| _version_ | 1866901210924056576 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | We investigate arithmetic obstructions to the Hasse principle, also known as the local-global principle, in the context of higher-dimensional varieties over p-adic fields. The Hasse principle, which asserts that the existence of solutions to a Diophantine equation in all local fields implies the existence of a global solution, often fails for varieties beyond curves and quadrics. Our focus is on understanding the nature and structure of these obstructions, particularly those arising from Brauer-Manin obstructions and etale Brauer-Manin obstructions. We analyze specific families of higher-dimensional p-adic varieties, including geometrically rational surfaces and rationally connected varieties, to determine the extent to which these obstructions account for the failure of the Hasse principle. Furthermore, we explore connections between the arithmetic of these varieties and the geometry of their reductions modulo p, providing insights into the interplay between local and global arithmetic properties. The computations and theoretical results presented contribute to a deeper understanding of the arithmetic of higher-dimensional varieties over p-adic fields and related number fields. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17829957 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Arithmetic Obstructions to Local-Global Principles in Higher Dimensional p-adic Varieties Revista, Zen MATH, 10 We investigate arithmetic obstructions to the Hasse principle, also known as the local-global principle, in the context of higher-dimensional varieties over p-adic fields. The Hasse principle, which asserts that the existence of solutions to a Diophantine equation in all local fields implies the existence of a global solution, often fails for varieties beyond curves and quadrics. Our focus is on understanding the nature and structure of these obstructions, particularly those arising from Brauer-Manin obstructions and etale Brauer-Manin obstructions. We analyze specific families of higher-dimensional p-adic varieties, including geometrically rational surfaces and rationally connected varieties, to determine the extent to which these obstructions account for the failure of the Hasse principle. Furthermore, we explore connections between the arithmetic of these varieties and the geometry of their reductions modulo p, providing insights into the interplay between local and global arithmetic properties. The computations and theoretical results presented contribute to a deeper understanding of the arithmetic of higher-dimensional varieties over p-adic fields and related number fields. |
| title | Arithmetic Obstructions to Local-Global Principles in Higher Dimensional p-adic Varieties |
| url | https://doi.org/10.5281/zenodo.17829957 |