Algebraic numbers, transcendental numbers and elliptic curves derived from infinite products

Abstract

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