The Ramanujan Journal

, Volume 22, Issue 3, pp 293–313 | Cite as

An explicit height bound for the classical modular polynomial

  • Reinier Bröker
  • Andrew V. Sutherland


For a prime l, let Φ l be the classical modular polynomial, and let h l ) denote its logarithmic height. By specializing a theorem of Cohen, we prove that \(h(\Phi_{l})\le 6l\log l+16l+14\sqrt{l}\log l\). As a corollary, we find that h l )≤6llog l+18l also holds. A table of h l ) values is provided for l≤3600.


Modular polynomials Height bounds j-function 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, D.R., Apostol, T.M.: The evaluation of Ramanujan’s sum and generalizations. Duke Math. J. 11, 211–216 (1953) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1974) MATHGoogle Scholar
  3. 3.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, Berlin (1976) MATHGoogle Scholar
  4. 4.
    Bateman, P.T., Diamond, H.G.: Analytic Number Theory. World Scientific, Singapore (2004) MATHGoogle Scholar
  5. 5.
    Brisebarre, N., Philibert, G.: Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Ramanujan Math. Soc. 20(4), 255–282 (2005) MATHMathSciNetGoogle Scholar
  6. 6.
    Bröker, R., Lauter, K., Sutherland, A.V.: Modular polynomials via isogeny volcanoes. arXiv:1001.0402
  7. 7.
    Charles, D., Lauter, K.: Computing modular polynomials. LMS J. Comput. Math. 8, 195–204 (2005) MATHMathSciNetGoogle Scholar
  8. 8.
    Cohen, H., Dress, F., El Marraki, M.: Explicit estimates for summatory functions linked to the Möbius μ-function. Funct. Approx. 37, 51–63 (2007) MATHGoogle Scholar
  9. 9.
    Cohen, P.: On the coefficients of the transformation polynomials for the elliptic modular function. Math. Proc. Camb. Philos. Soc. 95, 389–402 (1984) MATHCrossRefGoogle Scholar
  10. 10.
    Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory: Proceedings of a Conference Honor of A.O.L. Atkin. AMS/IP Studies in Advances Mathematics, vol. 7, pp. 21–76 (1998) Google Scholar
  11. 11.
    Enge, A.: Computing modular polynomials in quasi-linear time. Math. Comput. 78, 1809–1824 (2009) Google Scholar
  12. 12.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading (1994) MATHGoogle Scholar
  13. 13.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford (1979) MATHGoogle Scholar
  14. 14.
    Hölder, O.: Zur Theorie der Kreisteilungsgleichung k m(x)=0. Prace Mat. Fiz. 43, 13–23 (1936) Google Scholar
  15. 15.
    Lang, S.: Elliptic Functions, 2nd edn. Springer, Berlin (1987) MATHGoogle Scholar
  16. 16.
    Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62, 26–29 (1955) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV. VEB Deutscher Verlag der Wissenschaften (1963) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Brown UniversityProvidenceUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations