This article gives an introduction to arithmetic motivic integration in the context of p-adic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives.

We show this Chow ring is Z ⊕ Z. We do this by partitioning the space into 2n subvarieties each of which is fibered over Gl(2n − 2, C)/SO(2n − 2, C).

Let k be a field and X a smooth projective variety of dimension d over k. Generalizing a construction of Kato and Somekawa, we define a Milnortype group Ks(k; CH0(X);Gm) which is isomorphic to the ordinary Milnor K-group Km s (k) in the case X = Spec k. We prove that Ks(k; CH0(X);Gm) is isomorphic to both the higher Chow group CHd+s(X, s) and the Zariski cohomology group Hd Zar(X,K d+s).

We prove an absolute Chow–Künneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surfaces. We furthermore prove the Hodge conjecture for general fibres At of A, extending results of Ribet.

This article shows that under general conditions, p-adic orbital integrals of definable functions are represented by virtual Chow motives. This gives an explicit example of the philosophy of Denef and Loeser, which predicts that all “naturally occurring” p-adic integrals are motivic.

Abstract. We give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, ...

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not...