The main object of this thesis is the motivic zeta function for Calabi-Yau varieties defined over a non Archimedean valued field, focusing on Hilbert schemes of points on surfaces. We use the tools from logarithmic geometry for the construction of semistable models of the surfaces with trivial canonical sheaf. We exploit such construction in order to give a recipe for constructing weak Néron models of their Hilbert schemes of points and deduce from this a formula for
computing their motivic zeta function. We use the formula developed in this way in order to prove that the Hilbert schemes of points have the monodromy property if their underlying surfaces have it.