Determinant Lines and the Birch–Swinnerton–Dyer Conjecture

This project consists of two companion papers proposing a determinant-theoretic perspective on the Birch–Swinnerton–Dyer conjecture for elliptic curves over \mathbb{Q}. The first paper develops a programmatic and conceptual framework in which the conjecture is interpreted as the existence of a canonical trivialization of a determinant line attached to the Selmer complex. This perspective emphasizes functoriality, independence from auxiliary choices, and compatibility with arithmetic, p-adic, and motivic realizations. The discussion is intentionally programmatic and heuristic in nature, focusing on structural organization rather than on new arithmetic or analytic results. The second paper provides a precise and conservative structural reformulation of the conjecture by constructing arithmetic and analytic determinant lines and formulating the Birch–Swinnerton–Dyer conjecture as the existence of a canonical comparison isomorphism between these determinant torsors. This formulation is shown to be equivalent, under standard identifications of regulators, periods, and local factors, to the classical Birch–Swinnerton–Dyer conjecture. The two papers are logically independent but conceptually complementary. Together, they aim to clarify the structural content of the conjecture and to organize its arithmetic and analytic components within a unified determinant framework. No claim of a proof of the Birch–Swinnerton–Dyer conjecture is made in either paper. Paper I (Programmatic Framework). This paper develops a determinant-theoretic and categorical perspective on the Birch–Swinnerton–Dyer conjecture, interpreting the conjecture as the existence of a canonical trivialization of a determinant line associated with the Selmer complex. Paper II (Structural Reformulation). This paper constructs arithmetic and analytic determinant lines associated with an elliptic curve over \mathbb{Q} and formulates the Birch–Swinnerton–Dyer conjecture as the existence of a canonical comparison isomorphism between these determinant torsors, shown to be equivalent to the classical formulation under standard identifications.