The Iwasawa $\mu$-invariant of certain elliptic curves of analytic rank zero

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 $E\mathrm{/}\mathbb{Q}$ 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 ${\mathbb{Q}}^{\text{cyc}}$ 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.