Motivic zeta functions of the Hilbert schemes of points on a surface

Abstract

Let $K$ be a discretely-valued field. Let $X\to \mathrm{Spec}K$ be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes ${\mathrm{Hilb}}^n(X)$ over the ring of integers $R\subseteq K$. We exploit this construction in order to compute the motivic zeta function of ${\mathrm{Hilb}}^n(X)$ in terms of the motivic zeta functions of $X$ and of its base-changes with respect to the totally ramified extensions of $K$. We determine the poles of $Z_{{\mathrm{Hilb}}^n(X)}$ and study its monodromy property, showing that if the monodromy conjecture holds for $X$ then it holds for ${\mathrm{Hilb}}^n(X)$ too.