Moonshine for all finite groups

  • Samuel DeHority
  • Xavier Gonzalez
  • Neekon Vafa
  • Roger Van Peski


In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O’Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional \(\mathbb {C}[G]\)-modules where the McKay–Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on \(\varGamma _0({{\mathrm{ord}}}(g))\). As there are only finitely many normalized Hauptmoduln, groups whose McKay–Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level \({{\mathrm{ord}}}(g)\) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include \((\mathbb {Z}/ 5 \mathbb {Z})^5\) and \((\mathbb {Z}/ 7 \mathbb {Z})^4\), which are not subgroups of the Monster.


Moonshine Modular forms Finite groups 

Mathematics Subject Classification

11F11 11F30 11F33 20C05 



The authors would like to thank Ken Ono and John Duncan for advising this project and for their many helpful conversations and suggestions. We also thank Hannah Larson for helpful conversations and edits, and Robert Wilson for answering our question about subgroups of the Monster. Finally, we thank Emory University, Princeton University, and the NSF (via Grant DMS-1557690) for their support.


  1. 1.
    Beneish, L., Larson, H.: Traces of singular values of Hauptmoduln. arXiv:1407.4479 [math.NT]
  2. 2.
    Borcherds, R.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Natl. Acad. Sci. U.S.A. 83(10), 3068–3071 (1986)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Borcherds, R.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications. Colloquium Publications, Amer. Math. Soc., Ann Arbor (2017)MATHGoogle Scholar
  5. 5.
    Bringmann, K., Mahlburg, K.: Asymptotic formulas for stacks and unimodal sequences. J. Comb. Theory A(126), 194–215 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Carnahan, S.: Generalized moonshine I: genus zero functions. Algebra Number Theory 4(6), 649–679 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Carnahan, S.: Generalized Moonshine IV: monstrous Lie algebras (2012). arXiv:1208.6254
  8. 8.
    Cheng, M., Duncan, J., Harvey, J.: Umbral Moonshine, Commun. Number Theory Phys. 8(2), 101–242 (2014)Google Scholar
  9. 9.
    Cheng, M., Duncan, J., Harvey, J.: Umbral moonshine and the Niemeier lattices. Res. Math. Sci. 1(1), 3 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cummins, C.J., Norton, S.P.: Rational Hauptmodul are replicable. Can. J. Math. 47, 1201–1218 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cotron, T., Dicks, R., Fleming, S.: Asymptotics and congruences for partition functions which arise from finitary permutation groups. arXiv:1606.09074 [math.NT] (2016)
  13. 13.
    Dong, C., Li, H., Mason, G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dummit, D.S., Foote, R.M.: Abstract Algebra. Wiley, New York (2004)MATHGoogle Scholar
  15. 15.
    Duncan, J.F.R., Griffin, M., Ono, K.: Moonshine. Res. Math. Sci. 2, A11 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Duncan, J.F.R., Griffin, M., Ono, K.: Proof of the umbral moonshine conjecture. Res. Math. Sci. 2, A26 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Duncan, J.F.R., Frenkel, I.: Rademacher sums, moonshine and gravity. Commun. Number Theory Phys. 5, 849–976 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Duncan, J.F.R., Mertens, M.H., Ono, K.: O’Nan moonshine and arithmetic. arXiv:1702.03516
  19. 19.
    Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 surface and the Mathieu group M24. Exper. Math. 20, 91–96 (2011)CrossRefMATHGoogle Scholar
  20. 20.
    Frenkel, I., Lepowsky, J., Meurman, A.: A natural representation of the Fischer–Griess Monster with the modular function J as character. Proc. Natl. Acad. Sci. U.S.A. 81(10), 3256–3260 (1984)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Frenkel, I., Lepowsky, J., Meurman, A.: A moonshine module for the Monster. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.) Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., vol. 3, pp. 231–273. Springer, New York (1985)Google Scholar
  22. 22.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press Inc., Boston (1988)MATHGoogle Scholar
  23. 23.
    Laforgia, A., Natalini, P.: Some inequalities for modified Bessel functions. J. Inequal. Appl. Art. ID 253035, 10 pp (2010)Google Scholar
  24. 24.
    Gannon, T.: Much ado about Matthieu (2012)Google Scholar
  25. 25.
    Ingham, A.E.: A Tauberian theorem for partitions. Ann. Math. 42(5), 1075–1090 (1941)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Iwaniec, H.: Topics in Classical Automorphic Forms, Grad. Studies in Math., vol. 17. AMS, Providence (1997)MATHGoogle Scholar
  27. 27.
    Lam, C.H., Lin, X.: A holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra \(F_{4,6}A_{2,2}\). arXiv:1612.08123v1 [math.QA] (2016)
  28. 28.
    Larson, H.: Coefficients of Mckay–Thompson series and distributions of the moonshine module. Proc. Am. Math. Soc. 144(10), 4183–4197 (2016)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Mason, G.: Finite groups and modular functions. In: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proceedings of Symposium on Pure Mathematics, vol. 47, Amer. Math. Soc., Providence, RI, 1987, With an appendix by S. P. Norton, pp. 181–210Google Scholar
  30. 30.
    McKay, J., Sebbar, A.: Replicable functions: an introduction. In: Cartier, P.E., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics and Geometry II, pp. 373–386. Springer, Berlin (2007)Google Scholar
  31. 31.
    Norton, S.P.: Generalized moonshine. Proc. Symp. Pure Math 47, 208–209 (1987)Google Scholar
  32. 32.
    Ogg, A.: Automorphisms de courbes modulaires. Sem. Delange-Pisot-Poitou, Théorie des nombres, 16, no. 1, exp. no. 7, 1–8 (1974–1975)Google Scholar
  33. 33.
    Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, 102, Amer. Math. Soc., Providence (2004)Google Scholar
  34. 34.
    Queen, L.: Modular functions arising from some finite groups. Math. Comput. 37(156), 547–580 (1981)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Smith, S.D.: On the head characters of the Monster simple group. In: McKay, J. (ed.) Finite Groups—Coming of Age (Montreal, Que, 1982), Contemp. Math., vol. 45, pp. 303–313. Amer. Math. Soc., Providence (1985)Google Scholar
  36. 36.
    Stein, W.A., et al.: Sage Mathematics Software (Version 7.6). The Sage Development Team (2017).
  37. 37.
    Sturm, J.: On the congruence of modular forms. In: Alladi, K. (ed.) Number Theory (New York, 1984–1985), Lecture Notes in Math., vol. 1240, pp. 275–280. Springer, Berlin (1987)Google Scholar
  38. 38.
    Thompson, J.G.: Finite groups and modular functions. Bull. Lond. Math. Soc. 11(3), 347–351 (1979)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Thompson, J.G.: Some numerology between the Fischer–Griess Monster and the elliptic modular function. Bull. Lond. Math. Soc. 11(3), 352–353 (1979)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    van Ekeren, J., Möller, S., Scheithauer, N.R.: Construction and classification of holomorphic vertex operator algebras. arXiv:1507.08142v2 [math.RT] (2015)
  41. 41.
    Wilson, R.A.: The odd-local subgroups of the Monster. J. Austral. Math. Soc. Ser. A 44, 1–16 (1988)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Wilson, R.A.: Personal communication (June 2017)Google Scholar
  43. 43.
    Zagier, D.: Traces of Singular Moduli, Motives, Polylogarithms and Hodge Theory, Lecture Series 3, pp. 209–244. International Press, Somerville (2002)Google Scholar
  44. 44.
    Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUNC-Chapel HillChapel HillUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations