[{"department":[{"_id":"TiBr"}],"intvolume":"        58","page":"1179-1242","_id":"22929","article_processing_charge":"No","date_updated":"2026-09-17T08:22:39Z","status":"public","project":[{"call_identifier":"H2020","grant_number":"893012","name":"A motivic circle method","_id":"05A4F6F0-7A3F-11EA-A408-12923DDC885E"},{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory","grant_number":"P36278"}],"corr_author":"1","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."},{"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.","lang":"fre"}],"language":[{"iso":"eng"}],"oa_version":"Preprint","researchdata_availability":"no","year":"2025","volume":58,"keyword":["Circle method","moduli spaces of curves","hypersurfaces","Grothendieck ring of varieties","motivic integration"],"external_id":{"arxiv":["2304.09645"]},"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.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"supplementarymaterial":"yes","publisher":"Société Mathématique de France","type":"journal_article","ec_funded":1,"author":[{"first_name":"Margaret","id":"98C47862-10D5-11EA-BEDD-0F6F3DDC885E","last_name":"Bilu","full_name":"Bilu, Margaret"},{"orcid":"0000-0002-8314-0177","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","full_name":"Browning, Timothy D"}],"month":"05","date_created":"2026-09-13T22:01:57Z","day":"01","OA_type":"green","fulldoi":"https://doi.org/10.24033/asens.2628","article_type":"original","publication_status":"published","date_published":"2025-05-01T00:00:00Z","scopus_import":"1","das_tickbox":"0","OA_place":"repository","issue":"5","quality_controlled":"1","publication_identifier":{"issn":["0012-9593"],"eissn":["1873-2151"]},"citation":{"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>.","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>","ista":"Bilu M, Browning TD. 2025. A motivic circle method. Annales Scientifiques de l’École Normale Supérieure. 58(5), 1179–1242.","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>","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.","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>.","short":"M. Bilu, T.D. Browning, Annales Scientifiques de l’École Normale Supérieure 58 (2025) 1179–1242."},"arxiv":1,"doi":"10.24033/asens.2628","title":"A motivic circle method","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2304.09645","open_access":"1"}],"publication":"Annales Scientifiques de l’École Normale Supérieure"},{"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Societe Mathematique de France","status":"public","date_updated":"2026-07-06T13:57:35Z","article_processing_charge":"No","_id":"8539","page":"663-671","intvolume":"        53","department":[{"_id":"TaHa"}],"external_id":{"isi":["000592182600004"],"arxiv":["1708.08013"]},"volume":53,"year":"2020","oa_version":"Preprint","language":[{"iso":"eng"}],"abstract":[{"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":"eng"},{"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. ","lang":"fre"}],"isi":1,"quality_controlled":"1","issue":"3","publication":"Annales Scientifiques de l'Ecole Normale Superieure","main_file_link":[{"url":"https://arxiv.org/abs/1708.08013","open_access":"1"}],"title":"On the K-theory stable bases of the springer resolution","doi":"10.24033/asens.2431","arxiv":1,"citation":{"short":"C. Su, G. Zhao, C. Zhong, Annales Scientifiques de l’Ecole Normale Superieure 53 (2020) 663–671.","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.","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>.","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>","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>","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."},"publication_identifier":{"issn":["0012-9593"]},"date_created":"2020-09-20T22:01:38Z","month":"06","author":[{"first_name":"C.","last_name":"Su","full_name":"Su, C."},{"last_name":"Zhao","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-7361-2882","first_name":"Gufang","full_name":"Zhao, Gufang"},{"full_name":"Zhong, C.","last_name":"Zhong","first_name":"C."}],"type":"journal_article","das_tickbox":"1","scopus_import":"1","date_published":"2020-06-01T00:00:00Z","publication_status":"published","article_type":"original","fulldoi":"https://doi.org/10.24033/asens.2431","day":"01"}]