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The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of Deligne-Mumford stacks.

We introduce the notion of twisted jets. For a Deligne-Mumford stack X of finite type over C, a twisted ∞-jet on X is a representable morphism D → X such that D is a smooth Deligne-Mumford stack with the coarse moduli space Spec C [[t]]. We study a motivic measure on the space of the twisted ∞-jets on a smooth Deligne-Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology as a Hodge structure.

Abstract. We study a relation between the Artin conductor and the weight coming from the motivic integration over wild Deligne-Mumford stacks. As an application, we prove some version of the McKay correspondence, which relates Bhargava's mass formula for extensions of a local field and the Hilbert scheme of points.

We study the McKay correspondence for representations of the cyclic group of order p in characteristic p. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic p as in the tame case. Also, we link a crepant resolution with a count of Artin-Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

For each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n = 1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

Abstract. We prove a version of the wild McKay correspondence by using padic measures. This result provides new proofs of mass formulas for extensions of a local field by Serre, Bhargava and Kedlaya. The aim of this paper is to prove a version of the wild McKay correspondence, the McKay correspondence in positive or mixed characteristic where a given finite group may have order dividing the characteristic of the base field or the residue field. Our main tool is the p-adic measure.By the McKay correspondence, we mean an equality between a certain invariant of a G-variety V with G a finite group and a similar invariant of the quotient variety V /G or a desingularization of it. There are different versions for different invariants. Our concern is the one using motivic invariants or their realizations. In characteristic zero, such a version was studied by Batyrev [Bat99] and . Recently, after examining a special case in [Yas14], the author started to try to generalize it to positive or mixed characteristic, and formulated a conjecture in [Yasa] for linear actions on affine spaces over a complete discrete valuation ring with algebraically closed residue field. Later, variants and generalizations were formulated in [WY15,Yasb]. In [WY15], the situation was considered where the residue field is only perfect. Moreover, when the residue field is finite, the pointcounting realization was discussed. In [Yasb], non-linear actions on affine normal varieties were treated. In the present paper, we consider non-linear actions on normal quasi-projective varieties over a complete discrete valuation ring with finite residue field and prove a version of the wild McKay correspondence at the level of point-counting realization, with a little dissatisfaction at the formulation in the non-affine case.Let O K be a complete discrete valuation ring, K its fraction field and k its residue field, which is supposed to be finite. For the pair (X, D) of an O K -variety X and a Q-divisor D on X such that K X + D is Q-Cartier with K X the canonical divisor of X over O K , we define the stringy point count of (X, D),as the volume of X(O K ) with respect to a certain p-adic measure. When D = 0, identifying the pair (X, 0) with the variety X itself, we write ♯ st (X, 0) = ♯ st X. Roughly, the stringy point count is the point-count realization of the motivic counterpart of the stringy E-function introduced by Batyrev [Bat98, Bat99]. Its principal properties are as follows.• When X is O K -smooth, we have ♯ st X = ♯X(k).• There exists a decomposition into contributions of k-points,• If f : Y → X is a proper birational morphism of normal O K -varieties which induces a crepant map (Y, E) → (X, D) of pairs, then we haveWe generalize the invariant to pairs having a finite group action. Let (V, E) be a pair as above and suppose that a finite group G faithfully acts on V and the divisor E is stable under the action. Let M be a G-étale K-algebra, that is, Spec M → Spec K is an étale G-torsor and let O M be its integer ring. We define the M -stringy point ...

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, which was earlier proved in a joint work of Tonini and the author by a different method. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.