p-adic and Motivic Measure on Artin n-stacks

Balwe, Chetan

Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1219-1246

Résumé

We define a notion of $p$ -adic measure on Artin $n$ -stacks that 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 Toën. As a consequence, we obtain the result that the Poincaré 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 $p$ .

Détail

DOI : 10.4153/CJM-2014-021-7

Mots-clés : 14E18, 14A20, p-adic integration, motivic integration, Artin stacks

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