Modular equations for congruence subgroups of genus zero

  • Bumkyu ChoEmail author


It is well known that the modular equation for the full modular group satisfies several properties such as Kronecker’s congruence relation. In this article, we prove that the modular equation for any congruence subgroup \(\varGamma _1(m) \cap \varGamma _0(mN)\) of genus zero satisfies similar properties including Kronecker’s congruence relation.


Modular equations Modular polynomials Kronecker’s congruence relation 

Mathematics Subject Classification




The author would like to express his sincere thanks to the anonymous referee for a careful reading of the manuscript.


  1. 1.
    Cais, B., Conrad, B.: Modular curves and Ramanujan continued fraction. J. Reine Angew. Math. 597, 27–104 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chen, I., Yui, N.: Singular values of Thompson series. In: Arasu, K.T., Dillon, J.F., Harada, K., Sehgal, S., Solomon, R. (eds.) Groups, Difference Sets, and the Monster (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications, vol. 4, pp. 255–326. de Gruyter, Berlin (1996)Google Scholar
  3. 3.
    Cho, B.: Recurrence relations satisfied by the traces of singular moduli for \(\Gamma _0(N)\) (submitted for publication)Google Scholar
  4. 4.
    Cho, B., Koo, J.K., Park, Y.K.: Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction. J. Number Theory 129, 922–948 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cho, B., Koo, J.K., Park, Y.K.: On Ramanujan’s cubic continued fraction as a modular function. Tohoku Math. J. 62, 579–603 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cox, D.A.: Primes of the form \(x^2 + ny^2\), 2nd edn. Wiley, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Diamond, F., Im, J.: Modular forms and modularcurves. In: Seminar on Fermat’s Last Theorem. CMS Conf. Proc., vol. 17. Am. Math. Soc., Providence, RI (1995)Google Scholar
  8. 8.
    Lang, S.: Elliptic Functions. Graduate Texts in Mathematics, vol. 112, 2nd edn. Springer, New York (1987)CrossRefGoogle Scholar
  9. 9.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. In: Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, vol. 11. Iwanami Shoten Publishers/Princeton University Press, Tokyo/Princeton, NJ (1971)Google Scholar
  10. 10.
    Washington, L.C.: Introduction to cyclotomic fields, Graduate Texts in Mathematics 83. Springer, New York (1982)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsDongguk University-SeoulSeoulRepublic of Korea

Personalised recommendations