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