| _version_ | 1866901609106112512 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper explores the intricate relationship between Pellian lattices and the fine structure of ideal class groups, particularly in real quadratic number fields. The study posits that the geometric properties encoded within Pellian lattices, which arise from the fundamental solutions of Pell's equation, offer profound insights into the arithmetic invariants of these fields. We delve into how the units group, governed by Pell's equation, influences the lattice structure and, consequently, the generators and relations within the ideal class group. By analyzing the action of fundamental units on ideals, we establish a framework for understanding the distribution of ideal classes and their orders. The investigation focuses on identifying specific signatures within Pellian lattices that correspond to torsion elements or specific cyclic structures within the class group, moving beyond mere class number calculations to uncover deeper structural properties. Through a blend of algebraic number theory and geometric methods, we aim to elucidate how these seemingly disparate areas of mathematics converge, providing new avenues for characterizing the arithmetic complexity of number fields. The results shed light on long-standing problems concerning class group exponents and the capitulation problem. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17714570 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Pellian Lattices and the Fine Structure of Ideal Class Groups Revista, Zen MATH, 10 This paper explores the intricate relationship between Pellian lattices and the fine structure of ideal class groups, particularly in real quadratic number fields. The study posits that the geometric properties encoded within Pellian lattices, which arise from the fundamental solutions of Pell's equation, offer profound insights into the arithmetic invariants of these fields. We delve into how the units group, governed by Pell's equation, influences the lattice structure and, consequently, the generators and relations within the ideal class group. By analyzing the action of fundamental units on ideals, we establish a framework for understanding the distribution of ideal classes and their orders. The investigation focuses on identifying specific signatures within Pellian lattices that correspond to torsion elements or specific cyclic structures within the class group, moving beyond mere class number calculations to uncover deeper structural properties. Through a blend of algebraic number theory and geometric methods, we aim to elucidate how these seemingly disparate areas of mathematics converge, providing new avenues for characterizing the arithmetic complexity of number fields. The results shed light on long-standing problems concerning class group exponents and the capitulation problem. |
| title | Pellian Lattices and the Fine Structure of Ideal Class Groups |
| url | https://doi.org/10.5281/zenodo.17714570 |