The Fine Structure of Local-Global Principles in p-adic Fields
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper investigates the intricate details and nuances, referred to as the "fine structure," of local-global principles within the context of p-adic fields. While classical local-global principles, such as the Hasse principle, provide a fundamental framework for relating arithmetic properties over a global field to those over its completions, their behavior in the specific setting of p-adic fields and their various extensions reveals a richer and more complex landscape. We move beyond a simple dichotomy of "holds" or "fails" to explore the precise conditions, obstructions, and quantitative measures that characterize the validity or breakdown of these principles. Employing advanced tools from Galois cohomology, algebraic K-theory, and local class field theory, this study examines the depth of congruence relations and their implications for Diophantine equations, quadratic forms, and algebraic groups over p-adic fields. We present a theoretical framework for analyzing the failure of local-global principles, identifying key invariants that quantify the discrepancy between local and global solvability. The discussion illuminates the profound interplay between arithmetic and topology in these non-Archimedean settings, offering new perspectives on the challenges and successes of applying local information to infer global properties. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17834132 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Fine Structure of Local-Global Principles in p-adic Fields Revista, Zen MATH, 10 This paper investigates the intricate details and nuances, referred to as the "fine structure," of local-global principles within the context of p-adic fields. While classical local-global principles, such as the Hasse principle, provide a fundamental framework for relating arithmetic properties over a global field to those over its completions, their behavior in the specific setting of p-adic fields and their various extensions reveals a richer and more complex landscape. We move beyond a simple dichotomy of "holds" or "fails" to explore the precise conditions, obstructions, and quantitative measures that characterize the validity or breakdown of these principles. Employing advanced tools from Galois cohomology, algebraic K-theory, and local class field theory, this study examines the depth of congruence relations and their implications for Diophantine equations, quadratic forms, and algebraic groups over p-adic fields. We present a theoretical framework for analyzing the failure of local-global principles, identifying key invariants that quantify the discrepancy between local and global solvability. The discussion illuminates the profound interplay between arithmetic and topology in these non-Archimedean settings, offering new perspectives on the challenges and successes of applying local information to infer global properties. |
| title | The Fine Structure of Local-Global Principles in p-adic Fields |
| url | https://doi.org/10.5281/zenodo.17834132 |