\tau)
-$$
+</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>
-* '''what is $N$?'''
-* '''what is $\det Q$?'''
 ==related items==
 * {{수학노트|url=리만_세타_함수}}
+==references==
+* Iwaniec, Topics in classical automorphic forms 174p. Equation (10.39)
 ==computational resource==
 * https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q7783550 Q7783550]
+===Spacy 패턴 목록===
+* [{'LOWER': 'theta'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'lattice'}]