Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

Abstract — v1

We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over $\mathbb Q$ whose full $q$-power Selmer group has corank zero or one at some prime $q$. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.