A Discrete Normalization Approach to the Birch–swinnerton–dyer Conjecture

We present a normalization-first and audit-oriented framework for elliptic curves over Q that recasts the strong Birch–Swinnerton–Dyer conjecture (BSD) in a split form and separates formal determinant-line logic from genuinely deep arithmetic inputs. First, we fix a canonical normalization package (global minimal model, Néron differential, real period Ω+E, canonical height pairing, regulator) and define the analytic rank ran = ords=1 L(E,s) and leading coefficient L∗(E,1) non circularly, yielding a normalized defect ∆(E) for the BSD leading-term identity. Second, we work at the correct discrete level by developing the full Selmer tower at ℓn and ℓ∞, and we close the standard Mordell–Weil compensation loophole by introducing MW-inert primes and reduced Selmer groups Selℓn(E) ∼=Sha(E)ℓn, so that one-place Selmer modifications cannot be absorbed by the visible Mordell–Weil quotient. Third, we package the BSD identities in a global determinant line equipped with canonical archimedean and ℓ-adic realizations and lattices, and we show that, once the bridge evaluations are granted, the leading-term identity becomes a formal consequence of rational reconstruction from all ℓ-adic valuations, while the rank equality becomes a formal consequence of a single Bloch–Kato rank bridge together with ℓ∞-finiteness. Accordingly, the manuscript isolates the only deep inputs as three explicit bridge statements (RT, LTℓ, RB), and provides a complete navigation/audit index for referees.