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


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.


Modular functions Class fields Conductor 

Mathematics Subject Classification (2000)

11F11 11R04 11R37 14H55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen, I., Yui, N.: Singular values of Thompson series. In: Groups, Difference Sets and Monster, pp. 255–326. de Gruyter (1995) Google Scholar
  2. 2.
    Cox David, A.: Primes of the Form x 2+ny 2. Wiley, New York (1989) Google Scholar
  3. 3.
    Harada, K.: Moonshine of finite groups. Ohio State University (Lecture note) Google Scholar
  4. 4.
    Hong, K.J., Koo, J.K.: Generation of class fields by the modular function j 1,12. Acta Arith. 93, 257–291 (2000) MATHMathSciNetGoogle Scholar
  5. 5.
    Hong, K.J., Koo, J.K.: The modular function j 1,6 and its application to quadratic forms and class fields. Preprint Google Scholar
  6. 6.
    Ishida, N., Ishii, N.: The equations for modular function fields of principal congruence subgroups of prime level. Manuscripta Math. 90, 271–285 (1996) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ishida, N., Ishii, N.: The equation of the modular curve X 1(N) derived from the equation of the modular curve X(N). Tokyo J. Math. 22(1), 167–175 (1999) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ishii, N.: Construction of generators of modular function fields. Math. Jpn. 28, 65–68 (1983) Google Scholar
  9. 9.
    Kim, C.H., Koo, J.K.: Arithmetic of the modular function j 1,4. Acta Arith. 84, 129–143 (1998) MATHMathSciNetGoogle Scholar
  10. 10.
    Kim, C.H., Koo, J.K.: Arithmetic of the modular function j 1,8. Ramanujan J. 4(3), 317–338 (2000) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lang, S.: Algebra. Addison-Wesley, Reading (1984) MATHGoogle Scholar
  12. 12.
    Lang, S.: Elliptic Functions. Springer, Berlin (1987) MATHGoogle Scholar
  13. 13.
    Lang, S.: Algebraic Number Theory. Springer, Berlin (1994) MATHGoogle Scholar
  14. 14.
    Rankin, R.: Modular Forms and Functions. Cambridge University Press, Cambridge (1977) MATHGoogle Scholar
  15. 15.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Jpn. No. 11. Tokyo Princeton (1971) Google Scholar
  16. 16.
    Walker Peter, L.: Elliptic Functions. Wiley, New York (1996) MATHGoogle Scholar

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

Personalised recommendations