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

TL;DR

This paper proves new cases of Greenberg's conjecture on the Iwasawa $mbda$-invariant being zero for certain elliptic curves of rank zero, using $p$-adic $L$-functions and Rankin-Selberg methods.

Contribution

It introduces a novel technique employing Rankin-Selberg methods to establish the $mbda=0$ case of Greenberg's conjecture for specific elliptic curves.

Findings

Proved new cases of Greenberg's conjecture for rank zero elliptic curves.

Developed a new approach using Rankin-Selberg methods in Iwasawa theory.

Connected $p$-adic $L$-functions with the $mbda$-invariant results.

Abstract

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.