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

TL;DR

This paper develops a version of geometric motivic integration that extends to more general sets and specializes to p-adic integration through point counting, involving a new topology on the Grothendieck ring.

Contribution

It introduces a stronger topology on the Grothendieck ring to enable limits in motivic integration, extending the framework to broader sets beyond stable sets.

Findings

Defined a new topology on the Grothendieck ring for motivic integration

Extended motivic integration to more general sets

Showed standard constructions remain valid in the new setting

Abstract

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.