Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves

Abstract

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ be an odd prime number at which $E$ has good ordinary reduction. Let $Sel_{p^\infty}(\mathbb{Q}_\infty, E)$ denote the $p$-primary Selmer group of $E$ considered over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. The (algebraic) \emph{$\mu$-invariant} of $Sel_{p^\infty}(\mathbb{Q}_\infty, E)$ is denoted $\mu_p(E)$. Denote by $\bar{\rho}_{E, p}:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{Z}/p\mathbb{Z})$ the Galois representation on the $p$-torsion subgroup of $E(\bar{\mathbb{Q}})$. Greenberg conjectured that if $\bar{\rho}_{E, p}$ is reducible, then there is a rational isogeny $E\rightarrow E'$ whose degree is a power of $p$, and such that $\mu_p(E')=0$. In this article, we study this conjecture by showing that it is satisfied provided some purely Galois theoretic conditions hold that are expressed in terms of the representation $\bar{\rho}_{E,p}$. In establishing our results, we leverage a theorem of Coates and Sujatha on the algebraic structure of the fine Selmer group. Furthermore, in the case when $\bar{\rho}_{E, p}$ is irreducible, we show that our hypotheses imply that $\mu_p(E)=0$ provided the classical Iwasawa $\mu$-invariant vanishes for the splitting field $\mathbb{Q}(E[p]):=\bar{\mathbb{Q}}^{ker\bar{\rho}_{E,p}}$.