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

In this paper, we study Greenberg’s conjecture from a new perspective. Before discussing the main results, let us introduce some further notation. Let $L_{\infty}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of $L\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{Q}(E[p])$ and set $G\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(L_{\infty}/\mathbb{Q}_{\infty})$. Choose an embedding $\iota_{p}\mathrel{\mathop{\mathchar 58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$, and let $\tilde{\beta}$ be the prime of $\bar{\mathbb{Q}}$ that lies above $p$, prescribed by this embedding. Let $\beta$ be the prime of $L_{\infty}$ that lies below $\tilde{\beta}$, and $I_{\beta}$ be the inertia group at $\beta$. Let $\Sigma$ be a finite set of prime numbers that contains $p$ and the primes of bad reduction of $E$, and let $\mathbb{Q}_{\Sigma}$ be the maximal algebraic extension of $\mathbb{Q}$ in which the primes $\ell\notin\Sigma$ are unramified. Since $E$ is assumed to be ordinary at $p$, there is a unique $\operatorname{G}_{p}$-invariant line $\bar{C}$ contained in $E[p]$, such that the inertia group at $p$ acts on $\bar{C}$ via the mod-$p$ cyclotomic character. Note that the Galois group $\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty})$ acts trivially on $E[p]$. At a prime $\eta$ of $L_{\infty}$ that lies above a prime in $\Sigma$, let $f_{\eta}$ be the restriction of $f$ the decomposition group at $\eta$. The residual Selmer group over $L_{\infty}$ is defined as follows

A homomorphism $f\in\mathrm{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),E[p]\right)$ is $G$-equivariant if for any $g\in G$,

where $\tilde{g}$ is a lift of $g$ to $\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty})$. Let $\operatorname{Sel}(L_{\infty},E[p])^{G}$ be the subgroup of $\operatorname{Sel}(L_{\infty},E[p])$ consisting Selmer homomorphisms that are $G$-equivariant.

The above is a Galois theoretic criterion expressed purely in terms of the residual representation, which as we shall see gives a different formulation of Greenberg’s conjectures. It follows from our arguments that the above condition is in fact in many situations equivalent to Greenberg’s conjecture. In establishing our result, we crucially leverage results of Coates and Sujatha on the vanishing of the $\mu$-invariant of the fine Selmer group. We prove the first part of Conjecture 1.1, provided Conjecture 1.2 holds for the $\mathbb{Q}$-isogeny class of $E$.

Moreover, we show that the property that $\mu_{p}(E)$ vanishes can be detected precisely from the structure of the representation $\bar{\rho}_{E,p}$, see Theorem 6.4. We prove that the second part of this conjecture follows from certain additional conditions.

The main novelty in the above results, is that the algebraic structure of the fine Selmer group plays an important role in establishing them. Indeed, the Conjecture 1.2 gives a precise criterion for the fine Selmer group and the Greenberg Selmer group to have the same $\mu$-invariant. We explain this in greater detail below.

The results are proven by proving an explicit relationship between the Selmer group over $\mathbb{Q}_{\infty}$ and the fine Selmer group. This latter Selmer group is defined by imposing strict conditions at the prime above $p$. The algebraic properties of the fine Selmer group closely resemble those of class groups, and the $\mu$-invariants of these Selmer groups are always conjectured to vanish. Suppose that the Conjecture 1.2 holds for all elliptic curve that are isogenous to $E$. Then, we show that if either $\bar{\rho}_{E,p}$ is irreducible, or if $\bar{\rho}_{E,p}$ is reducible and satisfies some further conditions, then the $\mu$-invariant of the Greenberg Selmer group vanishes if and only if the $\mu$-invariant of the fine Selmer group vanishes. We show that any elliptic curve $E$ for which $\bar{\rho}_{E,p}$ is reducible is isogenous to another elliptic curve $E^{\prime}$ for which these additional conditions on $\bar{\rho}_{E^{\prime},p}$ are satisfied. The results of Coates-Sujatha [CS05] and Ferrero-Washington [FW79] together imply that the $\mu$-invariant of the fine Selmer group vanishes when the residual representation is reducible (cf. Theorem 6.2). We leverage the vanishing of the $\mu$-invariant of the fine Selmer group to deduce that the $\mu$-invariant of $E^{\prime}$ vanishes, thus proving Theorem 1. The Theorem 2 is proven via a similar technique. It is shown that if $\bar{\rho}_{E,p}$ is irreducible, then Greenberg’s conjecture is equivalent to a conjecture of Coates and Sujatha on the vanishing of the $\mu$-invariant of the fine Selmer group (cf. Corollary 5.5). The results establishing the relationship between the Greenberg Selmer group and the fine Selmer group hold in a more general context, namely, for ordinary Galois representations (cf. Theorem 5.4).

The methods developed in this paper should lead to interesting generalizations of Greenberg’s conjecture for ordinary Galois representations associated to modular forms and abelian varieties. Such questions have not been pursued in this paper, however would certainly be of interest to study in the future. It is of natural interest to ascertain if the statement Theorem 2 generalizes in some sense to the case of Kobayashi’s signed Selmer groups [Kob03] associated to elliptic curves with supersingular reduction at $p$. The main difficulty here is that Kobayashi’s Selmer groups have not been defined over fields other that the rational numbers, and we work with a suitably defined primitive residual Selmer group considered over the cyclotomic $\mathbb{Z}_{p}$-extension of $L$.

Let us discuss related work towards Greenberg’s conjecture in the residually reducible case. Greenberg and Vatsal [GV00] showed that if $\bar{\rho}_{E,p}$ is reducible and $E[p]$ contains a $1$-dimensional line which is $\operatorname{G}_{\mathbb{Q}}$-stable on which the action is via a character which is either

1. (1)

   unramified at $p$ and odd,

2. (2)

   ramified at $p$ and even,

then, $\mu_{p}(E)=0$. The case that remains is when the Galois representation $\bar{\rho}_{E,p}$ is indecomposable of the form $\left({\begin{array}[]{cc}\varphi_{1}&0\\ \ast&\varphi_{2}\\ \end{array}}\right)$, where $\varphi_{2}$ is unramified at $p$ and even. In this case, Greenberg conjectures that $\mu_{p}(E)=0$. The conjecture has been studied for $p=3$ by Hachimori in the case when $E(\mathbb{Q})[3]\neq 0$ (i.e., $\varphi_{2}=1$), cf. [Hac04, Theorem 1.1]. On the other hand, Trifkovic [Tri05] constructs infinitely many examples for $p=3,5$ of elliptic curves $E_{/\mathbb{Q}}$ for which $\bar{\rho}_{E,p}$ is of the form $\left({\begin{array}[]{cc}\varphi_{1}&0\\ \ast&\varphi_{2}\\ \end{array}}\right)$ described above, and such that $\mu_{p}(E)=0$. The case of interest falls under "type 3" under the classification in section 6.

We note here that the methods developed in this paper do not immediately generalize to elliptic curves over arbitrary number fields, since the arguments make use of the fact that there is only one prime of $\mathbb{Q}_{\infty}$ that lies above $p$. For a number field in which $p$ splits into $2$ or more primes, the same reasoning no longer applies. In fact, Drinen [Dri03] proved that there are large enough number fields over which the analogue of part (1) of Greenberg’s conjecture does not hold.

Including the introduction, the article consists of 6 sections. In section 2, we introduce basic notation and conventions, introduce the Greenberg Selmer group associated with an ordinary Galois representations. These Selmer groups are modules over the Iwasawa algebra, the structure theory of such modules leads to the definition of the $\mu$-invariant. When the Selmer group is cotorsion over the Iwasawa algebra, the $\mu$-invariant vanishes precisely when it is cofinitely generated as a $\mathbb{Z}_{p}$-module. We conclude section 2 by discussing the relationship between the Bloch-Kato Selmer group and the Greenberg Selmer group. It turns out that for elliptic curves $E_{/\mathbb{Q}}$, the Greenberg Selmer group coincides with the classical Selmer group for which the local conditions are defined via Kummer maps. The Greenberg Selmer groups are however more convenient to work with when employing Galois cohomological arguments.

In section 3, we introduce a Selmer group associated to the residual representation. Such Selmer groups were considered by Greenberg and Vatsal in [GV00] in studying the role of congruences between elliptic curves in Iwasawa theory. In loc. cit., certain imprimitive residual Selmer groups are defined, i.e., the local conditions at a number of primes are not imposed on these Selmer groups. It is necessary to work with such imprimitive conditions when studing the effect of congruences on the $\lambda$-invariant. In this article however, we only study the $\mu$-invariant, and therefore work with primitive residual Selmer groups. The section ends with Proposition 3.3, which shows that the $\mu$-invariant vanishes for the Greenberg Selmer group of an ordinary representation if and only if the (primitive) residual Selmer group is finite. Such a result is well known to the experts, however, we do include it for completeness.

The section 4 is devoted to the definition and basic properties of the fine Selmer group associated to a continuous Galois representation. We discuss a conjecture of Coates and Sujatha on the vanishing of the $\mu$-invariant. We end section 4 by recalling a key result on the vanishing of this $\mu$-invariant.

In section 5 we introduce a key assumption on the residual representation. This assumption allows us to relate the Greenberg Selmer group to the fine Selmer group. It is shown that the residual fine Selmer group is of finite index in the residual Greenberg Selmer group, provided the residual representation satisfies the additional hypothesis. It the end of this section 5, we give a proof of Theorem 2.

Finally, section 6 is devoted to the proof of Theorem 1. We give a classification of the residual representation into three types. It is shown that the $\mu$-invariant is positive when the residual representation is of type 1, and is zero when it is of type 2 or 3. It is in the case when the representation is of type 3 that the results from section 5 are applied. We then recall a theorem of Schneider, which describes the difference between the $\mu$-invariants of isogenous elliptic curves. We use this result, along with our classification theorem to prove Theorem 1.

The author’s research is supported by the CRM Simons postdoctoral fellowship. He thanks Jeffrey Hatey, Antonio Lei and Shaunak V. Deo for some helpful comments.

In this section, we introduce some standard notation.

• •

  Throughout $p$ will denote an odd prime number and $K/\mathbb{Q}_{p}$ a finite extension.

• •

  Let $\mathbb{F}_{p}$ be the field with $p$ elements.

• •

  Denote by $\mathcal{O}$ the valuation ring in $K$. Let $\varpi$ be a uniformizer of $\mathcal{O}$ and set $\mathbb{F}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{O}/\varpi$.

• •

  Let $\bar{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. For a subfield $F$ of $\bar{\mathbb{Q}}$, we set $\operatorname{G}_{F}$ to denote the absolute Galois group $\operatorname{Gal}(\bar{\mathbb{Q}}/F)$. We make the convention that all algebraic extensions of $\mathbb{Q}$ considered are contained in $\bar{\mathbb{Q}}$.

• •

  For each prime $\ell$, set $\operatorname{G}_{\ell}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(\bar{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$. Choose an embedding $\iota_{\ell}\mathrel{\mathop{\mathchar 58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$; set $\iota_{\ell}^{*}\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{\ell}\hookrightarrow\operatorname{G}_{\mathbb{Q}}$ to denote the induced inclusion.

• •

  Given a finite set of prime numbers $\Sigma$, set $\mathbb{Q}_{\Sigma}$ to be the maximal algebraic extension of $\mathbb{Q}$ in which all primes $\ell\notin\Sigma$ are unramified.

• •

  Let $\mathcal{F}$ be an extension of $\mathbb{Q}$ contained in $\mathbb{Q}_{\Sigma}$, set $H^{i}(\mathbb{Q}_{\Sigma}/\mathcal{F},\cdot)\mathrel{\mathop{\mathchar 58\relax}}=H^{i}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathcal{F}),\cdot\right)$.

• •

  Let $\mu_{p^{n}}\subset\bar{\mathbb{Q}}$ denote the $p^{n}$-th roots of unity, and $\mathbb{Q}(\mu_{p^{n}})$ be the cyclotomic field generated by $\mu_{p^{n}}$. Set $\mathbb{Q}(\mu_{p^{\infty}})$ to denote the union of number fields $\mathbb{Q}(\mu_{p^{n}})$ for $n\in\mathbb{Z}_{\geq 1}$.

• •

  Set $\mathcal{G}_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$ and $\chi\mathrel{\mathop{\mathchar 58\relax}}\mathcal{G}_{\infty}\xrightarrow{\sim}\mathbb{Z}_{p}^{\times}$ be the $p$-adic cyclotomic character.

• •

  Denote by $\mathbb{Q}_{\infty}$ the unique $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ that is contained in $\mathbb{Q}(\mu_{p^{\infty}})$, this is the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$.

• •

  Set $\Gamma\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ and $\Delta\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(\mathbb{Q}(\mu_{p})/\mathbb{Q})$, note that $\mathcal{G}_{\infty}\simeq\Delta\times\Gamma$.

• •

  For $n\in\mathbb{Z}_{\geq 1}$, let $\mathbb{Q}_{n}$ be the subfield of $\mathbb{Q}_{\infty}$ with $\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q}_{n})=\Gamma^{p^{n}}$. Identify $\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})$ with $\Gamma_{n}\mathrel{\mathop{\mathchar 58\relax}}=\Gamma/\Gamma^{p^{n}}$.

• •

  Given a prime $\eta$ of $\mathbb{Q}_{\infty}$, denote by $\mathbb{Q}_{\infty,\eta}$ the union of completions of the finite layers $\mathbb{Q}_{n}$ at $\eta$.

• •

  Let $\eta_{p}$ be the unique prime of $\mathbb{Q}_{\infty}$ that lies above $p$.

• •

  The Iwasawa algebra $\Lambda$ over $\mathcal{O}$ is the completed group algebra $\varprojlim_{n}\mathcal{O}[\Gamma_{n}]$.

• •

  Let $\gamma$ be a topological generator of $\Gamma$, setting $T\mathrel{\mathop{\mathchar 58\relax}}=(\gamma-1)$ we identify $\Lambda$ with the formal power series ring $\mathcal{O}\llbracket T\rrbracket$.

We recall the definition of the Greenberg Selmer group associated to an ordinary Galois representation. These Selmer groups are considered over $\mathbb{Q}_{\infty}$, and were introduced in [G⁺89]. We follow the notation and conventions in [GV00]. Let $n\geq 2$ be an integer, and $M$ be a free $\mathcal{O}$-module of rank $n$, equipped with a continuous $\mathcal{O}$-linear action of $\operatorname{G}_{\mathbb{Q}}$. Choose an $\mathcal{O}$-basis for $M$, and identify the group of $\mathcal{O}$-linear automorphisms of $M$ with $\operatorname{GL}_{n}(\mathcal{O})$. The Galois action on $M$ is encoded by a continuous Galois representation

Tensoring with $K$, we obtain the representation on the $K$-vector space $M_{K}\mathrel{\mathop{\mathchar 58\relax}}=M\otimes_{\mathcal{O}}K$, denoted

Set $d\mathrel{\mathop{\mathchar 58\relax}}=\dim_{K}M_{K}$, and $d^{\pm}$ to denote the $K$-dimensions of the plus and minus eigenspaces of $M_{K}$ for the action of complex conjugation. We set $A\mathrel{\mathop{\mathchar 58\relax}}=M_{K}/M$; we take note of the fact that $A\simeq(K/\mathcal{O})^{d}$.

A Galois representation satisfying the conditions of [G⁺89, p.98] is referred to as ordinary and satisfies the above assumption. Let $C$ to be the image of $W$ in $A$, and note that $C\simeq(K/\mathcal{O})^{d^{+}}$. Setting $D\mathrel{\mathop{\mathchar 58\relax}}=A/C$, we find that $D\simeq(K/\mathcal{O})^{d^{-}}$. Following [GV00, section 2], we recall the definition of the Greenberg Selmer group associated with the pair $(A,C)$. For $\ell\neq p$, set

where $\eta$ runs through the primes of $\mathbb{Q}_{\infty}$ that lie above $\ell$. The set of primes $\eta$ of $\mathbb{Q}_{\infty}$ that lie above any given rational prime is finite. Denote by $I_{\eta_{p}}$ the inertia group of $\operatorname{Gal}\left(\mkern 2.5mu\overline{\mkern-2.5mu\mathbb{Q}_{\infty,\eta_{p}}\mkern-2.5mu}\mkern 2.5mu/\mathbb{Q}_{\infty,\eta_{p}}\right)$. We let $L_{\eta_{p}}$ be defined as follows

where $\kappa_{p}$ is the kernel of the composite of the natural maps

The first of the above maps is the restriction map and the second map is induced by the map $A\rightarrow D$. The local condition at $p$ is prescribed as follows $\mathcal{H}_{p}(\mathbb{Q}_{\infty},A)\mathrel{\mathop{\mathchar 58\relax}}=H^{1}(\mathbb{Q}_{\infty},A)/L_{\eta_{p}}$.

The Selmer group over $\mathbb{Q}_{\infty}$ is defined as follows

where $\Sigma$ is a finite set of prime numbers containing $p$ and all prime numbers at which $\rho_{M}$ is ramified. As is well known, the Selmer group defined above is independent of the choice of primes $\Sigma$. Given a module $\mathbf{M}$ over $\Lambda$, set $\mathbf{M}^{\vee}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Hom}_{\mathbb{Z}_{p}}\left(\mathbf{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$ to denote its Pontryagin dual. We say that $\mathbf{M}$ is cofinitely generated (resp. cotorsion) over $\Lambda$ if $\mathbf{M}^{\vee}$ is finitely generated (resp. torsion) as a $\Lambda$-module.

The Selmer group $S_{A}(\mathbb{Q}_{\infty})$ is cofinitely generated as a module over $\Lambda$. Throughout we make the following assumption.

This assumption is known to hold in the main case of interest, namely for ordinary Galois representations associated with rational elliptic curves, cf. [G⁺89, Theorem 1.5].

Let $\mathbf{M}$ be a cofinitely generated and cotorsion $\Lambda$-module. We recall the definition of the Iwasawa $\mu$-invariant associated to $\mathbf{M}$.

A map of $\Lambda$-modules $\mathbf{M}_{1}\rightarrow\mathbf{M}_{2}$ is said to be a pseudo-isomorphism if its kernel and cokernel are both finite. A polynomial $f(T)\in\Lambda$ is distinguished if it is a monic polynomial and all non-leading coefficients are divisible by $\varpi$. According to the structure theorem of $\Lambda$-modules [Was97, Chapter 13], there is a pseudoisomorphism of the form

where $\mu_{i}$ are positive integers and $f_{j}(T)$ are irreducible distinguished polynomials. The $\mu$ invariant of $\mathbf{M}$ is defined as follows

where we set $\mu(\mathbf{M})=0$ if $s=0$. From the definition of the $\mu$-invariant, it is clear that $\mu(\mathbf{M})=0$ if and only if $M^{\vee}$ is finitely generated as a $\mathbb{Z}_{p}$-module.

Let $\tau$ be a variable in the complex upper half plane and set $q\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{exp}(2\pi i\tau)$. We take $f=\sum_{n=1}^{\infty}a_{n}(f)q^{n}$ be a normalized Hecke eigencuspform of weight $k\geq 2$. We assume that with respect to the embedding $\iota_{p}$, the modular form $f$ is ordinary. This means that $\iota_{p}(a_{p}(f))$ is a unit of $\mathcal{O}$. Let $K$ be the extension of $\mathbb{Q}_{p}$ generated by $\{\iota_{p}(a_{n}(f))\mid n\in\mathbb{Z}_{\geq 1}\}$. We note that $K$ is a finite extension of $\mathbb{Q}_{p}$. Let $\rho_{f,\iota_{p}}\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(K)$ be the Galois representation associated to $(f,\iota_{p})$. Let $V=V_{f,\iota_{p}}$ be the underlying $2$-dimensional $K$-vector space on which $\operatorname{G}_{\mathbb{Q}}$ acts by $K$-linear automorphisms. We choose a Galois stable $\mathcal{O}$-lattice $M$ in $V$, and set $A\mathrel{\mathop{\mathchar 58\relax}}=V/M$. As a module over $\operatorname{G}_{p}$, $M$ fits into a short exact sequence

The modules $M_{0}$ and $M_{1}$ are uniquely determined by the property that $M_{0}\simeq\mathcal{O}(\alpha\chi^{k-1})$ and $M_{1}\simeq\mathcal{O}(\alpha^{\prime})$, where $\alpha$ and $\alpha^{\prime}$ are unramified characters. We set $W\mathrel{\mathop{\mathchar 58\relax}}=(M_{0})\otimes_{\mathcal{O}}K$ and $C\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{image}\left(W\rightarrow A\right)$. Then the Greenberg Selmer group $S_{A}(\mathbb{Q}_{\infty})$ associated to $(A,C)$ clearly satisfies the Assumption 2.1. As is well known, in this context, the Greenberg Selmer group is pseudo-isomorphic to the Bloch-Kato Selmer group considered over $\mathbb{Q}_{\infty}$ (cf. [Och00, Corollary 4.3] for further details). It follows from results of Kato [K⁺04] that this Selmer groups are cotorsion over $\Lambda$, i.e., Assumption 2.2 is satisfied. We note that the Selmer group $S_{A}(\mathbb{Q}_{\infty})$ depends on the choice of embedding $\iota_{p}$ and the choice of Galois stable $\mathcal{O}$-lattice $M$.

Let us now consider Galois representations associated with elliptic curves over $\mathbb{Q}$. Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at $p$, and let $M\mathrel{\mathop{\mathchar 58\relax}}=T_{p}(E)$ be its $p$-adic Tate-module. The $p$-divisible Galois module $A$ is identified with $E[p^{\infty}]$, the $p$-power torsion points in $E(\bar{\mathbb{Q}})$. Since $E$ has ordinary reduction at $p$, there is a unique $\mathbb{Z}_{p}[\operatorname{G}_{p}]$-submodule $C\simeq\mathbb{Q}_{p}/\mathbb{Z}_{p}(\alpha\chi)$ of $E[p^{\infty}]$, where $\alpha\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{p}\rightarrow\mathbb{Z}_{p}^{\times}$ is an unramified character. The quotient $D\mathrel{\mathop{\mathchar 58\relax}}=A/C$ is unramified. The Greenberg Selmer group associated to $(A,C)$ is then denoted $\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$. Since $E$ arises from a Hecke eigencuspform of weight $2$, it follows from results of Kato [K⁺04] that $\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)^{\vee}$ is a torsion $\Lambda$-module, i.e., the Assumption 2.2 is satisfied. In this setting, there is no ambiguity in the definition of the Selmer group, since the field of coefficients equals $\mathbb{Q}_{p}$, and the $\mathbb{Z}_{p}$-Galois module is prescribed to be the $p$-adic Tate module of $E$. The Greenberg Selmer group coincides with the classical Selmer group, where the local conditions are defined via Kummer maps. We refer to [G⁺89, section 2] for further details. Throughout, we shall set $\mu_{p}(E)$ to denote the $\mu$-invariant of the Selmer group $\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$.

If $E^{\prime}$ is another elliptic curve over $\mathbb{Q}$ which is $\mathbb{Q}$-isogenous to $E^{\prime}$, then, $T_{p}(E^{\prime})\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ is isomorphic to $T_{p}(E)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ as a $\mathbb{Q}_{p}[\operatorname{G}_{\mathbb{Q}}]$-module, however, $T_{p}(E)$ is not isomorphic to $T_{p}(E^{\prime})$. It is possible that $\mu_{p}(E^{\prime})=0$, while $\mu_{p}(E)>0$, cf. [RS21, section 7].

Let $M$ be a module over $\operatorname{G}_{\mathbb{Q}}$ for which Assumption 2.1 is satisfied. Associated with $S_{A}(\mathbb{Q}_{\infty})$ is the residual Selmer group associated to the pair $(A,C)$. Stipulate that the cotorsion Assumption 2.2 is also satisfied. Set $A[\varpi^{n}]$ to denote the kernel of the multiplication by $\varpi^{n}$ endomorphism of $A$. We denote by $\bar{A}\mathrel{\mathop{\mathchar 58\relax}}=A[\varpi]$, and refer to the representation

as the residual representation. This is because we may identify $\bar{A}\mathrel{\mathop{\mathchar 58\relax}}=A[\varpi]$ with $M/\varpi M$, and thus think of $\rho_{\bar{A}}$ as the mod-$\varpi$ reduction of $\rho_{M}$. We let $\bar{C}\mathrel{\mathop{\mathchar 58\relax}}=C[\varpi]$, and $\bar{D}\mathrel{\mathop{\mathchar 58\relax}}=\bar{A}/\bar{C}$; note that the vector spaces $\bar{A},\bar{D}$ and $\bar{C}$ are $\mathbb{F}[\operatorname{G}_{p}]$-modules and they fit into a short exact sequence

We now introduce the residual Selmer group associated to $(\bar{A},\bar{C})$. For $\ell\neq p$, set

where $\eta$ runs over all primes of $\mathbb{Q}_{\infty}$ that lie above $\ell$. At $p$, the local condition is defined by setting $\mathcal{H}_{p}(\mathbb{Q}_{\infty},\bar{A})\mathrel{\mathop{\mathchar 58\relax}}=H^{1}(\mathbb{Q}_{\infty,\eta_{p}},\bar{A})/\bar{L}_{\eta_{p}}$, where

the map $\bar{\kappa}_{p}$ is the mod-$\varpi$ reduction of $\kappa_{p}$.

We note in passing that this Selmer group depends not only on the residual representation, but also the choice $\bar{C}$. For an elliptic curve $E_{/\mathbb{Q}}$ with good ordinary reduction at $p$, the space $\bar{A}$ is identified with $E[p]$, and there is a unique one dimensional subspace $\bar{C}$ on which the inertia group at $p$ acts via the mod-$p$ cyclotomic character. Therefore, there is no ambiguity in the definition when it is specialized to an elliptic curve with good ordinary reduction at $p$. We now study the relationship between the residual Selmer group and the $\mu$-invariant of $S_{A}(\mathbb{Q}_{\infty})$.