From Pellian Equations to Class Group Symmetries: A New Arithmetic-Geometric Framework

This paper introduces a novel arithmetic-geometric framework designed to establish deeper and more systematic connections between the solutions of Pellian equations and the inherent symmetries within ideal class groups of real quadratic number fields. While classical number theory has long recognized a foundational link between the fundamental solution of a Pellian equation $x^2 - Dy^2 = 1$ and the fundamental unit of the quadratic field $mathbb{Q}(sqrt{D})$, this work transcends mere existence by exploring how the structure and properties of Pellian solutions induce specific automorphisms and reveal previously underexplored symmetries within the class group structure itself. Our methodology involves defining a "Pellian map" that describes the action of units, derived from Pellian solutions, on the ideal class group. We demonstrate that this map constitutes a group automorphism, whose characteristics are intimately tied to the arithmetic properties of the underlying Pellian equation and the geometry of the quadratic field's ring of integers. Theoretical results reveal how the order and parity of Pellian solutions can dictate specific cyclic or dihedral symmetries within the class group. The discussion interprets these findings, positioning the framework as a powerful lens for understanding the intricate interplay between Diophantine analysis and algebraic number theory, offering new avenues for classifying class groups and elucidating the arithmetic structure of quadratic fields. This research not only extends the classical understanding but also proposes new computational strategies for probing class group structures through the lens of Diophantine solutions.