Journal of the
Korean Mathematical Society JKMS

Since the modular curves $X(N)=\Gamma(N)\backslash\frak H^*$ ($N=1,2,3$) have genus $0$, we have field isomorphisms $K(X(1))\approx\Bbb C(J)$, $K(X(2))\approx\Bbb C(\lambda)$ and $K(X(3))\approx\Bbb C(j_3)$ where $J$, $\lambda$ are the classical modular functions of level 1 and 2, and $j_3$ can be represented as the quotient of reduced Eisenstein series. When $N=4$, we see from the genus formula that the curve $X(4)$ is of genus $0$ too. Thus the field $K(X(4))$ is a rational function field over $\Bbb C$. We find such a field generator $j_4(z)=x(z)/y(z)$ ($x(z)=\theta_3(\frac z2), \, y(z)=\theta_4(\frac z2)$ Jacobi theta functions). We also investigate the structures of the spaces \hskip0.1cm $M_k(\Gamma(4)), \, S_k(\Gamma(4)),$ $M_{\frac{k}{2}}(\widetilde{\Gamma}(4))$ and $S_{\frac{k}{2}}(\widetilde{\Gamma}(4))$ in terms of $x(z)$ and $y(z)$. As its application, we apply the above results to quadratic forms.

Keywords: modular functions, Jacobi theta functions, half integral modular forms, reduced $\wp$-division values, Fricke functions, quadratic forms

MSC numbers: 11F11, 11E12, 11R04, 14H55