Kontsevich-Zagier conjecture for Hensel-minimal fields with sections

Abstract

We generalise Yin's motivic integration with section to the Hensel-minimal framework, with and without volume forms, building on Stout and Vermeulen's work on a Hrushovski-Kazhdan style motivic integral for $1$-h-minimal valued fields. In particular, we get a bijective motivic integral \emph{\`a la} Hrushovski-Kazhdan between certain Grothendieck groups of definable sets on the valued field and the coarse Grothendieck groups on $\mathrm{RV}$ in the Denef-Pas language, with angular component. In the second half of the article, we build a Cluckers-Loeser-like framework and show that our constructions indeed give back Cluckers-Loeser motivic integral. Thus Cluckers-Loeser motivic integration derives from an isomorphism of Hrushovski-Kazhdan motivic integration, but in the Denef-Pas language with a section of $\mathrm{RV}$.