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p-adic and Motivic Measure on Artin n-stacks
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- Journal:
- Canadian Journal of Mathematics / Volume 67 / Issue 6 / 01 December 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1219-1246
- Print publication:
- 01 December 2015
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- Article
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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 $p$.
