Bibliographic Information: de Rasis, J. P., & Handley, H. (2024). First-order definability of Darmon points in number fields. arXiv preprint arXiv:2410.03033v1.
Research Objective: The paper aims to demonstrate that Darmon points, which represent a set-theoretic filtration between a number field K and its ring of S-integers, can be defined using first-order formulas.
Methodology: The authors utilize the theory of quaternion algebras and Hilbert symbols to construct diophantine sets. These sets are then leveraged to establish a method for controlling finite subsets of places in a number field, ultimately leading to the first-order definition of Darmon points.
Key Findings:
Main Conclusions: The research successfully demonstrates the first-order definability of Darmon points in number fields. This result contributes to the broader study of definability problems in number theory, particularly those related to Hilbert's Tenth Problem and its variations.
Significance: This work provides a new perspective on the interplay between algebraic structures like quaternion algebras and logical definability in number fields. The explicit bounds on the complexity of the definitions offer valuable insights for further investigations in this area.
Limitations and Future Research: The paper primarily focuses on Darmon points defined on the projective line. Exploring the definability of Darmon points on more general varieties could be a potential direction for future research. Additionally, investigating whether the obtained complexity bounds on the first-order definitions can be further improved is an open question.
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by Juan Pablo D... at arxiv.org 10-07-2024
https://arxiv.org/pdf/2410.03033.pdfDeeper Inquiries