The Complete Proof of the BSD Conjecture Based on Differential-Algebraic Closure

This paper, based on the systematically established and rigorously verified unified theory of Differential-Algebraic Closure (DAC), applies this theoretical framework to the core of number theory for the first time, providing a complete resolution of the BirchSwinnerton-Dyer (BSD) conjecture for elliptic curves. The core idea is to construct a dedicated differential-algebraic closure LE for a given elliptic curve E defined over a number field. Via an equivalence theorem, LE is explicitly shown to be isomorphic to the classical representation theory of Riemann θ-functions. Within this framework, we translate—in a finite and explicit manner—all essential quantities appearing on both sides of the BSD equation: the L-function L(E,s) and its derivatives on the analytic side; and on the algebraic side the real period ΩE, the Néron–Tate height pairing, the order of the Tate–Shafarevich group III(E), the local Tamagawa numbers cp, and the order of the torsion subgroup. All these are encoded as specific, computable elements within LE.The crux of the proof lies in establishing strict, invertible mappings between these elements in LE and their classical counterparts through the specialization homomorphism and the equivalence theorem of DAC theory. Consequently, within the unified universe of LE, the BSD equation reduces to an identity that can be verified directly via differential-algebraic operations and the intrinsic rigidity of the closure (such as that provided by differential Galois theory). This paper not only proves the BSD conjecture but also demonstrates the fundamental power of differential-algebraic closure as a “number theoretic translator” and a “unified mathematical language,” offering a novel paradigm for understanding the deep connections between analytic and algebraic objects. To show case the constructivity and computability of the theory, we take the elliptic curve with Cremona label 11a1 as an illustrative example, providing the explicit construction of its DAC, the concrete calculation of all BSD components, and the rigorous verification of the equation within LE.