Journal of the
Korean Mathematical Society JKMS

Let $k$ be an imaginary quadratic field, $\frak h$ the complex upper half plane, and let $\tau\in \frak h \cap k$, $p=e^{\pi i \tau}$. In this article, using the infinite product formulas for $g_2$ and $g_3$, we prove that values of certain infinite products are transcendental whenever $\tau$ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of $\prod_{n=1}^\infty (\frac{1-p^{2n-1}}{1+p^{2n-1}})^8$ and $p\prod_{n=1}^\infty (1+p^{2n})^{12}$ when we know $j(\tau)$. And we construct an elliptic curve $ E:y^2 =x^3 + 3x^2 +(3-\frac{j}{256})x+1$ with $j=j(\tau)\neq 0$ and $P=(16^2 p^2 \prod_{n=1}^\infty (1+p^{2n})^{24},0)\in E$.

Keywords: infinite product, transcendental number, elliptic curve

MSC numbers: 11Jxx, 11R04, 11F11