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

수학노트
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imported>Pythagoras0
(새 문서: ==introduction== * {{수학노트|url=리만_세타_함수}} ==computational resource== * https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view)
 
 
(사용자 2명의 중간 판 8개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==introduction==
 
==introduction==
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* 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>
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* <math>r(Q, m)</math> : number of <math>X\in \Z^n</math> such that <math>Q(X) = m</math>
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* theta function of <math>Q</math>
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:<math>
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\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}}
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</math>
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* we can use the Riemann theta function to evaluate the above
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:<math>
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\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)}}
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</math>
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:<math>
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\theta_Q(\tau)=\Theta(0,2A_{Q}\tau)
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</math>
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;thm
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* set <math>\det Q := \det (2A_Q)</math>
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* 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
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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>,
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:<math>
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\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)
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</math>
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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>
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 +
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==related items==
 
* {{수학노트|url=리만_세타_함수}}
 
* {{수학노트|url=리만_세타_함수}}
  
 +
 +
==references==
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* Iwaniec, Topics in classical automorphic forms 174p. Equation (10.39)
  
  
 
==computational resource==
 
==computational resource==
 
* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view
 
* https://drive.google.com/file/d/1EKNk3vLdkHiOFxTldZ5cwrSP6mab8-3D/view
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[[분류:migrate]]
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 +
==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q7783550 Q7783550]
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===Spacy 패턴 목록===
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* [{'LOWER': 'theta'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'lattice'}]

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


references

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


computational resource

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'theta'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'lattice'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Theta_function_of_a_quadratic_form&oldid=51885"