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

Abstract:

This paper is about the Iwasawa theory of elliptic curves over the cyclotomic $\mathbf {Z}_p$-extension $\mathbf {Q}_{\operatorname {cyc}}$ of $\mathbf {Q}$. We discuss a deep conjecture of Greenberg that if $E/\mathbf {Q}$ is an elliptic curve with good ordinary reduction at an odd prime $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over $\mathbf {Q}_{\operatorname {cyc}}$ has $\mu$-invariant zero. Given an elliptic curve $E/\mathbf {Q}$ of analytic rank zero satisfying certain extra conditions, we give an explicit lower bound for primes $p$ for which $\mu =0$ and $\lambda =0$. The proof involves studying the $p$-adic $L$-function of $E$. The crucial input is a new technique using the Rankin-Selberg method.