Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves
Anwesh Ray
TL;DR
This work analyzes Greenberg's conjecture for Galois representations attached to elliptic curves over $\mathbb{Q}$ at odd primes with good ordinary reduction, focusing on the algebraic $\mu$-invariant of the Greenberg Selmer group over the cyclotomic $\mathbb{Z}_p$-extension. It develops a representation-theoretic criterion, expressed purely in terms of the residual representation $\bar{\rho}_{E,p}$, that ensures $\mu_p(E')=0$ after passing to a $\mathbb{Q}$-isogenous curve $E'$, by relating the Greenberg Selmer group to the fine Selmer group and invoking Coates–Sujatha vanishing results for the latter. For irreducible $\bar{\rho}_{E,p}$, the conjecture follows when the classical Iwasawa $\mu$-invariant for the splitting field $L=\mathbb{Q}(E[p])$ vanishes; for reducible $\bar{\rho}_{E,p}$ the paper identifies concrete conditions under which $\mu_p(E)=0$, linking to the residual and fine Selmer structures. Overall, the results provide a concrete, residual-representation–driven framework to verify Greenberg's conjecture in many cases and point toward extensions to modular forms and abelian varieties, as well as connections to isogeny-based reductions of $\mu$-invariants.
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{$μ$-invariant} of $Sel_{p^\infty}(\mathbb{Q}_\infty, E)$ is denoted $μ_p(E)$. Denote by $\barρ_{E, p}:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{Z}/p\mathbb{Z})$ the Galois representation on the $p$-torsion subgroup of $E(\bar{\mathbb{Q}})$. Greenberg conjectured that if $\barρ_{E, p}$ is reducible, then there is a rational isogeny $E\rightarrow E'$ whose degree is a power of $p$, and such that $μ_p(E')=0$. 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 $\barρ_{E,p}$. 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 $\barρ_{E, p}$ is irreducible, we show that our hypotheses imply that $μ_p(E)=0$ provided the classical Iwasawa $μ$-invariant vanishes for the splitting field $\mathbb{Q}(E[p]):=\bar{\mathbb{Q}}^{ker\barρ_{E,p}}$.