The Ramanujan Journal

, Volume 33, Issue 1, pp 121–130 | Cite as

Special values of generalized \(\mathbf{\lambda}\) functions at imaginary quadratic points

  • Noburo Ishii


We study a modular function Λ k, that is one of generalized λ functions. We show that Λ k, and the modular invariant function j generate the modular function field with respect to the modular subgroup Γ 1(N). Further, we prove that Λ k, is integral over Z[j]. From this result we obtain that a value of Λ k, at an imaginary quadratic point is an algebraic integer and generates a ray class field over a Hilbert class field.


Modular function Special value 

Mathematics Subject Classification (2010)

11F03 11G15 


  1. 1.
    Cox, D.: Primes of the Form x 2+ny 2. A Wiley-Interscience Publication. Wiley, New York (1989) Google Scholar
  2. 2.
    Gee, A.: Class invariants by Shimura’s reciprocity law. J. Théor. Nr. Bordx. 11, 45–72 (1999) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    Ishii, N., Kobayashi, M.: Singular values of some modular functions. Ramanujan J. 24, 67–83 (2011). doi: 10.1007/s11139-010-9249-y CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Lang, S.: Elliptic Functions. Addison-Wesley, London (1973) MATHGoogle Scholar
  6. 6.
    Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami-Shoten and Princeton University Press, Tokyo (1971) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.KyotoJapan

Personalised recommendations