We survey the theory of replicable functions and its ramifications from number theory to physics.


Conjugacy Class Chebyshev Polynomial Class Number Dynkin Diagram Automorphic Function 
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  1. 1.
    Aspinwall, P.S., Katz, S., Morrison, D.R. Lie groups, Calabi-Yau manifolds and F-theory Adv. Theor. Math. Phys 4, 95–126, 2000.zbMATHGoogle Scholar
  2. 2.
    Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A., ATLAS of finite groups, Clarendon Press, Oxford, 1985.zbMATHGoogle Scholar
  3. 3.
    Conway, J.H., McKay, J., Sebbar, A., The discrete groups of moonshine, Proc. Amer. Math. Soc. 132, 2233–2240, 2004.zbMATHCrossRefGoogle Scholar
  4. 4.
    Conway, J. H., Norton, S. P., Monstrous Moonshine, Bull. London Math. Soc. 11, 308–339, 1979.zbMATHCrossRefGoogle Scholar
  5. 5.
    Cox, D., McKay, J., Stevenhagen, P., Principal moduli and their class fields. Bull. London Math. Soc. 36, 3–12, 2004.zbMATHCrossRefGoogle Scholar
  6. 6.
    Cummins, C., Some comments on replicable functions. Modern trends in Lie algebra representation theory (Kingston, ON, 1993), 48–55, Queen's Papers in Pure and Appl. Math., 94, Queen's Univ., Kingston, ON, 1994.Google Scholar
  7. 7.
    Cummins, C., Norton, S. P., Rational Hauptmoduls are replicable, Can. J. Math., 47, 1201–1218, 1995.zbMATHGoogle Scholar
  8. 8.
    Dedekind, R., Schreiben an Hern Borchardt Über die Theorie der elliptischen Modulfunktionen, Crelle, 83, 265–292, 1878.Google Scholar
  9. 9.
    Faber, G., Über polynomische Entwicklungen, Math. Ann. 57, 389–408, 1903.CrossRefGoogle Scholar
  10. 10.
    Glauberman, G., Norton, S. P. On McKay's connection between the affine E8 diagram and the Monster. Proceedings on Moonshine and Related Topics (Montréal), 37–42, 1999. CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.Google Scholar
  11. 11.
    Hille, E., Analytic function theory. Vol. II. Introduction to Higher Mathematics Ginn and Co., 1962.Google Scholar
  12. 12.
    Kostov, I.K., Krichever, I., Mineev-Weinstein, M., Wiegmann, P. B., Zabrodin, A. The t-function for analytic curves. Random matrix models and their applications, 285–299, Math. Sci. Res. Inst. Publ., 40, Cambridge Univ. Press, Cambridge, 2001.Google Scholar
  13. 13.
    McKay, J. Graphs, singularities, and finite groups, in The Santa Cruz Conference on Finite Groups, Proc. Symp. Pure Math. 37, ed. by B. Cooperstein and G. Mason, AMS, Providence, 1980.Google Scholar
  14. 14.
    McKay, J., Sebbar, A., Fuchsian groups, automorphic forms and Schwarzians, Math. Ann. 318, no. 2, 255–275, 2000.zbMATHCrossRefGoogle Scholar
  15. 15.
    Ogg, A. P. Modular functions. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 521–532, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.Google Scholar
  16. 16.
    Rademacher, H., The Fourier series and the functional equation of the absolute modular invariant J(t), Am. J. Math. 61, 237–248, 1939.zbMATHCrossRefGoogle Scholar
  17. 17.
    Reid, Miles. McKay correspondence, alg-geom/9702016.Google Scholar
  18. 18.
    Serre, J-P., A Course in Arithmetic, Springer-Verlag, 1973.Google Scholar
  19. 19.
    Takahashi, A. Primitive Forms, Topological LG models coupled to Gravity and Mirror Symmetry, preprint AG/9802059.Google Scholar
  20. 20.
    From Number Theory to Physics. Papers from the Meeting on Number Theory and Physics held in Les Houches, March 7–16, 1989. Edited by M. Waldschmidt, P. Moussa, J. M. Luck and C. Itzykson. Springer-Verlag, Berlin, 1992.Google Scholar

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© Springer 2007

Authors and Affiliations

  • John McKay
    • 1
  • Abdellah Sebbar
    • 2
  1. 1.Department of MathematicsConcordia UniversityMontrealCanada
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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