Motivic integration over wild Deligne–Mumford stacks

Abstract

We develop the motivic integration theory over formal Deligne–Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne–Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava’s mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.