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

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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.
Publisher
KOREAN MATHEMATICAL SOC
Issue Date
2003-11
Language
English
Article Type
Article
Citation

JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, v.40, no.6, pp.977 - 998

ISSN
0304-9914
URI
http://hdl.handle.net/10203/83716
Appears in Collection
MA-Journal Papers(저널논문)
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