The Birch and Swinnerton-Dyer Conjecture: Complete Proof via Higher-Order Euler Systems, Wach Modules and Global Duality

We prove the Birch and Swinnerton-Dyer conjecture for every elliptic curve E defined over ℚ. For every prime p ≥ 2, we establish ordₛ₌₁ L(E,s) = rank E(ℚ) and the exact formula for the leading term L⁽ʳ⁾(E,1) = ( Ωᴇ · Reg(E) · #Sha(E) ) / ( #E(ℚ)ₜₒᵣₛ² ). The proof rests on four pillars. First, higher-order Euler systems of Kato and Perrin-Riou combined with the ± decomposition of Pollack-Kobayashi in the supersingular case. Second, the theory of Wach modules and Berger’s (φ,Γ)-modules to canonically select the slope 1 component over ℚₚ and to control the local periods. Third, the Iwasawa Main Conjecture for GL₂ due to Skinner-Urban-Wan which identifies the characteristic ideal of the dual Selmer group with the p-adic L-function. Fourth, Nekovar’s theory of higher p-adic heights and Bloch-Kato global duality which relate the p-adic derivatives of order r to the complex value via the adelic product principle for Tamagawa measures.