Iwasawa (1969a) was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety.
Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fields.
[3] As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite.
These results were used by Manjul Bhargava, Skinner, and Wei Zhang to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.
[4][5] The main conjecture of Iwasawa theory proved by Mazur and Wiles states that if i is an odd integer not congruent to 1 mod p–1 then the ideals of