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

Let K be a discretely-valued field. Let X ! SpecK be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes Hilbn.X/over the ring of integers R & K. We exploit this construction in order to compute the motivic zeta function of Hilbn.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 ZHilbn.X/and study its monodromy property, showing that if the monodromy conjecture holds for X then it holds for Hilbn.X/too.