--- _id: '12072' abstract: - lang: eng 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. " 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. alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Alec L full_name: Shute, Alec L id: 440EB050-F248-11E8-B48F-1D18A9856A87 last_name: Shute orcid: 0000-0002-1812-2810 citation: ama: 'Shute AL. Existence and density problems in Diophantine geometry: From norm forms to Campana points. 2022. doi:10.15479/at:ista:12072' apa: 'Shute, A. L. (2022). Existence and density problems in Diophantine geometry: From norm forms to Campana points. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:12072' chicago: 'Shute, Alec L. “Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:12072.' ieee: 'A. L. Shute, “Existence and density problems in Diophantine geometry: From norm forms to Campana points,” Institute of Science and Technology Austria, 2022.' ista: 'Shute AL. 2022. Existence and density problems in Diophantine geometry: From norm forms to Campana points. Institute of Science and Technology Austria.' mla: 'Shute, Alec L. Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:12072.' short: 'A.L. Shute, Existence and Density Problems in Diophantine Geometry: From Norm Forms to Campana Points, Institute of Science and Technology Austria, 2022.' date_created: 2022-09-08T21:53:03Z date_published: 2022-09-08T00:00:00Z date_updated: 2023-02-21T16:37:35Z day: '08' ddc: - '512' degree_awarded: PhD department: - _id: GradSch - _id: TiBr doi: 10.15479/at:ista:12072 ec_funded: 1 file: - access_level: open_access checksum: bf073344320e05d92c224786cec2e92d content_type: application/pdf creator: ashute date_created: 2022-09-08T21:50:34Z date_updated: 2022-09-08T21:50:34Z file_id: '12073' file_name: Thesis_final_draft.pdf file_size: 1907386 relation: main_file success: 1 - access_level: closed checksum: b054ac6baa09f70e8235403a4abbed80 content_type: application/octet-stream creator: ashute date_created: 2022-09-08T21:50:42Z date_updated: 2022-09-12T11:24:21Z file_id: '12074' file_name: athesis.tex file_size: 495393 relation: source_file - access_level: closed checksum: 0a31e905f1cff5eb8110978cc90e1e79 content_type: application/x-zip-compressed creator: ashute date_created: 2022-09-09T12:05:00Z date_updated: 2022-09-12T11:24:21Z file_id: '12078' file_name: qfcjsfmtvtbfrjjvhdzrnqxfvgjvxtbf.zip file_size: 944534 relation: source_file file_date_updated: 2022-09-12T11:24:21Z has_accepted_license: '1' language: - iso: eng license: https://creativecommons.org/licenses/by-nc-sa/4.0/ month: '09' oa: 1 oa_version: Published Version page: '208' project: - _id: 2564DBCA-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '665385' name: International IST Doctoral Program publication_identifier: isbn: - 978-3-99078-023-7 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '12076' relation: part_of_dissertation status: public - id: '12077' relation: part_of_dissertation status: public status: public supervisor: - first_name: Timothy D full_name: Browning, Timothy D id: 35827D50-F248-11E8-B48F-1D18A9856A87 last_name: Browning orcid: 0000-0002-8314-0177 title: 'Existence and density problems in Diophantine geometry: From norm forms to Campana points' tmp: image: /images/cc_by_nc_sa.png legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) short: CC BY-NC-SA (4.0) type: dissertation user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ...