1. CJM Online first
|$p$-adic and Motivic Measure on Artin $n$-stacks|
We define a notion of $p$-adic measure on Artin $n$-stacks which are of strongly finite type over the ring of $p$-adic integers. $p$-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo $p^n$. We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting measure defined by ToÃ«n. As a consequence, we obtain the result that the PoincarÃ© and Serre series of such stacks are rational functions, thus extending Denef's result for varieties. Finally, using motivic integration we show that as $p$ varies, the rationality of the Serre series of an Artin stack defined over the integers is uniform with respect to $p$.
Keywords:p-adic integration, motivic integration, Artin stacks
2. CJM 2006 (vol 58 pp. 93)
|Motivic Haar Measure on Reductive Groups |
We define a motivic analogue of the Haar measure for groups of the form $G(k\llp t\rrp)$, where~$k$ is an algebraically closed field of characteristic zero, and $G$ is a reductive algebraic group defined over $k$. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M.~Kontsevich to define an additive function on a certain natural Boolean algebra of subsets of $G(k\llp t\rrp)$. This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of varieties over the base field. It is invariant under translations by all elements of $G(k\llp t\rrp)$, and therefore we call it a motivic analogue of Haar measure. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process.
Keywords:motivic integration, reductive group