Journal of the
Korean Mathematical Society JKMS

Let $Q(n,1)$ be the set of even unimodular positive definite integral quadratic forms in $n$-variables. Then $n$ is divisible by $8$. For $A[X]$ in $Q(n,1)$, the theta series $\theta_{A}(z)=\sum_{X\in\mathbb Z^n}e^{\pi izA[X]}$ ($z\in\mathfrak H$ the complex upper half plane) is a modular form of weight $n/2$ for the congruence group $\Gamma_{1}(8)=\{\delta\in SL_{2}(\mathbb Z) ~|~ \delta\equiv \left(\begin{smallmatrix} 1 & * \\ 0 & 1 \end{smallmatrix} \right)~mod~ 8\}$. If $n\geq 24$ and $A[X]$, $B[X]$ are two quadratic forms in $Q(n,1)$, the quotient $\theta_{A}(z)/\theta_{B}(z)$ is a modular function for $\Gamma_{1}(8)$. Since we identify the field of modular functions for $\Gamma_{1}(8)$ with the function field $K(X_{1}(8))$ of the modular curve $X_{1}(8)=\Gamma_{1}(8)\backslash\mathfrak H^{*}$ ($\mathfrak H^{*}$ the extended plane of $\mathfrak H$) with genus $0$, we can express it as a rational function of $j_{1,8}$ over $\mathbb C$ which is a field generator of $K(X_{1}(8))$ and defined by $j_{1,8}(z)=\theta_{3}(2z)/\theta_{3}(4z)$. Here, $\theta_{3}$ is the classical Jacobi theta series.

Keywords: positive definite quadratic forms, theta series, modular forms

MSC numbers: 11F11, 11E12