Skip to main content
SpringerLink
Log in

Completely replicable functions and symmetries

  • Published:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Aims and scope Submit manuscript

Abstract

Completely replicable functions play an important role in number theory and finite group theory, in particular the Monstrous Moonshine. In this paper, we give a characterization of completely replicable functions by certain symmetries.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander, D., Cummins, C., McKay, J., Simons, C.: Completely replicable functions. In: Groups, Combinatorics and Geometry, London Mathematical Society Lecture Note Series, vol. 165, pp. 87–95 (1992)

  2. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405–444 (1992)

    Article  MathSciNet  Google Scholar 

  3. Carnahan, S.: Generalized moonshine I: genus-zero functions. Algebra Number Theory 4, 649–679 (2010)

    Article  MathSciNet  Google Scholar 

  4. Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)

    Article  MathSciNet  Google Scholar 

  5. Cummins, C.J.: Some comments on replicable functions. In: Modern Trends in Lie Algebra Representation Theory, pp. 48–55. Kingston, ON (1993)

  6. Cummins, C.J., Gannon, T.: Modular equations and the genus zero property of moonshine functions. Invent. Math. 129(3), 413–443 (1997)

    Article  MathSciNet  Google Scholar 

  7. Cummins, C.J., Norton, S.P.: Rational Hauptmoduls are replicable. Can. J. Math. 47(6), 1201–1218 (1995)

    Article  MathSciNet  Google Scholar 

  8. Faber, G.: Über Tschebycheffsche Polynome. Crelle 150, 79–106 (1920)

    Google Scholar 

  9. Heim, B., Kaiser, C., Murase, A.: Modular curves and symmetries of Hecke type. Int. J. Math. 29(07), 1850045 (2018)

    Article  MathSciNet  Google Scholar 

  10. Kozlov, D.N.: On Completely Replicable Functions and Extremal Poset Theory. M. Sc. Thesis, Department of Mathematics, University of Lund, Sweden (1994)

  11. Kozlov, D.N.: On functions satisfying modular equations for infinitely many primes. Can. J. Math. 51, 1020–1034 (1999)

    Article  MathSciNet  Google Scholar 

  12. Martin, Y.: On modular invariance of completely replicable functions. Contemp. Math. 193, 263–286 (1996)

    Article  MathSciNet  Google Scholar 

  13. Norton, S.P.: More on Moonshine, Computational Group Theory (Durham, 1982), pp. 185–193. Academic Press, London (1984)

  14. Schiffer, M.: Faber polynomials in the theory of univalent functions. Bull. AMS 54, 503–517 (1948)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for carefully reading our manuscript and for giving valuable suggestions. The first author thanks the Max Planck Institute for Mathematics for support and an invitation in July and August 2017. He also thanks Kyoto Sangyo University and Professor Murase for an invitation in January 2018 to extend our collaboration. He also thanks Prof. Krieg for useful discussions at the Graduate school: Experimental and constructive algebra, at the RWTH Aachen in 2018 and the German University of Technology in Oman for work leave. The second author is partially supported by Grants-in-Aids from JSPS (17K05186).

Author information

Authors and Affiliations

  1. Department of Mathematics and Science, Faculty of Science, German University of Technology in Oman, Muscat, Sultanate of Oman

    Bernhard Heim

  2. Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan

    Atsushi Murase

Authors
  1. Bernhard Heim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Atsushi Murase
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atsushi Murase.

Additional information

Communicated by Jens Funke.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heim, B., Murase, A. Completely replicable functions and symmetries. Abh. Math. Semin. Univ. Hambg. 89, 169–177 (2019). https://doi.org/10.1007/s12188-019-00212-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12188-019-00212-9

Keywords

Mathematics Subject Classification

Search

Navigation