[{"citation":{"ista":"Wang V, Xu M. 2026. Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics., 1–15.","ama":"Wang V, Xu M. Average sizes of mixed character sums. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. 2026:1-15. doi:<a href=\"https://doi.org/10.1017/prm.2026.10123\">10.1017/prm.2026.10123</a>","ieee":"V. Wang and M. Xu, “Average sizes of mixed character sums,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press, pp. 1–15, 2026.","apa":"Wang, V., &#38; Xu, M. (2026). Average sizes of mixed character sums. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2026.10123\">https://doi.org/10.1017/prm.2026.10123</a>","short":"V. Wang, M. Xu, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2026) 1–15.","chicago":"Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press, 2026. <a href=\"https://doi.org/10.1017/prm.2026.10123\">https://doi.org/10.1017/prm.2026.10123</a>.","mla":"Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, Cambridge University Press, 2026, pp. 1–15, doi:<a href=\"https://doi.org/10.1017/prm.2026.10123\">10.1017/prm.2026.10123</a>."},"tmp":{"short":"CC BY (4.0)","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"},"article_processing_charge":"Yes (via OA deal)","publication":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","has_accepted_license":"1","year":"2026","department":[{"_id":"TiBr"}],"type":"journal_article","doi":"10.1017/prm.2026.10123","ec_funded":1,"publisher":"Cambridge University Press","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"epub_ahead","PlanS_conform":"1","oa_version":"Published Version","external_id":{"arxiv":["2411.14181"]},"page":"1-15","oa":1,"title":"Average sizes of mixed character sums","status":"public","arxiv":1,"month":"01","project":[{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413"}],"_id":"21385","date_published":"2026-01-01T00:00:00Z","date_updated":"2026-07-16T08:39:57Z","corr_author":"1","quality_controlled":"1","OA_place":"publisher","license":"https://creativecommons.org/licenses/by/4.0/","date_created":"2026-03-02T10:09:23Z","acknowledgement":"We thank Ofir Gorodetsky, Andrew Granville, Adam Harper, Youness Lamzouri,\r\nKannan Soundararajan, Ping Xi, and Matt Young for their interest, helpful discussions, and comments. Special thanks are due to Jonathan Bober, Oleksiy Klurman,\r\nand Besfort Shala for sending us a letter about Question 1.3, and to Hung Bui\r\nfor informing us of [7]. V.W. thanks Stanford University for its hospitality and is supported by the European Union’s Horizon 2020 research and innovation program\r\nunder the Marie Skłodowska–Curie Grant Agreement No. 101034413. M.X. is supported by a Simons Junior Fellowship from the Simons Society of Fellows at the\r\nSimons Foundation.","das_tickbox":"0","fulldoi":"https://doi.org/10.1017/prm.2026.10123","abstract":[{"text":"We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. ","lang":"eng"}],"article_type":"original","publication_identifier":{"eissn":["1473-7124"],"issn":["0308-2105"]},"ddc":["510"],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1017/prm.2026.10123"}],"researchdata_availability":"no","author":[{"first_name":"Victor","orcid":"0000-0002-0704-7026","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","last_name":"Wang","full_name":"Wang, Victor"},{"first_name":"Max","full_name":"Xu, Max","last_name":"Xu"}],"OA_type":"hybrid","supplementarymaterial":"no"},{"project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"month":"10","arxiv":1,"volume":155,"status":"public","title":"Common valuations of division polynomials","oa":1,"file_date_updated":"2025-12-30T06:45:47Z","page":"1646-1660","quality_controlled":"1","corr_author":"1","date_updated":"2026-07-16T08:43:40Z","scopus_import":"1","date_published":"2025-10-01T00:00:00Z","_id":"12311","article_type":"original","publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"abstract":[{"lang":"eng","text":"In this note, we prove a formula for the cancellation exponent  kv,n between division polynomials  ψn  and  ϕn  associated with a sequence  {nP}n∈N of points on an elliptic curve  E  defined over a discrete valuation field  K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields."}],"fulldoi":"https://doi.org/10.1017/prm.2024.7","file":[{"date_updated":"2025-12-30T06:45:47Z","file_size":477624,"checksum":"c5ec6e29aca2fb4533cb95fac409a0b2","success":1,"content_type":"application/pdf","creator":"dernst","access_level":"open_access","file_id":"20878","date_created":"2025-12-30T06:45:47Z","relation":"main_file","file_name":"2025_ProceedingsRoyalSocEdinburghA_Naskrecki.pdf"}],"acknowledgement":"Silverman, and Paul Voutier for the comments on the earlier version of this paper. The first author acknowledges the support by Dioscuri programme initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research. The second author has been supported by MIUR (Italy) through PRIN 2017 ‘Geometric, algebraic and analytic methods in arithmetic’ and has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413.","das_tickbox":"0","OA_place":"publisher","date_created":"2023-01-16T11:45:22Z","supplementarymaterial":"no","author":[{"first_name":"Bartosz","full_name":"Naskręcki, Bartosz","last_name":"Naskręcki"},{"first_name":"Matteo","orcid":"0000-0002-0854-0306","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","last_name":"Verzobio","full_name":"Verzobio, Matteo"}],"OA_type":"hybrid","researchdata_availability":"no","ddc":["510"],"has_accepted_license":"1","publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","article_processing_charge":"Yes (via OA deal)","tmp":{"short":"CC BY (4.0)","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"},"citation":{"mla":"Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol. 155, no. 5, Cambridge University Press, 2025, pp. 1646–60, doi:<a href=\"https://doi.org/10.1017/prm.2024.7\">10.1017/prm.2024.7</a>.","ama":"Naskręcki B, Verzobio M. Common valuations of division polynomials. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. 2025;155(5):1646-1660. doi:<a href=\"https://doi.org/10.1017/prm.2024.7\">10.1017/prm.2024.7</a>","ieee":"B. Naskręcki and M. Verzobio, “Common valuations of division polynomials,” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol. 155, no. 5. Cambridge University Press, pp. 1646–1660, 2025.","chicago":"Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/prm.2024.7\">https://doi.org/10.1017/prm.2024.7</a>.","short":"B. Naskręcki, M. Verzobio, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 155 (2025) 1646–1660.","apa":"Naskręcki, B., &#38; Verzobio, M. (2025). Common valuations of division polynomials. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2024.7\">https://doi.org/10.1017/prm.2024.7</a>","ista":"Naskręcki B, Verzobio M. 2025. Common valuations of division polynomials. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 155(5), 1646–1660."},"keyword":["Elliptic curves","Néron models","division polynomials","height functions","discrete valuation rings"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"publisher":"Cambridge University Press","ec_funded":1,"doi":"10.1017/prm.2024.7","isi":1,"type":"journal_article","issue":"5","department":[{"_id":"TiBr"}],"year":"2025","day":"01","oa_version":"Published Version","PlanS_conform":"1","intvolume":"       155","external_id":{"isi":["001174907100001"],"arxiv":["2203.02015"]}},{"OA_type":"hybrid","author":[{"first_name":"Francesco","last_name":"Ballini","full_name":"Ballini, Francesco"},{"last_name":"Lombardo","full_name":"Lombardo, Davide","first_name":"Davide"},{"full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","last_name":"Verzobio","orcid":"0000-0002-0854-0306","first_name":"Matteo"}],"supplementarymaterial":"no","ddc":["500"],"main_file_link":[{"url":"https://doi.org/10.1017/prm.2025.7","open_access":"1"}],"researchdata_availability":"no","acknowledgement":"We thank Umberto Zannier for bringing the problem to our attention, for many useful suggestions, and especially for pointing out the relevance of the equidistribution results of Katz–Sarnak, noting that they imply the case  q≫g0 of theorem 1.4. In addition, the first author would like to thank Umberto Zannier for his guidance during his undergraduate studies, on a topic that ultimately inspired much of the work in this article. We are grateful to J. Kaczorowski and A. Perelli for sharing their work [Reference Kaczorowski and Perelli28] before publication. We thank Christophe Ritzenthaler and Elisa Lorenzo García for their interesting comments on the first version of this article, Zhao Yu Ma for a comment about remark 3.12, and the anonymous referees for their helpful suggestions.","das_tickbox":"0","fulldoi":"https://doi.org/10.1017/prm.2025.7","abstract":[{"lang":"eng","text":"We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over  Fq . Among other results, this allows us to prove that the  Q-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields."}],"article_type":"original","publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"OA_place":"publisher","date_created":"2025-03-16T23:01:25Z","date_updated":"2026-07-16T09:31:02Z","corr_author":"1","quality_controlled":"1","_id":"19407","date_published":"2025-02-06T00:00:00Z","scopus_import":"1","month":"02","oa":1,"title":"On the L-polynomials of curves over finite fields","status":"public","external_id":{"isi":["001414690400001"]},"day":"06","oa_version":"Published Version","doi":"10.1017/prm.2025.7","publisher":"Cambridge University Press","publication_status":"epub_ahead","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","department":[{"_id":"TiBr"}],"year":"2025","type":"journal_article","isi":1,"article_processing_charge":"Yes (via OA deal)","publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","citation":{"mla":"Ballini, Francesco, et al. “On the L-Polynomials of Curves over Finite Fields.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, Cambridge University Press, 2025, doi:<a href=\"https://doi.org/10.1017/prm.2025.7\">10.1017/prm.2025.7</a>.","ieee":"F. Ballini, D. Lombardo, and M. Verzobio, “On the L-polynomials of curves over finite fields,” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press, 2025.","ama":"Ballini F, Lombardo D, Verzobio M. On the L-polynomials of curves over finite fields. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. 2025. doi:<a href=\"https://doi.org/10.1017/prm.2025.7\">10.1017/prm.2025.7</a>","short":"F. Ballini, D. Lombardo, M. Verzobio, Proceedings of the Royal Society of Edinburgh Section A: Mathematics (2025).","apa":"Ballini, F., Lombardo, D., &#38; Verzobio, M. (2025). On the L-polynomials of curves over finite fields. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2025.7\">https://doi.org/10.1017/prm.2025.7</a>","chicago":"Ballini, Francesco, Davide Lombardo, and Matteo Verzobio. “On the L-Polynomials of Curves over Finite Fields.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/prm.2025.7\">https://doi.org/10.1017/prm.2025.7</a>.","ista":"Ballini F, Lombardo D, Verzobio M. 2025. On the L-polynomials of curves over finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics."}},{"citation":{"short":"L. Grafakos, M. Vişan, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 154 (2024) 1504–1517.","chicago":"Grafakos, Loukas, and Monica Vişan. “Remarks on Countable Subadditivity.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press, 2024. <a href=\"https://doi.org/10.1017/prm.2023.77\">https://doi.org/10.1017/prm.2023.77</a>.","apa":"Grafakos, L., &#38; Vişan, M. (2024). Remarks on countable subadditivity. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2023.77\">https://doi.org/10.1017/prm.2023.77</a>","ama":"Grafakos L, Vişan M. Remarks on countable subadditivity. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. 2024;154(5):1504-1517. doi:<a href=\"https://doi.org/10.1017/prm.2023.77\">10.1017/prm.2023.77</a>","ieee":"L. Grafakos and M. Vişan, “Remarks on countable subadditivity,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, vol. 154, no. 5. Cambridge University Press, pp. 1504–1517, 2024.","mla":"Grafakos, Loukas, and Monica Vişan. “Remarks on Countable Subadditivity.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, vol. 154, no. 5, Cambridge University Press, 2024, pp. 1504–17, doi:<a href=\"https://doi.org/10.1017/prm.2023.77\">10.1017/prm.2023.77</a>.","ista":"Grafakos L, Vişan M. 2024. Remarks on countable subadditivity. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 154(5), 1504–1517."},"publication":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","article_processing_charge":"No","type":"journal_article","issue":"5","year":"2024","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publisher":"Cambridge University Press","doi":"10.1017/prm.2023.77","intvolume":"       154","day":"01","oa_version":"Preprint","external_id":{"arxiv":["2304.07831"]},"extern":"1","volume":154,"status":"public","title":"Remarks on countable subadditivity","oa":1,"page":"1504-1517","month":"10","arxiv":1,"scopus_import":"1","date_published":"2024-10-01T00:00:00Z","_id":"22074","quality_controlled":"1","date_updated":"2026-06-30T11:03:28Z","date_created":"2026-06-19T08:22:51Z","OA_place":"repository","publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"article_type":"original","abstract":[{"text":"We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.","lang":"eng"}],"fulldoi":"https://doi.org/10.1017/prm.2023.77","das_tickbox":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2304.07831","open_access":"1"}],"OA_type":"green","author":[{"first_name":"Loukas","full_name":"Grafakos, Loukas","last_name":"Grafakos"},{"first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","full_name":"Visan, Monica"}]}]