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

TL;DR

This paper explicitly computes the zeta function of the Hilbert scheme of points on a 2D torus, revealing a functional equation and providing combinatorial and arithmetical interpretations of associated polynomials.

Contribution

It derives an explicit formula for the zeta function of the Hilbert scheme on a 2D torus and explores its properties and special values, including arithmetical interpretations.

Findings

Derived explicit formula for the zeta function.

Proved the zeta function satisfies a functional equation.

Connected polynomial coefficients to divisor counting and arithmetical values.

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