Abstract
Apply weight 0 Hecke operators to the modular function j and express the result as a polynomial in j. These polynomials were considered long ago in analysis, and recently attracted the attention of number theorists primarily for their connection with Borcherds’ infinite products. In particular, Ken Ono conjectured that all of them are irreducible. We prove a partial result towards this conjecture by presenting infinite families of these polynomials which are proved to be irreducible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asai, T., Kaneko, M., Ninomiya, H.: Zeros of certain modular functions and an application. Comment. Math. Univ. St. Paul. 46(1), 93–101 (1997)
Bruinier, J.H., Kohnen, W., Ono, K.: The arithmetic of the values of modular functions and the divisors of modular forms. Compos. Math. 140(3), 552–566 (2004)
Eholzer, W., Skoruppa, N.-P.: Product expansions of conformal characters. Phys. Lett. B 388(1), 82–89 (1996)
Faber, G.: Über polynomische Entwickelungen. Math. Ann. 57, 389–408 (1903)
Guerzhoy, P.: A remark on congruences for coefficients of Faber polynomials. Ramanujan J. 12(3), 349–358 (2006)
Ono, K.: The Web of Modularity: Arithmetics of the Coefficients of Modular Forms and q-Series. CBMS Regional Conference Series in Mathematics, vol. 102. Amer. Math. Soc., Providence (2004)
Zagier, D.: Traces of singular moduli. In: Motives, Polylogarithms and Hodge Theory, Part I. Irvine, CA, 1998. International Press Lecture Series, vol. 3, pp. 211–244. International Press, Somerville (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSF grant DMS-0501225.
Rights and permissions
About this article
Cite this article
Guerzhoy, P. Irreducibility of some Faber polynomials. Ramanujan J 16, 53–57 (2008). https://doi.org/10.1007/s11139-007-9092-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-007-9092-y