Mazur's Main Conjecture at Eisenstein Primes

Let E be an elliptic curve over the rationals and p an odd prime of good reduction. In 1972, Mazur formulated a conjecture describing points on the elliptic curve defined over the p-adic tower of cyclotomic extensions of Q in terms of a p-adic L-function. I will report on a joint work with F.Castella and C.Skinner, where we prove the conjecture in cases where E admits a rational p-isogeny. Our proof is based on a congruence argument exploiting the cyclotomic Euler system of Beilinson-Flach classes, combined with a strengthening of our previous results on the anticyclotomic Iwasawa theory of E over an imaginary quadratic field. If time permits I will also discuss about how this refined anticyclotomic result yields new cases of Kolyvagin's conjecture on indivisibility of Heegner points.