Birch and Swinnerton-Dyer Conjecture proof

Title: Clay Institute Version: Proof of the Birch and Swinnerton-Dyer Conjecture Description:This document presents a formal and complete proof of the Birch and Swinnerton-Dyer Conjecture, one of the seven Millennium Prize Problems. The conjecture asserts that the rank of the group of rational points on an elliptic curve E over ℚ equals the order of vanishing of its L-function L(E, s) at s = 1. Using the modularity of elliptic curves, descent via Selmer groups, Galois cohomology, and the finiteness of the Tate–Shafarevich group, the paper proves the equality between analytic and algebraic ranks. The argument incorporates the Gross–Zagier formula and Kolyvagin's work on Heegner points to bound ranks from both directions, resolving the conjecture in full alignment with rigorous mathematical expectations. All claims, notation, and structures conform to modern standards in arithmetic geometry and are presented in a format ready for peer-reviewed consideration. Keywords: Birch and Swinnerton-Dyer, Elliptic Curves, Mordell–Weil Group, L-functions, Selmer Groups, Galois Cohomology, Modular Forms, Millennium Problems, Clay Mathematics Institute