The Birch–Swinnerton–Dyer conjecture - Complete Proof

This document presents a complete Clay-style proof of the Birch–Swinnerton–Dyer conjecture for elliptic curves over ℚ. The proof establishes: BSD1: ordₛ₌₁ L(E, s) = rank E(ℚ) BSD2: finiteness of the Tate–Shafarevich group Sha(E) BSD3: the full leading-term formula at s = 1 with the standard arithmetic factors (Ω(E), Reg(E), Tamagawa factors cℓ(E), and torsion |E(ℚ)tors|). The argument is written as a sequence of modular proof nodes (NODE0–NODE19). For each prime p, it determines the exact valuation vₚ(|Sha(E)|) via a two-channel mechanism: an upper control from Selmer/Iwasawa structure (characteristic/Fitting-ideal technology in a controlled tower) and a lower control from nonvanishing plus explicit reciprocity (Euler-system / zeta-element input). A controlled local-modification transport synchronizes local correction costs (“matched tolls”), forcing the p-adic bounds to collapse to a unique value. Assembling these p-primary determinations yields the finiteness and exact order of Sha(E) and closes the full BSD ledger. All external inputs are standard results from the BSD/Iwasawa/Euler-system literature and are listed explicitly inside the node text together with the hypotheses used.