Mazur's main conjecture at Eisenstein primes

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel ${\mathbb{F}}_p(\phi )$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $\phi {\mathrm{\mid }}_{G_p}\mathrm{\ne }1,\omega$, where $G_p$ is a decomposition group at $p$ and $\omega$ is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson–Flach classes.