---
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'
...
---
_id: '8539'
abstract:
- lang: eng
  text: Cohomological and K-theoretic stable bases originated from the study of quantum
    cohomology and quantum K-theory. Restriction formula for cohomological stable
    bases played an important role in computing the quantum connection of cotangent
    bundle of partial flag varieties. In this paper we study the K-theoretic stable
    bases of cotangent bundles of flag varieties. We describe these bases in terms
    of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar.
    Using this algebraic description and the method of root polynomials, we give a
    restriction formula of the stable bases. We apply it to obtain the restriction
    formula for partial flag varieties. We also build a relation between the stable
    basis and the Casselman basis in the principal series representations of the Langlands
    dual group. As an application, we give a closed formula for the transition matrix
    between Casselman basis and the characteristic functions.
- lang: fre
  text: "Les bases stables cohomologiques et K-théoriques proviennent de l’étude de
    la cohomologie quantique et de la K-théorie quantique. La formule de restriction
    pour les bases stables cohomologiques a joué un rôle important dans le calcul
    de la connexion quantique du fibré cotangent de variétés de drapeaux partielles.
    Dans cet article, nous étudions les bases stables K-théoriques de fibré cotangents
    des variétés de drapeaux. Nous décrivons ces bases en fonction de l’action de
    l’algèbre de Hecke affine et de l’algèbre de Kostant-Kumar. En utilisant cette
    description algébrique et la méthode des polynômes de racine, nous donnons une
    formule de restriction des bases stables. Nous l’appliquons\r\npour obtenir la
    formule de restriction pour les variétés de drapeaux partielles. Nous construisons
    également une relation entre la base stable et la base de Casselman dans les représentations
    de la série principale du groupe dual de Langlands p-adique. Comme une application,
    nous donnons une formule close pour la matrice de transition entre la base de
    Casselman et les fonctions caractéristiques. "
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: C.
  full_name: Su, C.
  last_name: Su
- first_name: Gufang
  full_name: Zhao, Gufang
  id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
  last_name: Zhao
  orcid: 0000-0001-7361-2882
- first_name: C.
  full_name: Zhong, C.
  last_name: Zhong
citation:
  ama: Su C, Zhao G, Zhong C. On the K-theory stable bases of the springer resolution.
    <i>Annales Scientifiques de l’Ecole Normale Superieure</i>. 2020;53(3):663-671.
    doi:<a href="https://doi.org/10.24033/asens.2431">10.24033/asens.2431</a>
  apa: Su, C., Zhao, G., &#38; Zhong, C. (2020). On the K-theory stable bases of the
    springer resolution. <i>Annales Scientifiques de l’Ecole Normale Superieure</i>.
    Societe Mathematique de France. <a href="https://doi.org/10.24033/asens.2431">https://doi.org/10.24033/asens.2431</a>
  chicago: Su, C., Gufang Zhao, and C. Zhong. “On the K-Theory Stable Bases of the
    Springer Resolution.” <i>Annales Scientifiques de l’Ecole Normale Superieure</i>.
    Societe Mathematique de France, 2020. <a href="https://doi.org/10.24033/asens.2431">https://doi.org/10.24033/asens.2431</a>.
  ieee: C. Su, G. Zhao, and C. Zhong, “On the K-theory stable bases of the springer
    resolution,” <i>Annales Scientifiques de l’Ecole Normale Superieure</i>, vol.
    53, no. 3. Societe Mathematique de France, pp. 663–671, 2020.
  ista: Su C, Zhao G, Zhong C. 2020. On the K-theory stable bases of the springer
    resolution. Annales Scientifiques de l’Ecole Normale Superieure. 53(3), 663–671.
  mla: Su, C., et al. “On the K-Theory Stable Bases of the Springer Resolution.” <i>Annales
    Scientifiques de l’Ecole Normale Superieure</i>, vol. 53, no. 3, Societe Mathematique
    de France, 2020, pp. 663–71, doi:<a href="https://doi.org/10.24033/asens.2431">10.24033/asens.2431</a>.
  short: C. Su, G. Zhao, C. Zhong, Annales Scientifiques de l’Ecole Normale Superieure
    53 (2020) 663–671.
das_tickbox: '1'
date_created: 2020-09-20T22:01:38Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2026-07-06T13:57:35Z
day: '01'
department:
- _id: TaHa
doi: 10.24033/asens.2431
external_id:
  arxiv:
  - '1708.08013'
  isi:
  - '000592182600004'
fulldoi: https://doi.org/10.24033/asens.2431
intvolume: '        53'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1708.08013
month: '06'
oa: 1
oa_version: Preprint
page: 663-671
publication: Annales Scientifiques de l'Ecole Normale Superieure
publication_identifier:
  issn:
  - 0012-9593
publication_status: published
publisher: Societe Mathematique de France
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the K-theory stable bases of the springer resolution
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 53
year: '2020'
...