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

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 $\mu \le 1$ for all but finitely many primes $p$ of good ordinary reduction.

The previous version claiming to prove "mu=0 for almost all primes" has a mistake, which was kindly pointed out by Lei and Gajek-Leonard. As such, the article is being updated to claim only "mu <= 1 for almost all primes"