On Greenberg's conjecture for rational elliptic curves at Eisenstein primes

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg’s conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic ${\mathbb{Z}}_p$-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens’s conjecture on the behavior of analytic $\mu$-invariants.