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

Abstract

Let k be an imaginary quadratic field, h the complex upper half plane, and let tau is an element of h boolean AND k, p = e(piitau). 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 Pi(n=1)(infinity)(1-p(2n-1)/1+p(2n-1))(8) and pPi(n=1)(infinity) (1 + p(2n))(12) when we know j (tau). And we construct an elliptic curve E : y(2) = x(3) + 3x(2) + (3 - j/2563)x + 1 with i = j(tau) not equal 0 and P = (16(2)P(2) Pi(n=1)(infinity)(1 + p(2n)) (24), 0) is an element of E.