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

Abstract : We compute the coefficients of the polynomials $C_n(q)$ defined by the equation \begin{equation*} 1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{equation*} 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 $3$, $4$, $6$.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-02880770
Contributor : Christian Kassel <>
Submitted on : Thursday, June 25, 2020 - 11:28:47 AM
Last modification on : Friday, June 26, 2020 - 3:33:36 AM

### Citation

Christian Kassel, Christophe Reutenauer. Complete determination of the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus. Ramanujan Journal, Springer Verlag, 2018, 46 (3), pp.633-655. ⟨10.1007/s11139-018-0011-1⟩. ⟨hal-02880770⟩

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