---
res:
bibo_abstract:
- "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.
@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Alec L
foaf_name: Shute, Alec L
foaf_surname: Shute
foaf_workInfoHomepage: http://www.librecat.org/personId=440EB050-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1812-2810
bibo_doi: 10.15479/at:ista:12072
dct_date: 2022^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/2663-337X
- http://id.crossref.org/issn/978-3-99078-023-7
dct_language: eng
dct_publisher: Institute of Science and Technology Austria@
dct_title: 'Existence and density problems in Diophantine geometry: From norm forms
to Campana points@'
...