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

  • Christian KasselEmail author
  • Christophe Reutenauer


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).


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Copyright information

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Authors and Affiliations

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

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