[{"degree_awarded":"PhD","article_processing_charge":"No","license":"https://creativecommons.org/licenses/by-nc/4.0/","oa_version":"Published Version","language":[{"iso":"eng"}],"page":"195","file_date_updated":"2024-09-25T14:08:57Z","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"}],"status":"public","type":"dissertation","tmp":{"image":"/images/cc_by_nc.png","short":"CC BY-NC (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)"},"related_material":{"record":[{"id":"18293","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18294"},{"id":"18295","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18173"}]},"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","date_published":"2024-09-23T00:00:00Z","department":[{"_id":"GradSch"},{"_id":"TiBr"}],"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278","name":"Rational curves via function field analytic number theory"}],"OA_place":"publisher","supervisor":[{"full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","first_name":"Timothy D","orcid":"0000-0002-8314-0177"}],"day":"23","_id":"18132","date_created":"2024-09-23T18:58:08Z","ddc":["512"],"oa":1,"file":[{"file_size":5382106,"relation":"source_file","access_level":"closed","date_created":"2024-09-23T18:49:22Z","checksum":"2f8cf5cefdab108b1979caa8146cae9a","creator":"jglas","content_type":"application/x-zip-compressed","file_id":"18133","file_name":"PhDthesis (3).zip","date_updated":"2024-09-23T18:49:22Z"},{"file_size":2380127,"relation":"main_file","access_level":"open_access","date_created":"2024-09-25T14:08:57Z","content_type":"application/pdf","creator":"jglas","checksum":"08bb6f14c42b47ff25882a2ce3ea0d8a","success":1,"file_id":"18140","file_name":"example-phd.pdf","date_updated":"2024-09-25T14:08:57Z"}],"publication_status":"published","citation":{"ista":"Glas J. 2024. Counting rational points over function fields. Institute of Science and Technology Austria.","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>","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>.","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>.","ieee":"J. Glas, “Counting rational points over function fields,” Institute of Science and Technology Austria, 2024.","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>","short":"J. Glas, Counting Rational Points over Function Fields, Institute of Science and Technology Austria, 2024."},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","month":"09","doi":"10.15479/at:ista:18132","author":[{"last_name":"Glas","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","full_name":"Glas, Jakob","first_name":"Jakob"}],"corr_author":"1","year":"2024","date_updated":"2026-04-07T12:53:54Z","publisher":"Institute of Science and Technology Austria","title":"Counting rational points over function fields","publication_identifier":{"issn":["2663-337X"]}},{"OA_place":"publisher","project":[{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385"}],"supervisor":[{"first_name":"Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","last_name":"Browning"}],"has_accepted_license":"1","acknowledgement":"I acknowledge the received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska Curie Grant Agreement No. 665385.","date_published":"2022-09-08T00:00:00Z","department":[{"_id":"GradSch"},{"_id":"TiBr"}],"type":"dissertation","status":"public","tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)"},"related_material":{"record":[{"status":"public","id":"12076","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"12077"}]},"alternative_title":["ISTA Thesis"],"degree_awarded":"PhD","article_processing_charge":"No","oa_version":"Published Version","license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","language":[{"iso":"eng"}],"page":"208","file_date_updated":"2022-09-12T11:24:21Z","abstract":[{"text":"In this thesis, we study two of the most important questions in Arithmetic geometry: that of the existence and density of solutions to Diophantine equations. In order for a Diophantine equation to have any solutions over the rational numbers, it must have solutions everywhere locally, i.e., over R and over Qp for every prime p. The converse, called the Hasse principle, is known to fail in general. However, it is still a central question in Arithmetic geometry to determine for which varieties the Hasse principle does hold. In this work, we establish the Hasse principle for a wide new family of varieties of the form f(t) = NK/Q(x) ̸= 0, where f is a polynomial with integer coefficients and NK/Q denotes the norm\r\nform associated to a number field K. Our results cover products of arbitrarily many linear, quadratic or cubic factors, and generalise an argument of Irving [69], which makes use of the beta sieve of Rosser and Iwaniec. We also demonstrate how our main sieve results can be applied to treat new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.\r\nIn the second question, about the density of solutions, one defines a height function and seeks to estimate asymptotically the number of points of height bounded by B as B → ∞. Traditionally, one either counts rational points, or\r\nintegral points with respect to a suitable model. However, in this thesis, we study an emerging area of interest in Arithmetic geometry known as Campana points, which in some sense interpolate between rational and integral points.\r\nMore precisely, we count the number of nonzero integers z1, z2, z3 such that gcd(z1, z2, z3) = 1, and z1, z2, z3, z1 + z2 + z3 are all squareful and bounded by B. Using the circle method, we obtain an asymptotic formula which agrees in\r\nthe power of B and log B with a bold new generalisation of Manin’s conjecture to the setting of Campana points, recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [96]. However, in this thesis we also provide the first known counterexamples to leading constant predicted by their conjecture. ","lang":"eng"}],"year":"2022","corr_author":"1","publisher":"Institute of Science and Technology Austria","date_updated":"2026-04-07T14:13:35Z","title":"Existence and density problems in Diophantine geometry: From norm forms to Campana points","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-023-7"]},"citation":{"apa":"Shute, A. L. (2022). <i>Existence and density problems in Diophantine geometry: From norm forms to Campana points</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:12072\">https://doi.org/10.15479/at:ista:12072</a>","short":"A.L. Shute, Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points, Institute of Science and Technology Austria, 2022.","ieee":"A. L. Shute, “Existence and density problems in Diophantine geometry: From norm forms to Campana points,” Institute of Science and Technology Austria, 2022.","mla":"Shute, Alec L. <i>Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points</i>. Institute of Science and Technology Austria, 2022, doi:<a href=\"https://doi.org/10.15479/at:ista:12072\">10.15479/at:ista:12072</a>.","chicago":"Shute, Alec L. “Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points.” Institute of Science and Technology Austria, 2022. <a href=\"https://doi.org/10.15479/at:ista:12072\">https://doi.org/10.15479/at:ista:12072</a>.","ama":"Shute AL. Existence and density problems in Diophantine geometry: From norm forms to Campana points. 2022. doi:<a href=\"https://doi.org/10.15479/at:ista:12072\">10.15479/at:ista:12072</a>","ista":"Shute AL. 2022. Existence and density problems in Diophantine geometry: From norm forms to Campana points. Institute of Science and Technology Austria."},"month":"09","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","author":[{"orcid":"0000-0002-1812-2810","first_name":"Alec L","full_name":"Shute, Alec L","id":"440EB050-F248-11E8-B48F-1D18A9856A87","last_name":"Shute"}],"doi":"10.15479/at:ista:12072","file":[{"file_name":"Thesis_final_draft.pdf","file_id":"12073","success":1,"date_updated":"2022-09-08T21:50:34Z","content_type":"application/pdf","creator":"ashute","checksum":"bf073344320e05d92c224786cec2e92d","date_created":"2022-09-08T21:50:34Z","access_level":"open_access","relation":"main_file","file_size":1907386},{"creator":"ashute","content_type":"application/octet-stream","checksum":"b054ac6baa09f70e8235403a4abbed80","relation":"source_file","file_size":495393,"date_created":"2022-09-08T21:50:42Z","access_level":"closed","date_updated":"2022-09-12T11:24:21Z","file_id":"12074","file_name":"athesis.tex"},{"relation":"source_file","file_size":944534,"date_created":"2022-09-09T12:05:00Z","access_level":"closed","checksum":"0a31e905f1cff5eb8110978cc90e1e79","creator":"ashute","content_type":"application/x-zip-compressed","file_id":"12078","file_name":"qfcjsfmtvtbfrjjvhdzrnqxfvgjvxtbf.zip","date_updated":"2022-09-12T11:24:21Z"}],"publication_status":"published","ec_funded":1,"day":"08","_id":"12072","date_created":"2022-09-08T21:53:03Z","ddc":["512"],"oa":1}]