An Informational-Geometric Interpretation of the Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer (BSD) conjecture establishes a deep connection between the arithmetic structure of elliptic curves and the analytic behavior of their associated L-functions, yet its conceptual interpretation remains elusive despite extensive partial results. In this work, we propose an informational–geometric reinterpretation of the conjecture within the framework of Viscous Time Theory (VTT), in which arithmetic invariants are mapped to measurable quantities governing informational coherence. Within this framework, the canonical height is interpreted as a coherence potential, the Mordell–Weil rank as the dimensionality of stable coherence directions, and the BSD regulator as an informational volume. The analytic behavior of the L-function near the critical point s = 1 is reinterpreted as a global coherence response of the underlying informational manifold. This leads naturally to a regime-dependent conservation principle for informational coherence, under which the classical BSD identity emerges as a stable balance condition. To test this formulation, we perform independent numerical validation on benchmark elliptic curves with established BSD data. An informational L-function is fitted to empirical analytic profiles, from which informational curvature and volume are derived. The results show strong quantitative agreement between informational volumes and arithmetic regulators, with correlation coefficients exceeding 0.99, and demonstrate robust, rank-dependent stability behavior across curves of varying complexity. The framework is further explored under increasing geometric and topological complexity through higher-genus and synthetic informational models, which exhibit systematic coherence suppression consistent with known arithmetic phenomena. While this work does not claim a proof of the Birch and Swinnerton-Dyer conjecture, it offers a coherent explanatory framework that clarifies its internal structure, geometrically aligns analytic and arithmetic invariants, and opens new avenues for numerical and conceptual investigation.
