The Ramanujan Journal

, Volume 46, Issue 3, pp 633–655 | Cite as

Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus



We compute the coefficients of the polynomials \(C_n(q)\) defined by the equation
$$\begin{aligned} 1 + \sum _{n\ge 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod _{i\ge 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{aligned}$$
As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of n points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials \(C_n(q)\) are divisible by \((q-1)^2\). We also compute the coefficients of the polynomials \(P_n(q) = C_n(q)/(q-1)^2\): each coefficient counts the divisors of n in a certain interval; it is thus a non-negative integer. Finally we give arithmetical interpretations for the values of \(C_n(q)\) and of \(P_n(q)\) at \(q = -1\) and at roots of unity of order 3, 4, 6.


Infinite product Modular forms Zeta function Hilbert scheme 

Mathematics Subject Classification

Primary 05A17 14C05 14G10 14N10 Secondary 05A30 11P84 14G15 



We are grateful to Giuseppe Ancona, Pierre Baumann, Olivier Benoist, François Bergeron, Mark Haiman, Günter Köhler, Emmanuel Letellier, and Luca Migliorini for helpful discussions. We thank Michael Somos for various comments and for the idea on which Appendix A is based. We also thank the referee and the editor for their remarks. Christophe Reutenauer is grateful to the Université de Strasbourg for the invited professorship which allowed him to spend the month of June 2014 at IRMA; he was also supported by NSERC (Canada).


  1. 1.
    Andrews, G.: Hecke modular forms and the Kac–Peterson identities. Trans. Am. Math. Soc. 283(2), 451–458 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, G., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. (N.S.) 18(2), 167–171 (1988)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chapman, R., Ericksson, K., Stanley, R.P., Martin, R.: The American Mathematical Monthly 109(1), 80 (2002)Google Scholar
  4. 4.
    Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups (with additional contributions by E. Bannai, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen and B.B. Venkov). Grundlehren der mathematischen Wissenschaften, 290, Springer, New York (1988)Google Scholar
  5. 5.
    Deligne, P.: La conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dickson, L.E.: Introduction to the Theory of Numbers. The University of Chicago Press, Sixth impression, Chicago (1946)Google Scholar
  7. 7.
    Dwork, B.: On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82, 631–648 (1960)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdős, P., Nicolas, J.-L.: Méthodes probabilistes et combinatoires en théorie des nombres. Bull. Sci. Math. (2) 100(4), 301–320 (1976)MathSciNetMATHGoogle Scholar
  9. 9.
    Fine, N.J.: Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, 27. American Mathematical Society, Providence (1988)CrossRefMATHGoogle Scholar
  10. 10.
    Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod \(5,7\) and \(11\). Trans. Am. Math. Soc. 305(1), 47–77 (1988)MathSciNetMATHGoogle Scholar
  11. 11.
    Grothendieck, A.: Formule de Lefschetz et rationalité des fonctions \(L\). Séminaire Bourbaki, vol. 9, Exp. No. 279, pp. 41–55. W. A. Benjamin, New York (1966)Google Scholar
  12. 12.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 3rd edn. Clarendon Press, Oxford (1954)MATHGoogle Scholar
  13. 13.
    Kassel, C., Reutenauer, C.: Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables. Michigan Math. J. arXiv:1505.07229v4 (2016, to appear)
  14. 14.
    Kassel, C., Reutenauer, C.: The Fourier expansion of \(\eta (z) \eta (2z) \eta (3z) /\eta (6z)\). Arch. Math. (Basel) 108(5), 453–463 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Heidelberg (2011)CrossRefMATHGoogle Scholar
  16. 16.
    Rodríguez Caballero, J. M.: Middle divisors and \(\sigma \)-palindromic Dyck words. arXiv:1709.05333
  17. 17.
    Rodríguez Caballero, J. M.: Factorization of Dyck words and the distribution of the divisors of an integer. arXiv:1709.05334
  18. 18.
    Rodríguez Caballero, J. M.: On a function introduced by Erdős and Nicolas. arXiv:1709.09639
  19. 19.
    Rutherford, J.S.: Sublattice enumeration IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type. Acta Crystallogr. Sect. A 65, 156–163 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sloane, N.J.A.: Theta series and magic numbers for diamond and certain ionic crystal structures. J. Math. Phys. 28(7), 1653–1657 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Somos, M.: A multisection of \(q\)-series. (2014)
  22. 22.
    The On-Line Encyclopedia of Integer Sequences. published electronically at
  23. 23.
    Vatne, J.E.: The sequence of middle divisors is unbounded. J. Number Theory 172, 413–415 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Strasbourg, CNRS, IRMA UMR 7501StrasbourgFrance
  2. 2.Mathématiques, Université du Québec à MontréalMontrealCanada

Personalised recommendations