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

TL;DR

This paper investigates the Iwasawa $$-invariants of elliptic curves over $Q$, providing evidence that the $$-invariant is zero or at most one for most primes, supporting Greenberg's conjecture.

Contribution

The authors prove that for elliptic curves over Q, the Iwasawa $$-invariant is at most one for all but finitely many primes of good ordinary reduction.

Findings

$$-invariant is zero for almost all primes

$$-invariant is at most one for all but finitely many primes

Supports Greenberg's conjecture in a broad setting

Abstract

In this paper, we discuss a longstanding conjecture of Greenberg in the Iwasawa theory of elliptic curves. Greenberg's conjecture states 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 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 $\mu \leq 1$ for all but finitely many primes $p$ of good ordinary reduction.