Local solubility of a family of quadric surfaces over a biprojective base

Wilson, Cameron (2025) Local solubility of a family of quadric surfaces over a biprojective base. PhD thesis, University of Glasgow.

Abstract

We prove an asymptotic formula for the number of everywhere locally soluble diagonal quadric surfaces 𝑦₀𝑥²₀+𝑦₁𝑥²₁+𝑦₂𝑥²₂+𝑦₃𝑥²₃=0 parametrised by points 𝑦 ∈ ℙ³ (ℚ) lying on the split quadric surface 𝑦₀𝑦₁ = 𝑦₂𝑦₃ which do not satisfy −𝑦₀𝑦₂ = □ nor −𝑦₀𝑦₃ = □. Our methods involve proving asymptotic formulae for character sums with a hyperbolic height condition and proving variations of large sieve inequalities for quadratic characters.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Funding for the Ph.D has been provided by the Carnegie Trust for the Universities of Scotland.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Carnegie Trust for the Universities of Scotland (CARNEGTR)
• Supervisor's Name: Sofos, Dr. Efthymios and Bartel, Professor Alex
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85541
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 27 Oct 2025 11:52
• Last Modified: 27 Oct 2025 11:52
• Thesis DOI: 10.5525/gla.thesis.85541
• URI: https://theses.gla.ac.uk/id/eprint/85541