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

Let k be an imaginary quadratic field, h-fraktur sign the complex upper half plane, and let τ ∈ h-fraktur sign ∩ k, p = e ^πiτ. In this article, using the infinite product formulas for g₂ and g₃, we prove that values of certain infinite products are transcendental whenever τ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of Π_n=1^∞(1-p²ⁿ-1/1+p²ⁿ-1) ⁸ and pΠ_n=1^∞(1 + p²ⁿ) ¹² when we know j(τ). And we construct an elliptic curve E : y² = x³ + 3x² + (3 - j/256)x + 1 with j = j(τ) ≠ 0 and P = (16²p²Π_n=1 ^∞(1 + p²ⁿ)²⁴, 0) ∈ E.