Mazur's main conjecture at Eisenstein primes

Let E be a rational elliptic curve, and p an odd prime of good ordinary reduction for E. In 1972, Mazur formulated an analogue of Iwasawa's main conjecture for the p-primary Selmer group of E over the cyclotomic Z_p-extension of Q. In this talk, after discussing some background and motivation, I will outline a proof of Mazur's main conjecture in the case where E admits a rational p-isogeny. Based on a joint work with Giada Grossi and Chris Skinner.