{"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","doi":"10.15479/at:ista:18132","title":"Counting rational points over function fields","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"18173"},{"id":"18293","relation":"part_of_dissertation","status":"public"},{"id":"18294","relation":"part_of_dissertation","status":"public"},{"id":"18295","status":"public","relation":"part_of_dissertation"}]},"day":"23","citation":{"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>.","short":"J. Glas, Counting Rational Points over Function Fields, Institute of Science and Technology Austria, 2024.","ista":"Glas J. 2024. Counting rational points over function fields. Institute of Science and Technology Austria.","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>","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>.","ieee":"J. Glas, “Counting rational points over function fields,” Institute of Science and Technology Austria, 2024.","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>"},"has_accepted_license":"1","type":"dissertation","publisher":"Institute of Science and Technology Austria","department":[{"_id":"GradSch"},{"_id":"TiBr"}],"author":[{"first_name":"Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","full_name":"Glas, Jakob","last_name":"Glas"}],"article_processing_charge":"No","_id":"18132","page":"195","date_created":"2024-09-23T18:58:08Z","year":"2024","status":"public","supervisor":[{"first_name":"Timothy D","orcid":"0000-0002-8314-0177","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D"}],"publication_identifier":{"issn":["2663-337X"]},"file":[{"date_updated":"2024-09-23T18:49:22Z","checksum":"2f8cf5cefdab108b1979caa8146cae9a","date_created":"2024-09-23T18:49:22Z","file_id":"18133","file_size":5382106,"creator":"jglas","access_level":"closed","content_type":"application/x-zip-compressed","relation":"source_file","file_name":"PhDthesis (3).zip"},{"creator":"jglas","file_size":2380127,"relation":"main_file","file_name":"example-phd.pdf","access_level":"open_access","content_type":"application/pdf","date_updated":"2024-09-25T14:08:57Z","file_id":"18140","checksum":"08bb6f14c42b47ff25882a2ce3ea0d8a","success":1,"date_created":"2024-09-25T14:08:57Z"}],"alternative_title":["ISTA Thesis"],"tmp":{"short":"CC BY-NC (4.0)","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode"},"date_updated":"2024-10-11T09:44:23Z","language":[{"iso":"eng"}],"oa":1,"oa_version":"Published Version","month":"09","degree_awarded":"PhD","date_published":"2024-09-23T00:00:00Z","ddc":["512"],"file_date_updated":"2024-09-25T14:08:57Z","project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"corr_author":"1","publication_status":"published","abstract":[{"lang":"eng","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"}]}