[{"date_published":"2026-01-06T00:00:00Z","publication":"Journal of the London Mathematical Society","quality_controlled":"1","date_updated":"2026-07-16T08:33:56Z","intvolume":"       113","date_created":"2026-01-18T23:02:44Z","publication_status":"published","article_type":"original","year":"2026","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","fulldoi":"https://doi.org/10.1112/jlms.70371","doi":"10.1112/jlms.70371","corr_author":"1","day":"06","month":"01","status":"public","OA_type":"hybrid","OA_place":"publisher","has_accepted_license":"1","title":"The Davenport–Heilbronn method: 80 years on","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"21002","abstract":[{"text":"The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.","lang":"eng"}],"PlanS_conform":"1","oa_version":"Published Version","department":[{"_id":"TiBr"}],"project":[{"grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory"}],"oa":1,"language":[{"iso":"eng"}],"researchdata_availability":"no","ddc":["510"],"supplementarymaterial":"no","publisher":"Wiley","file":[{"success":1,"file_id":"21004","creator":"dernst","date_updated":"2026-01-19T08:19:46Z","file_name":"2026_JourLondonMathSoc_Browning.pdf","file_size":235238,"content_type":"application/pdf","access_level":"open_access","date_created":"2026-01-19T08:19:46Z","relation":"main_file","checksum":"3b05bd625c81d038259a14f7e2ddd57c"}],"acknowledgement":"The author is very grateful to Jörg Brüdern, Simon Rydin Myerson and Trevor Wooley for their help and advice with preparing this survey, in addition to Vinay Kumaraswamy, Victor Wang and the anonymous referee for useful comments on an earlier draft. This work was supported by a FWF Grant (DOI 10.55776/P36278).\r\nOpen Access funding provided by Institute of Science and Technology Austria/KEMÖ.","volume":113,"publication_identifier":{"issn":["0024-6107"],"eissn":["1469-7750"]},"article_number":"e70371","issue":"1","type":"journal_article","das_tickbox":"0","license":"https://creativecommons.org/licenses/by/4.0/","author":[{"last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","first_name":"Timothy D"}],"file_date_updated":"2026-01-19T08:19:46Z","citation":{"chicago":"Browning, Timothy D. “The Davenport–Heilbronn Method: 80 Years On.” <i>Journal of the London Mathematical Society</i>. Wiley, 2026. <a href=\"https://doi.org/10.1112/jlms.70371\">https://doi.org/10.1112/jlms.70371</a>.","apa":"Browning, T. D. (2026). The Davenport–Heilbronn method: 80 years on. <i>Journal of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/jlms.70371\">https://doi.org/10.1112/jlms.70371</a>","mla":"Browning, Timothy D. “The Davenport–Heilbronn Method: 80 Years On.” <i>Journal of the London Mathematical Society</i>, vol. 113, no. 1, e70371, Wiley, 2026, doi:<a href=\"https://doi.org/10.1112/jlms.70371\">10.1112/jlms.70371</a>.","ista":"Browning TD. 2026. The Davenport–Heilbronn method: 80 years on. Journal of the London Mathematical Society. 113(1), e70371.","short":"T.D. Browning, Journal of the London Mathematical Society 113 (2026).","ama":"Browning TD. The Davenport–Heilbronn method: 80 years on. <i>Journal of the London Mathematical Society</i>. 2026;113(1). doi:<a href=\"https://doi.org/10.1112/jlms.70371\">10.1112/jlms.70371</a>","ieee":"T. D. Browning, “The Davenport–Heilbronn method: 80 years on,” <i>Journal of the London Mathematical Society</i>, vol. 113, no. 1. Wiley, 2026."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"}},{"fulldoi":"https://doi.org/10.1017/fms.2026.10259","doi":"10.1017/fms.2026.10259","article_processing_charge":"Yes","scopus_import":"1","year":"2026","date_created":"2026-08-02T22:01:52Z","publication_status":"published","article_type":"original","intvolume":"        14","quality_controlled":"1","date_updated":"2026-08-03T12:13:59Z","publication":"Forum of Mathematics Sigma","date_published":"2026-07-22T00:00:00Z","department":[{"_id":"TiBr"},{"_id":"GradSch"}],"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278","name":"Rational curves via function field analytic number theory"},{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"_id":"22618","ec_funded":1,"abstract":[{"text":"We use a function field version of the circle method to prove that a positive proportion of elements in 𝔽𝑞⁡[𝑡] are representable as a sum of three cubes of minimal degree from 𝔽𝑞⁡[𝑡], assuming a suitable form of the Ratios Conjecture and that char⁡(𝔽𝑞) >3. The analogue of this conjecture for quadratic Dirichlet L-functions is known for large fixed q, via recent developments in homological stability.","lang":"eng"}],"oa_version":"Published Version","PlanS_conform":"1","external_id":{"arxiv":["2402.07146"]},"title":"Sums of three cubes over a function field","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","OA_type":"gold","OA_place":"publisher","month":"07","day":"22","status":"public","corr_author":"1","publication_identifier":{"eissn":["2050-5094"]},"volume":14,"acknowledgement":"While working on this paper the first two authors were supported by FWF grant (DOI 10.55776/P36278) and the third author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413, and by the National Science and Technology Council Project Grant 114-2115-M-001-010-MY2.","file":[{"file_size":810718,"date_updated":"2026-08-03T12:12:03Z","file_name":"2026_ForumMathematics_Browning.pdf","file_id":"22636","creator":"dernst","success":1,"checksum":"e92a762e03f832bdca8c106a9a88ef9b","relation":"main_file","date_created":"2026-08-03T12:12:03Z","access_level":"open_access","content_type":"application/pdf"}],"publisher":"Cambridge University Press","supplementarymaterial":"no","researchdata_availability":"no","ddc":["500"],"arxiv":1,"language":[{"iso":"eng"}],"oa":1,"citation":{"ieee":"T. D. Browning, J. Glas, and V. Wang, “Sums of three cubes over a function field,” <i>Forum of Mathematics Sigma</i>, vol. 14. Cambridge University Press, 2026.","ama":"Browning TD, Glas J, Wang V. Sums of three cubes over a function field. <i>Forum of Mathematics Sigma</i>. 2026;14. doi:<a href=\"https://doi.org/10.1017/fms.2026.10259\">10.1017/fms.2026.10259</a>","short":"T.D. Browning, J. Glas, V. Wang, Forum of Mathematics Sigma 14 (2026).","apa":"Browning, T. D., Glas, J., &#38; Wang, V. (2026). Sums of three cubes over a function field. <i>Forum of Mathematics Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2026.10259\">https://doi.org/10.1017/fms.2026.10259</a>","ista":"Browning TD, Glas J, Wang V. 2026. Sums of three cubes over a function field. Forum of Mathematics Sigma. 14, e112.","mla":"Browning, Timothy D., et al. “Sums of Three Cubes over a Function Field.” <i>Forum of Mathematics Sigma</i>, vol. 14, e112, Cambridge University Press, 2026, doi:<a href=\"https://doi.org/10.1017/fms.2026.10259\">10.1017/fms.2026.10259</a>.","chicago":"Browning, Timothy D, Jakob Glas, and Victor Wang. “Sums of Three Cubes over a Function Field.” <i>Forum of Mathematics Sigma</i>. Cambridge University Press, 2026. <a href=\"https://doi.org/10.1017/fms.2026.10259\">https://doi.org/10.1017/fms.2026.10259</a>."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"DOAJ_listed":"1","author":[{"first_name":"Timothy D","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning"},{"last_name":"Glas","first_name":"Jakob","full_name":"Glas, Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb"},{"first_name":"Victor","orcid":"0000-0002-0704-7026","full_name":"Wang, Victor","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","last_name":"Wang"}],"file_date_updated":"2026-08-03T12:12:03Z","type":"journal_article","das_tickbox":"0","article_number":"e112"},{"oa":1,"language":[{"iso":"eng"}],"arxiv":1,"supplementarymaterial":"no","publisher":"Cambridge: Alliance of Diamond Open Access Journals","researchdata_availability":"no","ddc":["510"],"acknowledgement":"Supported by FWF grant (DOI 10.55776/P36278), Supported by European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant\r\nAgreement No. 101034413.","file":[{"relation":"main_file","checksum":"3d38e850b40f3e1abbfd30073bd4388a","date_created":"2026-02-12T07:50:47Z","access_level":"open_access","content_type":"application/pdf","date_updated":"2026-02-12T07:50:47Z","file_name":"2025_DiscreteAnalysis_Browning.pdf","file_size":393625,"success":1,"creator":"dernst","file_id":"21214"}],"publication_identifier":{"eissn":["2397-3129"]},"volume":2025,"das_tickbox":"0","type":"journal_article","article_number":"12","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"citation":{"short":"T.D. Browning, M. Verzobio, Discrete Analysis 2025 (2025).","ieee":"T. D. Browning and M. Verzobio, “Counting integer points on affine surfaces with a side condition,” <i>Discrete Analysis</i>, vol. 2025. Cambridge: Alliance of Diamond Open Access Journals, 2025.","ama":"Browning TD, Verzobio M. Counting integer points on affine surfaces with a side condition. <i>Discrete Analysis</i>. 2025;2025. doi:<a href=\"https://doi.org/10.19086/da.143787\">10.19086/da.143787</a>","chicago":"Browning, Timothy D, and Matteo Verzobio. “Counting Integer Points on Affine Surfaces with a Side Condition.” <i>Discrete Analysis</i>. Cambridge: Alliance of Diamond Open Access Journals, 2025. <a href=\"https://doi.org/10.19086/da.143787\">https://doi.org/10.19086/da.143787</a>.","ista":"Browning TD, Verzobio M. 2025. Counting integer points on affine surfaces with a side condition. Discrete Analysis. 2025, 12.","mla":"Browning, Timothy D., and Matteo Verzobio. “Counting Integer Points on Affine Surfaces with a Side Condition.” <i>Discrete Analysis</i>, vol. 2025, 12, Cambridge: Alliance of Diamond Open Access Journals, 2025, doi:<a href=\"https://doi.org/10.19086/da.143787\">10.19086/da.143787</a>.","apa":"Browning, T. D., &#38; Verzobio, M. (2025). Counting integer points on affine surfaces with a side condition. <i>Discrete Analysis</i>. Cambridge: Alliance of Diamond Open Access Journals. <a href=\"https://doi.org/10.19086/da.143787\">https://doi.org/10.19086/da.143787</a>"},"file_date_updated":"2026-02-12T07:50:47Z","author":[{"full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning"},{"first_name":"Matteo","orcid":"0000-0002-0854-0306","full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","last_name":"Verzobio"}],"publication":"Discrete Analysis","date_published":"2025-09-01T00:00:00Z","intvolume":"      2025","date_updated":"2026-07-16T08:48:35Z","year":"2025","publication_status":"published","article_type":"original","date_created":"2026-01-18T23:02:44Z","doi":"10.19086/da.143787","fulldoi":"https://doi.org/10.19086/da.143787","scopus_import":"1","article_processing_charge":"No","status":"public","month":"09","day":"01","corr_author":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Counting integer points on affine surfaces with a side condition","has_accepted_license":"1","OA_type":"diamond","OA_place":"publisher","external_id":{"arxiv":["2408.11453"]},"department":[{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"},{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"oa_version":"Published Version","ec_funded":1,"_id":"21003","abstract":[{"lang":"eng","text":"We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers."}]},{"supplementarymaterial":"no","publisher":"Instytut Matematyczny","researchdata_availability":"no","arxiv":1,"oa":1,"language":[{"iso":"eng"}],"page":"141-151","publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"volume":221,"acknowledgement":"The author would like to thank Tim Browning, Jakob Glas and Simon Rydin Myerson for useful suggestions and conversations. Finally, he would like to thank the anonymous referees for their helpful comments. The author was supported by the NWO Veni Grant 016.Veni.192.047 during his time at Utrecht University and by the FWF grant P 36278 at the Institute of Science and Technology Austria while working on this article.","type":"journal_article","issue":"2","das_tickbox":"0","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2304.02620","open_access":"1"}],"citation":{"chicago":"Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2025. <a href=\"https://doi.org/10.4064/aa241029-19-8\">https://doi.org/10.4064/aa241029-19-8</a>.","apa":"Yamagishi, S. (2025). Birch’s theorem on forms in many variables with a Hessian condition. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href=\"https://doi.org/10.4064/aa241029-19-8\">https://doi.org/10.4064/aa241029-19-8</a>","mla":"Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” <i>Acta Arithmetica</i>, vol. 221, no. 2, Instytut Matematyczny, 2025, pp. 141–51, doi:<a href=\"https://doi.org/10.4064/aa241029-19-8\">10.4064/aa241029-19-8</a>.","ista":"Yamagishi S. 2025. Birch’s theorem on forms in many variables with a Hessian condition. Acta Arithmetica. 221(2), 141–151.","short":"S. Yamagishi, Acta Arithmetica 221 (2025) 141–151.","ama":"Yamagishi S. Birch’s theorem on forms in many variables with a Hessian condition. <i>Acta Arithmetica</i>. 2025;221(2):141-151. doi:<a href=\"https://doi.org/10.4064/aa241029-19-8\">10.4064/aa241029-19-8</a>","ieee":"S. Yamagishi, “Birch’s theorem on forms in many variables with a Hessian condition,” <i>Acta Arithmetica</i>, vol. 221, no. 2. Instytut Matematyczny, pp. 141–151, 2025."},"author":[{"id":"0c3fbc5c-f7a6-11ec-8d70-9485e75b416b","full_name":"Yamagishi, Shuntaro","first_name":"Shuntaro","last_name":"Yamagishi"}],"intvolume":"       221","quality_controlled":"1","date_updated":"2026-07-16T09:04:43Z","publication":"Acta Arithmetica","date_published":"2025-10-28T00:00:00Z","fulldoi":"https://doi.org/10.4064/aa241029-19-8","doi":"10.4064/aa241029-19-8","article_processing_charge":"No","scopus_import":"1","year":"2025","date_created":"2026-04-26T22:01:48Z","publication_status":"published","article_type":"original","title":"Birch’s theorem on forms in many variables with a Hessian condition","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","OA_type":"green","OA_place":"repository","month":"10","day":"28","status":"public","keyword":["Diophantine equations","homogeneous forms"],"corr_author":"1","department":[{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"_id":"21768","abstract":[{"text":"Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes p. We give a different formulation of this result. More precisely, we replace dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch if HF<dimV∗F. We also prove the corresponding result for systems of forms of equal degree.","lang":"eng"}],"oa_version":"Preprint","external_id":{"arxiv":["2304.02620"]}},{"doi":"10.4171/jems/1704","fulldoi":"https://doi.org/10.4171/jems/1704","article_processing_charge":"No","year":"2025","publication_status":"epub_ahead","date_created":"2026-02-17T07:46:26Z","article_type":"original","quality_controlled":"1","date_updated":"2026-07-16T09:00:02Z","publication":"Journal of the European Mathematical Society","date_published":"2025-09-17T00:00:00Z","project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"department":[{"_id":"TiBr"}],"oa_version":"Published Version","_id":"21266","abstract":[{"lang":"eng","text":"For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E \r\nD have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve."}],"external_id":{"arxiv":["2401.04375"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Almost all quadratic twists of an elliptic curve have no integral points","OA_place":"publisher","OA_type":"diamond","status":"public","month":"09","day":"17","corr_author":"1","publication_identifier":{"issn":["1435-9855"],"eissn":["1435-9863"]},"acknowledgement":"The authors are grateful to Roger Heath-Brown and to the anonymous referees for useful comments. The first author was supported by an FWF grant (DOI 10.55776/P36278).","publisher":"EMS Press","supplementarymaterial":"no","ddc":["510"],"researchdata_availability":"no","language":[{"iso":"eng"}],"oa":1,"arxiv":1,"DOAJ_listed":"1","citation":{"short":"T.D. Browning, S. Chan, Journal of the European Mathematical Society (2025).","ieee":"T. D. Browning and S. Chan, “Almost all quadratic twists of an elliptic curve have no integral points,” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025.","ama":"Browning TD, Chan S. Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. 2025. doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>","chicago":"Browning, Timothy D, and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>.","ista":"Browning TD, Chan S. 2025. Almost all quadratic twists of an elliptic curve have no integral points. Journal of the European Mathematical Society.","mla":"Browning, Timothy D., and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>, EMS Press, 2025, doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>.","apa":"Browning, T. D., &#38; Chan, S. (2025). Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. EMS Press. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>"},"author":[{"full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning"},{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","first_name":"Yik Tung","last_name":"Chan"}],"das_tickbox":"0","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://doi.org/10.4171/JEMS/1704"}]},{"corr_author":"1","month":"05","day":"23","isi":1,"status":"public","OA_type":"hybrid","OA_place":"publisher","has_accepted_license":"1","title":"Optimal sums of three cubes in Fq[t]","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"arxiv":["2408.03668 "],"isi":["001494367000001"]},"ec_funded":1,"_id":"19776","abstract":[{"lang":"eng","text":"We use the circle method to prove that a density 1 of elements in Fq[t] are representable as a sum of three cubes of essentially minimal degree from Fq[t], assuming the Ratios Conjecture and that char(Fq)>3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting."}],"oa_version":"Published Version","project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"department":[{"_id":"TiBr"}],"date_published":"2025-05-23T00:00:00Z","publication":"Mathematische Zeitschrift","date_updated":"2026-07-16T10:50:51Z","quality_controlled":"1","intvolume":"       310","publication_status":"published","date_created":"2025-06-03T07:30:21Z","article_type":"original","year":"2025","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","fulldoi":"https://doi.org/10.1007/s00209-025-03765-z","doi":"10.1007/s00209-025-03765-z","article_number":"65","type":"journal_article","issue":"4","das_tickbox":"0","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","first_name":"Timothy D","last_name":"Browning"},{"first_name":"Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","full_name":"Glas, Jakob","last_name":"Glas"},{"last_name":"Wang","orcid":"0000-0002-0704-7026","full_name":"Wang, Victor","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","first_name":"Victor"}],"file_date_updated":"2025-06-03T08:28:14Z","citation":{"apa":"Browning, T. D., Glas, J., &#38; Wang, V. (2025). Optimal sums of three cubes in Fq[t]. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-025-03765-z\">https://doi.org/10.1007/s00209-025-03765-z</a>","mla":"Browning, Timothy D., et al. “Optimal Sums of Three Cubes in Fq[T].” <i>Mathematische Zeitschrift</i>, vol. 310, no. 4, 65, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00209-025-03765-z\">10.1007/s00209-025-03765-z</a>.","ista":"Browning TD, Glas J, Wang V. 2025. Optimal sums of three cubes in Fq[t]. Mathematische Zeitschrift. 310(4), 65.","chicago":"Browning, Timothy D, Jakob Glas, and Victor Wang. “Optimal Sums of Three Cubes in Fq[T].” <i>Mathematische Zeitschrift</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00209-025-03765-z\">https://doi.org/10.1007/s00209-025-03765-z</a>.","ieee":"T. D. Browning, J. Glas, and V. Wang, “Optimal sums of three cubes in Fq[t],” <i>Mathematische Zeitschrift</i>, vol. 310, no. 4. Springer Nature, 2025.","ama":"Browning TD, Glas J, Wang V. Optimal sums of three cubes in Fq[t]. <i>Mathematische Zeitschrift</i>. 2025;310(4). doi:<a href=\"https://doi.org/10.1007/s00209-025-03765-z\">10.1007/s00209-025-03765-z</a>","short":"T.D. Browning, J. Glas, V. Wang, Mathematische Zeitschrift 310 (2025)."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"arxiv":1,"oa":1,"language":[{"iso":"eng"}],"ddc":["510"],"researchdata_availability":"no","publisher":"Springer Nature","supplementarymaterial":"no","file":[{"relation":"main_file","checksum":"6f71e25740c28257bf89b8bf116c2b4d","access_level":"open_access","date_created":"2025-06-03T08:28:14Z","content_type":"application/pdf","file_name":"2025_MathZeitschrift_Browning.pdf","date_updated":"2025-06-03T08:28:14Z","file_size":461622,"success":1,"file_id":"19782","creator":"dernst"}],"acknowledgement":"We thank Alexandra Florea for discussions on cubic Gauss sums over function fields, in addition to the anonymous referee for helpful comments. While working on this paper the first two authors were supported by a FWF grant (DOI 10.55776/P36278) and the third author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Open access funding provided by Institute of Science and Technology (IST Austria).","volume":310,"publication_identifier":{"issn":["0025-5874"],"eissn":["1432-1823"]}},{"author":[{"first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","last_name":"Browning"},{"full_name":"Sawin, Will","first_name":"Will","last_name":"Sawin"},{"last_name":"Wang","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","orcid":"0000-0002-0704-7026","full_name":"Wang, Victor","first_name":"Victor"}],"file_date_updated":"2026-01-05T13:15:44Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"citation":{"short":"T.D. Browning, W. Sawin, V. Wang, Mathematische Annalen 393 (2025) 1863–1880.","ama":"Browning TD, Sawin W, Wang V. Pairs of commuting integer matrices. <i>Mathematische Annalen</i>. 2025;393:1863–1880. doi:<a href=\"https://doi.org/10.1007/s00208-025-03285-5\">10.1007/s00208-025-03285-5</a>","ieee":"T. D. Browning, W. Sawin, and V. Wang, “Pairs of commuting integer matrices,” <i>Mathematische Annalen</i>, vol. 393. Springer Nature, pp. 1863–1880, 2025.","chicago":"Browning, Timothy D, Will Sawin, and Victor Wang. “Pairs of Commuting Integer Matrices.” <i>Mathematische Annalen</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00208-025-03285-5\">https://doi.org/10.1007/s00208-025-03285-5</a>.","ista":"Browning TD, Sawin W, Wang V. 2025. Pairs of commuting integer matrices. Mathematische Annalen. 393, 1863–1880.","mla":"Browning, Timothy D., et al. “Pairs of Commuting Integer Matrices.” <i>Mathematische Annalen</i>, vol. 393, Springer Nature, 2025, pp. 1863–1880, doi:<a href=\"https://doi.org/10.1007/s00208-025-03285-5\">10.1007/s00208-025-03285-5</a>.","apa":"Browning, T. D., Sawin, W., &#38; Wang, V. (2025). Pairs of commuting integer matrices. <i>Mathematische Annalen</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00208-025-03285-5\">https://doi.org/10.1007/s00208-025-03285-5</a>"},"type":"journal_article","das_tickbox":"0","file":[{"relation":"main_file","checksum":"1e94da1a67306e03c8e0086518faf4bc","content_type":"application/pdf","access_level":"open_access","date_created":"2026-01-05T13:15:44Z","date_updated":"2026-01-05T13:15:44Z","file_name":"2025_MathAnnalen_Browning.pdf","file_size":337505,"success":1,"file_id":"20950","creator":"dernst"}],"acknowledgement":"The authors are very grateful to Alina Ostafe, Matthew Satriano and Igor Shparlinski for drawing their attention to this problem and for useful comments, and to Michael Larsen and Peter Sarnak for their helpful correspondence. We also thank the referee for their valuable input. While working on this paper the first author was supported by a FWF grant (DOI 10.55776/P36278), the second author by a Sloan Research Fellowship, and the third author by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Open access funding provided by Institute of Science and Technology (IST Austria).","volume":393,"publication_identifier":{"eissn":["1432-1807"],"issn":["0025-5831"]},"page":"1863–1880","arxiv":1,"language":[{"iso":"eng"}],"oa":1,"ddc":["510"],"researchdata_availability":"no","supplementarymaterial":"no","publisher":"Springer Nature","external_id":{"isi":["001567740200001"],"arxiv":["2409.01920"]},"ec_funded":1,"_id":"20367","abstract":[{"lang":"eng","text":"We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket."}],"PlanS_conform":"1","oa_version":"Published Version","department":[{"_id":"TiBr"}],"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278","name":"Rational curves via function field analytic number theory"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"corr_author":"1","month":"10","day":"01","isi":1,"status":"public","has_accepted_license":"1","OA_place":"publisher","OA_type":"hybrid","title":"Pairs of commuting integer matrices","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2025-09-21T22:01:31Z","publication_status":"published","article_type":"original","year":"2025","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","fulldoi":"https://doi.org/10.1007/s00208-025-03285-5","doi":"10.1007/s00208-025-03285-5","date_published":"2025-10-01T00:00:00Z","publication":"Mathematische Annalen","date_updated":"2026-07-17T12:01:12Z","quality_controlled":"1","intvolume":"       393"},{"type":"journal_article","das_tickbox":"0","citation":{"chicago":"Browning, Timothy D, and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique, 2025. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>.","ista":"Browning TD, Chan S. 2025. Solubility of a resultant equation and applications. Journal de l’Ecole Polytechnique - Mathematiques. 12, 1677–1691.","mla":"Browning, Timothy D., and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 12, Ecole Polytechnique, 2025, pp. 1677–91, doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>.","apa":"Browning, T. D., &#38; Chan, S. (2025). Solubility of a resultant equation and applications. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>","short":"T.D. Browning, S. Chan, Journal de l’Ecole Polytechnique - Mathematiques 12 (2025) 1677–1691.","ama":"Browning TD, Chan S. Solubility of a resultant equation and applications. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. 2025;12:1677-1691. doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>","ieee":"T. D. Browning and S. Chan, “Solubility of a resultant equation and applications,” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 12. Ecole Polytechnique, pp. 1677–1691, 2025."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"DOAJ_listed":"1","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","first_name":"Timothy D","last_name":"Browning"},{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","first_name":"Yik Tung","last_name":"Chan"}],"file_date_updated":"2026-02-24T07:56:34Z","supplementarymaterial":"no","publisher":"Ecole Polytechnique","ddc":["510"],"researchdata_availability":"no","arxiv":1,"language":[{"iso":"eng"}],"oa":1,"page":"1677-1691","publication_identifier":{"issn":["2429-7100"],"eissn":["2270-518X"]},"volume":12,"acknowledgement":"While working on this paper, the first author was supported by a FWF grant (DOI 10.55776/P36278).","file":[{"access_level":"open_access","content_type":"application/pdf","date_created":"2026-02-24T07:56:34Z","checksum":"828577ea48ac6109d3e9dd1aeddd45c4","relation":"main_file","file_id":"21356","creator":"dernst","success":1,"file_size":1003689,"date_updated":"2026-02-24T07:56:34Z","file_name":"2025_JEP_Browning.pdf"}],"title":"Solubility of a resultant equation and applications","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"publisher","OA_type":"gold","has_accepted_license":"1","day":"21","month":"10","status":"public","corr_author":"1","department":[{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"abstract":[{"text":"The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x],\r\nsuch that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q.\r\nGiven a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all\r\nquadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common\r\nroot in Fp. Using recent work of Landesman, an application to the average size of the odd part\r\nof the class group of quadratic number fields is also given","lang":"eng"},{"text":" Le grand crible est utilisé pour estimer la densité des polynômes quadratiques Q ∈ Z[x] tels qu’il existe un polynôme de degré impair défini sur Z dont le résultant avec Q est égal à ±1. Étant donné un polynôme unitaire R ∈ Z[x] de degré impair, on s’en sert pour montrer que, pour presque tous les polynômes quadratiques Q ∈ Z[x], il existe un nombre premier p tel que Q et R aient une racine commune dans Fp. En utilisant des travaux récents de Landesman, on obtient également une application concernant la taille moyenne de la partie impaire du groupe de classe des corps quadratiques.","lang":"fre"}],"_id":"21343","oa_version":"Published Version","PlanS_conform":"1","external_id":{"arxiv":["2411.09264"]},"intvolume":"        12","date_updated":"2026-08-12T11:17:50Z","quality_controlled":"1","publication":"Journal de l'Ecole Polytechnique - Mathematiques","date_published":"2025-10-21T00:00:00Z","fulldoi":"https://doi.org/10.5802/jep.320","doi":"10.5802/jep.320","scopus_import":"1","article_processing_charge":"Yes","year":"2025","article_type":"original","date_created":"2026-02-22T23:01:36Z","publication_status":"published"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Integral points on cubic surfaces: heuristics and numerics","has_accepted_license":"1","OA_place":"publisher","OA_type":"hybrid","isi":1,"status":"public","day":"01","related_material":{"record":[{"id":"22234","relation":"research_data","status":"public"}],"link":[{"url":"https://github.com/fwilsch/cubicpts","relation":"software"}]},"month":"09","corr_author":"1","department":[{"_id":"TiBr"}],"project":[{"call_identifier":"FWF","name":"New frontiers of the Manin conjecture","grant_number":"P32428","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425"},{"name":"Rational curves via function field analytic number theory","grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"oa_version":"Published Version","PlanS_conform":"1","abstract":[{"text":"We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.","lang":"eng"}],"_id":"20249","external_id":{"isi":["001552779800001"],"arxiv":["2407.16315"]},"dataavailabilitystatement":"The data used in Sections 6 and 9 is hosted on the Göttingen Research Online Data repository [https://doi.org/10.25625/4FLFH8]. The code used to determine the data in Sections 6.2, 9.1 and 9.2 is found on the second author’s github page.","intvolume":"        31","date_updated":"2026-08-12T12:15:32Z","quality_controlled":"1","publication":"Selecta Mathematica New Series","date_published":"2025-09-01T00:00:00Z","doi":"10.1007/s00029-025-01074-1","fulldoi":"https://doi.org/10.1007/s00029-025-01074-1","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","year":"2025","date_created":"2025-08-31T22:01:31Z","article_type":"original","publication_status":"published","das_tickbox":"1","type":"journal_article","issue":"4","article_number":"81","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"citation":{"chicago":"Browning, Timothy D, and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00029-025-01074-1\">https://doi.org/10.1007/s00029-025-01074-1</a>.","apa":"Browning, T. D., &#38; Wilsch, F. A. (2025). Integral points on cubic surfaces: heuristics and numerics. <i>Selecta Mathematica New Series</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00029-025-01074-1\">https://doi.org/10.1007/s00029-025-01074-1</a>","mla":"Browning, Timothy D., and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>, vol. 31, no. 4, 81, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00029-025-01074-1\">10.1007/s00029-025-01074-1</a>.","ista":"Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 31(4), 81.","short":"T.D. Browning, F.A. Wilsch, Selecta Mathematica New Series 31 (2025).","ieee":"T. D. Browning and F. A. Wilsch, “Integral points on cubic surfaces: heuristics and numerics,” <i>Selecta Mathematica New Series</i>, vol. 31, no. 4. Springer Nature, 2025.","ama":"Browning TD, Wilsch FA. Integral points on cubic surfaces: heuristics and numerics. <i>Selecta Mathematica New Series</i>. 2025;31(4). doi:<a href=\"https://doi.org/10.1007/s00029-025-01074-1\">10.1007/s00029-025-01074-1</a>"},"file_date_updated":"2025-09-03T06:44:44Z","author":[{"first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","last_name":"Browning"},{"orcid":"0000-0001-7302-8256","full_name":"Wilsch, Florian Alexander","id":"560601DA-8D36-11E9-A136-7AC1E5697425","first_name":"Florian Alexander","last_name":"Wilsch"}],"publisher":"Springer Nature","supplementarymaterial":"yes","ddc":["500"],"researchdata_availability":"yes","language":[{"iso":"eng"}],"oa":1,"arxiv":1,"publication_identifier":{"eissn":["1420-9020"],"issn":["1022-1824"]},"volume":31,"acknowledgement":"The authors owe a debt of thanks to Yonatan Harpaz for asking about circle method heuristics for log K3 surfaces. His contribution to the resulting discussion is gratefully acknowledged. Thanks are also due to Andrew Sutherland for help with numerical data for the equation x^3 + y^3 + z^3 = 1, together with Alex Gamburd, Amit Ghosh, Peter Sarnak and Matteo Verzobio for their interest in this paper. Special thanks are due to Victor Wang for helpful conversations about the circle method heuristics and to the anonymous referee for several useful comments. While working on this paper, the authors were supported by a FWF grant (DOI 10.55776/P32428), and the first author was supported by a further FWF grant (DOI 10.55776/P36278) and a grant from the School of Mathematics at the Institute for Advanced Study in Princeton.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).","file":[{"checksum":"89352f1f7e8d2b367ae5f4e9bf9eb1f5","relation":"main_file","access_level":"open_access","date_created":"2025-09-03T06:44:44Z","content_type":"application/pdf","file_size":2484757,"date_updated":"2025-09-03T06:44:44Z","file_name":"2025_SelectaMathematica_Browning.pdf","creator":"dernst","file_id":"20281","success":1}]},{"day":"01","month":"05","status":"public","keyword":["Circle method","moduli spaces of curves","hypersurfaces","Grothendieck ring of varieties","motivic integration"],"corr_author":"1","title":"A motivic circle method","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"repository","OA_type":"green","external_id":{"arxiv":["2304.09645"]},"project":[{"grant_number":"893012","_id":"05A4F6F0-7A3F-11EA-A408-12923DDC885E","call_identifier":"H2020","name":"A motivic circle method"},{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"department":[{"_id":"TiBr"}],"_id":"22929","ec_funded":1,"abstract":[{"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":"eng"},{"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"}],"oa_version":"Preprint","publication":"Annales Scientifiques de l’École Normale Supérieure","date_published":"2025-05-01T00:00:00Z","intvolume":"        58","quality_controlled":"1","date_updated":"2026-09-17T08:22:39Z","year":"2025","publication_status":"published","date_created":"2026-09-13T22:01:57Z","article_type":"original","fulldoi":"https://doi.org/10.24033/asens.2628","doi":"10.24033/asens.2628","article_processing_charge":"No","scopus_import":"1","issue":"5","type":"journal_article","das_tickbox":"0","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2304.09645"}],"citation":{"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>.","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>","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>.","ista":"Bilu M, Browning TD. 2025. A motivic circle method. Annales Scientifiques de l’École Normale Supérieure. 58(5), 1179–1242.","short":"M. Bilu, T.D. Browning, Annales Scientifiques de l’École Normale Supérieure 58 (2025) 1179–1242.","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>","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."},"author":[{"last_name":"Bilu","first_name":"Margaret","full_name":"Bilu, Margaret","id":"98C47862-10D5-11EA-BEDD-0F6F3DDC885E"},{"last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","first_name":"Timothy D"}],"arxiv":1,"language":[{"iso":"eng"}],"oa":1,"page":"1179-1242","publisher":"Société Mathématique de France","supplementarymaterial":"yes","researchdata_availability":"no","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.","publication_identifier":{"eissn":["1873-2151"],"issn":["0012-9593"]},"volume":58},{"OA_place":"publisher","has_accepted_license":"1","OA_type":"hybrid","title":"Strong divisibility sequences and sieve methods","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","corr_author":"1","month":"10","day":"01","isi":1,"status":"public","_id":"17323","ec_funded":1,"abstract":[{"lang":"eng","text":"We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences."}],"oa_version":"Published Version","department":[{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"external_id":{"isi":["001273912800001"]},"date_updated":"2026-07-29T10:01:24Z","quality_controlled":"1","intvolume":"        70","date_published":"2024-10-01T00:00:00Z","publication":"Mathematika","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","fulldoi":"https://doi.org/10.1112/mtk.12269","doi":"10.1112/mtk.12269","article_type":"original","date_created":"2024-07-28T22:01:08Z","publication_status":"published","year":"2024","article_number":"e12269","issue":"4","type":"journal_article","das_tickbox":"0","author":[{"first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","last_name":"Browning"},{"last_name":"Verzobio","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","full_name":"Verzobio, Matteo","orcid":"0000-0002-0854-0306","first_name":"Matteo"}],"file_date_updated":"2025-01-13T11:06:25Z","citation":{"short":"T.D. Browning, M. Verzobio, Mathematika 70 (2024).","ieee":"T. D. Browning and M. Verzobio, “Strong divisibility sequences and sieve methods,” <i>Mathematika</i>, vol. 70, no. 4. London Mathematical Society, 2024.","ama":"Browning TD, Verzobio M. Strong divisibility sequences and sieve methods. <i>Mathematika</i>. 2024;70(4). doi:<a href=\"https://doi.org/10.1112/mtk.12269\">10.1112/mtk.12269</a>","chicago":"Browning, Timothy D, and Matteo Verzobio. “Strong Divisibility Sequences and Sieve Methods.” <i>Mathematika</i>. London Mathematical Society, 2024. <a href=\"https://doi.org/10.1112/mtk.12269\">https://doi.org/10.1112/mtk.12269</a>.","apa":"Browning, T. D., &#38; Verzobio, M. (2024). Strong divisibility sequences and sieve methods. <i>Mathematika</i>. London Mathematical Society. <a href=\"https://doi.org/10.1112/mtk.12269\">https://doi.org/10.1112/mtk.12269</a>","ista":"Browning TD, Verzobio M. 2024. Strong divisibility sequences and sieve methods. Mathematika. 70(4), e12269.","mla":"Browning, Timothy D., and Matteo Verzobio. “Strong Divisibility Sequences and Sieve Methods.” <i>Mathematika</i>, vol. 70, no. 4, e12269, London Mathematical Society, 2024, doi:<a href=\"https://doi.org/10.1112/mtk.12269\">10.1112/mtk.12269</a>."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"researchdata_availability":"no","ddc":["510"],"publisher":"London Mathematical Society","supplementarymaterial":"no","oa":1,"language":[{"iso":"eng"}],"volume":70,"publication_identifier":{"issn":["0025-5793"],"eissn":["2041-7942"]},"file":[{"file_name":"2024_Mathematika_Browning.pdf","date_updated":"2025-01-13T11:06:25Z","file_size":273006,"success":1,"file_id":"18842","creator":"dernst","relation":"main_file","checksum":"0b1518bdc1a901413005c19202cfa497","content_type":"application/pdf","access_level":"open_access","date_created":"2025-01-13T11:06:25Z"}],"acknowledgement":"The authors are very grateful to Andrew Granville, Dimitris Koukoulopoulos, Davide Lombardo,Florian Luca, Igor Shparlinski and Joni Teräväinen for useful comments. While working on thispaper, the first author was supported by a FWF Grant (DOI 10.55776/P36278) and the secondauthor was supported by the European Union’s Horizon 2020 research and innovation programunder the Marie Skłodowska-Curie Grant Agreement Number 101034413."},{"corr_author":"1","day":"23","related_material":{"record":[{"status":"public","id":"18173","relation":"part_of_dissertation"},{"status":"public","id":"18295","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"18294","status":"public"},{"relation":"part_of_dissertation","id":"18293","status":"public"}]},"month":"09","status":"public","has_accepted_license":"1","degree_awarded":"PhD","OA_place":"publisher","title":"Counting rational points over function fields","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","alternative_title":["ISTA Thesis"],"abstract":[{"text":"In this thesis, we are dealing with both arithmetic and geometric problems coming from the\r\nstudy of rational points with a particular focus on function fields over finite fields:\r\n(1) Using the circle method we produce upper bounds for the number of rational points of\r\nbounded height on diagonal cubic surfaces and fourfolds over Fq(t). This is based on\r\njoint work with Leonhard Hochfilzer.\r\n(2) We study rational points on smooth complete intersections X defined by cubic and\r\nquadratic hypersurfaces over Fq(t). We refine the Farey dissection of the “unit square”\r\ndeveloped by Vishe [202] and use the circle method with a Kloosterman refinement to\r\nestablish an asymptotic formula for the number of rational points of bounded height on\r\nX when dim(X) ≥ 23. Under the same hypotheses, we also verify weak approximation.\r\n(3) In joint work with Hochfilzer, we obtain upper bounds for the number of rational points of\r\nbounded height on del Pezzo surfaces of low degree over any global field. Our approach\r\nis to take hyperplane sections, which reduces the problem to uniform estimates for the\r\nnumber of rational points on curves.\r\n(4) We develop a version of the circle method capable of counting Fq-points on jet schemes\r\nof moduli spaces of rational curves on hypersurfaces. Combining this with a spreading\r\nout argument and a result of Mustaţă [150], this allows us to show that these moduli\r\nspaces only have canonical singularities under suitable assumptions on the degree and the\r\ndimension.\r\nIn addition, we give an overview of guiding questions and conjectures in the field of rational\r\npoints and explain the basic mechanism underlying the circle method.\r\n","lang":"eng"}],"_id":"18132","oa_version":"Published Version","department":[{"_id":"GradSch"},{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"date_published":"2024-09-23T00:00:00Z","doi_confirm":"1","date_updated":"2026-10-02T11:44:29Z","date_created":"2024-09-23T18:58:08Z","publication_status":"published","year":"2024","article_processing_charge":"No","supervisor":[{"last_name":"Browning","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D"}],"fulldoi":"https://doi.org/10.15479/at:ista:18132","doi":"10.15479/at:ista:18132","type":"dissertation","das_tickbox":"0","license":"https://creativecommons.org/licenses/by-nc/4.0/","author":[{"first_name":"Jakob","full_name":"Glas, Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","last_name":"Glas"}],"file_date_updated":"2024-09-25T14:08:57Z","citation":{"apa":"Glas, J. (2024). <i>Counting rational points over function fields</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18132\">https://doi.org/10.15479/at:ista:18132</a>","ista":"Glas J. 2024. Counting rational points over function fields. Institute of Science and Technology Austria.","mla":"Glas, Jakob. <i>Counting Rational Points over Function Fields</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:18132\">10.15479/at:ista:18132</a>.","chicago":"Glas, Jakob. “Counting Rational Points over Function Fields.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:18132\">https://doi.org/10.15479/at:ista:18132</a>.","ama":"Glas J. Counting rational points over function fields. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:18132\">10.15479/at:ista:18132</a>","ieee":"J. Glas, “Counting rational points over function fields,” Institute of Science and Technology Austria, 2024.","short":"J. Glas, Counting Rational Points over Function Fields, Institute of Science and Technology Austria, 2024."},"tmp":{"name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","short":"CC BY-NC (4.0)","image":"/images/cc_by_nc.png"},"page":"195","oa":1,"language":[{"iso":"eng"}],"researchdata_availability":"no","ddc":["512"],"publisher":"Institute of Science and Technology Austria","supplementarymaterial":"no","file":[{"date_created":"2024-09-23T18:49:22Z","content_type":"application/x-zip-compressed","access_level":"closed","checksum":"2f8cf5cefdab108b1979caa8146cae9a","relation":"source_file","file_id":"18133","creator":"jglas","file_size":5382106,"file_name":"PhDthesis (3).zip","date_updated":"2024-09-23T18:49:22Z"},{"access_level":"open_access","content_type":"application/pdf","date_created":"2024-09-25T14:08:57Z","relation":"main_file","checksum":"08bb6f14c42b47ff25882a2ce3ea0d8a","success":1,"file_id":"18140","creator":"jglas","date_updated":"2024-09-25T14:08:57Z","file_name":"example-phd.pdf","file_size":2380127}],"publication_identifier":{"issn":["2663-337X"]}},{"_id":"18173","abstract":[{"text":"Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided char (Fq)>3. Under the same hypotheses, we also verify weak approximation.","lang":"eng"}],"oa_version":"Published Version","department":[{"_id":"TiBr"}],"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"external_id":{"arxiv":["2306.02718"]},"has_accepted_license":"1","title":"Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t)","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","corr_author":"1","month":"10","day":"01","related_material":{"record":[{"status":"public","id":"18132","relation":"dissertation_contains"}]},"status":"public","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","fulldoi":"https://doi.org/10.1112/jlms.12991","doi":"10.1112/jlms.12991","article_type":"original","date_created":"2024-10-06T22:01:11Z","publication_status":"published","year":"2024","quality_controlled":"1","date_updated":"2026-10-02T11:44:29Z","intvolume":"       110","date_published":"2024-10-01T00:00:00Z","publication":"Journal of the London Mathematical Society","author":[{"last_name":"Glas","first_name":"Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","full_name":"Glas, Jakob"}],"file_date_updated":"2024-10-07T08:51:01Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"citation":{"ieee":"J. Glas, “Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t),” <i>Journal of the London Mathematical Society</i>, vol. 110, no. 4. London Mathematical Society, 2024.","ama":"Glas J. Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). <i>Journal of the London Mathematical Society</i>. 2024;110(4). doi:<a href=\"https://doi.org/10.1112/jlms.12991\">10.1112/jlms.12991</a>","short":"J. Glas, Journal of the London Mathematical Society 110 (2024).","mla":"Glas, Jakob. “Rational Points on Complete Intersections of Cubic and Quadric Hypersurfaces over Fq(T).” <i>Journal of the London Mathematical Society</i>, vol. 110, no. 4, e12991, London Mathematical Society, 2024, doi:<a href=\"https://doi.org/10.1112/jlms.12991\">10.1112/jlms.12991</a>.","ista":"Glas J. 2024. Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). Journal of the London Mathematical Society. 110(4), e12991.","apa":"Glas, J. (2024). Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). <i>Journal of the London Mathematical Society</i>. London Mathematical Society. <a href=\"https://doi.org/10.1112/jlms.12991\">https://doi.org/10.1112/jlms.12991</a>","chicago":"Glas, Jakob. “Rational Points on Complete Intersections of Cubic and Quadric Hypersurfaces over Fq(T).” <i>Journal of the London Mathematical Society</i>. London Mathematical Society, 2024. <a href=\"https://doi.org/10.1112/jlms.12991\">https://doi.org/10.1112/jlms.12991</a>."},"article_number":"e12991","issue":"4","type":"journal_article","das_tickbox":"0","volume":110,"publication_identifier":{"eissn":["1469-7750"],"issn":["0024-6107"]},"file":[{"creator":"dernst","file_id":"18181","success":1,"file_size":579601,"file_name":"2024_JLondonMathSoc_Glas.pdf","date_updated":"2024-10-07T08:51:01Z","content_type":"application/pdf","access_level":"open_access","date_created":"2024-10-07T08:51:01Z","checksum":"11ebf690363151026ce81f91f2220855","relation":"main_file"}],"acknowledgement":"The author would like to thank his supervisor Tim Browning for suggesting this project and many helpful conversations and Pankaj Vishe for useful comments. Moreover, he is grateful to Dante Bonolis and Julian Lyczak for sharing their expertise in exponential sums and geometry. While working on this paper, the author was supported by FWF grant (DOI 10.55776/P36278).","researchdata_availability":"no","ddc":["510"],"publisher":"London Mathematical Society","supplementarymaterial":"no","arxiv":1,"language":[{"iso":"eng"}],"oa":1},{"external_id":{"arxiv":["2405.16648"]},"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"department":[{"_id":"TiBr"}],"_id":"18295","abstract":[{"lang":"eng","text":"By developing a suitable version of the circle method, we show that the space of degree e rational curves on a smooth hypersurface of degree d has only canonical singularities provided its dimension is sufficiently large with respect to e and d."}],"oa_version":"Preprint","month":"05","related_material":{"record":[{"id":"19013","relation":"later_version","status":"public"},{"relation":"dissertation_contains","id":"18132","status":"public"}]},"day":"26","status":"public","corr_author":"1","title":"Canonical singularities on moduli spaces of rational curves via the  circle method","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","OA_place":"repository","year":"2024","date_created":"2024-10-10T13:15:43Z","publication_status":"submitted","fulldoi":"https://doi.org/10.48550/arXiv.2405.16648","doi":"10.48550/arXiv.2405.16648","article_processing_charge":"No","publication":"arXiv","date_published":"2024-05-26T00:00:00Z","date_updated":"2026-10-02T11:44:29Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"citation":{"chicago":"Glas, Jakob. “Canonical Singularities on Moduli Spaces of Rational Curves via the  Circle Method.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2405.16648\">https://doi.org/10.48550/arXiv.2405.16648</a>.","mla":"Glas, Jakob. “Canonical Singularities on Moduli Spaces of Rational Curves via the  Circle Method.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/arXiv.2405.16648\">10.48550/arXiv.2405.16648</a>.","ista":"Glas J. Canonical singularities on moduli spaces of rational curves via the  circle method. arXiv, <a href=\"https://doi.org/10.48550/arXiv.2405.16648\">10.48550/arXiv.2405.16648</a>.","apa":"Glas, J. (n.d.). Canonical singularities on moduli spaces of rational curves via the  circle method. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2405.16648\">https://doi.org/10.48550/arXiv.2405.16648</a>","short":"J. Glas, ArXiv (n.d.).","ama":"Glas J. Canonical singularities on moduli spaces of rational curves via the  circle method. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2405.16648\">10.48550/arXiv.2405.16648</a>","ieee":"J. Glas, “Canonical singularities on moduli spaces of rational curves via the  circle method,” <i>arXiv</i>. ."},"author":[{"last_name":"Glas","first_name":"Jakob","full_name":"Glas, Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb"}],"type":"preprint","das_tickbox":"0","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2405.16648"}],"arxiv":1,"oa":1,"language":[{"iso":"eng"}],"supplementarymaterial":"no","researchdata_availability":"no"}]