  This paper is about the Iwasawa theory of elliptic curves over the cyclotomic
$\mathbb{Z}_p$-extension $\mathbb{Q}^{\text{cyc}}$ of $\mathbb{Q}$. We discuss
a deep conjecture of Greenberg that if $E/\mathbb{Q}$ 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 $\mathbb{Q}^{\text{cyc}}$ has $\mu$-invariant
zero. We prove new cases of Greenberg's conjecture for some elliptic curves of
analytic rank $0$. The proof involves studying the $p$-adic $L$-function of
$E$. The crucial input is a new technique using the Rankin-Selberg method.
