On a Determinant-Line Formulation of the Birch–Swinnerton–Dyer Conjecture

This paper presents a conceptual reformulation of the Birch–Swinnerton–Dyer conjecture using the formalism of determinant functors and Picard groupoids. Rather than proposing a new analytic or arithmetic proof, the work isolates the logical core of the conjecture by expressing it as a canonical identification problem between two one-dimensional determinant torsors. To an elliptic curve E/\mathbb{Q}, we associate: • an arithmetic determinant line, constructed from the Selmer complex under standard finiteness assumptions, and • an analytic determinant line, built from the leading term of the Hasse–Weil L-function together with period data. Under standard conjectural assumptions (including Bloch–Kato and the expected finiteness properties of Selmer groups), we show that the Birch–Swinnerton–Dyer conjecture is equivalent to the existence of a canonical and functorial isomorphism between these two determinant lines. The paper illustrates this perspective in the cases of analytic rank 0 and 1, demonstrating how classical BSD invariants naturally assemble into determinant torsors, free from auxiliary normalization choices. We further clarify the relationship between this formulation and the frameworks of Fukaya–Kato and the Equivariant Tamagawa Number Conjecture, emphasizing that the present approach isolates a non-equivariant structural core of BSD without introducing additional Iwasawa-theoretic data. This work does not claim a proof of the Birch–Swinnerton–Dyer conjecture, but provides a logically coherent and functorial framework in which its arithmetic and analytic components are unified at a structural level.