A DISCRETE NORMALIZATION APPROACH TO THE BIRCH–SWINNERTON–DYER CONJECTURE

Abstract. We present a normalization-first approach to the Birch Swinnerton–Dyer conjecture for elliptic curves over Q. By fixing from the outset a global minimal model, the N´eron differential, the real N´eron period, and the N´eron–Tate canonical height pairing, all continuous and real-analytic contributions are isolated explicitly in the period and the regulator. With this canonical normalization in place, the conjectural formula separates naturally into continuous and purely discrete arithmetic components.We show that the resulting normalized formulation is rigid under Qisogenies, forcing exact compensation among regulators, torsion, Tamagawa numbers, and the Tate–Shafarevich group. Quadratic twists are then used to vary arithmetic data in a controlled way, with all variation localized at finitely many places. Selmer groups provide the mechanism that translates these local conditions into global arithmetic information and allow precise control of the Tate–Shafarevich group.This reorganization does not introduce new invariants or conjectures. Rather, it exposes the internal coherence of the Birch–Swinnerton–Dyer formula and reduces it to a finite, local arithmetic problem once canonical normalization is fixed.