Integral points on a del Pezzo surface over imaginary quadratic fields

authored by

Judith Lena Ortmann

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Research in Number Theory

Volume

10

No. of pages

32

Publication date

08.11.2024

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Research Area (based on ÖFOS 2012)

Number theory

Electronic version(s)

https://doi.org/10.48550/arXiv.2307.12877 (Access: Open)
https://doi.org/10.1007/s40993-024-00572-z (Access: Open)