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

Abstract

Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \), and let \( p \) be an odd prime at which \( E \) has good ordinary reduction. Consider the \( p \)-primary Selmer group \( \operatorname{Sel}_{p^\infty}(\mathbb{Q}_\infty, E) \) over the cyclotomic \( \mathbb{Z}_p \)-extension of \( \mathbb{Q} \), and denote its (algebraic) \(\mu\)-invariant by \( \mu_p(E) \). Let \( \bar{\rho}_{E, p} : \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{Z}/p\mathbb{Z}) \) be the Galois representation arising from the action of the absolute Galois group on the \( p \)-torsion of \( E \). Greenberg conjectured that if \( \bar{\rho}_{E, p} \) is reducible, then there exists a rational isogeny \( E \to E' \) of \( p \)-power degree such that \( \mu_p(E') = 0 \). In this article, we investigate this conjecture by establishing sufficient Galois-theoretic criteria for its validity, formulated in terms of the representation \( \bar{\rho}_{E, p} \). Our approach relies on a fundamental result of Coates and Sujatha concerning the fine Selmer group. Furthermore, in the case where \( \bar{\rho}_{E, p} \) is irreducible, we show that our conditions imply \( \mu_p(E) = 0 \) under the additional assumption that the classical Iwasawa \(\mu\)-invariant vanishes for the splitting field \( \mathbb{Q}(E[p]) = \bar{\mathbb{Q}}^{\ker \bar{\rho}_{E, p}} \).