The Birch and Swinnerton-Dyer conjecture is a bridge between algebraic invariants of an elliptic curve and its (complex analytic) L-function. In the case of low ranks, we prove this conjecture up to the finitely many bad primes and the prime 2, by proving the Iwasawa main conjecture in full generality. The ideas in the proof and formulation also lead us to new and mysterious phenomena. This talk assumes no specialized background in number theory.