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 $3$, $4$, $6$.

Complements arXiv:1505.07229v4. Version 2: minor corrections

References in corpus (2)

• The Fourier expansion of η(z)η(2z)η(3z)/η(6z)
• Factorization of Dyck words and the distribution of the divisors of an integer

Cited by in corpus (2)

• Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables
• A relationship between the ideals of ${\mathbb{F}}_q\left[x,y,x^{-1},y^{-1}\right]$ and the Fibonacci numbers