---
OA_place: publisher
_id: '18132'
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"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Jakob
  full_name: Glas, Jakob
  id: d6423cba-dc74-11ea-a0a7-ee61689ff5fb
  last_name: Glas
citation:
  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>
  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.
  ista: Glas J. 2024. Counting rational points over function fields. Institute of
    Science and Technology Austria.
  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.
corr_author: '1'
date_created: 2024-09-23T18:58:08Z
date_published: 2024-09-23T00:00:00Z
date_updated: 2026-04-07T12:53:54Z
day: '23'
ddc:
- '512'
degree_awarded: PhD
department:
- _id: GradSch
- _id: TiBr
doi: 10.15479/at:ista:18132
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  date_created: 2024-09-25T14:08:57Z
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has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: '195'
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '18293'
    relation: part_of_dissertation
    status: public
  - id: '18294'
    relation: part_of_dissertation
    status: public
  - id: '18295'
    relation: part_of_dissertation
    status: public
  - id: '18173'
    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: Counting rational points over function fields
tmp:
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  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)
  short: CC BY-NC (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...