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# $p$-adic and Motivic Measure on Artin $n$-stacks

Published:2014-07-03

• Chetan Balwe,
School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India
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## Abstract

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 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 $p$.
 Keywords: p-adic integration, motivic integration, Artin stacks
 MSC Classifications: 14E18 - Arcs and motivic integration 14A20 - Generalizations (algebraic spaces, stacks)

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