Details
Originalsprache | Englisch |
---|---|
Aufsatznummer | 90 |
Seitenumfang | 32 |
Fachzeitschrift | Research in Number Theory |
Jahrgang | 10 |
Ausgabenummer | 4 |
Publikationsstatus | Veröffentlicht - 8 Nov. 2024 |
Abstract
Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q. In this paper, we generalise the torsor method for integral points from Q to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3 over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
Fachgebiet (basierend auf ÖFOS 2012)
- NATURWISSENSCHAFTEN
- Mathematik
- Mathematik
- Zahlentheorie
Zitieren
- Standard
- Harvard
- Apa
- Vancouver
- BibTex
- RIS
in: Research in Number Theory, Jahrgang 10, Nr. 4, 90, 08.11.2024.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Integral points on a del Pezzo surface over imaginary quadratic fields
AU - Ortmann, Judith Lena
N1 - Publisher Copyright: © The Author(s) 2024.
PY - 2024/11/8
Y1 - 2024/11/8
N2 - Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q. In this paper, we generalise the torsor method for integral points from Q to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3 over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.
AB - Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q. In this paper, we generalise the torsor method for integral points from Q to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3 over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.
KW - 11R11
KW - 14G05
KW - 14J26
KW - Del Pezzo surface
KW - Imaginary quadratic field
KW - Integral points
KW - Manin’s conjecture
KW - Primary: 11D45
KW - Secondary: 11G35
KW - Universal torsor
UR - http://www.scopus.com/inward/record.url?scp=85209585111&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2307.12877
DO - 10.48550/arXiv.2307.12877
M3 - Article
AN - SCOPUS:85209585111
VL - 10
JO - Research in Number Theory
JF - Research in Number Theory
IS - 4
M1 - 90
ER -