A version of geometric motivic integration that specializes to p-adic integration via point counting

Details A version of geometric motivic integration that specializes to p-adic integration via point counting A version of geometric motivic integration that specializes to p-adic integration via point counting

2008-10-24

arxiv.org/abs/0810.4496v1

Mathematics

Algebraic Geometry

10 pages

Scientific paper

We give a version of geometric motivic integration that specializes to p-adic integration via point counting. This has been done before for stable sets; we extend this to more general sets. The main problem in doing this is that it requires to take limits, hence the measure will have to take values in a completion of the (localized) Grothendieck ring of varieties. The standard choice is to complete with respect to the dimension filtration; however, since the point counting homomorphism is not continuous with respect to this topology we have to use a stronger one. The first part of the paper is devoted to defining this topology; in the second part we will then see that many of the standard constructions of geometric motivic integration work also in this setting.

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