---
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. Annales Scientifiques de l’École
Normale Supérieure. 2025;58(5):1179-1242. doi:10.24033/asens.2628
apa: Bilu, M., & Browning, T. D. (2025). A motivic circle method. Annales
Scientifiques de l’École Normale Supérieure. Société Mathématique de France.
https://doi.org/10.24033/asens.2628
chicago: Bilu, Margaret, and Timothy D Browning. “A Motivic Circle Method.” Annales
Scientifiques de l’École Normale Supérieure. Société Mathématique de France,
2025. https://doi.org/10.24033/asens.2628.
ieee: M. Bilu and T. D. Browning, “A motivic circle method,” Annales Scientifiques
de l’École Normale Supérieure, 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.” Annales
Scientifiques de l’École Normale Supérieure, vol. 58, no. 5, Société Mathématique
de France, 2025, pp. 1179–242, doi:10.24033/asens.2628.
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'
...
---
_id: '15279'
abstract:
- lang: eng
text: We formulate and prove an analog of Poonen’s finite-field Bertini theorem
with Taylor conditions that holds in the Grothendieck ring of varieties. This
gives a broad generalization of the work of Vakil and Wood, who treated the case
of smooth hypersurface sections, and is made possible by the use of motivic Euler
products to write down candidate motivic probabilities. As applications, we give
motivic analogs of many results in arithmetic statistics that have been proven
using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections
and Erman and Wood on semiample Bertini theorems.
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: Sean
full_name: Howe, Sean
last_name: Howe
citation:
ama: Bilu M, Howe S. Motivic Euler products in motivic statistics. Algebra &
Number Theory. 2021;15(9):2195-2259. doi:10.2140/ant.2021.15.2195
apa: Bilu, M., & Howe, S. (2021). Motivic Euler products in motivic statistics.
Algebra & Number Theory. Mathematical Sciences Publishers. https://doi.org/10.2140/ant.2021.15.2195
chicago: Bilu, Margaret, and Sean Howe. “Motivic Euler Products in Motivic Statistics.”
Algebra & Number Theory. Mathematical Sciences Publishers, 2021. https://doi.org/10.2140/ant.2021.15.2195.
ieee: M. Bilu and S. Howe, “Motivic Euler products in motivic statistics,” Algebra
& Number Theory, vol. 15, no. 9. Mathematical Sciences Publishers, pp.
2195–2259, 2021.
ista: Bilu M, Howe S. 2021. Motivic Euler products in motivic statistics. Algebra
& Number Theory. 15(9), 2195–2259.
mla: Bilu, Margaret, and Sean Howe. “Motivic Euler Products in Motivic Statistics.”
Algebra & Number Theory, vol. 15, no. 9, Mathematical Sciences Publishers,
2021, pp. 2195–259, doi:10.2140/ant.2021.15.2195.
short: M. Bilu, S. Howe, Algebra & Number Theory 15 (2021) 2195–2259.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-04-03T08:12:59Z
date_published: 2021-12-23T00:00:00Z
date_updated: 2026-08-06T11:10:09Z
day: '23'
department:
- _id: TiBr
doi: 10.2140/ant.2021.15.2195
external_id:
arxiv:
- '1910.05207'
fulldoi: https://doi.org/10.2140/ant.2021.15.2195
intvolume: ' 15'
issue: '9'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.1910.05207
month: '12'
oa: 1
oa_version: Preprint
page: 2195-2259
publication: Algebra & Number Theory
publication_identifier:
eissn:
- 1944-7833
issn:
- 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Motivic Euler products in motivic statistics
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 15
year: '2021'
...