Theta function of a quadratic form

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

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