Mazur’s main conjecture at Eisenstein primes

Let be an elliptic curve and let p be an odd prime of good reduction for E. Assume that E admits a rational p-isogeny, and let be the character by which acts on. In this paper, we prove the Iwasawa main conjecture for E, as formulated by B. Mazur in 1972, when, where is a decomposition group at p and is the Teichmüller character. Two key innovations in our proof are a Kolyvagin system argument for the Selmer group of E twisted by anticyclotomic Hecke characters arbitrarily close to the trivial character, and a congruence argument exploiting Beilinson–Flach classes and their explicit reciprocity laws.