# Anticyclotomic $\mu$-invariants of residually reducible Galois
Representations

Debanjana Kundu  Department of Mathematics
University of British Columbia
Vancouver BC, V6T 1Z2, Canada<EMAIL_ADDRESS>and  Anwesh Ray  Department
of Mathematics
University of British Columbia
Vancouver BC, V6T 1Z2, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be
an odd prime such that the residual representation $E[p]$ is reducible. The
$\mu$-invariant of the fine Selmer group of $E$ over the anticyclotomic
$\mathbb{Z}_{p}$-extension of $K$ is studied. We do not impose the Heegner
hypothesis on $E$, thus allowing certain primes of bad reduction to decompose
infinitely in the anticyclotomic $\mathbb{Z}_{p}$-extension. It is shown that
the fine $\mu$-invariant vanishes if certain explicit conditions are
satisfied. Further, a partial converse is proven.

##  1\. Introduction

Iwasawa theory is the study of objects of arithmetic interest over infinite
towers of number fields. K. Iwasawa conjectured that over the cyclotomic
$\mathbb{Z}_{p}$-extension of a number field $F$, the $p$-primary part of the
Hilbert class group is a finitely generated $\mathbb{Z}_{p}$-module (see
[17]). This is known as Iwasawa’s $\mu=0$ conjecture for the number field $F$.
In [9], B. Ferrero and L. Washington proved this conjecture for all finite
abelian extensions $F$ over of $\mathbb{Q}$. The Iwasawa theory of abelian
varieties (in particular, elliptic curves) was initiated by B. Mazur in [25].
The main object of study is the $p$-primary Selmer group of an elliptic curve
$E$ defined over a number field $F$, with good ordinary reduction at $p$. The
Selmer group over a $\mathbb{Z}_{p}$-extension of $F$ is a cofinitely
generated module over the Iwasawa algebra, $\Lambda$. In [11], R. Greenberg
analyzed the algebraic structure of these Selmer groups. For elliptic curves
$E$ over $\mathbb{Q}$, it is conjectured that if the residual representation
on $E[p]$ is irreducible, then $\mu$-invariant of the $p$-primary Selmer group
over the cyclotomic $\mathbb{Z}_{p}$-extension, denoted
$\mu(E/\mathbb{Q}^{\operatorname{cyc}})$, vanishes (see [12, Conjecture
1.11]).

The fine Selmer group is a subgroup of the classical Selmer group obtained by
imposing vanishing conditions at primes above $p$. It had first been studied
by K. Rubin [32] and B. Perrin-Riou [28, 29], under various guises. The
analysis of the fine Selmer groups is an essential part of K. Kato’s seminal
work on the Iwasawa Main Conjecture for elliptic curves and modular forms (see
[19]). In recent years, their study has gained considerable momentum (see for
instance [5, 35, 34, 18, 1]). J. Coates and R. Sujatha conjectured that the
the fine Selmer group of $E$ over $F^{\operatorname{cyc}}$ is
$\Lambda$-cotorsion with associated $\mu$-invariant,
$\mu^{\operatorname{fine}}(E/F^{\operatorname{cyc}})$, equal to zero [5,
Conjecture A]. The formulation of this conjecture makes no hypothesis on the
reduction type at primes above $p$ or the residual representation on $E[p]$.
However, such a statement need not be true for the (classical) Selmer group
even for elliptic curves over $\mathbb{Q}$ with good ordinary reduction at
$p$. In fact, Mazur provided examples of elliptic curves $E_{/\mathbb{Q}}$ for
which the residual representation $E[p]$ is reducible and
$\mu(E/\mathbb{Q}^{\operatorname{cyc}})$ is non-zero. There is a systematic
approach towards finding examples of elliptic curves with positive
$\mu(E/F^{\operatorname{cyc}})$ (see [6, 7]).

On the other hand, the conjecture by Coates and Sujatha predicts a close
relationship in the growth of ideal class groups and fine Selmer groups in
cyclotomic $\mathbb{Z}_{p}$-extensions. Some evidence towards this is provided
in [5, Theorem 3.4]. In particular, the result of Ferrero and Washington
implies that for an elliptic curve over an abelian number field,
$\mu^{\operatorname{fine}}(E/F^{\operatorname{cyc}})$ is zero when $E[p]$ is
reducible (see [5, Corollary 3.6]). Subsequently, the relation in the growth
of ideal class groups and fine Selmer groups has been studied in more general
settings (see [23, 20, 21, 22]). In this paper, we study the relationship in
the growth of fine Selmer groups and class groups in anticyclotomic
$\mathbb{Z}_{p}$-extensions of imaginary quadratic fields.

Several authors have studied classical Selmer groups of elliptic curves (more
generally, abelian varieties or modular forms) in anticyclotomic
$\mathbb{Z}_{p}$-extensions of imaginary quadratic fields (see for example
[33, 2, 30]). In line with the conjecture of Coates and Sujatha, for an
elliptic curve defined over the imaginary quadratic field $K$, its fine Selmer
group over the anticyclotomic $\mathbb{Z}_{p}$-extension is expected to be
$\Lambda$-cotorsion with
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$ (see [24, Conjecture
B]). Most results in literature focus on the case when the residual
representation is irreducible. In contrast, this paper primarily studies the
case when the residual representation is reducible. The first main result (see
Theorem 3.2) shows that if certain conditions are satisfied then the
$\mu$-invariant of the $p$-primary fine Selmer group may be detected by a
certain analogous fine Selmer group associated to the residual representation,
$E[p]$. Further, if $E[p]$ is reducible then there are characters

$\varphi_{1},\varphi_{2}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muK}/K)\rightarrow\operatorname{GL}_{1}(\mathbb{F}_{p})$

which fit into a short exact sequence

$0\rightarrow\mathbb{F}_{p}(\varphi_{1})\rightarrow
E[p]\rightarrow\mathbb{F}_{p}(\varphi_{2})\rightarrow 0.$

Note that $\varphi_{2}=\overline{\chi}\varphi_{1}^{-1}$, where
$\overline{\chi}$ denotes the mod-$p$ cyclotomic character. This short exact
sequence makes it possible to analyze the structure of the fine Selmer group
associated to $E[p]$. Next, we prove a partial converse to the above theorem
(see Theorem 3.4). A key difference between the cyclotomic and anticyclotomic
$\mathbb{Z}_{p}$-extensions is the following: in the cyclotomic extension, all
primes are finitely decomposed, whereas in the anticyclotomic extension, there
are infinitely many primes which split completely. For the aforementioned
result of Coates and Sujatha in the cyclotomic extension case, this fact is
crucially used (in the proofs of [5, Lemmas 3.2, 3.7, Theorem 3.4]). The
elliptic curve $E$ is said to satisfy the Heegner hypothesis if all primes
$v\nmid p$ of $K$ at which it has bad reduction are split completely in $K$.
By Fact 2.3, such a a prime number is finitely decomposed in the
anticyclotomic $\mathbb{Z}_{p}$-extension, $K^{\operatorname{ac}}$. We work in
a more general setting, where primes of bad reduction of $E$ may be infinitely
decomposed in $K^{\operatorname{ac}}$, and thereby rely on a different
strategy (from that of Coates and Sujatha) to prove our results.

Two elliptic curves, $E_{1}$ and $E_{2}$, are said to be $p$-congruent if the
$p$-torsion subgroups $E_{1}[p]$ and $E_{2}[p]$ are isomorphic as Galois
modules. In [14], R. Greenberg and V. Vatsal studied the relation between
cyclotomic invariants of elliptic curves over $\mathbb{Q}$ which are
$p$-congruent. In [13, Proposition 4.1.6], Greenberg reformulated the
conjecture of Coates and Sujatha in terms of the vanishing of a second
cohomology group with coefficients in $E[p]$. From this, it is immediate that
if elliptic curves $E_{1}$, $E_{2}$ are $p$-congruent then
$\mu^{\operatorname{fine}}(E_{1}/\mathbb{Q}^{\operatorname{cyc}})=0$ if and
only if $\mu^{\operatorname{fine}}(E_{2}/\mathbb{Q}^{\operatorname{cyc}})=0$.
In the same spirit, we prove a result for fine $\mu$-invariants over
$K^{\operatorname{ac}}$ (see Theorem 5.2), without imposing any hypothesis on
the reducibility of the residual representations.

The authors expect that the methods in this paper should generalize to
residually reducible Galois representations arising from abelian varieties.
The results shall however be more technical and the arguments more cumbersome
in the higher dimensional setting. The authors choose a less general framework
in which the inherent simplicity of the underlying ideas come across easily.

The paper is organized into six sections. Preliminary notions are discussed in
§2. In §3 and §4, we state and prove the main results: we establish a
criterion for the vanishing of the $\mu$-invariant of the fine Selmer group in
the anticyclotomic $\mathbb{Z}_{p}$-extension. We also prove a partial
converse to the above theorem. In §5, we compare the anticyclotomic
$\mu$-invariant for two elliptic curves which are $p$-congruent. Finally, in
§6 we list examples illustrating the results in this article.

##  2\. Preliminaries

Throughout, let $p$ be an odd prime and $K$ be an imaginary quadratic field.
Let $S_{p}$ be the set of primes of $K$ above $p$. Fix an elliptic curve $E$
over $K$. Set $\operatorname{G}_{K}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}({}\mkern 3.0mu\overline{\mkern-3.0muK}/K)$ to be
the absolute Galois group of $K$. Denote by $E[p^{n}]$ (resp. $E[p^{\infty}]$)
the $p^{n}$-torsion (resp. $p$-primary torsion) subgroup of $E({}\mkern
3.0mu\overline{\mkern-3.0muK})$. The Tate module $\operatorname{T}_{p}(E)$ is
the inverse limit with respect to multiplication by $p$ maps,

$\operatorname{T}_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}E[p^{n}].$

The $\mathbb{Z}_{p}$-module $\operatorname{T}_{p}(E)$ is free of rank $2$ and
the group of $\mathbb{Z}_{p}$-linear automorphisms of
$\operatorname{T}_{p}(E)$ is identified with
$\operatorname{GL}_{2}(\mathbb{Z}_{p})$. The action of $\operatorname{G}_{K}$
on $\operatorname{T}_{p}(E)$ induces a continuous Galois representation,
$\rho_{E}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$.
Set ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ to denote the mod-$p$
reduction of $\rho_{E}$, as depicted

$\operatorname{G}_{K}$$\operatorname{GL}_{2}(\mathbb{F}_{p})$.$\operatorname{GL}_{2}(\mathbb{Z}_{p})$${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$$\rho_{E}$

The residual representation ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is
induced by the action of $\operatorname{G}_{K}$ on $E[p]$. Let $\mathcal{N}$
be the conductor of $E$ and $S$ the set of primes that divide $\mathcal{N}p$.
It is known that $\rho_{E}$ is unramified at all primes $v\notin S$. Let
$K_{S}$ denote the maximal algebraic extension of $K$ in which all primes
$v\notin S$ are unramified and set
$\operatorname{G}_{K,S}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K_{S}/K)$. Throughout this paper, we primarily
focus on the case when the Galois representation $\rho_{E}$ is residually
reducible. However, some of the results will apply to the residually
irreducible case as well. We shall be careful to make precise when the
following hypothesis is required.

######  Hypothesis 2.1.

Assume that the residual representation ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible, i.e. $E[p]$ contains a
proper non-zero $\operatorname{G}_{K}$-stable submodule.

This is indeed the case when $E(K)[p]\neq 0$. Though the condition is in fact
far more general. In this setting, there are characters

$\varphi_{1},\ \varphi_{2}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{1}(\mathbb{F}_{p})$

and a $1$-cocycle

$\beta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\mathbb{F}_{p}(\varphi_{1}\varphi_{2}^{-1})$

such that

${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}\simeq\left({\begin{array}[]{cc}\varphi_{1}&\varphi_{2}\beta\\\
&\varphi_{2}\\\ \end{array}}\right).$

The residual representation ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is
said to be indecomposable if the cohomology class $[\beta]\in
H^{1}(\operatorname{G}_{K,S},\mathbb{F}_{p}(\varphi_{1}\varphi_{2}^{-1}))$ is
non-zero, and split otherwise. If ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is split, then ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}\simeq\left({\begin{array}[]{cc}\varphi_{1}&\\\
&\varphi_{2}\\\ \end{array}}\right)$, and the characters $\varphi_{1}$ and
$\varphi_{2}$ may be interchanged. If ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is indecomposable, then $\varphi_{1}$ is
the unique character such that $E[p]$ contains a Galois submodule isomorphic
to $\mathbb{F}_{p}(\varphi_{1})$.

######  Remark 2.2.

If $E_{/\mathbb{Q}}$ is an elliptic curve such that $E[p]$ is reducible as a
$\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0mu\mathbb{Q}}/\mathbb{Q})$-module then $E$ has either
good ordinary reduction or bad reduction at $p$. By a result of Fontaine, if
$E_{/\mathbb{Q}}$ is an elliptic curve with good supersingular reduction at
$p$, then ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E\restriction\operatorname{G}_{\mathbb{Q}_{p}}}$
is irreducible.

Suppose $\mathcal{K}$ is a $\mathbb{Z}_{p}$-extension of $K$ which is Galois
over $\mathbb{Q}$. Then, there are exactly two cases to consider: either
$\mathcal{K}$ is the cyclotomic $\mathbb{Z}_{p}$-extension or the
anticyclotomic $\mathbb{Z}_{p}$-extension. Note that
$\Gamma_{\mathcal{K}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathcal{K}/K)$ is an index two normal subgroup
of $\operatorname{Gal}(\mathcal{K}/\mathbb{Q})$. The group
$\operatorname{Gal}(K/\mathbb{Q})$ acts on $\Gamma_{\mathcal{K}}$, as is
explained. Let $\tau\in\operatorname{Gal}(K/\mathbb{Q})$ and
$x\in\Gamma_{\mathcal{K}}$; choose a lift
$\tilde{\tau}\in\operatorname{Gal}(\mathcal{K}/\mathbb{Q})$ of $\tau$ and set
$\tau\cdot x\mathrel{\mathop{\mathchar
58\relax}}=\tilde{\tau}x\tilde{\tau}^{-1}$. Since $\Gamma_{\mathcal{K}}$ is
abelian, $\tau\cdot x$ does not depend on the choice of the lift
$\tilde{\tau}$. If the action of $\operatorname{Gal}(K/\mathbb{Q})$ on
$\Gamma_{\mathcal{K}}$ is via the trivial (resp. non-trivial) character, then
$\mathcal{K}$ is the cyclotomic (resp. anticyclotomic) extension of $K$. Thus,
the cyclotomic extension is pro-cyclic and the anticyclotomic extension is
pro-dihedral. We only consider the anticyclotomic extension in this article,
and denote it by $K^{\operatorname{ac}}$. Set $\Gamma$ to denote
$\operatorname{Gal}(K^{\operatorname{ac}}/K)$. For $n\geq 0$, the $n$-th layer
is the unique number field $K_{n}$ such that $K\subseteq K_{n}\subset
K^{\operatorname{ac}}$ and $[K_{n}\mathrel{\mathop{\mathchar
58\relax}}K]=p^{n}$. Note that $K_{n}$ is Galois over $\mathbb{Q}$ and its
Galois group $\operatorname{Gal}(K_{n}/\mathbb{Q})$ is (isomorphic to) the
dihedral group of order $2p^{n}$.

For a set of primes $S^{\prime}$, set $S^{\prime}(K^{\operatorname{ac}})$
(resp. $S^{\prime}(K_{n})$) to denote the primes of $K^{\operatorname{ac}}$
(resp. $K_{n}$) which lie above some prime $v\in S^{\prime}$. For instance,
$v(K^{\operatorname{ac}})$ (resp. $v(K_{n})$) will denote the primes $\eta|v$
of $K^{\operatorname{ac}}$ (resp. of $K_{n}$). The set of primes above a given
prime of $K$ in the cyclotomic $\mathbb{Z}_{p}$-extension
$K^{\operatorname{cyc}}$ is finite. This is not the case for
$K^{\operatorname{ac}}$; the following characterization is well known.

######  Fact 2.3.

Let $v$ be a prime of $K$ and $l$ the prime number such that $v|l$. The set of
primes $v(K^{\operatorname{ac}})$ is finite if and only if $l=p$ or $l$ splits
in $K$.

Indeed, there is a large enough value of $n$ such that all primes of $K_{n}$
above $p$ are totally ramified in $K^{\operatorname{ac}}$, see the last three
lines of [3, p. 2131]. A prime $l\neq p$ which does not split in $K$ must
split completely in $K^{\operatorname{ac}}$, see the first paragraph of p.
2132 of loc. cit. Prime numbers $l$ which split in $K$ must be finitely
decomposed in $K^{\operatorname{ac}}$, see for example, Corollary 1 of loc.
cit.

Recall that $\mathcal{N}$ is the conductor of $E$. Denote by ${}\mkern
3.0mu\overline{\mkern-3.0mu\mathcal{N}}$ the Artin conductor of ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$. Note that ${}\mkern
3.0mu\overline{\mkern-3.0mu\mathcal{N}}$ divides $\mathcal{N}$, denote by
$(\mathcal{N}/{}\mkern 3.0mu\overline{\mkern-3.0mu\mathcal{N}})$ the quotient.
We make an assumption on a certain subset of primes in $S$ which is described
below.

######  Definition 2.4.

Denote by $\Sigma\subset S$ the set of primes $v\nmid p$ at which all of the
following conditions are satisfied.

  1. (1)

$v|(\mathcal{N}/{}\mkern 3.0mu\overline{\mkern-3.0mu\mathcal{N}})$.

  2. (2)

If $p\geq 5$ and $\mu_{p}\subset K_{v}$, then $E$ has split multiplicative
reduction at $v$.

  3. (3)

If $p=3$ and $\mu_{3}\subset K_{v}$, then $E$ has split multiplicative
reduction or additive reduction at $v$.

######  Hypothesis 2.5.

Let $E$ be an elliptic curve over an imaginary quadratic field $K$. Let
$v\in\Sigma$ and $l$ be the prime number such that $v|l$. Then $l$ is split in
$K$.

######  Definition 2.6.

Suppose that ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible and
indecomposable. Let $\Sigma(\varphi_{2})$ be the set of primes $v\in
S\setminus S_{p}$ such that
$\varphi_{2\restriction\operatorname{G}_{\mathbb{Q}_{v}}}=1$.

######  Hypothesis 2.7.

Suppose that ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible and
indecomposable. Let $v\in\Sigma(\varphi_{2})$ and $l$ be the prime number in
such that $v|l$. Then $l$ is split in $K$.

######  Remark 2.8.

The above hypotheses are all weaker than the Heegner hypothesis. Some of our
results will require the above hypotheses; they will be assumed only when
explicitly stated. For instance, Theorem 3.2 will require Hypothesis 2.5 (and
Hypothesis 2.7 when ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is
indecomposable). On the other hand, Theorem 3.4 does not require these
hypotheses.

We now introduce the fine Selmer group. At each prime $v\in S$, set

$\mathcal{H}_{v}(K^{\operatorname{ac}},E[p^{\infty}])\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta\in
v(K^{\operatorname{ac}})}H^{1}(K^{\operatorname{ac}}_{\eta},E[p^{\infty}]).$

######  Definition 2.9.

The fine Selmer group associated to $E[p^{\infty}]$ is defined as follows

$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(K_{S}/K^{\operatorname{ac}},E[p^{\infty}])\longrightarrow\bigoplus_{v\in
S}\mathcal{H}_{v}(K^{\operatorname{ac}},E[p^{\infty}])\right\\}.$

Recall that $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K^{\operatorname{ac}}/K)\simeq\mathbb{Z}_{p}$.
The Iwasawa algebra $\Lambda$ is the completed group algebra
$\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. After fixing
a topological generator $\gamma$ of $\Gamma$, there is an isomorphism of rings
$\Lambda\cong\mathbb{Z}_{p}\llbracket X\rrbracket$, by sending $\gamma-1$ to
the formal variable $X$. The fine Selmer group is a cofinitely generated
$\Lambda$-module.

The Pontryagin dual $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})^{\vee}$
is, up to pseudo-isomorphism, a finite direct sum of cyclic $\Lambda$-modules:

$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})^{\vee}\sim\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{s}\Lambda/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(f_{j}(T))\right).$

Here, $\mu_{i}>0$ and $f_{j}(T)$ is a distinguished polynomial (i.e. a monic
polynomial with non-leading coefficients divisible by $p$). Here, $r$ is the
$\Lambda$-corank of the fine Selmer group. The $\mu$-invariant of the fine
Selmer group is defined as follows,

$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}0&\textrm{ if }s=0\\\ \sum_{i=1}^{s}\mu_{i}&\textrm{
if }s>0.\end{cases}$

The number of summands $s$ is a well-defined invariant and we refer to it as
the $\mu$-multiplicity. We now introduce the fine Selmer group associated to
the residual representation on $E[p]$. At each prime $v$ of $K$, set

$\mathcal{H}_{v}(K^{\operatorname{ac}},E[p])\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\eta\in
v(K^{\operatorname{ac}})}H^{1}(K^{\operatorname{ac}}_{\eta},E[p]).$

######  Definition 2.10.

Let $T$ be a finite set of primes containing $S$. The fine Selmer group
associated to $E[p]$ and the set of primes $T$ is defined as follows

$\mathcal{R}^{T}(E[p]/K^{\operatorname{ac}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(K_{T}/K^{\operatorname{ac}},E[p])\longrightarrow\bigoplus_{v\in
T}\mathcal{H}_{v}(K^{\operatorname{ac}},E[p])\right\\}.$

Set
$\mathcal{R}(E[p]/K^{\operatorname{ac}})=\mathcal{R}^{S}(E[p]/K^{\operatorname{ac}})$;
this is the mod-$p$ fine Selmer group.

Set $\Omega$ to denote the mod-$p$ Iwasawa algebra,
$\Omega\mathrel{\mathop{\mathchar
58\relax}}=\Lambda/(p)\simeq\mathbb{F}_{p}\llbracket X\rrbracket$. Note that
both $\mathcal{R}(E[p]/K^{\operatorname{ac}})$ and
$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]$ are $\Omega$-modules.
Since $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is cofinitely
generated over $\Lambda$, it follows that
$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]$ is cofinitely generated
over $\Omega$. The following is an easy consequence of the structure theory of
$\Lambda$-modules.

######  Lemma 2.11.

The $\Omega$-corank of $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]$
is equal to $r+s$, where $r$ and $s$ are defined above. In particular,

$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0\Leftrightarrow\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]\text{
is finite.}$

###### Proof.

Note that
$\left(\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]\right)^{\vee}$ is
isomorphic to $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})^{\vee}/p$.
Let $\Phi$ be a pseudo-isomorphism of $\Lambda$-modules, i.e. a homomorphism,

$\Phi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})^{\vee}\rightarrow\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{s}\Lambda/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(f_{j}(T))\right)$

with finite kernel and cokernel. The mod-$p$ reduction $\overline{\Phi}$ is
the following map

${}\mkern 3.0mu\overline{\mkern-3.0mu\Phi}\mathrel{\mathop{\mathchar
58\relax}}\left(\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]\right)^{\vee}\rightarrow\Omega^{r+s}\oplus\left(\bigoplus_{j=1}^{t}\Omega/({}\mkern
3.0mu\overline{\mkern-3.0muf}_{j}(T))\right),$

where ${}\mkern 3.0mu\overline{\mkern-3.0muf}_{j}(T)$ is the mod-$p$ reduction
of $f_{j}(T)$. Since $f_{j}(T)$ is a distinguished polynomial, ${}\mkern
3.0mu\overline{\mkern-3.0muf}_{j}(T)=T^{\operatorname{deg}f_{j}}$ and
$\Omega/({}\mkern 3.0mu\overline{\mkern-3.0muf}_{j}(T))$ is finite. The result
follows. ∎

##  3\. A Criterion for the Vanishing of the fine $\mu$-invariant

In this section, we establish a criterion for the vanishing of the fine
$\mu$-invariant in the anticyclotomic $\mathbb{Z}_{p}$-extension. Before
stating the result, let us introduce some notation. We shall not assume that
${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible in this section,
unless when explicitly stated. However, when it is reducible, recall that the
characters $\varphi_{1}$ and $\varphi_{2}$ are such that $E[p]$ fits into a
short exact sequence of Galois-modules,

$0\rightarrow\mathbb{F}_{p}(\varphi_{1})\rightarrow
E[p]\rightarrow\mathbb{F}_{p}(\varphi_{2})\rightarrow 0.$

Let $K(\varphi_{i})$ be an extension of $K$ fixed by $\ker\varphi_{i}$. For an
algebraic extension $\mathcal{F}$ of $K$, let $\mathcal{F}(\varphi_{i})$ be
the composite $\mathcal{F}\cdot K(\varphi_{i})$. Let $\mathcal{L}_{n}^{(i)}$
be the maximal abelian unramified $p$-extension of $K_{n}(\varphi_{i})$ split
at primes above $S(K_{n}(\varphi_{i}))$ (called the $p$-Hilbert $S$-class
field extension). Class field theory prescribes a natural isomorphism

$\operatorname{Gal}(\mathcal{L}_{n}^{(i)}/K_{n}(\varphi_{i}))\simeq\operatorname{Cl}_{S}(K_{n}(\varphi_{i}))[p^{\infty}],$

where $\operatorname{Cl}_{S}(K_{n}(\varphi_{i}))[p^{\infty}]$ is the
$p$-primary part of the $S$-class group of $K_{n}(\varphi_{i})$.

Let $\rm{M}$ be any $\mathbb{Z}_{p}$-module on which
$\operatorname{Gal}(K(\varphi_{i})/K)$ acts by $\mathbb{Z}_{p}$-linear
automorphisms. Let $\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(K(\varphi_{i})/K)\rightarrow\mathbb{F}_{p}^{\times}$
be a character, and consider its Teichmüller lift,
$\tilde{\psi}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(K(\varphi_{i})/K)\rightarrow\mathbb{Z}_{p}^{\times}$.
Set

$\textrm{M}_{\psi}\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\textrm{M}|g\cdot x=\tilde{\psi}(g)x\\}.$

There is a unique field extension $\mathcal{E}_{n}^{(i)}$ of
$K_{n}(\varphi_{i})$ which is Galois over $K$ with the additional property
that $\operatorname{Gal}(\mathcal{E}_{n}^{(i)}/K_{n}(\varphi_{i}))$ is
identified with
$\left(\operatorname{Cl}_{S}(K_{n}(\varphi_{i}))[p^{\infty}]\right)_{\varphi_{i}}$.
Set $\mathcal{L}^{(i)}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{n}\mathcal{L}^{(i)}_{n}$ and
$\mathcal{E}^{(i)}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{n}\mathcal{E}^{(i)}_{n}$. Further, setting
$\mathcal{X}^{(i)}=\operatorname{Gal}(\mathcal{L}^{(i)}/K^{\operatorname{ac}}(\varphi_{i}))$,
standard arguments show that $\mathcal{X}^{(i)}$ is a finitely generated
torsion $\Lambda$-module. Note that $\mathcal{X}^{(i)}$ decomposes into a
direct sum of $\Lambda$-submodules

$\mathcal{X}^{(i)}=\bigoplus_{\psi}\mathcal{X}^{(i)}_{\psi},$

where $\psi$ ranges over the characters
$\operatorname{Gal}(K(\varphi_{i})/K)\rightarrow\mathbb{F}_{p}^{\times}$.
Recall that the $\psi$-eigenspace $\mathcal{X}^{(i)}_{\psi}$ is defined as
follows

$\mathcal{X}^{(i)}_{\psi}\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathcal{X}^{(i)}\mid g\cdot x=\tilde{\psi}(g)x\\}.$

Let $Y_{i}$ be the $\varphi_{i}$-eigenspace $\mathcal{X}^{(i)}_{\varphi_{i}}$
and $\widetilde{Y}_{i}$ be the $\varphi_{i}$-component of the $p$-Hilbert
class field extension $\mathcal{M}^{(i)}$ of
$K^{\operatorname{ac}}(\varphi_{i})$. Denote their $\mu$-invariants (with
respect to the $\mathbb{Z}_{p}$-extension
$K^{\operatorname{ac}}(\varphi)/K(\varphi_{i})$) by $\mu(Y_{i})$ and
$\mu(\widetilde{Y_{i}})$, respectively. Note that $Y_{i}$ is the quotient of
$\widetilde{Y}_{i}$ where all the primes in
$S(K^{\operatorname{ac}}(\varphi_{i}))$ are split. We identify $Y_{i}$ with
the Galois group
$\operatorname{Gal}(\mathcal{E}^{(i)}/K^{\operatorname{ac}})$. The next result
follows from the work of H. Hida [16, Theorem I] (see also [10, Theorem 1.1])
and Rubin [31, Theorem 4.1].

######  Theorem 3.1.

Suppose that $p=v\overline{v}$ in $K$. Let
$\operatorname{G}_{v}=\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muK_{v}}/K_{v})$ and assume that
$\varphi_{1\restriction\operatorname{G}_{v}}\neq 1,\
\overline{\chi}_{\restriction\operatorname{G}_{v}}$ (or equivalently,
$\varphi_{2\restriction\operatorname{G}_{v}}\neq 1,\
\overline{\chi}_{\restriction\operatorname{G}_{v}}$). Then,

$\mu(\widetilde{Y}_{1})=\mu(\widetilde{Y}_{2})=\mu(Y_{1})=\mu(Y_{2})=0.$

###### Proof.

Since $\varphi_{1}\varphi_{2}=\overline{\chi}$, the condition on $\varphi_{1}$
is equivalent to that on $\varphi_{2}$. Let $\varphi_{i}$ be either of the
characters, $\varphi_{1}$ or $\varphi_{2}$. Let $\mathcal{Q}^{(i)}$ be the
maximal abelian pro-$p$ extension of $K^{\operatorname{ac}}(\varphi_{i})$
which is unramified away from $v(K^{\operatorname{ac}})$. Rubin identifies
$\mathcal{Q}^{(i)}$ with an anticyclotomic $p$-adic L-function, and Hida (also
T. Finis) proves that the $\mu$-invariant of such an anticyclotomic $p$-adic
L-function vanishes. Combining their results, we deduce that
$\mu(\mathcal{Q}^{(i)})=0$ for $i=1,2$. Since $\widetilde{Y_{i}}$ is a
quotient of $\mathcal{Q}^{(i)}$, it follows that $\mu(\widetilde{Y}_{i})=0$
for $i=1,2$. Hence, $\mu(Y_{i})=0$ for $i=1,2$ as well. ∎

The most explicit examples of reducible Galois representations arise from
elliptic curves with $p$-torsion points over the base field. In this setting,
the conditions of Theorem 3.1 are not satisfied since
$\\{\varphi_{1},\varphi_{2}\\}=\\{1,\overline{\chi}\\}$. We now state the main
theorems of this article.

######  Theorem 3.2.

Suppose the following conditions hold.

  1. (1)

${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible.

  2. (2)

Hypothesis 2.5 is satisfied.

  3. (3)

If ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is indecomposable, then
Hypothesis 2.7 is satisfied.

  4. (4)

For $i=1,2$, the $\mu$-invariant $\mu(Y_{i})=0$ (for instance, if the
conditions of Theorem 3.1 are satisfied).

Then, the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is $\Lambda$-cotorsion
with $\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$.

This result is illustrated by Example 6.1. For elliptic curves over
$\mathbb{Q}$ where hypotheses of Theorem 3.1 are satisfied, a similar result
is proven in [4].

######  Remark 3.3.

When all primes $v\in S\setminus S_{p}$ are split in $K$, the vanishing of the
$\mu$-invariant for $Y_{i}$ is equivalent to the vanishing of the
$\mu$-invariant of $\widetilde{Y}_{i}$ (see the proof of Proposition 4.3).
However, if a prime $v\in S\setminus S_{p}$ does not split in $K$, Fact 2.3
asserts that there are infinitely many primes above $v$. Hence, there are
infinitely many splitting conditions cutting out $Y_{i}$ as a quotient of
$\widetilde{Y}_{i}$. The condition $\mu(Y_{i})=0$ is therefore a more optimal
condition than requiring $\mu(\widetilde{Y}_{i})=0$.

The next result is a (partial) converse to Theorem 3.2.

######  Theorem 3.4.

Let $E$ be an elliptic curve defined over $K$. Assume that

  1. (1)

${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible.

  2. (2)

$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is a cotorsion
$\Lambda$-module with $\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$.

If ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is indecomposable, then
$\mu(Y_{1})=0$. If ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is split,
then $\mu(Y_{1})=\mu(Y_{2})=0$.

The next corollary takes note of a special case of interest.

######  Corollary 3.5.

Let $E$ be an elliptic curve defined over $K$ and $S$ be the set of primes
dividing $\mathcal{N}p$, where $\mathcal{N}$ is the conductor of $E$. Assume
that

  1. (1)

$E(K)[p]\neq 0$.

  2. (2)

$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is a cotorsion
$\Lambda$-module with $\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$.

  3. (3)

the Heegner hypothesis is satisfied: for $v\in S\setminus S_{p}$, let $l$ be
the prime number such that $v|l$, then $l$ splits in $K$.

Then the (classical) Iwasawa $\mu$-invariant,
$\mu(K^{\operatorname{ac}}/K)=0$.

###### Proof.

Note that $E(K)[p]\neq 0$ when the residual representation is of the form
${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}\simeq\left({\begin{array}[]{cc}1&\ast\\\
&\overline{\chi}\\\ \end{array}}\right)$. The result follows from Theorem 3.4
and Proposition 4.3. ∎

Example 6.2 illustrates Corollary 3.5. Before proving the above results, a
brief sketch of the method is provided. The Kummer sequence

(3.1) $0\rightarrow E[p]\rightarrow E[p^{\infty}]\xrightarrow{\times
p}E[p^{\infty}]\rightarrow 0$

induces a comparison map

(3.2) $\Psi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}(E[p]/K^{\operatorname{ac}})\rightarrow\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p].$

The main theorem is deduced from the following statements.

  1. (1)

The kernel of $\Psi$ is finite. If Hypothesis 2.5 is satisfied, then
$\operatorname{cok}\Psi$ is also finite. This step does not require any
assumption on ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$.

  2. (2)

Suppose that ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible.
Further, if the conditions of Theorem 3.2 are satisfied, then the mod-$p$ fine
Selmer group $\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite.

We now analyze the map $\Psi$ and deduce a criterion for the vanishing of the
fine $\mu$-invariant. To show that $\operatorname{cok}\Psi$ is finite when
Hypothesis 2.5 is satisfied, we prove the following result.

######  Lemma 3.6.

Let $v\in S\setminus\Sigma$ be a prime such that $v\nmid p$ and $\eta$ be a
prime of $K^{\operatorname{ac}}$ above $v$. Then, the group
$E(K^{\operatorname{ac}}_{\eta})[p^{\infty}]$ is $p$-divisible.

###### Proof.

Since $v\notin\Sigma$, (at least) one of the following conditions is
satisfied:

  1. (1)

$v\nmid(\mathcal{N}/{}\mkern 3.0mu\overline{\mkern-3.0mu\mathcal{N}})$,

  2. (2)

$p\geq 5$, $\mu_{p}\subset K_{v}$, and $E$ has non-split multiplicative
reduction or additive reduction at $v$,

  3. (3)

$p=3$, $\mu_{3}\subset K_{v}$, and $E$ has non-split multiplicative reduction
at $v$.

The claim follows from the proof of [8, Lemma 4.1.2] and [15, Proposition 5.1
(iii)]. ∎

For $v\in S$, let $h_{v}$ denote the natural map

$h_{v}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{v}(K^{\operatorname{ac}},E[p])\rightarrow\mathcal{H}_{v}(K^{\operatorname{ac}},E[p^{\infty}])[p]$

induced from $\eqref{kummersequence}$. For each prime $\eta|v$ of
$K^{\operatorname{ac}}$, set $h_{\eta}$ to denote the natural map

$h_{\eta}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(K^{\operatorname{ac}}_{\eta},E[p])\rightarrow
H^{1}(K^{\operatorname{ac}}_{\eta},E[p^{\infty}])[p],$

also induced from $\eqref{kummersequence}$.

######  Corollary 3.7.

If Hypothesis 2.5 is satisfied, then $\operatorname{ker}h_{v}$ is finite for
$v\in S$.

###### Proof.

The kernel of $h_{v}$ is the product of $\ker h_{\eta}$, as $\eta$ ranges over
$v(K^{\operatorname{ac}})$. By Kummer theory, $\ker h_{\eta}$ is isomorphic to
$H^{0}(K^{\operatorname{ac}}_{\eta},E[p^{\infty}])/p$; hence, it is finite.

First, consider the case when $v\in\Sigma\cup S_{p}$. Let $l$ be the prime
number for which $v|l$. By Hypothesis 2.5, if $v\in\Sigma$, then $l$ is split
in $K$. Therefore, by Fact 2.3, the set of primes $v(K^{\operatorname{ac}})$
is finite for $v\in\Sigma\cup S_{p}$. Since there are only finitely many
primes $\eta\in v(K^{\operatorname{ac}})$, $\operatorname{ker}h_{v}$ is finite
in this case.

Next, consider the case when $l\in S\setminus(\Sigma\cup S_{p})$. Lemma 3.6
asserts that the group $E(K^{\operatorname{ac}}_{\eta})[p^{\infty}]$ is
$p$-divisible for all $\eta\in v(K^{\operatorname{ac}})$. In this case,

$\operatorname{ker}h_{\eta}=H^{0}(K^{\operatorname{ac}}_{\eta},E[p^{\infty}])/p=0.$

Therefore, $h_{v}$ is injective. ∎

######  Proposition 3.8.

The kernel of $\Psi$ is finite. If Hypothesis 2.5 is satisfied, then
$\operatorname{cok}\Psi$ is also finite.

###### Proof.

The Kummer sequence $\eqref{kummersequence}$ induces the commutative diagram:

${0}$${\mathcal{R}(E[p]/K^{\operatorname{ac}})}$${H^{1}(K_{S}/K^{\operatorname{ac}},E[p])}$${\operatorname{im}({}\mkern
3.0mu\overline{\mkern-3.0mu\Phi}_{E})}$${0}$${0}$${\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})[p]}$${H^{1}(K_{S}/K^{\operatorname{ac}},E[p^{\infty}])[p]}$${\bigoplus_{v\in
S}\mathcal{H}_{v}(K^{\operatorname{ac}},E[p^{\infty}])[p].}$$\scriptstyle{\Psi}$$\scriptstyle{g}$$\scriptstyle{h}$

Here, ${}\mkern 3.0mu\overline{\mkern-3.0mu\Phi}_{E}$ is the natural map

${}\mkern 3.0mu\overline{\mkern-3.0mu\Phi}_{E}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(K_{S}/K^{\operatorname{ac}},E[p])\rightarrow\bigoplus_{v\in
S}\mathcal{H}_{v}(K^{\operatorname{ac}},E[p]).$

Clearly, the map $g$ is surjective. The snake lemma yields an exact sequence,

(3.3)
$0\rightarrow\operatorname{ker}\Psi\rightarrow\operatorname{ker}g\rightarrow\operatorname{ker}h\rightarrow\operatorname{cok}\Psi\rightarrow
0.$

Since $\ker g\simeq H^{0}(K^{\operatorname{ac}},E[p^{\infty}])/p$, it is
finite (with cardinality at most $p^{2}$). From $\eqref{ses}$, we deduce that
$\operatorname{ker}\Psi$ is finite. It follows from Corollary 3.7 that $\ker
h$ is finite. Therefore, $\operatorname{cok}\Psi$ is finite as well. ∎

######  Remark 3.9.

The proof of the above proposition shows that $\ker h$ must be finite if
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$ (even if Hypothesis 2.5
is not satisfied). This point is of considerable interest to the authors,
since there is no apparent reason to suggest why this should be true in
general.

######  Proposition 3.10.

Let $E_{/K}$ be an elliptic curve. The following assertions hold.

  1. (1)

If $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is $\Lambda$-cotorsion
with $\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$, then
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite.

  2. (2)

Assume that Hypothesis 2.5 is satisfied. Then, the fine Selmer group is
$\Lambda$-cotorsion with
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$ if and only if
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite.

###### Proof.

The assertion is a consequence of Lemma 2.11 and Proposition 3.8. ∎

Let $F=K(E[p])$ be the field extension of $K$ left fixed by the kernel of
${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$. Note that $F$ is a Galois
extension of $K$ with Galois group isomorphic to the image of ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$. In particular, if ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible, then it is a $2$-step
solvable extension of $K$. Set $F_{\infty}$ to denote the compositum
$K^{\operatorname{ac}}\cdot F$. In the remainder of this section, let
$\mathcal{X}_{E}^{S}$ be the $S$-Hilbert class field extension of
$F_{\infty}$. Note that there is a natural action of $\operatorname{Gal}(F/K)$
on $\mathcal{X}_{E}^{S}$. The $\mu$-invariant of $\mathcal{X}_{E}^{S}$ is
taken with respect to the $\mathbb{Z}_{p}$-extension $F_{\infty}/F$. Recall
that in this section, there is no hypothesis on (the reducibility) of the
residual representation ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$.

######  Corollary 3.11.

Let $E_{/K}$ be an elliptic curve such that

  1. (1)

Hypothesis 2.5 is satisfied.

  2. (2)

$\mu(\mathcal{X}_{E}^{S})=0$.

Then, the fine $\mu$-invariant,
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$.

###### Proof.

By inflation-restriction, we have

$0\rightarrow
H^{1}\left(\operatorname{Gal}(F_{\infty}/K^{\operatorname{ac}}),E[p]\right)\rightarrow
H^{1}(K^{\operatorname{ac}},E[p])\rightarrow\operatorname{Hom}\left(F,E[p]\right)^{\operatorname{Gal}(F/K)}.$

This induces the following map with finite kernel

(3.4)
$\mathcal{R}(E[p]/K^{\operatorname{ac}})\rightarrow\operatorname{Hom}\left(\mathcal{X}_{E}^{S}/p,E[p]\right)^{\operatorname{Gal}(F/K)}.$

Since $\mu(\mathcal{X}_{E}^{S})=0$, the group $\mathcal{X}_{E}^{S}/p$ is
finite. By $\eqref{inflationrestriction}$, the mod-$p$ fine Selmer group
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite. The result follows from
Proposition 3.10. ∎

##  4\. Finiteness of the residual fine Selmer group

Throughout this section ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is
assumed to be reducible. Therefore, $E[p]$ fits into the short exact sequence,

$0\rightarrow\mathbb{F}_{p}(\varphi_{1})\rightarrow
E[p]\rightarrow\mathbb{F}_{p}(\varphi_{2})\rightarrow 0.$

This allows the analysis of the algebraic structure of the mod-$p$ fine Selmer
group $\mathcal{R}(E[p]/K^{\operatorname{ac}})$ in terms of the mod-$p$ fine
Selmer groups $\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})$
for $i=1,2$ (definitions given below). This completes the proof of Theorems
3.2 and 3.4.

######  Definition 4.1.

Let $S$ be the set of primes dividing $\mathcal{N}p$, where $\mathcal{N}$ is
the conductor of $E$. The fine Selmer group
$\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})$ is defined as
follows

$\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{i}))\rightarrow\prod_{\eta\in
S(K^{\operatorname{ac}})}H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{i}))\right\\}.$

The dependence on $S$ is suppressed in the notation.

######  Proposition 4.2.

Suppose that ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible and
let $\varphi_{i}$ be one of the characters on the diagonal of ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$. Then,
$\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})$ is finite if
and only if $\mu(Y_{i})=0$.

###### Proof.

Set $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K(\varphi_{i})/K)$. Since the order of $\Delta$
is coprime to $p$,

$\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K)\simeq\Delta\times\operatorname{Gal}(K^{\operatorname{ac}}/K).$

The group $\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K)$ acts on
$\mathcal{X}^{(i)}$. Let
$g\in\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K)$ and
$x\in\mathcal{X}^{(i)}$. Choose a lift
$\tilde{g}\in\operatorname{Gal}(\mathcal{L}^{(i)}/K)$, and set $g\cdot
x\mathrel{\mathop{\mathchar 58\relax}}=\tilde{g}x\tilde{g}^{-1}$. Since
$\mathcal{X}^{(i)}$ is abelian, $g\cdot x$ is independent of the choice of
lift, $\tilde{g}$. This induces the action of $\Delta$ on $\mathcal{X}^{(i)}$.
There is a natural isomorphism

$\operatorname{Hom}(Y_{i},\mathbb{F}_{p})\simeq\operatorname{Hom}(\mathcal{X}^{(i)},\mathbb{F}_{p}(\varphi_{i}))^{\Delta}.$

Since the order of
$\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K^{\operatorname{ac}})$
is prime to $p$, it follows (from restriction-corestriction) that the
cohomology group
$H^{j}(\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K^{\operatorname{ac}}),\mathbb{F}_{p}(\varphi_{i}))=0$
for $j>0$. It follows from the inflation-restriction sequence that

$H^{1}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{i}))\xrightarrow{\sim}\operatorname{Hom}(\operatorname{Gal}(K_{S}/K^{\operatorname{ac}}(\varphi_{i})),\mathbb{F}_{p}(\varphi_{i}))^{\Delta},$

where $\Delta$ is identified with
$\operatorname{Gal}(K^{\operatorname{ac}}(\varphi_{i})/K^{\operatorname{ac}})$.
This induces an isomorphism

$\iota\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})\xrightarrow{\sim}\operatorname{Hom}(Y_{i}/pY_{i},\mathbb{F}_{p}).$

It is an easy consequence of the structure theory of $\Lambda$-modules (see
the argument in the proof of Lemma 2.11) that the $\mu$-invariant of $Y_{i}$
is zero if and only if $Y_{i}/pY_{i}$ is finite. The result follows. ∎

######  Proposition 4.3.

Suppose that ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is reducible and
$\varphi_{i}$ is one of the characters on the diagonal. Suppose that the
following assumptions hold.

  1. (1)

$\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})$ is finite.

  2. (2)

The Heegner hypothesis is satisfied: let $v\in S\setminus S_{p}$ and $l$ be
the prime number such that $v|l$, then $l$ splits in $K$.

Then, we have that $\mu(\widetilde{Y}_{i})=0$.

This result is used in the proof of Corollary 3.5 where it is further assumed
that $\varphi_{i}=1$. In this special case, the assertion is that the
(classical) Iwasawa invariant $\mu(K^{\operatorname{ac}}/K)$ vanishes.

###### Proof.

Recall that the Galois extensions $\mathcal{M}^{(i)}$ and $\mathcal{X}^{(i)}$
of $K^{\operatorname{ac}}(\varphi_{i})$ are defined on p. 7. Consider the
natural action of $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K(\varphi_{i})/K)$ on $\mathcal{M}^{(i)}$ (resp.
$\mathcal{X}^{(i)}$) and recall that $\widetilde{Y}_{i}$ (resp. $Y_{i}$)
denotes $\varphi_{i}$-component of $\mathcal{M}^{(i)}$ (resp.
$\mathcal{X}^{(i)})$ with respect to this action. At each prime $\eta$ of
$K^{\operatorname{ac}}$, set
$H^{1}_{\operatorname{nr}}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{i}))$
to denote the subspace of unramified cohomology classes in
$H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{i}))$. Set
$\mathcal{R}^{\prime}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})$ to
consist of cohomology classes $f\in
H^{1}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{i}))$ which are
unramified at every prime $\eta\in S(K^{\operatorname{ac}})$. By an
application of inflation-restriction (see the argument in the proof of
Proposition 4.2),

$\begin{split}&\mathcal{R}^{\prime}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})\simeq\operatorname{Hom}(\mathcal{M}^{(i)}/p,\mathbb{F}_{p}(\varphi_{i}))^{\Delta}\simeq\operatorname{Hom}(\widetilde{Y}_{i}/p,\mathbb{F}_{p}),\\\
&\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})\simeq\operatorname{Hom}(\mathcal{X}^{(i)}/p,\mathbb{F}_{p}(\varphi_{i}))^{\Delta}\simeq\operatorname{Hom}(Y_{i}/p,\mathbb{F}_{p}).\end{split}$

The groups fit into an exact sequence,

$0\rightarrow\mathcal{R}(\mathbb{F}_{p}(\varphi_{i})/K^{\operatorname{ac}})\rightarrow\operatorname{Hom}(\widetilde{Y}_{i}/p,\mathbb{F}_{p}(\varphi_{i}))\rightarrow\prod_{\eta\in
S(K^{\operatorname{ac}})}H^{1}_{\operatorname{nr}}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{i})).$

By Fact 2.3, the set $S(K^{\operatorname{ac}})$ is finite. Hence, the group on
the right is finite. We deduce that $\widetilde{Y}_{i}/p$ is finite, therefore
the $\mu$-invariant of $\widetilde{Y}_{i}$ is zero. ∎

###### Proof of Theorem 3.2.

By Proposition 3.10, it suffices to show that
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite. The short exact sequence

$0\rightarrow\mathbb{F}_{p}(\varphi_{1})\rightarrow
E[p]\rightarrow\mathbb{F}_{p}(\varphi_{2})\rightarrow 0,$

induces the long exact sequence

$\cdots\rightarrow
H^{0}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{2}))\xrightarrow{\delta_{\eta}^{0}}H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{1}))\rightarrow
H^{1}(K^{\operatorname{ac}}_{\eta},E[p])\rightarrow\cdots.$

Set $S_{\operatorname{ac}}\mathrel{\mathop{\mathchar
58\relax}}=S(K^{\operatorname{ac}})$ and

$U\mathrel{\mathop{\mathchar 58\relax}}=\prod_{\eta\in
S_{\operatorname{ac}}}\operatorname{image}\delta_{\eta}^{0}\subset\prod_{\eta\in
S_{\operatorname{ac}}}H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{1})).$

If ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is split, then
$\delta_{\eta}^{0}$ is the 0 map; hence, $U=0$.
Next, consider the case when ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$
is indecomposable. When $v\in S\setminus(\Sigma(\varphi_{2})\cup S_{p})$, note
that $H^{0}(K_{v},\mathbb{F}_{p}(\varphi_{2}))=0$. Let $\eta\in
v(K^{\operatorname{ac}})$. Since $K^{\operatorname{ac}}_{\eta}/K_{v}$ is a
$p$-extension, for all $v\in S\setminus(\Sigma(\varphi_{2})\cup S_{p})$, the
product $\prod_{\eta\in
v(K^{\operatorname{ac}})}H^{0}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{2}))=0$.
On the other hand, for all primes $v\in\Sigma(\varphi_{2})\cup S_{p}$, it
follows from Hypothesis 2.7 and Fact 2.3 that there are only finitely many
primes $\eta$ in $v(K^{\operatorname{ac}})$. Therefore, the assumptions imply
that $U$ is finite.

Consider the diagram

${H^{1}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{1}))}$${H^{1}(K_{S}/K^{\operatorname{ac}},E[p])}$${H^{1}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{2}))}$${0}$${(\displaystyle{\prod_{\eta\in
S_{\operatorname{ac}}}}H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{1})))/U}$${\displaystyle{\prod_{\eta\in
S_{\operatorname{ac}}}}H^{1}(K^{\operatorname{ac}}_{\eta},E[p])}$${\displaystyle{\prod_{\eta\in
S_{\operatorname{ac}}}}H^{1}(K^{\operatorname{ac}}_{\eta},\mathbb{F}_{p}(\varphi_{2}))}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

and the associated short exact sequence

(4.1)
$\operatorname{ker}\alpha\rightarrow\mathcal{R}(E[p]/K^{\operatorname{ac}})\rightarrow\mathcal{R}(\mathbb{F}_{p}(\varphi_{2})/K^{\operatorname{ac}}).$

The group $\operatorname{ker}\alpha$ fits into an exact sequence

(4.2)
$0\rightarrow\mathcal{R}(\mathbb{F}_{p}(\varphi_{1})/K^{\operatorname{ac}})\rightarrow\operatorname{ker}\alpha\rightarrow
U.$

Proposition 4.2 asserts that
$\mathcal{R}(\mathbb{F}_{p}(\varphi_{i}),K^{\operatorname{ac}})$ are finite
for $i=1,2$. This requires the assumption that $\mu(Y_{i})=0$ for $i=1,2$. It
follows from $\eqref{es1}$ and $\eqref{es2}$ that
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is also finite. By Lemma 2.11, we
have that $\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is
$\Lambda$-cotorsion with
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$. This completes the
proof. ∎

###### Proof of Theorem 3.4.

Refer to the argument in the proof of Theorem 3.2. Lemma 2.11 asserts that if
$\mathcal{R}_{p^{\infty}}(E/K^{\operatorname{ac}})$ is $\Lambda$-torsion with
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$, then
$\mathcal{R}(E[p]/K^{\operatorname{ac}})$ is finite. Refer to $\eqref{es1}$.
Since $H^{0}(K_{S}/K^{\operatorname{ac}},\mathbb{F}_{p}(\varphi_{1}))$ is
finite, finiteness of $\mathcal{R}(E[p]/K^{\operatorname{ac}})$ implies that
$\operatorname{ker}\alpha$ is finite. By $\eqref{es2}$, it is seen that
$\mathcal{R}(\mathbb{F}_{p}(\varphi_{1})/K^{\operatorname{ac}})$ is finite.
Therefore, by Proposition 4.2, it follows that $\mu(Y_{1})=0$. This completes
the proof in the case when ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is
indecomposable. When ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is split,
the proof is completed by interchanging the roles of $\varphi_{1}$ and
$\varphi_{2}$. ∎

##  5\. Congruent Galois representations

In this section, we consider two elliptic curves $E_{1}$ and $E_{2}$ defined
over $K$ which are $p$-congruent, i.e. their residual representations are
isomorphic,

${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E_{1}}\simeq{}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E_{2}}.$

We make no assumption on the reducibility of the residual representations.
However, it is assumed that both $E_{1}$ and $E_{2}$ satisfy the Heegner
hypothesis.

######  Hypothesis 5.1 (Heegner hypothesis).

Let $E_{/K}$ be an elliptic curve with conductor $\mathcal{N}$, and $S$ be the
set of primes dividing $\mathcal{N}p$. Let $v\in S$ and $l$ be the prime
number such that $v|l$. Then, $l$ splits in $K$ if $l\neq p$.

We prove the following main result in this section.

######  Theorem 5.2.

Let $E_{1}$ and $E_{2}$ be elliptic curves over $K$. Suppose that the
following assumptions hold.

  1. (1)

Both $E_{1}$ and $E_{2}$ satisfy the Heegner hypothesis.

  2. (2)

The residual Galois representations, ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E_{1}}$ and ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E_{2}}$, are isomorphic.

Then,

$\begin{split}&\mathcal{R}_{p^{\infty}}(E_{1}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{1}/K^{\operatorname{ac}})=0\\\
\Leftrightarrow&\mathcal{R}_{p^{\infty}}(E_{2}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{2}/K^{\operatorname{ac}})=0.\end{split}$

The key result used in the proof of the above theorem will be Proposition
3.10, which states that

$\mathcal{R}_{p^{\infty}}(E_{i}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{i}/K^{\operatorname{ac}})=0\Leftrightarrow\mathcal{R}(E_{i}[p]/K^{\operatorname{ac}})\text{
is finite.}$

In view of Lemma 2.11, it suffices to prove the following result.

######  Proposition 5.3.

Let $E_{1}$ and $E_{2}$ be elliptic curves satisfying the conditions of
Theorem 5.2. Then $\mathcal{R}(E_{1}[p]/K^{\operatorname{ac}})$ is finite if
and only if $\mathcal{R}(E_{2}[p]/K^{\operatorname{ac}})$ is finite.

###### Proof.

Let $\mathcal{N}_{i}$ be the conductor of $E_{i}$ and $S_{i}$ be the set of
primes dividing $\mathcal{N}_{i}p$. Denote by $T$ the union $S_{1}\cup S_{2}$.
Since $E_{1}[p]$ and $E_{2}[p]$ are isomorphic as Galois modules, the mod-$p$
fine Selmer groups are isomorphic, i.e.

$\mathcal{R}^{T}(E_{1}[p]/K^{\operatorname{ac}})\simeq\mathcal{R}^{T}(E_{2}[p]/K^{\operatorname{ac}})$

(recall $\mathcal{R}^{T}(E_{i}[p]/K^{\operatorname{ac}})$ from Definition
2.10). Let $v\in T$ and $\eta\in v(K^{\operatorname{ac}})$. Denote by
$\operatorname{G}_{\eta}$ the absolute Galois group of
$K^{\operatorname{ac}}_{\eta}$ and by
$\operatorname{I}_{\eta}\subset\operatorname{G}_{\eta}$ its inertia subgroup.
Set $H^{1}_{\operatorname{nr}}(K^{\operatorname{ac}}_{\eta},E_{i}[p])$ to
denote the kernel of the restriction map,

$\operatorname{res}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(K^{\operatorname{ac}}_{\eta},E_{i}[p])\longrightarrow
H^{1}(\operatorname{I}_{\eta},E_{i}[p])^{\operatorname{G}_{\eta}/\operatorname{I}_{\eta}}.$

It follows from inflation-restriction that
$H^{1}_{\operatorname{nr}}(K^{\operatorname{ac}}_{\eta},E_{i}[p])$ can be
identified with
$H^{1}(\operatorname{G}_{\eta}/\operatorname{I}_{\eta},E_{i}[p]^{\operatorname{I}_{\eta}})$.
Since $\operatorname{G}_{\eta}/\operatorname{I}_{\eta}\simeq\hat{\mathbb{Z}}$,
it follows that (see [26, Proposition 1.7.7])

$H^{1}(\operatorname{G}_{\eta}/\operatorname{I}_{\eta},E_{i}[p]^{\operatorname{I}_{\eta}})\simeq
H_{0}(\operatorname{G}_{\eta}/\operatorname{I}_{\eta},E_{i}[p]^{\operatorname{I}_{\eta}})$

and hence is finite. By the Heegner hypothesis and Fact 2.3, the set
$v(K^{\operatorname{ac}})$ is finite for $v\in T$. Let
$\mathcal{H}_{v}^{\operatorname{nr}}(K^{\operatorname{ac}},E[p])\subseteq\mathcal{H}_{v}(K^{\operatorname{ac}},E[p])$
denote the sum,

$\mathcal{H}_{v}^{\operatorname{nr}}(K^{\operatorname{ac}},E_{i}[p])\mathrel{\mathop{\mathchar
58\relax}}=\bigoplus_{\eta\in
v(K^{\operatorname{ac}})}H^{1}_{\operatorname{nr}}(K^{\operatorname{ac}}_{\eta},E_{i}[p]).$

It follows from the above discussion that
$\mathcal{H}_{v}^{\operatorname{nr}}(K^{\operatorname{ac}},E_{i}[p])$ is
finite for $v\in T$.

There is a natural exact sequence,

$0\rightarrow\mathcal{R}^{T}(E_{i}[p]/K^{\operatorname{ac}})\rightarrow\mathcal{R}(E_{i}[p]/K^{\operatorname{ac}})\rightarrow\bigoplus_{v\in
T\setminus
S_{i}}\mathcal{H}_{v}^{\operatorname{nr}}(K^{\operatorname{ac}},E_{i}[p]).$

Since the last term in the above sequence is finite, it follows that
$\mathcal{R}(E_{i}[p]/K^{\operatorname{ac}})$ is finite if and only if
$\mathcal{R}^{T}(E_{i}[p]/K^{\operatorname{ac}})$ is. The result follows from
this. ∎

###### Proof of Theorem 5.2.

Note that the Heegner hypothesis implies Hypothesis 2.5. Therefore,
Proposition 3.10 applies in this setting and asserts that

$\mathcal{R}_{p^{\infty}}(E_{i}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{i}/K^{\operatorname{ac}})=0\Leftrightarrow\mathcal{R}(E_{i}[p]/K^{\operatorname{ac}})\text{
is finite.}$

On the other hand, Proposition 5.3 states that

$\mathcal{R}(E_{1}[p]/K^{\operatorname{ac}})\text{ is finite
}\Leftrightarrow\mathcal{R}(E_{2}[p]/K^{\operatorname{ac}})\text{ is finite.}$

The result follows. ∎

##  6\. Examples

In this section, we list some examples to illustrate the results in the
article.

######  Example 6.1.

We begin with an example which illustrates Theorem 3.2. In this example, $p=3$
and $E$ is the elliptic curve 81.1-CMa1 over $K=\mathbb{Q}(\sqrt{-3})$. This
is the elliptic curve with Weierstass equation $y^{2}+y=x^{3}$ base changed to
$K$. It has complex multiplication and its endomorphism ring is
$\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$. The group $E(K)[3]$ is the full group
$E[3]$ (and isomorphic to $\mathbb{Z}/3\times\mathbb{Z}/3$). In particular,
residual representation ${}\mkern 3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is the
trivial representation $\left({\begin{array}[]{cc}1&\\\ &1\\\
\end{array}}\right)$. Therefore, ${}\mkern
3.0mu\overline{\mkern-3.0mu\rho}_{E}$ is split, $\varphi_{1}=\varphi_{2}=1$,
and $Y_{1}$ and $Y_{2}$ are both the $3$-Hilbert class field extension of
$K^{\operatorname{ac}}$. The prime $p=3$ is a prime of additive reduction and
3 is ramified in $K$. The conductor of $E$ is $\mathcal{N}=9\mathcal{O}_{K}$
and in particular is not divisible by any primes $v\nmid 3$. It follows that
$\Sigma=\emptyset$ and Hypothesis 2.5 is satisfied. This unique prime above
$(3)$ is totally ramified in the anticyclotomic $\mathbb{Z}_{3}$-extension of
$K$. Since the class number of $K$ is $1$, we know that
$\mu(K^{\operatorname{ac}}/K)=0$ [27, Proposition 4.2]. It follows from
Theorem 3.2 that $\mathcal{R}_{3^{\infty}}(E/K^{\operatorname{ac}})$ is a
cotorsion $\Lambda$-module with
$\mu^{\operatorname{fine}}(E/K^{\operatorname{ac}})=0$ (at $p=3$). In other
words, $\mathcal{R}_{3^{\infty}}(E/K^{\operatorname{ac}})$ is a cofinitely
generated $\mathbb{Z}_{3}$-module.

######  Example 6.2.

This example demonstrates Corollary 3.5. Let $K=\mathbb{Q}(\sqrt{-10})$,
$p=5$, and $E$ be the elliptic curve $11a1$ over $\mathbb{Q}$ defined by the
Weierstrass equation $y^{2}+y=x^{3}-x^{2}-10x-20$. The $5$-torsion group
$E(\mathbb{Q})[5]$, is equal to $\mathbb{Z}/5\mathbb{Z}$. In particular,
$E(K)[5]\neq 0$. Assume that
$\mathcal{R}_{5^{\infty}}(E/K^{\operatorname{ac}})$ is a cotorsion
$\Lambda$-module. The set $S$ consists of the primes above $11$ in $K$. Since
$11$ splits in $K$, it follows that condition $\eqref{corollory33c3}$ of
Corollary 3.5 is satisfied. Corollary 3.5 asserts that if the $\mu$-invariant
of the fine Selmer group $\mathcal{R}_{5^{\infty}}(E/K^{\operatorname{ac}})$
is zero, then the Iwasawa $\mu$-invariant of the $5$-Hilbert class field
extension of $K^{\operatorname{ac}}/K$ is zero as well.

######  Example 6.3.

This example demonstrates an application of Theorem 5.2. Consider the elliptic
curves $E_{1}=201c1$ and $E_{2}=469a1$. Both elliptic curves have rank $1$
with good ordinary reduction at the prime $p=5$. Further, there is an
isomorphism of $\operatorname{G}_{\mathbb{Q}}$-modules, $E_{1}[5]\simeq
E_{2}[5]$. Note that $E_{1}$ has bad reduction at the primes $3$, $67$ and
$E_{2}$ has bad reduction at the primes $7$, $67$. Set $M$ to be the product
of primes of bad reduction $3\cdot 7\cdot 67$. The primes of bad reduction of
$E_{1},E_{2}$ split in infinitely many imaginary quadratic number field $K$ of
the form $\mathbb{Q}(\sqrt{-Mk+1})$ where $k\in\mathbb{Z}_{\geq 1}$, i.e. the
Heegner hypothesis is satisfied. Theorem 5.2 applies; hence

$\begin{split}&\mathcal{R}_{p^{\infty}}(E_{1}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{1}/K^{\operatorname{ac}})=0\\\
\Leftrightarrow&\mathcal{R}_{p^{\infty}}(E_{2}/K^{\operatorname{ac}})\text{ is
}\Lambda\text{-cotorsion with
}\mu^{\operatorname{fine}}(E_{2}/K^{\operatorname{ac}})=0.\end{split}$

######  Example 6.4.

Through this final example we demonstrate that Theorem 5.2 can be used to
relate $\mu$-invariants of isogenous elliptic curves over
$K^{\operatorname{ac}}$. Consider the isogenous elliptic curves $E_{1}=17a1$
and $E_{2}=17a2$. Both elliptic curves have rank $0$ and good supersingular
reduction at the prime $p=11$. Further, there is an isomorphism of
$\operatorname{G}_{\mathbb{Q}}$-modules, $E_{1}[11]\simeq E_{2}[11]$. Consider
the imaginary quadratic number field, $K=\mathbb{Q}(\sqrt{-8})$. Note that the
prime 17 splits in $K$, i.e. both $E_{1}$ and $E_{2}$ satisfy the Heegner
hypothesis. It is known that
$\mu^{\operatorname{fine}}(E_{1}/K^{\operatorname{ac}})=0$ (see [24, Table in
§4]). It follows from Theorem 5.2 that
$\mu^{\operatorname{fine}}(E_{2}/K^{\operatorname{ac}})=0$, as well.

## Acknowledgement

The first named author is funded by the PIMS Postdoctoral Fellowship.

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