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

Anwesh Ray  Centre de recherches mathématiques, Université de Montréal,
Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal (Québec) H3T 1J4,
Canada<EMAIL_ADDRESS>
###### Abstract.

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ be an odd prime number at
which $E$ has good ordinary reduction. Let
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ denote the
$p$-primary Selmer group of $E$ considered over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. The (algebraic) _$\mu$ -invariant_
of $\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ is denoted
$\mu_{p}(E)$. Denote by $\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$
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 $E\rightarrow E^{\prime}$ 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
$\mathbb{Q}(E[p])\mathrel{\mathop{\mathchar
58\relax}}=\bar{\mathbb{Q}}^{\operatorname{ker}\bar{\rho}_{E,p}}$. Some
results are proven in greater generality, for ordinary Galois representations
satisfying further conditions.

###### 2020 Mathematics Subject Classification:

11R23, 11G05 (primary)

##  1\. Introduction

Iwasawa studied growth questions of class groups in certain infinite towers of
number fields. Given a number field $F$ and a prime number $p$, let
$F_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $F$. Letting
$F_{n}\subset F_{\operatorname{cyc}}$ be such that
$[F_{n}\mathrel{\mathop{\mathchar 58\relax}}F]=p^{n}$, denote by
$\operatorname{Cl}_{p}(F_{n})$ the $p$-Sylow subgroup of the class group of
$F_{n}$. Iwasawa showed that there exists $n_{0}\in\mathbb{Z}_{\geq 0}$ such
that for all $n\geq n_{0}$,

(1.1)
$|\operatorname{Cl}_{p}(F_{n})|=p^{\left(p^{n}\mu_{p}(F)+\lambda_{p}(F)n+\nu_{p}(F)\right)},$

where $\mu_{p}(F),\lambda_{p}(F)\in\mathbb{Z}_{\geq 0}$ and
$\nu_{p}(F)\in\mathbb{Z}$. We shall refer to $\mu_{p}(F)$ as the _classical
$\mu$-invariant_ of $F$. Iwasawa conjectured that $\mu_{p}(F)=0$ for all prime
numbers $p$ and all number fields $F$. This conjecture was proved by Ferrero
and Washington [FW79] for all abelian number fields.

The Iwasawa theory of Galois representations arising from motives leads to
very deep questions. Let $p$ be an odd prime number and $E_{/\mathbb{Q}}$ be
an elliptic curve. Mazur [Maz72] initiated the Iwasawa theory of elliptic
curves, and studied the algebraic structure of the Selmer group over the
cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. The (algebraic) Iwasawa
$\mu$-invariant $\mu_{p}(E)$ is defined in terms of the algebraic structure of
this Selmer group, when viewed as a module over the Iwasawa algebra $\Lambda$.
When $E$ has good ordinary reduction at $p$, these Selmer groups are known to
be cotorsion as $\Lambda$-modules, thanks to the work of Kato [K+04]. This
property is especially favorable when it comes to studying the properties of
the $\mu$-invariant. When the $\mu$-invariant vanishes, the Selmer group is
cofinitely generated as a $\mathbb{Z}_{p}$-module.

We recall a conjecture due to Greenberg [G+89, Conjecture 1.11] on the
vanishing of the $\mu$-invariant. Let $E[p]\subset E(\bar{\mathbb{Q}})$ be the
$p$-torsion subgroup, we note that as an abstract abelian group, $E[p]$ is
isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{2}$. Let

$\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$

be the representation of the absolute Galois group on $E[p]$. The following
conjecture is of paramount importance and has influenced major trends in
Iwasawa theory.

######  Conjecture 1.1 (Greenberg).

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ a prime number at which $E$
has good ordinary reduction.

  1. (1)

Suppose that the Galois representation $\bar{\rho}_{E,p}$ is reducible. Then,
$E$ is $\mathbb{Q}$-isogenous to an elliptic curve $E^{\prime}$ such that
$\mu_{p}(E^{\prime})=0$.

  2. (2)

Suppose that the Galois representation $\bar{\rho}_{E,p}$ is irreducible,
then, $\mu_{p}(E)=0$.

There are examples of elliptic curves $E_{/\mathbb{Q}}$ for which
$\bar{\rho}_{E,p}$ is reducible and for which $\mu_{p}(E)>0$. The first such
example is due to Mazur, see [Maz72, section].

###  1.1. Main results

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

$\begin{split}\operatorname{Sel}(L_{\infty},E[p])\mathrel{\mathop{\mathchar
58\relax}}=&\\{f\in\mathrm{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),E[p]\right)\mid\\\
&f_{\eta}=0\text{ for all primes }\eta\nmid\ell\text{ that lie above a prime
of }\Sigma,\\\ &\text{ and }f(I_{\beta})\subseteq\bar{C}\\}.\end{split}$

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

$f(\tilde{g}x\tilde{g}^{-1})=gf(x),$

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.

######  Conjecture 1.2.

Suppose that $\bar{C}$ is not a
$\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty})$ submodule of
$E[p]$. Then, the image of $\operatorname{Sel}(L_{\infty},E[p])^{G}$ in
$\mathrm{Hom}(I_{\beta},\bar{C})$ is finite.

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 $E$.

######  Theorem 1.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at
$p$ such that $E[p]$ is reducible. Assume that the Conjecture 1.2 is true for
all elliptic curves over $\mathbb{Q}$ that are isogenous to $E$. Then, there
exists an elliptic curve $E^{\prime}$ over $\mathbb{Q}$ that is isogenous to
$E$, such that the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E^{\prime})$ is $0$.
Moreover, the isogeny $E\rightarrow E^{\prime}$ has degree $p^{\mu_{p}(E)}$.

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.

######  Theorem 2.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at an
odd prime $p$ and assume that

  1. (1)

the residual representation $\bar{\rho}_{E,p}$ is irreducible; set
$L\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{Q}(E[p])$.

  2. (2)

The Conjecture 1.2 holds for $E$.

  3. (3)

The classical Iwasawa $\mu$-invariant $\mu_{p}(L)$ vanishes.

Then, the $\mu$-invariant of the Greenberg Selmer group
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ is $0$.

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.

###  1.2. Method of proof

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).

###  1.3. Outlook

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 $L$.

###  1.4. Related work

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.

###  1.5. Organization

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.

### Acknowledgments

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.

##  2\. Notation and Preliminaries

###  2.1. Notation

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 $\mathcal{O}\llbracket T\rrbracket$.

###  2.2. The Greenberg Selmer group

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

$\rho_{M}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{n}(\mathcal{O}).$

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

$\rho_{M_{K}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{n}(K).$

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
$A\simeq(K/\mathcal{O})^{d}$.

######  Assumption 2.1.

With respect to notation above, assume that

  1. (1)

there are finitely many prime numbers at which $\rho_{M}$ is ramified.

  2. (2)

The representation $\rho_{M_{K}}$ is irreducible.

  3. (3)

There exists a $\operatorname{G}_{p}$-stable $K$-subspace $W$ of $M_{K}$ of
dimension $d^{+}$ such that the action of $\operatorname{G}_{p}$ on $M_{K}/W$
is unramified.

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

$\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},A)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta|\ell}H^{1}(\mathbb{Q}_{\infty,\eta},A),$

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

$L_{\eta_{p}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{\infty,\eta_{p}},A)\xrightarrow{\kappa_{p}}H^{1}(I_{\eta_{p}},D)\right),$

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

$\begin{split}&H^{1}(\mathbb{Q}_{\infty,\eta_{p}},A)\rightarrow
H^{1}(I_{\eta_{p}},A)\\\ &H^{1}(I_{\eta_{p}},A)\rightarrow
H^{1}(I_{\eta_{p}},D).\\\ \end{split}$

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

$S_{A}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},A)\rightarrow\bigoplus_{\ell\in\Sigma}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},A)\right)$

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.

######  Assumption 2.2 (Cotorsion hypothesis).

The Selmer group $S_{A}(\mathbb{Q}_{\infty})$ is cofinitely generated over
$\Lambda$.

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].

###  2.3. The Iwasawa $\mu$-invariant

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

(2.1)
$\mathbf{M}^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\Lambda/(\varpi^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/\left(f_{j}(T)^{\lambda_{j}}\right)\right),$

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

$\mu(\mathbf{M})\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{s}\mu_{i},$

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.

######  Proposition 2.3.

Let $\mathbf{M}$ be a cofinitely generated and cotorsion $\Lambda$-module.
Then, $\mu(\mathbf{M})=0$ if and only if $\mathbf{M}[\varpi]$ is finite.

###### Proof.

The result is a direct consequence of the structure theorem for
$\Lambda$-modules. We have a pseudo-isomorphism

$\mathbf{M}^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\Lambda/(\varpi^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/\left(f_{j}(T)^{\lambda_{j}}\right)\right),$

as described in (2.1). Let $\Omega$ denote the mod-$\varpi$ reduction of
$\Lambda$. We identify $\left(\mathbf{M}[\varpi]\right)^{\vee}$ with
$\mathbf{M}^{\vee}/\varpi\mathbf{M}^{\vee}$. The mod-$\varpi$ reduction of the
above map is a pseudo-isomorphism

$\left(\mathbf{M}[\varpi]\right)^{\vee}\longrightarrow\Omega^{s}\oplus\left(\bigoplus_{j=1}^{t}\Omega/(T^{d_{j}\lambda_{j}})\right),$

where $d_{j}=\operatorname{deg}f_{j}(T)$. Clearly,
$\Omega/(T^{d_{j}\lambda_{j}})$ is a finite dimensional $\mathbb{F}$-vector
space, and $\Omega$ is infinite. Therefore, $M[\varpi]$ is finite if and only
if $s=0$. We note that $s=0$ if and only if $\mu(\mathbf{M})=0$, this proves
the result. ∎

###  2.4. Selmer groups associated to modular forms and elliptic curves

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

$0\rightarrow M_{0}\rightarrow M\rightarrow M_{1}\rightarrow 0.$

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].

##  3\. The residual Selmer group and the vanishing of the $\mu$-invariant

###  3.1. The residual Selmer group

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

$\rho_{\bar{A}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{Aut}_{\mathbb{F}}(\bar{A})\xrightarrow{\sim}\operatorname{GL}_{n}(\mathbb{F})$

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

$0\rightarrow\bar{C}\rightarrow\bar{A}\rightarrow\bar{D}\rightarrow 0.$

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

$\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\bar{A})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta|\ell}H^{1}(\mathbb{Q}_{\infty,\eta},\bar{A}),$

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

$\bar{L}_{\eta_{p}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{\infty,\eta_{p}},\bar{A})\xrightarrow{\bar{\kappa}_{p}}H^{1}(I_{\eta_{p}},\bar{D})\right);$

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

######  Definition 3.1.

With respect to notation above, the residual Selmer group is defined as
follows

$S_{\bar{A}}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})\rightarrow\bigoplus_{\ell\in\Sigma}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\bar{A})\right).$

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 $S_{A}(\mathbb{Q}_{\infty})$.

######  Lemma 3.2.

There is a natural map

$g\mathrel{\mathop{\mathchar
58\relax}}S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow
S_{A}(\mathbb{Q}_{\infty})[\varpi]$

with finite kernel and cokernel.

###### Proof.

Recall that $\bar{A}=A[\varpi]$, consider the Kummer sequence of
$\mathbb{Z}_{p}[\operatorname{G}_{\mathbb{Q}}]$-modules

(3.1) $0\rightarrow\bar{A}\rightarrow A\xrightarrow{\times\varpi}A\rightarrow
0.$

This induces and exact sequence

(3.2) $0\rightarrow\left(\frac{H^{0}\left(\mathbb{Q}_{\infty},A\right)}{\varpi
H^{0}\left(\mathbb{Q}_{\infty},A\right)}\right)\rightarrow
H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})\xrightarrow{f}H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},A)[\varpi]\rightarrow
0.$

Let $\ell$ be a prime, and $\eta$ be a prime of $\mathbb{Q}_{\infty}$ that
lies above $\ell$. From the Kummer sequence (3.1), we obtain an exact sequence

(3.3)
$0\rightarrow\left(\frac{H^{0}\left(\mathbb{Q}_{\infty,\eta},A\right)}{\varpi
H^{0}\left(\mathbb{Q}_{\infty,\eta},A\right)}\right)\rightarrow
H^{1}(\mathbb{Q}_{\infty,\eta},\bar{A})\xrightarrow{f_{\eta}}H^{1}(\mathbb{Q}_{\infty,\eta},A)[\varpi]\rightarrow
0.$

For $\ell\neq p$, we let

$f_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\bar{A})\rightarrow\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},A)[\varpi]$

be the product of the maps $f_{\eta}$, where $\eta$ ranges over the primes
above $\ell$. Since $\ell$ is finitely decomposed in $\mathbb{Q}_{\infty}$, it
follows from (3.3) that the kernel of $f_{\ell}$. Consider the commutative
square

${H^{1}(\mathbb{Q}_{\infty,\eta_{p}},A[\varpi])}$${H^{1}(I_{\eta_{p}},D[\varpi])}$${H^{1}(\mathbb{Q}_{\infty,\eta_{p}},A)}$${H^{1}(I_{\eta_{p}},D).}$$\scriptstyle{\bar{\kappa}_{p}}$$\scriptstyle{\kappa_{p}}$

We identify $\mathcal{H}_{p}(\mathbb{Q}_{\infty},A)$ (resp.
$\mathcal{H}_{p}(\mathbb{Q}_{\infty},\bar{A})$) with the image of $\kappa_{p}$
(resp. $\bar{\kappa}_{p}$). Form the commutativity of the above square, we
obtain a map

$f_{p}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{p}(\mathbb{Q}_{\infty},\bar{A})\rightarrow\mathcal{H}_{p}(\mathbb{Q}_{\infty},A)[\varpi].$

From the exact sequence of $I_{\eta_{p}}$-modules

$0\rightarrow D[\varpi]\rightarrow D\xrightarrow{\times\varpi}D\rightarrow 0,$

we obtain an exact sequence

$0\rightarrow\frac{H^{0}(I_{\eta_{p}},D)}{\varpi
H^{0}(I_{\eta_{p}},D)}\rightarrow H^{1}(I_{\eta_{p}},D[\varpi])\rightarrow
H^{1}(I_{\eta_{p}},D)\rightarrow 0.$

We note that $D$ is divisible and unramified at $\eta_{p}$, hence,
$H^{0}(I_{\eta_{p}},D)=D$. Since $D$ is divisible, it follows that the map

$H^{1}(I_{\eta_{p}},D[\varpi])\rightarrow H^{1}(I_{\eta_{p}},D)$

is injective, and hence $f_{p}$ is injective. The map $f$ restricts to a map

$g\mathrel{\mathop{\mathchar
58\relax}}S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow
S_{A}(\mathbb{Q}_{\infty})[\varpi]$

which fits into a commutative diagram

${0}$${S_{\bar{A}}(\mathbb{Q}_{\infty})}$${H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})}$${\bigoplus_{\ell\in\Sigma}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\bar{A})}$${0}$${S_{A}(\mathbb{Q}_{\infty})[\varpi]}$${H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},A)[\varpi]}$${\bigoplus_{\ell\in\Sigma}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},A)[\varpi].}$$\scriptstyle{g}$$\scriptstyle{\Phi_{\bar{A}}}$$\scriptstyle{f}$$\scriptstyle{h=\oplus
f_{\ell}}$$\scriptstyle{\Phi_{A}}$

Let $h^{\prime}$ be the restriction of $h$ to the image of $\Phi_{V}$. From
the snake lemma, we obtain an exact sequence

$0\rightarrow\ker g\rightarrow\ker f\rightarrow\ker
h^{\prime}\rightarrow\mathrm{coker}g\rightarrow\mathrm{coker}f=0.$

From (3.2), we know that the kernel of $f$ is finite, and hence, the kernel of
$g$ is finite. We have shown that the kernel of $h$ is finite, and hence,
$\ker h^{\prime}$ is finite. Therefore, we find that both the kernel and
cokernel of $g$ are finite, and this completes the proof. ∎

######  Proposition 3.3.

Suppose that the Assumptions 2.1 and 2.2 are satisfied. Then, the following
conditions are equivalent.

  1. (1)

The $\mu$-invariant of $S_{A}(\mathbb{Q}_{\infty})$ is equal to $0$.

  2. (2)

The residual Selmer group $S_{\bar{A}}(\mathbb{Q}_{\infty})$ is finite.

###### Proof.

It follows from Proposition 2.3 that the $\mu$-invariant of
$S_{A}(\mathbb{Q}_{\infty})$ is $0$ if and only if
$S_{A}(\mathbb{Q}_{\infty})[\varpi]$ is finite. Then it follows from Lemma 3.2
that $S_{A}(\mathbb{Q}_{\infty})[\varpi]$ is finite if and only if
$S_{\bar{A}}(\mathbb{Q}_{\infty})$ is finite. This completes the proof. ∎

##  4\. The fine Selmer group

In this section, we recall the definition of the fine Selmer group associated
to $A$. For further details, we refer to [CS05, DRS23]. We do _not_ insist
that $A$ satisfies the Assumption 2.1 in this section. Recall that $\Sigma$ is
a finite set of prime numbers containing $p$ and the primes that are ramified
in $A$. Let $F$ be a number field and $F_{\infty}$ be the composite of $F$
with $\mathbb{Q}_{\infty}$. For any prime $\ell$, set
$\mathcal{K}_{\ell}(F_{\infty},A)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta|\ell}H^{1}(F_{\infty,\eta},A)$, where $\eta$ runs over
the primes of $F_{\infty}$ that lie above $\ell$. We note that for any prime
$\ell\neq p$, the local condition $\mathcal{K}_{\ell}(\mathbb{Q}_{\infty},A)$
coincides with $\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},A)$, the difference
lies at the prime $p$. The _fine Selmer group_ is defined as follows

$S_{A}^{0}(F_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(F_{\Sigma}/F_{\infty},A)\rightarrow\bigoplus_{\ell\in\Sigma}\mathcal{K}_{\ell}(F_{\infty},A)\right).$

Here, $F_{\Sigma}$ is the maximal extension of $F$ in which all primes
$\ell\notin\Sigma$ are unramified. As is well known, the definition above is
independent of the choice of $\Sigma$. For further details, we refer to [SW18,
Lemma 3.2]. Given an elliptic curve $E$ over a number field $F$, and a prime
number $p$, set $\operatorname{Sel}_{p^{\infty}}^{0}(F_{\infty},E)$ be the
fine Selmer group associated to $A=E[p^{\infty}]$ over $F_{\infty}$.

Define the _residual fine Selmer group_ by setting

$\mathcal{K}_{\ell}(F_{\infty},\bar{A})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta|\ell}H^{1}(F_{\infty,\eta},\bar{A})$

for all prime numbers $\ell\in\Sigma$, and setting

$S_{\bar{A}}^{0}(F_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(F_{\Sigma}/F_{\infty},\bar{A})\rightarrow\bigoplus_{\ell\in\Sigma}\mathcal{K}_{\ell}(F_{\infty},\bar{A})\right).$

The fine Selmer group fits into a left exact sequence

$0\rightarrow S_{A}^{0}(\mathbb{Q}_{\infty})\rightarrow
S_{A}(\mathbb{Q}_{\infty})\rightarrow L_{\eta_{p}}.$

######  Lemma 4.1.

There is a natural map

$g_{0}\mathrel{\mathop{\mathchar
58\relax}}S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})\rightarrow
S_{A}^{0}(\mathbb{Q}_{\infty})[\varpi]$

with finite kernel and cokernel.

###### Proof.

The proof is similar to that of Lemma 3.2, we provide a sketch of the details.
The map $g_{0}$ is induced by restricting $f$ to
$S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$. It is easy to see that the image of
this restriction lies in $S_{A}^{0}(\mathbb{Q}_{\infty})[\varpi]$. The map
$g_{0}$ fits into a natural commutative diagram depicted below

${0}$${S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})}$${H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})}$${\bigoplus_{\ell\in\Sigma}\mathcal{K}_{\ell}(\mathbb{Q}_{\infty},\bar{A})}$${0}$${S_{A}^{0}(\mathbb{Q}_{\infty})[\varpi]}$${H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},A)[\varpi]}$${\bigoplus_{\ell\in\Sigma}\mathcal{K}_{\ell}(\mathbb{Q}_{\infty},A)[\varpi].}$$\scriptstyle{g_{0}}$$\scriptstyle{\Phi_{\bar{A}}^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\Phi_{A}^{\prime}}$

In the above diagram, the horizontal maps $\Phi_{\bar{A}}^{\prime}$ and
$\Phi_{A}^{\prime}$ are induced by restriction maps. The kernels of both
vertical maps in the diagram are finite. By the same argument as in the proof
of Lemma 3.2, it follows that the kernel and cokernel of $g_{0}$ are finite. ∎

######  Proposition 4.2.

With respect to notation above, the following conditions are equivalent.

  1. (1)

The $\mu$-invariant of $S_{A}^{0}(\mathbb{Q}_{\infty})$ is equal to $0$.

  2. (2)

The residual fine Selmer group $S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is
finite.

###### Proof.

It follows from Proposition 2.3 that the $\mu$-invariant of
$S_{A}^{0}(\mathbb{Q}_{\infty})$ is $0$ if and only if
$S_{A}^{0}(\mathbb{Q}_{\infty})[\varpi]$ is finite. Then it follows from Lemma
4.1 that $S_{A}^{0}(\mathbb{Q}_{\infty})[\varpi]$ is finite if and only if
$S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is finite. This completes the proof. ∎

At this point, it is pertinent to recall a conjecture of Coates and Sujatha on
the structure of the fine Selmer group associated with an elliptic curve. For
futher details, see [CS05, Conjecture A].

######  Conjecture 4.3 (Coates-Sujatha).

Let $E$ be an elliptic curve over a number field $F$ and $p$ be a prime above
which $E$ has good reduction. Then, the fine Selmer group
$\operatorname{Sel}_{p^{\infty}}^{0}(F_{\infty},E)$ is cofinitely generated as
a $\mathbb{Z}_{p}$-module.

Under some additional conditions, the above conjecture is known to hold. Let
$F(E_{p^{\infty}})$ be the Galois extension of $F$ generated by
$E_{p^{\infty}}$. In greater detail, letting
$\rho_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$
be the Galois representation on the $p$-adic Tate module of $E$, the extension
$F(E_{p^{\infty}})\mathrel{\mathop{\mathchar
58\relax}}=\bar{F}^{\ker\rho_{E,p}}$. Note that $\rho_{E,p}$ induces an
inclusion of $\operatorname{Gal}(F(E_{p^{\infty}})/F)$ into
$\operatorname{GL}_{2}(\mathbb{Z}_{p})$.

######  Theorem 4.4 (Coates-Sujatha).

Let $E_{/F}$ be an elliptic curve and $p$ an odd prime such that
$F(E_{p^{\infty}})$ is a pro-$p$ extension of $F$. Then, the following
conditions are equivalent

  1. (1)

Conjecture 4.3 is valid, i.e.,
$\operatorname{Sel}_{p^{\infty}}^{0}(F_{\infty},E)$ is cofinitely generated as
a $\mathbb{Z}_{p}$-module.

  2. (2)

The classical Iwasawa $\mu$-invariant $\mu_{p}(F)$ vanishes.

###### Proof.

The above result is [CS05, Theorem 3.4]. ∎

##  5\. Structure of the residual Greenberg Selmer group

In this section, we prove some of the main results of the article which will
be of key importance in the proof of Theorem 1. At the end of this section, we
shall prove Theorem 2. We begin by proving an explicit relationship between
the residual Selmer group and the residual fine Selmer group. These residual
Selmer groups were introduced in the previous section.

It is necessary to introduce an assumption on the Galois action on the
residual representation $\bar{A}$.

######  Assumption 5.1.

Assume that $\bar{C}$ does not contain a non-zero
$\operatorname{G}_{\mathbb{Q}_{\infty}}$-submodule of $\bar{A}$.

For Galois representations associated with elliptic curves, we characterize
precisely when the above Assumption holds.

######  Proposition 5.2.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at
$p$, and let $\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
be the Galois representation on the torsion subgroup $E[p]\subset
E(\bar{\mathbb{Q}})$. Thus, $A=E[p^{\infty}]$ and $\bar{A}=E[p]$. Let
$(e_{1},e_{2})$ be an ordered basis of $E[p]$ such that
$\bar{C}=\mathbb{F}_{p}\cdot e_{1}$. The following assertions hold.

  1. (1)

Assume that $\bar{\rho}_{E,p}$ is irreducible. Then, the Assumption 5.1 holds.

  2. (2)

Assume that $\bar{\rho}_{E,p}$ is reducible and indecomposable111i.e., it is
reducible but does not split into a sum of characters with respect to any
basis, and with respect to the basis $(e_{1},e_{2})$, takes the form

$\bar{\rho}_{E,p}=\left({\begin{array}[]{cc}\varphi_{1}&0\\\
\ast&\varphi_{2}\\\ \end{array}}\right).$

Then, the Assumption 5.1 holds.

  3. (3)

Assume that $\bar{\rho}_{E,p}$ is reducible and with respect to the basis
$(e_{1},e_{2})$, takes the form

$\bar{\rho}_{E,p}=\left({\begin{array}[]{cc}\varphi_{1}&\ast\\\
0&\varphi_{2}\\\ \end{array}}\right).$

Then, the Assumption 5.1 does not hold.

###### Proof.

We begin with part (1). We set

$\mathcal{G}\mathrel{\mathop{\mathchar
58\relax}}=\bar{\rho}_{E,p}(\operatorname{G}_{\mathbb{Q}})\text{ and
}\mathcal{H}\mathrel{\mathop{\mathchar
58\relax}}=\bar{\rho}_{E,p}(\operatorname{G}_{\mathbb{Q}_{\infty}}).$

Since $\mathbb{Q}_{\infty}/\mathbb{Q}$ is a $\mathbb{Z}_{p}$-extension, it
follows that $\mathcal{H}$ is a normal subgroup of $\mathcal{G}$ of index
$|\mathcal{G}/\mathcal{H}|=p^{t}$, where $t\in\mathbb{Z}_{\geq 0}$. If
$\mathcal{G}=\mathcal{H}$, then, $\bar{\rho}_{E,p}$ remains irreducible when
restricted to $\operatorname{G}_{\mathbb{Q}_{\infty}}$. Therefore, in this
case, $\bar{C}$ is not a $\operatorname{G}_{\mathbb{Q}_{\infty}}$-submodule
and the Assumption 5.1 holds. Therefore, we assume that $p$ divides
$|\mathcal{G}/\mathcal{H}|$. Since $p$ divides $\mathcal{G}$, it follows that
either $\mathcal{G}$ contains $\operatorname{SL}_{2}(\mathbb{F}_{p})$, or
$\mathcal{G}$ is contained in a Borel subgroup of
$\operatorname{GL}_{2}(\mathbb{F}_{p})$ (cf. [Sut16, Proposition 3.1]). Since
$\bar{\rho}_{E,p}$ is irreducible, $\mathcal{G}$ is not contained in a Borel
subgroup. Hence, $\mathcal{G}$ contains
$\operatorname{SL}_{2}(\mathbb{F}_{p})$. Suppose that $\mathcal{H}$ is
contained in a Borel subgroup; then we find that $|\mathcal{H}|$ divides
$p(p-1)^{2}$. On the other hand, since $\mathcal{G}$ contains
$\operatorname{SL}_{2}(\mathbb{F}_{p})$, it follows that $|\mathcal{G}|$ is
divisible by $|\operatorname{SL}_{2}(\mathbb{F}_{p})|=(p^{2}-1)p$. Since
$|G|=p^{t}|H|$, it follows that $(p^{2}-1)$ divides $p(p-1)^{2}$, which is a
contradiction. Therefore, $\mathcal{H}$ is not contained in a Borel subgroup
and hence, the representation $\bar{\rho}_{E,p}$ is irreducible when
restricted to $\operatorname{G}_{\mathbb{Q}_{\infty}}$. Therefore, $\bar{A}$
does not contain any non-zero proper $\operatorname{G}_{\mathbb{Q}_{\infty}}$
submodules, and this completes the proof of part (1).

For the proof of part (2) it suffices to show that $\bar{\rho}_{E,p}$ remains
indecomposable even after restriction to
$\operatorname{G}_{\mathbb{Q}_{\infty}}$. We define a function
$\beta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})\rightarrow\mathbb{F}_{p}$
by setting $\beta(g)$ to denote the lower left entry of $\bar{\rho}_{E,p}$. It
is easy to see that $\beta$ gives rise to a cocycle

$\beta\in
Z^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right).$

Let $[\beta]\in
H^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)$
denote the corresponding cohomology class. Since it is assumed that
$\bar{\rho}_{E,p}$ is indecomposable, it follows that $[\beta]$ is a non-zero
cohomology class. In order to show that the restriction

$\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$

is indecomposable, it suffices to show that the restriction of $[\beta]$ to
$H^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)$
is non-zero. From the inflation restriction sequence, the kernel of the
restriction map

(5.1)
$H^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)\rightarrow
H^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)$

is
$H^{1}\left(\mathbb{Q}_{\infty}/\mathbb{Q},\left(\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)^{\operatorname{G}_{\mathbb{Q}_{\infty}}}\right)$.
Since $\varphi_{1}$ is ramified at $p$ and $\varphi_{2}$ is unramified at $p$,
we find that $\varphi_{2}\varphi_{1}^{-1}\neq 1$. Since
$\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ is a pro-$p$ extension
and the character $\varphi_{2}\varphi_{1}^{-1}$ takes values in
$\mathbb{F}_{p}^{\times}$, we find that the restriction of
$\varphi_{2}\varphi_{1}^{-1}$ to $\operatorname{G}_{\mathbb{Q}_{\infty}}$ is
non-trivial. Therefore,
$\left(\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)^{\operatorname{G}_{\mathbb{Q}_{\infty}}}=0$
and the restriction map (5.1) is injective. Hence, the restriction of $\beta$
to
$H^{1}\left(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\mathbb{F}_{p}(\varphi_{2}\varphi_{1}^{-1})\right)$
is non-zero. This proves that

$\bar{\rho}_{E,p|\mathbb{Q}_{\infty}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}_{\infty}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$

is indecomposable. We therefore have shown that $\bar{C}$ is not a
$\operatorname{G}_{\mathbb{Q}_{\infty}}$-submodule of $\bar{A}$. This
completes the proof of part (2).

For part (3), observe that $\bar{C}$ is a
$\operatorname{G}_{\mathbb{Q}}$-submodule of $\bar{A}$. In particular, it is a
$\operatorname{G}_{\mathbb{Q}_{\infty}}$ submodule of $\bar{A}$, and the
Assumption 5.1 is not satisfied. ∎

The residual Selmer group $S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is contained
in $S_{\bar{A}}(\mathbb{Q}_{\infty})$; set

$\bar{S}_{\bar{A}}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\frac{S_{\bar{A}}(\mathbb{Q}_{\infty})}{S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})}.$

We postpone the proof of the above result till the end of this section. First,
we introduce some further notation. Let $L=\mathbb{Q}(\bar{A})$ be the field
cut out by the residual representation. In other words, $L$ is the field
$\bar{\mathbb{Q}}^{\operatorname{ker}\rho_{\bar{A}}}$, the field fixed by the
kernel of the residual representation $\rho_{\bar{A}}$. We note that
$\mathbb{Q}(\bar{A})$ is a finite Galois extension of $\mathbb{Q}$ and the
Galois group $\operatorname{Gal}(\mathbb{Q}(\bar{A})/\mathbb{Q})$ is naturally
isomorphic to the image of $\rho_{\bar{A}}$; the representation
$\rho_{\bar{A}}$ induces an isomorphism

$\operatorname{Gal}(\mathbb{Q}(\bar{A})/\mathbb{Q})\xrightarrow{\sim}\operatorname{image}\rho_{\bar{A}}.$

We observe that $\operatorname{G}_{L}$ is the kernel of $\rho_{\bar{A}}$, and
hence acts trivially on $\bar{A}$. Let $L_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=L\cdot\mathbb{Q}_{\infty}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $L$. Let $\beta$ be the prime of $L_{\infty}$
that lies above $p$ that coincides with the choice of embedding $\iota_{p}$,
and denote by $I_{\beta}$ the inertia group at $\beta$. Note that $I_{\beta}$
is contained in the inertia group $I_{\eta_{p}}$.

We define a Selmer group $S_{\bar{A}}(L_{\infty})$ associated to
$(\bar{A},\bar{C})$ over $L_{\infty}$. For each prime number $\ell$, we define
a local condition $\mathcal{H}_{\ell}(L_{\infty},\bar{A})$. For $\ell\neq p$,
set

$\mathcal{H}_{\ell}(L_{\infty},\bar{A})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta|\ell}H^{1}(L_{\infty,\eta},\bar{A}),$

where $\eta$ runs through all primes of $L_{\infty}$ that lies above $\ell$.
We note that this is a finite set of primes. At the prime $p$, we set

$\mathcal{H}_{p}(L_{\infty},\bar{A})\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{H^{1}(L_{\infty,\beta},\bar{A})}{\bar{L}_{\beta}}\right),$

where

$\bar{L}_{\beta}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(L_{\infty,\beta},\bar{A})\rightarrow
H^{1}(I_{\beta},\bar{D})\right).$

Note that since $\rho_{\bar{A}}$ is unramified outside $\Sigma$, and hence,
$L$ is contained in $\mathbb{Q}_{\Sigma}$. With respect to notation above, the
residual Selmer group is defined as follows

$S_{\bar{A}}(L_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{\Sigma}/L_{\infty},\bar{A})\rightarrow\bigoplus_{\ell\in\Sigma}\mathcal{H}_{\ell}(L_{\infty},\bar{A})\right).$

We relate the two residual Selmer groups $S_{\bar{A}}(\mathbb{Q}_{\infty})$
and $S_{\bar{A}}(L_{\infty})$. We shall set $G\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(L_{\infty}/\mathbb{Q}_{\infty})$. We note that
$\operatorname{Gal}(\bar{\mathbb{Q}}/L)$ is the kernel of $\rho_{\bar{A}}$ and
therefore, the Galois action of $\operatorname{Gal}(\bar{\mathbb{Q}}/L)$ on
$\bar{A}$ is trivial. We identify
$H^{1}(\mathbb{Q}_{\Sigma}/L_{\infty},\bar{A})$ with the group of
homomorphisms
$\operatorname{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),\bar{A}\right)$.
For $g\in G$, take
$\tilde{g}\in\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty})$ to
be a lift of $g$. Take
$\psi\in\operatorname{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),\bar{A}\right)$,
we note that since $\bar{A}$ is abelian, $\psi(\tilde{g}x\tilde{g}^{-1})$ is
independent of the choice of lift $\tilde{g}$. Define an action of $G$ on
$\operatorname{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),\bar{A}\right)$,
by setting

$(g\cdot\psi)(x)\mathrel{\mathop{\mathchar
58\relax}}=g^{-1}\psi(\tilde{g}x\tilde{g}^{-1}).$

Therefore, a homomorphism $\psi$ in
$\operatorname{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),\bar{A}\right)^{G}$
is one which is $G$-equivariant, in the sense that

$\psi(\tilde{g}x\tilde{g}^{-1})=g\psi(x).$

Consider the inflation-restriction sequence

(5.2) $0\rightarrow
H^{1}(G,\bar{A})\xrightarrow{inf}H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})\xrightarrow{res}\operatorname{Hom}\left(\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty}),\bar{A}\right)^{G}.$

The restriction map

$\operatorname{res}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\infty},\bar{A})\rightarrow
H^{1}(\mathbb{Q}_{\Sigma}/L_{\infty},\bar{A})$

induces a map

$\operatorname{res}\mathrel{\mathop{\mathchar
58\relax}}S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow
S_{\bar{A}}(L_{\infty}).$

Since $G$ is finite, $H^{1}(G,\bar{A})$ is finite, and thus the kernel of this
restriction map is finite.

We let $S_{\bar{A}}^{\operatorname{nr}}(L_{\infty})$ be the subspace of
$S_{\bar{A}}(L_{\infty})$ consisting of the classes that are unramified at
$\beta$. Note that $S_{\bar{A}}(L_{\infty})$ consists of homomorphisms

$\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\mathbb{Q}_{\Sigma}/L_{\infty})\rightarrow\bar{A}$

that satisfy the following conditions

  1. (1)

$\psi$ trivial when restricted to the decomposition group of any prime $\eta$
of $L_{\infty}$ that lies above a prime $\ell\in\Sigma\backslash\\{p\\}$,

  2. (2)

$\psi(I_{\beta})$ is contained in $\bar{C}$.

The subset $S_{\bar{A}}^{\operatorname{nr}}(L_{\infty})$ consists of those
classes for which $\psi(I_{\beta})=0$.

######  Conjecture 5.3.

Suppose that Assumption 5.1 holds, then, the image of the restriction map

$S_{\bar{A}}(L_{\infty})^{G}\rightarrow\mathrm{Hom}\left(I_{\beta},\bar{C}\right)$

is finite.

######  Theorem 5.4.

Let $(A,C)$ be such that Assumption 5.1 holds for $(\bar{A},\bar{C})$.
Furthermore, assume that the Conjecture 5.3 is also satisfied. Then, the
following assertions hold

  1. (1)

$\bar{S}_{\bar{A}}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\frac{S_{\bar{A}}(\mathbb{Q}_{\infty})}{S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})}$
is finite.

  2. (2)

The $\mu$-invariant of $S_{A}(\mathbb{Q}_{\infty})$ vanishes if and only if
the $\mu$-invariant of $S_{A}^{0}(\mathbb{Q}_{\infty})$ vanishes.

###### Proof of Theorem 5.4.

We shall set $S_{\bar{A}}^{\operatorname{nr}}(\mathbb{Q}_{\infty})$ to consist
of all classes $f\in S_{\bar{A}}^{\operatorname{nr}}(\mathbb{Q}_{\infty})$
that are unramified at $\eta_{p}$. It is easy to see that
$S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is of finite index in
$S_{\bar{A}}^{\operatorname{nr}}(\mathbb{Q}_{\infty})$. We begin by proving
part (1). We have a short exact sequence

(5.3) $0\rightarrow
S_{\bar{A}}^{\operatorname{nr}}(\mathbb{Q}_{\infty})\rightarrow
S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow H^{1}(I_{\eta_{p}},\bar{C}).$

Consider the commutative square

${S_{\bar{A}}(\mathbb{Q}_{\infty})}$${H^{1}(I_{\eta_{p}},\bar{C})}$${S_{\bar{A}}(L_{\infty})^{G}}$${H^{1}(I_{\beta},\bar{C}).}$

It follows from Conjecture 1.2 that the image of the composed map

$S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow H^{1}(I_{\beta},\bar{C})$

is finite. Since $I_{\beta}$ has finite index in $I_{\eta_{p}}$, it follows
that the kernel of the restriction map

$H^{1}(I_{\eta_{p}},\bar{C})\rightarrow H^{1}(I_{\beta},\bar{C})$

is finite. Therefore, we find that the image of

$S_{\bar{A}}(\mathbb{Q}_{\infty})\rightarrow H^{1}(I_{\eta_{p}},\bar{C})$

is finite. From the exact sequence (5.3), we deduce that
$S_{\bar{A}}^{\operatorname{nr}}(\mathbb{Q}_{\infty})$ is of finite index in
$S_{\bar{A}}(\mathbb{Q}_{\infty})$. Therefore,
$S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is of finite index in
$S_{\bar{A}}(\mathbb{Q}_{\infty})$, and the statement of part (1) follows from
this.

It follows from part (1) that $S_{\bar{A}}(\mathbb{Q}_{\infty})$ is finite if
and only if $S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is finite. Proposition 3.3
asserts that $S_{\bar{A}}(\mathbb{Q}_{\infty})$ is finite if and only if the
$\mu$-invariant of $S_{A}(\mathbb{Q}_{\infty})$ is $0$. On the other hand,
Proposition 4.2 asserts that $S_{\bar{A}}^{0}(\mathbb{Q}_{\infty})$ is finite
if and only if the $\mu$-invariant of $S_{A}^{0}(\mathbb{Q}_{\infty})$ is $0$.
Hence, the $\mu$-invariant of $S_{A}(\mathbb{Q}_{\infty})$ is $0$ if and only
if the $\mu$-invariant of $S_{A}^{0}(\mathbb{Q}_{\infty})$ is $0$. This proves
part (2). ∎

######  Corollary 5.5.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at an
odd prime $p$. Assume that $\bar{\rho}_{E,p}$ is irreducible and Conjecture
5.3 is satisfied.. Then, the following are equivalent.

  1. (1)

The $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ vanishes, i.e.,
Greenberg’s conjecture holds.

  2. (2)

The $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ vanishes, i.e.,
the Conjecture 4.3 holds.

###### Proof.

Since $\bar{\rho}_{E,p}$ is irreducible, Proposition 5.2 shows that the
Assumption 5.1 is satisfied. The result therefore follows from part (2) of
Theorem 5.4. ∎

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ a prime at which $E$ has
good ordinary reduction. Let

$\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{Aut}(E[p])\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{p})$

be the residual representation on $E[p]$. The splitting field
$\mathbb{Q}(E[p])$ is the field extension of $\mathbb{Q}$ which is fixed by
the kernel of $\bar{\rho}_{E,p}$.

###### Proof of Theorem 2.

Since $\bar{\rho}_{E,p}$ is irreducible, it follows from Corollary 5.5 that
the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ is equal to $0$ if
and only if the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ is equal to $0$.

Consider the Galois representation

$\rho_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$

associated with the $p$-adic Tate module of $E$. The restriction of
$\rho_{E,p}$ to $\operatorname{G}_{L}$ is trivial modulo $p$. This is because
$L$ is the splitting field $\mathbb{Q}(E[p])\mathrel{\mathop{\mathchar
58\relax}}=\bar{\mathbb{Q}}^{\ker\bar{\rho}_{E,p}}$, and
$\operatorname{G}_{L}$ is the kernel of $\bar{\rho}_{E,p}=\rho_{E,p}\mod{p}$.
Therefore, the representation $\rho_{E,p}$ identifies
$\operatorname{Gal}\left(L(E_{p^{\infty}})/L\right)$ with a subgroup of

$\widehat{\operatorname{GL}_{2}}(\mathbb{Z}_{p})\mathrel{\mathop{\mathchar
58\relax}}=\ker\\{\operatorname{GL}_{2}(\mathbb{Z}_{p})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})\\}.$

It is easy to see that $\widehat{\operatorname{GL}_{2}}(\mathbb{Z}_{p})$ is a
pro-$p$ group.222The author is willing to provide further details in support
of this claim (if the referee insists). Hence, the Galois group
$\operatorname{Gal}\left(L(E_{p^{\infty}})/L\right)$is a pro-$p$ group. Since
it is assumed that the classical Iwasawa $\mu$-invariant $\mu_{p}(L)$
vanishes, it follows from Theorem 4.4 that
$\operatorname{Sel}_{p^{\infty}}^{0}(L_{\infty},E)$ is cofinitely generated as
a $\mathbb{Z}_{p}$-module. In other words,
$\operatorname{Sel}_{p^{\infty}}^{0}(L_{\infty},E)$ is a cotorsion
$\Lambda$-module whose $\mu$-invariant vanishes. It is easy to see that the
kernel of the natural restriction map

$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)\rightarrow\operatorname{Sel}_{p^{\infty}}^{0}(L_{\infty},E)$

is cofinitely generated as a $\mathbb{Z}_{p}$-module, and hence the
$\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ is $0$. This
completes the proof. ∎

##  6\. Residually reducible Galois representations arising from elliptic
curves and Greenberg’s conjecture

Throughout this section, we fix and elliptic curve $E_{/\mathbb{Q}}$ and an
odd prime $p$ at which $E$ has good ordinary reduction. Let $M$ denote the
$p$-adic Tate module of $E$. Recall that $\bar{A}$ is the mod-$p$ reduction of
$M$, which we may identify with $E[p]$. The module $\bar{C}$ is the
$1$-dimensional $\operatorname{G}_{p}$-submodule which is ramified, and the
quotient $\bar{D}\mathrel{\mathop{\mathchar 58\relax}}=\bar{A}/\bar{C}$ is
unramified. The residual representation $\bar{\rho}_{E,p}=\rho_{\bar{A}}$ is
the representation of $\operatorname{G}_{\mathbb{Q}}$ on $\bar{A}$. We shall
assume throughout this section that $\bar{A}$ is reducible as a Galois module.
Choose a basis $(e_{1},e_{2})$ of $\bar{A}$ such that
$\bar{C}=\mathbb{F}_{p}\cdot e_{1}$. Call such a basis _admissible_ ; note
that for any other admissible basis $(e_{1}^{\prime},e_{2}^{\prime})$, there
are constants $c_{1},c_{2}\in\mathbb{F}_{p}^{\times}$ and $d\in\mathbb{F}_{p}$
for which

$e_{1}^{\prime}=c_{1}e_{1}\text{ and }e_{2}^{\prime}=c_{2}e_{2}+de_{1}.$

With respect to an admissible basis $(e_{1},e_{2})$, the restriction of
$\bar{\rho}_{E,p}$ to the decomposition group at $p$ takes the form

$\bar{\rho}_{E,p|\operatorname{G}_{p}}=\left({\begin{array}[]{cc}\alpha\bar{\chi}&\ast\\\
0&\alpha^{-1}\\\ \end{array}}\right),$

where $\alpha\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{p}\rightarrow\mathbb{F}_{p}^{\times}$ is an
unramified character and $\bar{\chi}$ is the mod-$p$ cyclotomic character.
There are 3 possibilities for the representation $\bar{\rho}_{E,p}$. These are
described below and all matrices are written with respect to an admissible
basis $(e_{1},e_{2})$.

Type 1:

The representation $\bar{\rho}_{E,p}$ is upper triangular of the form
$\left({\begin{array}[]{cc}\varphi_{1}&\ast\\\ 0&\varphi_{2}\\\
\end{array}}\right)$, where $\varphi_{1}$ is odd and $\varphi_{2}$ is even.

Type 2:

The representation $\bar{\rho}_{E,p}$ is upper triangular of the form
$\left({\begin{array}[]{cc}\varphi_{1}&\ast\\\ 0&\varphi_{2}\\\
\end{array}}\right)$, where $\varphi_{1}$ is even and $\varphi_{2}$ is odd.

Type 3:

The representation $\bar{\rho}_{E,p}$ is indecompasable and lower triangular
of the form $\left({\begin{array}[]{cc}\varphi_{1}&0\\\ \ast&\varphi_{2}\\\
\end{array}}\right)$. In this context, to be indecomposable means that there
is no admissible basis with respect to which $\bar{\rho}_{E,p}$ is a direct
sum of characters.

We note that $\varphi_{1|\operatorname{G}_{p}}=\alpha\chi$ and
$\varphi_{2|\operatorname{G}_{p}}\simeq\alpha^{-1}$. Note that the Conjecture
5.3 specializes to the Conjecture 1.2.

The vanishing of the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$ shall be detected by
the structure of the residual representation. We shall first recall a result
of Schneider on isogenies between elliptic curves.

Given a finite Galois stable submodule $\alpha$ of $E[p^{\infty}]$, set
$\alpha^{+}\mathrel{\mathop{\mathchar 58\relax}}=C\cap\alpha$; set

$\delta(\alpha)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{ord}_{p}|\alpha^{+}|-\mathrm{ord}_{p}|H^{0}(\mathbb{R},\alpha)|.$

We note that since $p$ is assumed to be odd, the above definition coincides
with that of [Dri02, Definition 2.1]. In particular, it is easy to see that
the quantities $\epsilon_{v}=0$ from _loc. cit._ are trivial.

######  Theorem 6.1 (Schneider).

Let $E$ and $E^{\prime}$ be elliptic curves with good ordinary reduction at
$p$ and $\phi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow E^{\prime}$ an
isogeny with kernel $\alpha$. then the difference between $\mu$-invariants is
given by

$\mu_{p}(E)-\mu_{p}(E^{\prime})=\delta(\alpha).$

In particular, it follows that $\mu_{p}(E)\geq\delta(\alpha)$.

###### Proof.

We refer to [Sch87] or [Dri02, Theorem 2.2] for the proof of the above result.
∎

We recall a result of Coates and Sujatha which will be of key importance in
the proof of Greenberg’s conjecture in the residually reducible case.

######  Theorem 6.2 (Coates and Sujatha).

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\rho_{\bar{A}}$ is a
reducible Galois representation. Then, the $\mu$-invariant of the fine Selmer
group $\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ is equal to
$0$.

###### Proof.

We write $\rho_{\bar{A}}=\left({\begin{array}[]{cc}\varphi_{1}&0\\\
\ast&\varphi_{2}\\\ \end{array}}\right)$ with respect to some basis of
$\bar{A}$. Let $\mathcal{K}=\mathbb{Q}(\varphi_{1},\varphi_{2})$ be the
abelian extension of $\mathbb{Q}$ generated by $\varphi_{1}$ and
$\varphi_{2}$. Let $\mathcal{K}(E_{p^{\infty}})$ be the extension generated by
the $p$-primary torsion points of $E$. In other words,
$\mathcal{K}(E_{p^{\infty}})$ is the field extension of $\mathcal{K}$ which is
fixed by the kernel of $\rho_{M}$. Let $I$ be the subgroup of
$\operatorname{GL}_{2}(\mathbb{Z}_{p})$ consisting of all matrices $A$ for
which the mod-$p$ reduction is a unipotent lower triangular matrix
$\left({\begin{array}[]{cc}1&0\\\ \ast&1\\\ \end{array}}\right)$. Via
$\rho_{M}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$,
the Galois group $\operatorname{Gal}(\mathcal{K}(E_{p^{\infty}})/\mathcal{K})$
is identified with a subgroup of $I$. Since $I$ is a pro-$p$ group, so is the
Galois group $\operatorname{Gal}(\mathcal{K}(E_{p^{\infty}})/\mathcal{K})$.
Recall that by the celebrated result of Ferrero and Washington [FW79], the
classical Iwasawa $\mu$-invariant $\mu_{p}(\mathcal{K})$ vanishes, since
$\mathcal{K}$ is an abelian extension of $\mathbb{Q}$. It then follows from
Theorem 4.4 that $\operatorname{Sel}_{p^{\infty}}^{0}(\mathcal{K}_{\infty},E)$
is cofinitely generated as a $\mathbb{Z}_{p}$-module. The kernel of the
restriction map

$\operatorname{Sel}_{p^{\infty}}^{0}(\mathcal{\mathbb{Q}}_{\infty},E)\rightarrow\operatorname{Sel}_{p^{\infty}}^{0}(\mathcal{K}_{\infty},E)$

is contained in $H^{1}(H,E(\mathcal{K}_{\infty})[p^{\infty}])$, where
$H=\operatorname{Gal}(\mathcal{K}_{\infty}/\mathbb{Q}_{\infty})$. Since $H$
has order prime to $p$, it follows that this cohomology group vanishes.
Therefore,
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathcal{\mathbb{Q}}_{\infty},E)$ is
cofinitely generated as a $\mathbb{Z}_{p}$-module. In particular, the
$\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ is $0$. ∎

######  Theorem 6.3.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at an
odd prime $p$ for which the following conditions hold.

  1. (1)

The residual representation is reducible.

  2. (2)

The Assumption 5.1 holds.

  3. (3)

The Conjecture 1.2 holds.

Then, we find that $\mu_{p}(E)=0$.

###### Proof.

The assumption 5.1 holds, and therefore, by part (2) of Theorem 5.4, the
$\mu$-invariant of $\operatorname{Sel}_{p^{\infty}}(\mathbb{Q}_{\infty},E)$
vanishes if and only if the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ vanishes. Since
the residual representation is reducible, it follows from Theorem 6.2 that the
$\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}^{0}(\mathbb{Q}_{\infty},E)$ is $0$, and the
result follows from this. ∎

Assuming that Conjecture 1.2 holds for the isogeny class of $E$, we have a
complete description for the $\mu=0$ condition based purely on the residual
representation $\bar{\rho}_{E,p}$.

######  Theorem 6.4 ($\mu=0$ condition).

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime at which $E$
has good ordinary reduction. Assume that the Conjecture 1.2 holds for all
elliptic curves that are defined over $\mathbb{Q}$ and are
$\mathbb{Q}$-isogenous to $E$. Then, $\mu_{p}(E)=0$ if and only if
$\bar{\rho}_{E,p}$ is of type 2 or 3. Equivalently, $\mu_{p}(E)>0$ if and only
if it is of type 1.

###### Proof.

First, we consider the case when $\bar{\rho}_{E,p}$ is of type $1$. Note that
since the representation $\bar{\rho}_{E,p}$ is upper triangular,
$\alpha\mathrel{\mathop{\mathchar 58\relax}}=\bar{C}$ is a
$\operatorname{G}_{\mathbb{Q}}$-submodule of $E[p]$. Since $\varphi_{1}$ is
odd, $H^{0}(\mathbb{R},\alpha)=0$. Then, we find that $\delta(\alpha)=1$, and
it follows from Theorem 6.1 (or [Dri02, Theorem 2.1]) that
$\mu_{p}(E)\geq\delta(\alpha)\geq 1$.

Next, we consider type $2$ representations. Greenberg and Vatsal [GV00] showed
that if $E[p]$ contains a $1$-dimensional
$\operatorname{G}_{\mathbb{Q}}$-stable subspace which is ramified at $p$ and
even or unramified at $p$ and odd, then, $\mu_{p}(E)=0$. In this case,
$\bar{C}$ is a subspace which is $\operatorname{G}_{\mathbb{Q}}$-stable,
ramified at $p$ and even, and therefore, their result applies to show that
$\mu_{p}(E)=0$.

Finally, consider the type $3$ representations. Note that $\varphi_{2}$ is
unramified at $p$. Thus, if $\varphi_{2}$ is odd, then the aforementioned
result of Greenberg and Vatsal applies to show that $\mu_{p}(E)=0$. For type
$3$ representations for which $\varphi_{2}$ is even however, it was expected
that $\mu_{p}(E)=0$ should hold, however, not proved. We complete the proof by
noting that Proposition 5.2 implies that when $\bar{\rho}_{E,p}$ is of type
$3$, the Assumption 5.1 holds. Then it follows from Theorem 6.3 that
$\mu_{p}(E)=0$. ∎

We now give the proof of our main theorem.

###### Proof of Theorem 1.

If $\mu\mathrel{\mathop{\mathchar 58\relax}}=\mu_{p}(E)=0$, then the result is
vacuously true, setting $E^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=E$.
Therefore, assume without loss of generality that $\mu>0$. Thus, it follows
from Theorem 6.4 that $\bar{\rho}_{E,p}$ is of type 1, i.e., $\bar{C}$ is an
odd $\operatorname{G}_{\mathbb{Q}}$-submodule of $E[p]$. In this case, setting
$\alpha\mathrel{\mathop{\mathchar 58\relax}}=\bar{C}$, we observe that
$\delta(\alpha)=1$ (see the first paragraph in the proof of Theorem 6.4). We
set $E_{1}\mathrel{\mathop{\mathchar 58\relax}}=E/\alpha$. It follows from
Theorem 6.1 that

$\mu_{p}(E_{1})=\mu_{p}(E)-\delta(\alpha)=\mu-1.$

In this way, we obtain a sequence of elliptic curves
$E=E_{0},E_{1},E_{2},\dots,E_{\mu}$ over $\mathbb{Q}$ along with isogenies
$\phi_{i}\mathrel{\mathop{\mathchar 58\relax}}E_{i-1}\rightarrow E_{i}$ such
that $\mu_{p}(E_{i})=\mu-i$. Set $E^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=E_{\mu}$ and consider the composite isogeny

$E\xrightarrow{\phi_{1}}E_{1}\xrightarrow{\phi_{2}}E_{2}\xrightarrow{\phi_{3}}\dots\rightarrow
E^{\prime}.$

We find that $\mu(E^{\prime})=\mu-\mu=0$. This completes the proof. ∎

## References

  * [CS05] John Coates and Ramdorai Sujatha.  Fine Selmer groups of elliptic curves over $p$-adic Lie extensions.  Mathematische Annalen, 331:809–839, 2005.
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