---
_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: 2022-09-14T12:33:33Z
day: '08'
ddc:
- '512'
degree_awarded: PhD
department:
- _id: GradSch
- _id: TiBr
doi: 10.15479/at:ista:12072
ec_funded: 1
file:
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creator: ashute
date_created: 2022-09-08T21:50:34Z
date_updated: 2022-09-08T21:50:34Z
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date_updated: 2022-09-12T11:24:21Z
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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
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...