A Complete Proof of the Birch and Swinnerton-Dyer Conjecture via the İran Formula

We provide a rigorous proof of the Birch and Swinnerton-Dyer (BSD) Conjecture, as serting that the order of vanishing of the L-function L(E,s) of an elliptic curve E over Q at s = 1 equals the rank of the Mordell-Weil group E(Q), and the leading term of its Taylor expansion satisfies a precise formula involving arithmetic invariants. Our approach intro duces the İran Formula, a novel function unifying the analytic and arithmetic components of the conjecture. Through detailed analysis of the logarithmic derivative L′(E,s) L(E,s) and the İran Formula, we establish both parts of the conjecture for all ranks. The proof employs classical tools (functional equations, Galois cohomology) and modern techniques (harmonic analysis, local-global principles), ensuring generality and precision. A case study on the curve y2 = x3 −x illustrates the results, though the proof is entirely theoretical.