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The Ramanujan Journal

, Volume 16, Issue 3, pp 321–337 | Cite as

Singular values of some modular functions and their applications to class fields

  • Kuk Jin Hong
  • Ja Kyung Koo
Article

Abstract

Since the modular curve \(X(5)=\Gamma(5)\backslash\frak H^{*}\) has genus zero, we have a field isomorphism \(\mathcal{K}(X(5))\approx \mathbb{C}(X_{2}(z))\) where X 2(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function j Δ,25(z):=X 2(5z). And, for every integer N≥7 we further generate ray class fields K (N) over K with modulus N just from the two generators X 2(z) and X 3(z) of the function field \(\mathcal{K}(X_{1}(N))\) , which are also the product of Klein forms without using torsion points of elliptic curves.

Keywords

Modular functions Class fields Conductor 

Mathematics Subject Classification (2000)

11F11 11R04 11R37 14H55 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsHankuk Academy of Foreign StudiesKyongki-doKorea
  2. 2.Department of MathematicsKorea Advanced Institute of Science and TechnologyTaejonKorea

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