Mazur's main conjecture at Eisenstein primes

Abstract

Let $E/\mathbb{Q}$ 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 $\mathbb{F}_p(ϕ)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $ϕ\vert_{G_p}\neq 1,ω$, where $G_p$ is a decomposition group at $p$ and $ω$ 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.