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 $Se{l}_{p^{\infty }}({\mathbb{Q}}_{\infty },E)$ 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 $Se{l}_{p^{\infty }}({\mathbb{Q}}_{\infty },E)$ is denoted ${\mu }_p(E)$. Denote by ${\bar{\rho }}_{E,p}:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to G{L}_2(\mathbb{Z}/p\mathbb{Z})$ 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 $E\to E^{\prime }$ whose degree is a power of $p$, and such that ${\mu }_p(E^{\prime })=0$. 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 $\mathbb{Q}(E[p]):={\bar{\mathbb{Q}}}^{ker{\bar{\rho }}_{E,p}}$.

Version 2: 21 pages, minor improvements and corrections. Accepted for publication in Journal of the Korean Math Society