On the Equality of Algebraic and Analytic Ranks for Algebraic Curves: A Rigorous Differential Algebraic Approach

This paper establishes a comprehensive and rigorous proof of the equality between algebraic and analytic ranks for algebraic curves defined over number fields, employing a novel differential algebraic framework. We construct specialized differential algebraic closures that simultaneously capture the algebraic structure of Jacobian varieties and the analytic properties of L-functions. Within these closures, we develop explicit parameterizations of algebraic curves and derive spectral representations of geometric partial differential equations governing the arithmetic and analytic structures. The main theorem demonstrates that the algebraic rank of the Jacobian variety equals the analytic rank of the associated Hasse-Weil L-function, providing a unified geometric-analytic perspective on this fundamental arithmetic relationship. Our approach synthesizes techniques from differential algebra, algebraic geometry, analytic number theory, and spectral theory, offering new insights into the geometric foundations of the Birch and Swinnerton-Dyer conjecture for curves. All constructions are carried out with complete mathematical rigor, and explicit computational frameworks with certified error bounds are provided.