The Ramanujan Journal

, Volume 10, Issue 1, pp 43–50 | Cite as

Congruences Concerning Values of the Modular Functions j n

  • Gwynneth H. Coogan


The j-function j(z) = q−1+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j m (ζ) = (j(ζ) – 744)∨T0(m), where T0(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL2(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function.


modular forms Hecke operators j-function σ function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Asai, M. Kaneko, and H. Ninomiya, “Zeros of certain modularfunctions and an application,” Comm. Math. Univ. St. Pauli 46 (1997) 93–101.MathSciNetGoogle Scholar
  2. 2.
    B. Berndt, P. Bialek, and A.J. Yee, “Formulas of Ramanujan for the power series coefficients of certainquotients of Eisenstein series,” IRMN 21 (2002) 1077–1109.MathSciNetGoogle Scholar
  3. 3.
    J. Bruinier, W. Kohnen, and K. Ono, “The arithmetic of the values of modular functions and the divisorsof modular forms,” Appearing in Compositio Mathematica.Google Scholar
  4. 4.
    N. Koblitz, An Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984.Google Scholar
  5. 5.
    F.K.C. Rankin and H.P.F. Swinnerton-Dyer, “On the zeros of Eisenstein series,” Bull. London Math. Soc. 2 (1970) 169–170.MathSciNetGoogle Scholar
  6. 6.
    J.-P. Serre, “Divisibilité des coefficients des formes modulaires de poids entier,” C.R. Acad. Sci.Paris ser A 279 (1974) 679–682.MathSciNetMATHGoogle Scholar
  7. 7.
    D. Zagier, “Traces of singular moduli,” preprint.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadison

Personalised recommendations