Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 15 (2019), 033, 35 pages      arXiv:1809.02913      https://doi.org/10.3842/SIGMA.2019.033
Contribution to the Special Issue on Moonshine and String Theory

$p$-Adic Properties of Hauptmoduln with Applications to Moonshine

Ryan C. Chen, Samuel Marks and Matthew Tyler
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

Received September 19, 2018, in final form April 10, 2019; Published online April 29, 2019

Abstract
The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the $j$-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the $j$-function satisfy congruences modulo $p^n$ for $p \in \{2, 3, 5, 7, 11\}$, which led to the theory of $p$-adic modular forms. We combine these two aspects of the $j$-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences.

Key words: modular forms congruences; $p$-adic modular forms; moonshine.

pdf (612 kb)   tex (46 kb)  

References

  1. Andersen N., Jenkins P., Divisibility properties of coefficients of level $p$ modular functions for genus zero primes, Proc. Amer. Math. Soc. 141 (2013), 41-53, arXiv:1106.1188.
  2. Atkin A.O.L., Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14-32.
  3. Atkin A.O.L., Lehner J., Hecke operators on $\Gamma_{0}(m)$, Math. Ann. 185 (1970), 134-160.
  4. Borcherds R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
  5. Borcherds R.E., Modular moonshine. III, Duke Math. J. 93 (1998), 129-154, arXiv:math.QA/9801101.
  6. Borcherds R.E., Ryba A.J.E., Modular moonshine. II, Duke Math. J. 83 (1996), 435-459.
  7. Calegari F., Congruences between modular forms, available at http://swc.math.arizona.edu/aws/2013/2013CalegariLectureNotes.pdf.
  8. Carnahan S., Generalized moonshine, IV: Monstrous Lie algebras, arXiv:1208.6254.
  9. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Oxford University Press, Eynsham, 1985.
  10. Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339.
  11. Conway J., McKay J., Sebbar A., On the discrete groups of Moonshine, Proc. Amer. Math. Soc. 132 (2004), 2233-2240.
  12. DeHority S., Gonzalez X., Vafa N., Van Peski R., Moonshine for all finite groups, Res. Math. Sci. 5 (2018), 14, 34 pages, arXiv:1707.05249.
  13. Diamond F., Shurman J., A first course in modular forms,Graduate Texts in Mathematics, Vol. 228, Springer-Verlag, New York, 2005.
  14. Duncan J.F.R., Griffin M.J., Ono K., Proof of the umbral moonshine conjecture, Res. Math. Sci. 2 (2015), 26, 47 pages, arXiv:1503.01472.
  15. Duncan J.F.R., Mack-Crane S., The moonshine module for Conway's group, Forum Math. Sigma 3 (2015), e10, 52 pages, arXiv:1409.3829.
  16. Elkies N., Ono K., Yang T., Reduction of CM elliptic curves and modular function congruences, Int. Math. Res. Not. 2005 (2005), 2695-2707, arXiv:math.NT/0512350.
  17. Ferenbaugh C.R., The genus-zero problem for $n|h$-type groups, Duke Math. J. 72 (1993), 31-63.
  18. Frenkel I.B., Lepowsky J., Meurman A., A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. USA 81 (1984), 3256-3260.
  19. Frenkel I.B., Lepowsky J., Meurman A., A moonshine module for the Monster, in Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., Vol. 3, Springer, New York, 1985 231-273.
  20. Gannon T., Much ado about Mathieu, Adv. Math. 301 (2016), 322-358, arXiv:1211.5531.
  21. GAP - Groups, Algorithms, and Programming, Version 4.9.2, 2018, https://www.gap-system.org.
  22. Gouvêa F.Q., Arithmetic of $p$-adic modular forms, Lecture Notes in Mathematics, Vol. 1304, Springer-Verlag, Berlin, 1988.
  23. Harada K., Lang M.L., The McKay-Thompson series associated with the irreducible characters of the Monster, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Contemp. Math., Vol. 193, Amer. Math. Soc., Providence, RI, 1996, 93-111, arXiv:q-alg/9412013.
  24. Harvey J.A., Rayhaun B.C., Traces of singular moduli and moonshine for the Thompson group, Commun. Number Theory Phys. 10 (2016), 23-62, arXiv:1504.08179.
  25. Hida H., Elementary theory of $L$-functions and Eisenstein series, London Mathematical Society Student Texts, Vol. 26, Cambridge University Press, Cambridge, 1993.
  26. Jenkins P., Thornton D.J., Congruences for coefficients of modular functions, Ramanujan J. 38 (2015), 619-628, arXiv:1404.0699.
  27. Jochnowitz N., Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), 269-285.
  28. Katz N.M., $p$-adic properties of modular schemes and modular forms, in Modular Functions of One Variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 350, Editors W. Kuijk, J.-P. Serre, Springer, Berlin, 1973, 69-190.
  29. Katz N.M., A result on modular forms in characteristic $p$, in Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., Vol. 601, Editors J.-P. Serre, D.B. Zagier, Springer, Berlin, 1977, 53-61.
  30. Larson H., Coefficients of McKay-Thompson series and distributions of the moonshine module, Proc. Amer. Math. Soc. 144 (2016), 4183-4197, arXiv:1508.03742.
  31. Lehner J., Divisibility properties of the Fourier coefficients of the modular invariant $j(\tau)$, Amer. J. Math. 71 (1949), 136-148.
  32. Lehner J., Further congruence properties of the Fourier coefficients of the modular invariant $j(\tau)$, Amer. J. Math. 71 (1949), 373-386.
  33. Norton S.P., Generalized moonshine, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., Vol. 47, Amer. Math. Soc., Providence, RI, 1987, 208-210.
  34. Pari/GP (Version 2.11.0), University of Bordeaux, 2018, http://pari.math.u-bordeaux.fr.
  35. Ryba A.J.E., Modular Moonshine?, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Contemp. Math., Vol. 193, Amer. Math. Soc., Providence, RI, 1996, 307-336.
  36. SageMath, the Sage Mathematics Software System (Version 8.3), 2018, http://www.sagemath.org.
  37. Serre J.-P., Formes modulaires et fonctions zêta $p$-adiques, in Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math., Vol. 350, Editors W. Kuijk, mboxJ.-P. Serre, Springer, Berlin, 1973, 191-268.
  38. Serre J.-P., Divisibilité de certaines fonctions arithmétiques, Enseignement Math. 22 (1976), 227-260.
  39. Sturm J., On the congruence of modular forms, in Number Theory (New York, 1984-1985), Lecture Notes in Math., Vol. 1240, Springer, Berlin, 1987, 275-280.
  40. Thompson J.G., Finite groups and modular functions, Bull. London Math. Soc. 11 (1979), 347-351.
  41. Thompson J.G., Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352-353.
  42. Wilson R.A., The odd-local subgroups of the Monster, J. Austral. Math. Soc. Ser. A 44 (1988), 1-16.

Previous article  Next article  Contents of Volume 15 (2019)