The Ramanujan Journal

, Volume 24, Issue 1, pp 67–83 | Cite as

Singular values of some modular functions



For an integer N greater than 5 and a triple \({\mathfrak{a}}=[a_{1},a_{2},a_{3}]\) of integers with the properties 0<a i N/2 and a i a j for ij, we consider a modular function \(W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}\) for the modular group Γ 1(N), where ℘(z;L τ ) is the Weierstrass ℘-function relative to the lattice L τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples \({\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]\) and a pair of non-negative integers F=[m,n], we define a modular function \(T_{{\mathfrak{A}},F}\) for the group Γ 0(N) as the trace of the product \(W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}\) to the modular function field of Γ 0(N). In this article, we study the integrality of singular values of the functions \(W_{\mathfrak{a}}\) and \(T_{{\mathfrak{A}},F}\) by using their modular equations. We prove that the functions \(T_{{\mathfrak{A}},F}\) for suitably chosen \({\mathfrak{A}}\) and F generate the modular function field of Γ 0(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values \(T_{{\mathfrak{A}},F}(\tau)\) for suitably chosen \({\mathfrak{A}}\) and F generate ring class fields. Further, we study the class polynomial of \(T_{{\mathfrak{A}},F}\) for Schertz N-system.


Singular value Modular form Modular equation 

Mathematics Subject Classification (2000)

11F03 11G15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cox, D.: Primes of the Form x 2+ny 2. Wiley-Interscience/Wiley, New York (1989) Google Scholar
  2. 2.
    Enge, A., Schertz, R.: Constructing elliptic curves over finite fields using double eta-quotients. J. Théor. Nr. Bordx. 16, 555–568 (2004) MATHMathSciNetGoogle Scholar
  3. 3.
    Gee, A.: Class invariants by Shimura’s reciprocity law. J. Théor. Nr. Bordx. 11, 45–72 (1999) MATHMathSciNetGoogle Scholar
  4. 4.
    Gee, A., Stevenhagen, P.: Generating class fields using Shimura reciprocity. In: Algorithmic Number Theory. Springer LNCS, vol. 1423, pp. 441–453. Springer, Berlin (1998) CrossRefGoogle Scholar
  5. 5.
    Ishida, N., Ishii, N.: Generators and defining equation of the modular function field of the group Γ 1(N). Acta Arith. 101.4, 303–320 (2002) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ishii, N.: Rational expression for J-invariant function in terms of generators of modular function fields. Int. Math. Forum 2(38), 1877–1894 (2007) Google Scholar
  7. 7.
    Lang, S.: Elliptic Functions. Addison-Wesley, Reading (1973) MATHGoogle Scholar
  8. 8.
    Schertz, R.: Weber’s class invariants revisited. J. Théor. Nr. Bordx. 14(1), 325–343 (2002) MATHMathSciNetGoogle Scholar
  9. 9.
    Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami-Shoten/Princeton University Press, Hiroshima/Princeton (1971) MATHGoogle Scholar
  10. 10.
    Silverman, J.: The Arithmetic of Elliptic Curves. Springer, New York, Berlin, Heidelberg, Tokyo (1986) MATHGoogle Scholar
  11. 11.
    Stevenhagen, P.: Hilbert’s 12th problem, complex multiplication and Shimura reciprocity. In: Class Field Theory—Its Centenary and Prospect, Tokyo, 1998. Adv. Stud. Pure Math., vol. 30, pp. 161–176. Math. Soc. Japan, Tokyo (2001) Google Scholar
  12. 12.
    Yoshimura, S., Comuta, A., Ishii, N.: N-systems, class polynomials of double eta-quotients and singular values of j-invariant function. Int. Math. Forum 4(8), 367–376 (2009) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Liberal Arts and SciencesOsaka Prefecture UniversityOsakaJapan
  2. 2.Graduate School of ScienceOsaka Prefecture UniversityOsakaJapan

Personalised recommendations