The Derived Adelic Cohomology Conjecture for Elliptic Curves

We propose a novel derived cohomological framework for the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves. In our approach, local arithmetic data are encoded in derived sheaves which, when glued via a mapping cone construction, yield an adelic complex. A natural Postnikov filtration on this complex gives rise to a spectral sequence whose first nonzero differential detects the analytic and algebraic rank of the curve. Moreover, the determinant of this differential equals the combination of classical invariants appearing in the BSD formula. We present rigorous constructions of the derived sheaves involved and establish their key properties, including explicit connections to L-functions through cohomological interpretations. Extensive numerical evidence across curves of various ranks, including those with non-trivial Tate-Shafarevich groups, supports these structural predictions. Our framework unifies several existing approaches to the BSD conjecture, providing a cohomological interpretation that explains both the rank equality and the precise formula where previous methods addressed only partial aspects of the conjecture.