---
OA_place: repository
OA_type: green
_id: '22929'
abstract:
- lang: eng
  text: The circle method has been successfully used over the last century to study
    rational points on hypersurfaces. More recently, a version of the method over
    function fields, combined with spreading out techniques, has led to a range of
    results about moduli spaces of rational curves on hypersurfaces. In this paper
    a version of the circle method is implemented in the setting of the Grothendieck
    ring of varieties. This allows us to approximate the classes of these moduli spaces
    directly, without relying on point counting, and leads to a deeper understanding
    of their geometry.
- lang: fre
  text: "La méthode du cercle a été utilisée avec succès au cours du siècle dernier
    pour l’étude\r\ndes points rationnels sur les hypersurfaces. Plus récemment, une
    version fonctionnelle de cette méthode,\r\ncombinée à des techniques d’étalement,
    a mené à une série de résultats sur les espaces de modules de\r\ncourbes sur les
    hypersurfaces. Dans cet article on implémente une version de la méthode du cercle
    dans\r\nle cadre de l’anneau de Grothendieck des variétés. Cela permet d’approximer
    les classes de ces espaces\r\nde modules directement, sans recours au comptage
    de points, ce qui donne accès à une compréhension\r\nplus profonde de leur géométrie."
acknowledgement: "The authors are grateful to Yohan Brunebarbe, Tom Burel, Antoine\r\nChambert-Loir,
  Loïs Faisant, Mirko Mauri and Will Sawin for useful comments. Thanks are\r\nalso
  due to the anonymous referees for numerous helpful remarks. M.B. received funding\r\nfrom
  the European Union’s Horizon 2020 research and innovation programme under the\r\nMarie
  Skłodowska-Curie Grant agreement No. 893012. T.B. was supported by a FWF grant\r\n(DOI
  10.55776/P36278) and by a grant from the Institute for Advanced Study School of\r\nMathematics."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Margaret
  full_name: Bilu, Margaret
  id: 98C47862-10D5-11EA-BEDD-0F6F3DDC885E
  last_name: Bilu
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
citation:
  ama: Bilu M, Browning TD. A motivic circle method. <i>Annales Scientifiques de l’École
    Normale Supérieure</i>. 2025;58(5):1179-1242. doi:<a href="https://doi.org/10.24033/asens.2628">10.24033/asens.2628</a>
  apa: Bilu, M., &#38; Browning, T. D. (2025). A motivic circle method. <i>Annales
    Scientifiques de l’École Normale Supérieure</i>. Société Mathématique de France.
    <a href="https://doi.org/10.24033/asens.2628">https://doi.org/10.24033/asens.2628</a>
  chicago: Bilu, Margaret, and Timothy D Browning. “A Motivic Circle Method.” <i>Annales
    Scientifiques de l’École Normale Supérieure</i>. Société Mathématique de France,
    2025. <a href="https://doi.org/10.24033/asens.2628">https://doi.org/10.24033/asens.2628</a>.
  ieee: M. Bilu and T. D. Browning, “A motivic circle method,” <i>Annales Scientifiques
    de l’École Normale Supérieure</i>, vol. 58, no. 5. Société Mathématique de France,
    pp. 1179–1242, 2025.
  ista: Bilu M, Browning TD. 2025. A motivic circle method. Annales Scientifiques
    de l’École Normale Supérieure. 58(5), 1179–1242.
  mla: Bilu, Margaret, and Timothy D. Browning. “A Motivic Circle Method.” <i>Annales
    Scientifiques de l’École Normale Supérieure</i>, vol. 58, no. 5, Société Mathématique
    de France, 2025, pp. 1179–242, doi:<a href="https://doi.org/10.24033/asens.2628">10.24033/asens.2628</a>.
  short: M. Bilu, T.D. Browning, Annales Scientifiques de l’École Normale Supérieure
    58 (2025) 1179–1242.
corr_author: '1'
das_tickbox: '0'
date_created: 2026-09-13T22:01:57Z
date_published: 2025-05-01T00:00:00Z
date_updated: 2026-09-17T08:22:39Z
day: '01'
department:
- _id: TiBr
doi: 10.24033/asens.2628
ec_funded: 1
external_id:
  arxiv:
  - '2304.09645'
fulldoi: https://doi.org/10.24033/asens.2628
intvolume: '        58'
issue: '5'
keyword:
- Circle method
- moduli spaces of curves
- hypersurfaces
- Grothendieck ring of varieties
- motivic integration
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2304.09645
month: '05'
oa: 1
oa_version: Preprint
page: 1179-1242
project:
- _id: 05A4F6F0-7A3F-11EA-A408-12923DDC885E
  call_identifier: H2020
  grant_number: '893012'
  name: A motivic circle method
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
publication: Annales Scientifiques de l’École Normale Supérieure
publication_identifier:
  eissn:
  - 1873-2151
  issn:
  - 0012-9593
publication_status: published
publisher: Société Mathématique de France
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: A motivic circle method
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 58
year: '2025'
...