The Iwasawa $\mu$-invariants of Elliptic Curves over $\mathbb{Q}$

In this paper, we discuss a longstanding conjecture of Greenberg in the Iwasawa theory of elliptic curves. Greenberg’s conjecture states that if $E\mathrm{/}\mathbb{Q}$ is an elliptic curve with good ordinary reduction at $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over the cyclotomic ${\mathbb{Z}}_p$ extension of $\mathbb{Q}$ has $\mu$-invariant zero. We prove that if $E$ is an elliptic curve over $\mathbb{Q}$, then we have $\mu \le 1$ for all but finitely many primes $p$ of good ordinary reduction.