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PreprintCC BY-NC-SA 4.0arXiv

Kontsevich-Zagier conjecture for Hensel-minimal fields with sections

  • Xavier Pigé
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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{à la} Hrushovski-Kazhdan between certain Grothendieck groups of definable sets on the valued field and the coarse Grothendieck groups on 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 RV.

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Cite

BibTeXRIS
BibTeX
@misc{pige2026kontsevichzagier,
  title = {{Kontsevich-Zagier conjecture for Hensel-minimal fields with sections}},
  author = {Xavier Pigé},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08417},
  eprint = {2609.08417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08417},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Kontsevich-Zagier conjecture for Hensel-minimal fields with sections
AU  - Xavier Pigé
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08417
UR  - https://arxiv.org/abs/2609.08417
AB  - 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{à la} Hrushovski-Kazhdan between certain Grothendieck groups of definable sets on the valued field and the coarse Grothendieck groups on 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 RV.
ER  -