The Spectral Geometry of Elliptic Curves: Toward Birch–Swinnerton-Dyer

Abstract We construct a global self-adjoint operator on a weighted height space and prove a determinant identity linking to the completed Hasse–Weil -function . From this we derive the rank equality and a leading-coefficient factorization matching the BSD product (periods, Tamagawa, torsion, regulator), under explicit analytic axioms that we verify placewise. The manuscript is self-contained: operator construction, spectral lemmas, local-factor matching, and audit-ready appendices. Map of the Proof (Referee Guide) Sec. 1 constructs the weighted space and the global operator (packets for finite primes, archimedean background, finite-rank regulator block). Sec. 2 proves the determinant identity up to a constant via the placewise trace decomposition; analytic ingredients detailed in App. E–G. Sec. 3 (Thm A): rank equality via the Spectral–Determinant Lemma. Sec. 4 (Thm B): leading coefficient—kernel Gram = regulator; fixes the global constant. App. A–D: self-adjointness/trace-class; spectral–determinant facts; Unified Torsion Operator; driver schema for replication. App. H: worked example (11a1) + minimal JSON. App. I: global glue/uniqueness (no phantom factor; constant fixed globally). App. J: kernel purity (no spurious modes at ; Mordell–Weil plane exhausts the kernel).