[{"month":"11","abstract":[{"text":"We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with respect to the largest admissible boundary divisor. The resulting Clemens complex is more complicated than usual, and leads to particularly interesting effective cone constants, associated with exponentially many polytopes whose volumes appear in the expected formula. Like a jigsaw puzzle, these polytopes fit together to one large polytope. The volume of this polytope appears in the asymptotic formula that we obtain using the universal torsor method via o-minimal structures.","lang":"eng"}],"das_tickbox":"1","oa_version":"Published Version","type":"journal_article","external_id":{"arxiv":["2601.08774"]},"date_published":"2026-11-01T00:00:00Z","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)","short":"CC BY (4.0)","image":"/images/cc_by.png"},"ddc":["510"],"citation":{"mla":"Bernert, Christian, et al. “Integral Points over Number Fields: A Clemens Complex Jigsaw Puzzle.” <i>Mathematische Zeitschrift</i>, vol. 314, no. 3, 37, Springer, 2026, doi:<a href=\"https://doi.org/10.1007/s00209-026-04140-2\">10.1007/s00209-026-04140-2</a>.","apa":"Bernert, C., Derenthal, U., Ortmann, J., &#38; Wilsch, F. A. (2026). Integral points over number fields: a Clemens complex jigsaw puzzle. <i>Mathematische Zeitschrift</i>. Springer. <a href=\"https://doi.org/10.1007/s00209-026-04140-2\">https://doi.org/10.1007/s00209-026-04140-2</a>","short":"C. Bernert, U. Derenthal, J. Ortmann, F.A. Wilsch, Mathematische Zeitschrift 314 (2026).","ieee":"C. Bernert, U. Derenthal, J. Ortmann, and F. A. Wilsch, “Integral points over number fields: a Clemens complex jigsaw puzzle,” <i>Mathematische Zeitschrift</i>, vol. 314, no. 3. Springer, 2026.","ista":"Bernert C, Derenthal U, Ortmann J, Wilsch FA. 2026. Integral points over number fields: a Clemens complex jigsaw puzzle. Mathematische Zeitschrift. 314(3), 37.","ama":"Bernert C, Derenthal U, Ortmann J, Wilsch FA. Integral points over number fields: a Clemens complex jigsaw puzzle. <i>Mathematische Zeitschrift</i>. 2026;314(3). doi:<a href=\"https://doi.org/10.1007/s00209-026-04140-2\">10.1007/s00209-026-04140-2</a>","chicago":"Bernert, Christian, Ulrich Derenthal, Judith Ortmann, and Florian Alexander Wilsch. “Integral Points over Number Fields: A Clemens Complex Jigsaw Puzzle.” <i>Mathematische Zeitschrift</i>. Springer, 2026. <a href=\"https://doi.org/10.1007/s00209-026-04140-2\">https://doi.org/10.1007/s00209-026-04140-2</a>."},"status":"public","OA_type":"hybrid","dataavailabilitystatement":"Our manuscript has no associated data.","doi":"10.1007/s00209-026-04140-2","article_number":"37","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","department":[{"_id":"TiBr"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","researchdata_availability":"no","issue":"3","publication_status":"published","fulldoi":"https://doi.org/10.1007/s00209-026-04140-2","date_updated":"2026-10-19T06:13:10Z","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"publisher":"Springer","file":[{"file_size":593086,"date_updated":"2026-10-19T06:11:23Z","date_created":"2026-10-19T06:11:23Z","file_name":"2026_MathemZeitschrift_Bernert.pdf","content_type":"application/pdf","checksum":"c2157633b4832a760336cd5d3f77b022","success":1,"creator":"dernst","relation":"main_file","access_level":"open_access","file_id":"23160"}],"scopus_import":"1","title":"Integral points over number fields: a Clemens complex jigsaw puzzle","quality_controlled":"1","author":[{"id":"aefba9fc-4694-11f0-b73e-8bd408e8f734","last_name":"Bernert","full_name":"Bernert, Christian","first_name":"Christian"},{"first_name":"Ulrich","last_name":"Derenthal","full_name":"Derenthal, Ulrich"},{"first_name":"Judith","last_name":"Ortmann","full_name":"Ortmann, Judith"},{"full_name":"Wilsch, Florian Alexander","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","orcid":"0000-0001-7302-8256","first_name":"Florian Alexander"}],"article_type":"original","file_date_updated":"2026-10-19T06:11:23Z","date_created":"2026-10-18T22:01:50Z","_id":"23149","supplementarymaterial":"no","year":"2026","OA_place":"publisher","language":[{"iso":"eng"}],"oa":1,"intvolume":"       314","volume":314,"publication":"Mathematische Zeitschrift","PlanS_conform":"1","acknowledgement":"Open Access funding enabled and organized by Projekt DEAL. We would like to thank the referee for their careful reading and helpful remarks that helped to improve this article. The first two authors were supported by the Deutsche Forschungsgemeinschaft (DFG) – 512730679 (RTG2965). The third author was partially supported by the Caroline Herschel Programme of Leibniz University Hannover. The fourth author was supported by the Deutsche Forschungsgemeinschaft (DFG) – 398436923 (RTG2491).","day":"01","arxiv":1},{"das_tickbox":"1","oa_version":"Preprint","external_id":{"arxiv":["2403.09989"]},"type":"journal_article","month":"07","abstract":[{"text":"We prove dispersive decay, pointwise in time, for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions d= 1, 2, 3.","lang":"eng"}],"status":"public","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2403.09989"}],"doi":"10.1007/s00209-025-03821-8","OA_type":"green","date_published":"2025-07-24T00:00:00Z","citation":{"ista":"Fan C, Killip R, Vişan M, Zhao Z. 2025. Dispersive decay for the mass-critical nonlinear Schrödinger equation. Mathematische Zeitschrift. 311, 21.","ama":"Fan C, Killip R, Vişan M, Zhao Z. Dispersive decay for the mass-critical nonlinear Schrödinger equation. <i>Mathematische Zeitschrift</i>. 2025;311. doi:<a href=\"https://doi.org/10.1007/s00209-025-03821-8\">10.1007/s00209-025-03821-8</a>","chicago":"Fan, Chenjie, Rowan Killip, Monica Vişan, and Zehua Zhao. “Dispersive Decay for the Mass-Critical Nonlinear Schrödinger Equation.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00209-025-03821-8\">https://doi.org/10.1007/s00209-025-03821-8</a>.","ieee":"C. Fan, R. Killip, M. Vişan, and Z. Zhao, “Dispersive decay for the mass-critical nonlinear Schrödinger equation,” <i>Mathematische Zeitschrift</i>, vol. 311. Springer Nature, 2025.","short":"C. Fan, R. Killip, M. Vişan, Z. Zhao, Mathematische Zeitschrift 311 (2025).","mla":"Fan, Chenjie, et al. “Dispersive Decay for the Mass-Critical Nonlinear Schrödinger Equation.” <i>Mathematische Zeitschrift</i>, vol. 311, 21, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00209-025-03821-8\">10.1007/s00209-025-03821-8</a>.","apa":"Fan, C., Killip, R., Vişan, M., &#38; Zhao, Z. (2025). Dispersive decay for the mass-critical nonlinear Schrödinger equation. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-025-03821-8\">https://doi.org/10.1007/s00209-025-03821-8</a>"},"article_processing_charge":"No","article_number":"21","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","fulldoi":"https://doi.org/10.1007/s00209-025-03821-8","date_updated":"2026-06-22T13:00:14Z","title":"Dispersive decay for the mass-critical nonlinear Schrödinger equation","quality_controlled":"1","author":[{"full_name":"Fan, Chenjie","last_name":"Fan","first_name":"Chenjie"},{"first_name":"Rowan","full_name":"Killip, Rowan","last_name":"Killip"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan","first_name":"Monica"},{"full_name":"Zhao, Zehua","last_name":"Zhao","first_name":"Zehua"}],"article_type":"original","publication_identifier":{"issn":["0025-5874"],"eissn":["1432-1823"]},"publisher":"Springer Nature","scopus_import":"1","extern":"1","date_created":"2026-06-19T07:44:05Z","_id":"22036","oa":1,"intvolume":"       311","year":"2025","OA_place":"repository","language":[{"iso":"eng"}],"day":"24","arxiv":1,"publication":"Mathematische Zeitschrift","volume":311},{"oa":1,"intvolume":"       310","year":"2025","OA_place":"repository","corr_author":"1","language":[{"iso":"eng"}],"day":"01","arxiv":1,"volume":310,"publication":"Mathematische Zeitschrift","title":"Scales","quality_controlled":"1","article_type":"original","author":[{"first_name":"Mathieu","last_name":"Helfter","id":"7d296fbe-e2c6-11ee-84d3-d5c2945f9a57","full_name":"Helfter, Mathieu"}],"publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"publisher":"Springer Nature","scopus_import":"1","_id":"19496","date_created":"2025-04-06T22:01:32Z","article_processing_charge":"No","department":[{"_id":"VaKa"}],"article_number":"15","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","date_updated":"2025-09-30T11:31:00Z","fulldoi":"https://doi.org/10.1007/s00209-025-03719-5","oa_version":"Preprint","type":"journal_article","external_id":{"arxiv":["2206.05231"],"isi":["001450830300001"]},"month":"05","abstract":[{"lang":"eng","text":"We introduce the notions of scale for sets and measures on metric space by generalizing the usual notions of dimension. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They are defined for different growth, allowing a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and from functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich–Lifshits (2005); the last refines Kolmogorov–Tikhomirov (1958) study on finitely differentiable functions."}],"status":"public","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2206.05231"}],"isi":1,"doi":"10.1007/s00209-025-03719-5","OA_type":"green","date_published":"2025-05-01T00:00:00Z","citation":{"ieee":"M. Helfter, “Scales,” <i>Mathematische Zeitschrift</i>, vol. 310. Springer Nature, 2025.","short":"M. Helfter, Mathematische Zeitschrift 310 (2025).","ama":"Helfter M. Scales. <i>Mathematische Zeitschrift</i>. 2025;310. doi:<a href=\"https://doi.org/10.1007/s00209-025-03719-5\">10.1007/s00209-025-03719-5</a>","chicago":"Helfter, Mathieu. “Scales.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00209-025-03719-5\">https://doi.org/10.1007/s00209-025-03719-5</a>.","ista":"Helfter M. 2025. Scales. Mathematische Zeitschrift. 310, 15.","mla":"Helfter, Mathieu. “Scales.” <i>Mathematische Zeitschrift</i>, vol. 310, 15, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00209-025-03719-5\">10.1007/s00209-025-03719-5</a>.","apa":"Helfter, M. (2025). Scales. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-025-03719-5\">https://doi.org/10.1007/s00209-025-03719-5</a>"}},{"oa_version":"Published Version","das_tickbox":"0","type":"journal_article","external_id":{"isi":["001494367000001"],"arxiv":["2408.03668 "]},"month":"05","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."}],"status":"public","project":[{"name":"Rational curves via function field analytic number theory","grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"},{"grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"OA_type":"hybrid","isi":1,"doi":"10.1007/s00209-025-03765-z","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)","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_published":"2025-05-23T00:00:00Z","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>.","short":"T.D. Browning, J. Glas, V. Wang, Mathematische Zeitschrift 310 (2025).","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>","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>.","ista":"Browning TD, Glas J, Wang V. 2025. Optimal sums of three cubes in Fq[t]. Mathematische Zeitschrift. 310(4), 65."},"ddc":["510"],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"TiBr"}],"article_number":"65","researchdata_availability":"no","ec_funded":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","fulldoi":"https://doi.org/10.1007/s00209-025-03765-z","date_updated":"2026-07-16T10:50:51Z","publication_status":"published","issue":"4","quality_controlled":"1","title":"Optimal sums of three cubes in Fq[t]","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","last_name":"Browning","first_name":"Timothy D","orcid":"0000-0002-8314-0177"},{"first_name":"Jakob","full_name":"Glas, Jakob","last_name":"Glas","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb"},{"id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","last_name":"Wang","full_name":"Wang, Victor","orcid":"0000-0002-0704-7026","first_name":"Victor"}],"article_type":"original","publisher":"Springer Nature","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"scopus_import":"1","file":[{"file_size":461622,"content_type":"application/pdf","file_name":"2025_MathZeitschrift_Browning.pdf","date_created":"2025-06-03T08:28:14Z","date_updated":"2025-06-03T08:28:14Z","success":1,"checksum":"6f71e25740c28257bf89b8bf116c2b4d","creator":"dernst","relation":"main_file","file_id":"19782","access_level":"open_access"}],"supplementarymaterial":"no","_id":"19776","date_created":"2025-06-03T07:30:21Z","file_date_updated":"2025-06-03T08:28:14Z","oa":1,"intvolume":"       310","year":"2025","language":[{"iso":"eng"}],"OA_place":"publisher","corr_author":"1","arxiv":1,"day":"23","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":"Mathematische Zeitschrift"},{"page":"2327-2352","_id":"12210","date_created":"2023-01-16T09:45:31Z","author":[{"last_name":"Akylzhanov","full_name":"Akylzhanov, Rauan","first_name":"Rauan"},{"last_name":"Kuznetsova","full_name":"Kuznetsova, Yulia","first_name":"Yulia"},{"last_name":"Ruzhansky","full_name":"Ruzhansky, Michael","first_name":"Michael"},{"first_name":"Haonan","full_name":"Zhang, Haonan","id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425","last_name":"Zhang"}],"article_type":"original","title":"Norms of certain functions of a distinguished Laplacian on the ax + b groups","quality_controlled":"1","scopus_import":"1","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"publisher":"Springer Nature","acknowledgement":"Yu. K. thanks Professor Waldemar Hebisch for valuable discussions on the general context of multipliers on Lie groups. This work was started during an ICL-CNRS fellowship of the second\r\nnamed author at the Imperial College London. Yu. K. is supported by the ANR-19-CE40-0002 grant of the French National Research Agency (ANR). H. Z. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. R. A. was supported by the EPSRC grant EP/R003025. M. R. is supported by the EPSRC grant EP/R003025/2 and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations.","arxiv":1,"day":"01","volume":302,"publication":"Mathematische Zeitschrift","intvolume":"       302","oa":1,"language":[{"iso":"eng"}],"year":"2022","isi":1,"doi":"10.1007/s00209-022-03143-z","status":"public","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"},{"grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis","_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6"}],"main_file_link":[{"url":"https://arxiv.org/abs/2101.00584","open_access":"1"}],"citation":{"short":"R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift 302 (2022) 2327–2352.","ieee":"R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain functions of a distinguished Laplacian on the ax + b groups,” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022.","chicago":"Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00209-022-03143-z\">https://doi.org/10.1007/s00209-022-03143-z</a>.","ama":"Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>. 2022;302(4):2327-2352. doi:<a href=\"https://doi.org/10.1007/s00209-022-03143-z\">10.1007/s00209-022-03143-z</a>","ista":"Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. 2022. Norms of certain functions of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift. 302(4), 2327–2352.","mla":"Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4, Springer Nature, 2022, pp. 2327–52, doi:<a href=\"https://doi.org/10.1007/s00209-022-03143-z\">10.1007/s00209-022-03143-z</a>.","apa":"Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., &#38; Zhang, H. (2022). Norms of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-022-03143-z\">https://doi.org/10.1007/s00209-022-03143-z</a>"},"date_published":"2022-12-01T00:00:00Z","external_id":{"arxiv":["2101.00584"],"isi":["000859680700001"]},"type":"journal_article","oa_version":"Preprint","abstract":[{"lang":"eng","text":"The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies\r\n∥kt∥1≍t,∥kt∥∞≍1,\r\nso that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s."}],"month":"12","keyword":["General Mathematics"],"publication_status":"published","issue":"4","date_updated":"2025-04-14T07:44:00Z","fulldoi":"https://doi.org/10.1007/s00209-022-03143-z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","ec_funded":1,"department":[{"_id":"JaMa"}],"article_processing_charge":"No"},{"oa":1,"intvolume":"       300","year":"2021","OA_place":"repository","language":[{"iso":"eng"}],"day":"17","arxiv":1,"volume":300,"publication":"Mathematische Zeitschrift","title":"Kuroda’s formula and arithmetic statistics","quality_controlled":"1","author":[{"last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","full_name":"Chan, Yik Tung","first_name":"Yik Tung","orcid":"0000-0001-8467-4106"},{"last_name":"Milovic","full_name":"Milovic, Djordjo","first_name":"Djordjo"}],"article_type":"original","publication_identifier":{"issn":["0025-5874"],"eissn":["1432-1823"]},"publisher":"Springer Nature","scopus_import":"1","extern":"1","date_created":"2025-04-05T10:51:04Z","_id":"19492","page":"1509-1527","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","issue":"2","date_updated":"2025-07-10T11:51:48Z","fulldoi":"https://doi.org/10.1007/s00209-021-02823-6","oa_version":"Preprint","external_id":{"arxiv":["1905.09745"]},"type":"journal_article","month":"08","abstract":[{"text":"Kuroda’s formula relates the class number of a multiquadratic number field K to the class numbers of its quadratic subfields ki. A key component in this formula is the unit group index (math formular). We study how Q(K) behaves on average in certain natural families of totally real biquadratic fields K parametrized by prime numbers.","lang":"eng"}],"status":"public","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1905.09745"}],"OA_type":"green","doi":"10.1007/s00209-021-02823-6","date_published":"2021-08-17T00:00:00Z","citation":{"chicago":"Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic Statistics.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00209-021-02823-6\">https://doi.org/10.1007/s00209-021-02823-6</a>.","ista":"Chan S, Milovic D. 2021. Kuroda’s formula and arithmetic statistics. Mathematische Zeitschrift. 300(2), 1509–1527.","ama":"Chan S, Milovic D. Kuroda’s formula and arithmetic statistics. <i>Mathematische Zeitschrift</i>. 2021;300(2):1509-1527. doi:<a href=\"https://doi.org/10.1007/s00209-021-02823-6\">10.1007/s00209-021-02823-6</a>","ieee":"S. Chan and D. Milovic, “Kuroda’s formula and arithmetic statistics,” <i>Mathematische Zeitschrift</i>, vol. 300, no. 2. Springer Nature, pp. 1509–1527, 2021.","short":"S. Chan, D. Milovic, Mathematische Zeitschrift 300 (2021) 1509–1527.","apa":"Chan, S., &#38; Milovic, D. (2021). Kuroda’s formula and arithmetic statistics. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-021-02823-6\">https://doi.org/10.1007/s00209-021-02823-6</a>","mla":"Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic Statistics.” <i>Mathematische Zeitschrift</i>, vol. 300, no. 2, Springer Nature, 2021, pp. 1509–27, doi:<a href=\"https://doi.org/10.1007/s00209-021-02823-6\">10.1007/s00209-021-02823-6</a>."}},{"language":[{"iso":"eng"}],"year":"2021","intvolume":"       299","oa":1,"publication":"Mathematische Zeitschrift","volume":299,"day":"05","acknowledgement":"While working on this paper the authors were both supported by EPSRC grant EP/P026710/1, and the second author received additional support from the NWO Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho Tanimoto for useful conversations related to this topic, and to the anonymous referee for numerous helpful suggestions.","scopus_import":"1","file":[{"file_size":492685,"date_updated":"2021-03-22T12:41:26Z","file_name":"2021_MathZeitschrift_Browning.pdf","date_created":"2021-03-22T12:41:26Z","content_type":"application/pdf","creator":"dernst","relation":"main_file","access_level":"open_access","file_id":"9279","checksum":"8ed9f49568806894744096dbbca0ad7b","success":1}],"publisher":"Springer Nature","publication_identifier":{"issn":["0025-5874"],"eissn":["1432-1823"]},"article_type":"original","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","full_name":"Browning, Timothy D","first_name":"Timothy D","orcid":"0000-0002-8314-0177"},{"first_name":"Shuntaro","full_name":"Yamagishi, Shuntaro","last_name":"Yamagishi"}],"quality_controlled":"1","title":"Arithmetic of higher-dimensional orbifolds and a mixed Waring problem","page":"1071–1101","_id":"9260","date_created":"2021-03-21T23:01:21Z","file_date_updated":"2021-03-22T12:41:26Z","supplementarymaterial":"no","department":[{"_id":"TiBr"}],"has_accepted_license":"1","article_processing_charge":"No","date_updated":"2026-08-06T11:14:26Z","fulldoi":"https://doi.org/10.1007/s00209-021-02695-w","publication_status":"published","researchdata_availability":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","abstract":[{"text":"We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ) when Δ is a Q-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.","lang":"eng"}],"month":"03","external_id":{"isi":["000625573800002"]},"type":"journal_article","das_tickbox":"0","oa_version":"Published Version","citation":{"apa":"Browning, T. D., &#38; Yamagishi, S. (2021). Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-021-02695-w\">https://doi.org/10.1007/s00209-021-02695-w</a>","mla":"Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>, vol. 299, Springer Nature, 2021, pp. 1071–1101, doi:<a href=\"https://doi.org/10.1007/s00209-021-02695-w\">10.1007/s00209-021-02695-w</a>.","ama":"Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. 2021;299:1071–1101. doi:<a href=\"https://doi.org/10.1007/s00209-021-02695-w\">10.1007/s00209-021-02695-w</a>","ista":"Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.","chicago":"Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00209-021-02695-w\">https://doi.org/10.1007/s00209-021-02695-w</a>.","short":"T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.","ieee":"T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds and a mixed Waring problem,” <i>Mathematische Zeitschrift</i>, vol. 299. Springer Nature, pp. 1071–1101, 2021."},"ddc":["510"],"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)","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_published":"2021-03-05T00:00:00Z","isi":1,"doi":"10.1007/s00209-021-02695-w","status":"public","project":[{"name":"Between rational and integral points","grant_number":"EP-P026710-2","_id":"26A8D266-B435-11E9-9278-68D0E5697425"}]},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.1007/s00209-017-1934-8","date_updated":"2026-06-25T07:36:26Z","issue":"3-4","publication_status":"published","article_processing_charge":"No","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1503.02716","open_access":"1"}],"status":"public","doi":"10.1007/s00209-017-1934-8","OA_type":"green","date_published":"2018-04-01T00:00:00Z","citation":{"mla":"Killip, R., et al. “Sobolev Spaces Adapted to the Schrödinger Operator with Inverse-Square Potential.” <i>Mathematische Zeitschrift</i>, vol. 288, no. 3–4, Springer Nature, 2018, pp. 1273–98, doi:<a href=\"https://doi.org/10.1007/s00209-017-1934-8\">10.1007/s00209-017-1934-8</a>.","apa":"Killip, R., Miao, C., Vişan, M., Zhang, J., &#38; Zheng, J. (2018). Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-017-1934-8\">https://doi.org/10.1007/s00209-017-1934-8</a>","ieee":"R. Killip, C. Miao, M. Vişan, J. Zhang, and J. Zheng, “Sobolev spaces adapted to the Schrödinger operator with inverse-square potential,” <i>Mathematische Zeitschrift</i>, vol. 288, no. 3–4. Springer Nature, pp. 1273–1298, 2018.","short":"R. Killip, C. Miao, M. Vişan, J. Zhang, J. Zheng, Mathematische Zeitschrift 288 (2018) 1273–1298.","chicago":"Killip, R., C. Miao, Monica Vişan, J. Zhang, and J. Zheng. “Sobolev Spaces Adapted to the Schrödinger Operator with Inverse-Square Potential.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2018. <a href=\"https://doi.org/10.1007/s00209-017-1934-8\">https://doi.org/10.1007/s00209-017-1934-8</a>.","ama":"Killip R, Miao C, Vişan M, Zhang J, Zheng J. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. <i>Mathematische Zeitschrift</i>. 2018;288(3-4):1273-1298. doi:<a href=\"https://doi.org/10.1007/s00209-017-1934-8\">10.1007/s00209-017-1934-8</a>","ista":"Killip R, Miao C, Vişan M, Zhang J, Zheng J. 2018. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. Mathematische Zeitschrift. 288(3–4), 1273–1298."},"oa_version":"Preprint","type":"journal_article","external_id":{"arxiv":["1503.02716"]},"month":"04","abstract":[{"lang":"eng","text":"We study the L p-theory for the Schrödinger operatorLa with inverse-square potential\r\na|x|^−2. Our main result describes when L p-based Sobolev spaces defined in terms of the\r\noperator (La)^s/2 agree with those defined via (−\u0002)^s/2.We consider all regularities 0 < s < 2.\r\nIn order to make the paper self-contained, we also review (with proofs) multiplier theorems,\r\nLittlewood–Paley theory, and Hardy-type inequalities associated to the operator La."}],"arxiv":1,"day":"01","publication":"Mathematische Zeitschrift","volume":288,"oa":1,"intvolume":"       288","year":"2018","mathsc":["35P25","35Q55"],"language":[{"iso":"eng"}],"OA_place":"repository","_id":"22042","page":"1273-1298","date_created":"2026-06-19T07:46:14Z","extern":"1","quality_controlled":"1","title":"Sobolev spaces adapted to the Schrödinger operator with inverse-square potential","article_type":"original","author":[{"last_name":"Killip","full_name":"Killip, R.","first_name":"R."},{"first_name":"C.","full_name":"Miao, C.","last_name":"Miao"},{"full_name":"Visan, Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica"},{"full_name":"Zhang, J.","last_name":"Zhang","first_name":"J."},{"first_name":"J.","full_name":"Zheng, J.","last_name":"Zheng"}],"publisher":"Springer Nature","publication_identifier":{"issn":["0025-5874"],"eissn":["1432-1823"]},"scopus_import":"1"}]