# On the $\mu$ equals zero Conjecture for Fine Selmer groups in Iwasawa theory

Shaunak V. Deo  Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India<EMAIL_ADDRESS>,  Anwesh Ray  Centre de
recherches mathématiques, Université de Montréal, Pavillon André-Aisenstadt,
2920 Chemin de la tour, Montréal (Québec) H3T 1J4, Canada
<EMAIL_ADDRESS>and  R. Sujatha  Department of Mathematics
University of British Columbia
Vancouver BC, Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

We study the Iwasawa theory of the fine Selmer group associated to Galois
representations arising from modular forms. The vanishing of the
$\mu$-invariant is shown to follow in some cases from a natural property
satisfied by Galois deformation rings. We outline conditions under which the
$\mu=0$ conjecture is shown to hold for various Galois representations of
interest.

###### Key words and phrases:

Iwasawa $\mu$-invariant, fine Selmer groups, adjoint representations,
deformations of Galois representations.

###### 2010 Mathematics Subject Classification:

11R23 (primary); 11F80, 11G05, 11F11 (secondary)

##  1\. Introduction

Let $F$ be a number field and $p$ be an odd prime. The $p$-primary roots of
unity, considered as a module over the absolute Galois group of $F$, is
denoted by $\mu_{p^{\infty}}.$ Set $F_{n}$ to be the unique field contained in
$F(\mu_{p^{\infty}})$ such that $[F_{n}\mathrel{\mathop{\mathchar
58\relax}}F]=p^{n}$. The cyclotomic $\mathbb{Z}_{p}$-extension $F_{cyc}$ of
$F$ is the subextension of $F(\mu_{p^{\infty}})$ obtained as the union of the
fields $F_{n}.$ In his seminal work [Iwa73], Iwasawa studied the growth of the
$p$-part of the class groups over certain towers of number fields. In
particular, for the cyclotomic $\mathbb{Z}_{p}$-extension of $F$, Iwasawa
proved that there are invariants $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and
$\nu\in\mathbb{Z}$ such that

$\\#\text{Cl}_{p}(F_{n})=p^{p^{n}\mu+\lambda n+\nu}\text{ for }n\gg 0,$

where $\text{Cl}_{p}(F_{n})$ is the $p$-Sylow subgroup of the class group of
$F_{n}$. The invariants $\mu$ and $\lambda$ are the Iwasawa invariants
associated to the $\mathbb{Z}_{p}$-extension $F_{\operatorname{cyc}}/F$.

The genesis of Iwasawa theory arose from the study of these invariants for a
broad class of $\mathbb{Z}_{p}$-extensions of a number field. Iwasawa
formulated various conjectures about these invariants. Amongst them is a
famous conjecture, henceforth referred to as Iwasawa’s $\mu=0$ conjecture,
which asserts that the $\mu$-invariant of the cyclotomic
$\mathbb{Z}_{p}$-extension $F_{\operatorname{\mathrm{cyc}}}/F$ of any number
field $F$ is zero. Let $\mathcal{F}$ be the maximal abelian, unramified,
pro-$p$ extension of $F_{\operatorname{\mathrm{cyc}}}$, let
$X(F_{\operatorname{\mathrm{cyc}}})=\operatorname{Gal}(\mathcal{F}/F_{\operatorname{\mathrm{cyc}}})$
and let $\Gamma=\operatorname{Gal}(F_{\operatorname{\mathrm{cyc}}}/F)$. Then
an equivalent formulation of this conjecture is the statement that
$X(F_{\operatorname{\mathrm{cyc}}})$ is a finitely generated
$\mathbb{Z}_{p}$-module. When $F$ is an abelian number field, the
$\mu$-invariant is known to vanish by the work of Ferrero and Washington
[FW79]. We remark that no general result in this direction is known for any
broader class of non-abelian number fields.

In the modern language of Galois representations and arithmetic geometry,
Iwasawa’s $\mu=0$ conjecture can be interpreted in terms of structural
invariants associated to certain modules over the Iwasawa algebra of
$\operatorname{Gal}(F_{\operatorname{cyc}}/F)$. These modules arise naturally
in the context of the Galois representation associated to the Tate motive
$\mu_{p^{\infty}}.$ Subsequently, Iwasawa theory evolved to cover the study of
a range of modules over Iwasawa algebras that arise from other Galois
representations. An important instance of this evolution is the work of Mazur
[Maz72] who initiated the Iwasawa theory of elliptic curves and abelian
varieties. In this pathbreaking work, Mazur studied the $p$-primary Selmer
group of an elliptic curve (or abelian variety) $E_{/F}$ which has good,
ordinary reduction at all primes dividing $p$ over $F_{\operatorname{cyc}}$,
and defined analogous Iwasawa $\mu$ and $\lambda$-invariants in this context.
The $\mu$-invariant of the $p$-primary Selmer group need not vanish. For
example, for $E_{/\mathbb{Q}}=X_{0}(11)$, the $\mu$-invariant of the
$5$-primary Selmer group is positive. Given the analogy between class groups
and Selmer groups of elliptic curves, it is natural to frame an analogue of
Iwasawa’s $\mu=0$ conjecture in this setting which we describe below.

Let $E$ be an elliptic curve defined over a number field $F$ and $p$ be a
prime above which $E$ has potential good reduction. Set
$\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(F_{\operatorname{cyc}}/F)$ and let $\Lambda$
denote the corresponding Iwasawa algebra, which is defined to be the completed
group algebra $\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. Then, the
$p$-primary _fine Selmer group_ consists of the Galois cohomology classes in
$H^{1}\left(\operatorname{Gal}(\bar{F}/F_{\operatorname{cyc}}),E[p^{\infty}]\right)$
that are trivial at all primes above $p$ and the primes at which $E$ has bad
reduction. The fine Selmer group is believed to be cotorsion as an Iwasawa
module and this is known to be true for elliptic curves over $\mathbb{Q}$ (see
[CS05]). In contrast, the usual Selmer group need not satisfy this condition
unless the elliptic curve is assumed to have ordinary reduction at all primes
above $p$.

The properties of the fine Selmer group were systematically studied by Coates
and the third named author who showed that the Iwasawa $\mu$-invariant of the
fine Selmer group is related to the classical Iwasawa $\mu$-invariant of the
cyclotomic $\mathbb{Z}_{p}$-extension of the number field $F(E[p])$ cut out by
the $p$-torsion in $E$. We refer to [CS05, Corollary 3.5] for the precise
statement of this result. This relationship shows that the vanishing of the
$\mu$-invariant of the fine Selmer group is a consequence of Iwasawa’s
classical $\mu=0$ conjecture. This led Coates and the third named author to
formulate the $\mu=0$ conjecture for the fine Selmer group of an elliptic
curve (see [CS05, Conjecture A]), as the analogue of Iwasawa’s $\mu=0$
conjecture for Galois representations associated to elliptic curves.

Iwasawa theory of $p$-adic Galois representations is currently a central theme
in number theory. The fine Selmer group can be defined in the more general
context of $p$-adic Galois representations (see Definition 2.1 and Definition
2.2). We remark that the $\mu=0$ conjecture can be formulated in this broader
framework (see Conjecture 2.6 and [LS18, Conjecture A, p.74]). It has
interesting applications to the study of Selmer groups associated to Galois
representations of interest. The goal of this article is to study the $\mu=0$
conjecture in this broader setting. The main results formulate explicit
conditions under which the $\mu$-invariant vanishes for the fine Selmer groups
associated to important classes of $p$-adic Galois representations. The cases
of interest are the Galois representations arising from elliptic curves
defined over number fields, Hecke eigencuspforms of weight at least $2$, Tate
twists of Artin representations and Tate twists of the adjoint representations
associated to modular Galois representations.

In the study of Galois representations, the theme of $p$-adic variation plays
a key role. More precisely, the deformation theory of Galois representations,
introduced and developed by Mazur in [Maz89], allows us to study $p$-adic
families of lifts of a fixed mod-$p$ Galois representation. Hida [Hid00]
studied the relationship between certain modular deformation rings defined
over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and the Iwasawa
theory of the classical Selmer group of the adjoint representation of a
modular form. In this article, the $\mu=0$ conjecture for the fine Selmer
group of the first Tate twist of the adjoint representation of a $p$-adic
Galois representation $\rho$ is related to _unobstructedness_ of the residual
representation of $\rho$ (see Definition 4.1). Note that the notion of
unobstructedness appears naturally in Galois deformation theory in connection
with the structure of the universal deformation ring (in the sense of Mazur).
This provides a new approach in investigating the fine Selmer group. It is our
belief that the linkage established here will help advance the current state
of art in both the Iwasawa theory of the fine Selmer group as well as
deformation theory of Galois representations. In particular, our methods
provide a large class of new examples of fine Selmer groups of Galois
representations for which the $\mu$-invariant vanishes.

Let us now briefly outline the results proved in this article. Section 2 is
preliminary in nature. In section 3, we establish a criterion for the
vanishing of the $\mu$-invariant of the fine Selmer group of a Galois
representation (see Theorem 3.7). To be precise, we prove that the
$\mu$-invariant of the fine Selmer group of a $p$-adic Galois representation
$\rho$ vanishes if the _second_ global Galois cohomology group of the first
Tate twist of the dual of the residual representation of $\rho$ vanishes.
Thus, this criterion is determined purely in terms of the residual
representation. Theorem 3.8 gives explicit conditions for the vanishing of the
$\mu$-invariant of the fine Selmer group, which we specialize to the case of
elliptic curves in Theorem 3.9.

Section 4 establishes the linkage between deformation theory of Galois
representations and the vanishing of the $\mu$-invariant of the fine Selmer
group. We study the fine Selmer group of the first Tate twist of the
$n^{2}$-dimensional adjoint Galois representation $\operatorname{Ad}\rho$
associated to an $n$-dimensional Galois representation $\rho$. An important
class of such Galois representations arises naturally from automorphic forms.
More precisely, Theorem 4.3 shows that the $\mu$-invariant vanishes for the
fine Selmer group of $\operatorname{Ad}\rho(1)$ if $\rho$ is _unobstructed_.
This concept was initially introduced by Mazur in [Maz89], and it was shown by
Weston in [Wes04] that Galois representations arising from modular forms are
unobstructed under some explicit conditions.

Let $f$ be a normalized Hecke newform of weight $k\geq 2$. Given a prime
$\mathfrak{p}$ in the field of Fourier coefficients of $f$, let
$\rho_{f,\mathfrak{p}}$ be the associated $2$-dimensional $\mathfrak{p}$-adic
Galois representation. Theorem 4.6 shows that the $\mu$-invariant of the fine
Selmer group of $\operatorname{Ad}\rho_{f,\mathfrak{p}}(1)$ vanishes for a set
of primes $\mathfrak{p}$ of Dirichlet density $1$. Moreover, if $k>2$, then it
is shown to vanish for all but finitely many primes $\mathfrak{p}$.
Furthermore, if the level of $f$ is squarefree, then there is an explicit set
of primes $\mathfrak{p}$ outside of which the associated $\mu$-invariant is
$0$. When $k=2$ and $f$ has trivial nebentypus, our results imply that the
$\mu$-invariant of the fine Selmer group of
$\operatorname{Sym}^{2}\rho_{f,\mathfrak{p}}$ vanishes for a set of primes
$\mathfrak{p}$ of Dirichlet density $1$. We also prove a similar result in the
setting of Hilbert modular forms (see Theorem 4.11). On the other hand, we
prove that if $E$ is a rational elliptic curve with squarefree conductor, then
there are infinitely many primes $p$ such that the $\mu$-invariant of the fine
Selmer group of $\operatorname{Ad}^{0}\rho_{E,p}$ vanishes (see Theorem 4.12).
Here, $\rho_{E,p}$ is the $2$-dimensional $p$-adic Galois representation
attached to $E$ and $\operatorname{Ad}^{0}\rho_{E,p}$ is the subrepresentation
of $\operatorname{Ad}\rho_{E,p}$ consisting of matrices with trace $0$.

We then turn our attention to Artin representations. Note that we can realize
an Artin representation as a $p$-adic Galois representation for all primes
$p$. We prove that if $\rho$ is an Artin representation such that
$H^{0}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),\rho)=0$, then the
$\mu$-invariants of the fine Selmer groups of the first Tate twists of the
$p$-adic realizations of the dual of $\rho$ vanish for all but finitely many
primes $p$ (see Theorem 4.14).

We also connect the $\mu=0$ conjecture with the notion of _neatness_
introduced by Mazur in [Maz89] to prove that the $\mu=0$ conjecture is true
for first Tate twist of the adjoint representation of a $p$-adic Galois
representation $\rho$ whose residual representation is _neat_ (see Corollary
4.19). Leveraging a construction of Mazur and Böckle, we end section 4 with
explicit families of odd and even $S_{3}$-representations such that the
$\mu=0$-conjecture holds for the fine Selmer groups of first Tate twists of
all their characteristic $0$ lifts.

In section 5, we study the vanishing of the $\mu$-invariant of the fine Selmer
group associated to $2$-dimensional residually dihedral Galois
representations. The conditions are stated purely in terms of the residual
representation and the set of primes at which the characteristic zero
representation ramifies. In Theorem 5.6 we give explicit conditions for the
vanishing of the $\mu$-invariant and obtain a large class of examples for
which our results apply. Finally, we remark that our results are illustrated
with concrete numerical examples.

### Acknowledgements

The first named author would like to thank Gabor Wiese for helping with
computation of examples presented in §5. The second named author is grateful
to Ravi Ramakrishna for introducing him to the fascinating subject of Galois
deformation theory, and would also like to thank Tom Weston and Jeffrey Hatley
for helpful discussions along the way. His research is supported by the CRM-
Simons bridge postdoctoral fellowship. The third named author gratefully
acknowledges support from NSERC Discovery grant 2019-03987. The first and
third named authors would also like to thank the online program on ‘Elliptic
curves and the special values of $L$-functions’ (code: ICTS/ECL2021/8) held at
ICTS Bangalore, for providing the motivation and opportunity to begin
discussions relevant to the work presented in this article. We would also like
to thank the anonymous referee for providing many comments and suggestions
which helped in improving the exposition.

##  2\. Preliminaries

Throughout, $p$ is an odd prime number and $F$ is a number field. Set $S_{p}$
to be the set of primes of $F$ above $p$. Let $S$ be a finite set of primes of
$F$ and assume that $S$ contains $S_{p}$ and all archimedean primes of $F$.
Throughout, fix an algebraic closure $\bar{F}$ of $F$. Let $F_{S}$ be the
maximal extension of $F$ contained in $\bar{F}$ in which all primes $v\notin
S$ are unramified. Set $\operatorname{G}_{F,S}$ to be the Galois group
$\operatorname{Gal}(F_{S}/F)$, identified with the maximal quotient of
$\operatorname{Gal}(\bar{F}/F)$ in which the primes $v\notin S$ are
unramified. Given a field $L\subset F_{S}$, we set

$H^{i}\left(F_{S}/L,\cdot\right)\mathrel{\mathop{\mathchar
58\relax}}=H^{i}\left(\operatorname{Gal}(F_{S}/L),\cdot\right).$

For a number field $L$ contained in $F_{S}$, set

$K^{i}_{v}(\cdot/L)\mathrel{\mathop{\mathchar
58\relax}}=\bigoplus_{w|v}H^{i}(L_{w},\cdot),$

where $w$ runs through the primes of $L$ above $v$. The fine Selmer group
associated with a Galois module is obtained by imposing the _strict local
condition_ at each prime that lies above the primes in $S$.

######  Definition 2.1.

Let $M$ be a discrete $p$-primary $\operatorname{G}_{F,S}$ module and $L$ be a
number field contained in $F_{S}$. The fine Selmer group over $L$ associated
to $M$ is defined as follows

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

Given an infinite extension $\mathcal{L}\subset F_{S}$, set

$\mathcal{R}_{p^{\infty}}(M/\mathcal{L})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{L}\mathcal{R}_{p^{\infty}}(M/L),$

where $L$ runs over all number fields contained in $\mathcal{L}$.

Note that, in general, the definition of
$\mathcal{R}_{p^{\infty}}(M/\mathcal{L})$ depends on the choice of $S$.
However, if $\mathcal{L}$ contains the cyclotomic $\mathbb{Z}_{p}$-extension
of $F$, then $\mathcal{R}_{p^{\infty}}(M/\mathcal{L})$ does not depend on the
choice of $S$ (see [SW18, Lemma 3.2]).

For every integer $n\geq 1$, let $\mu_{p^{n}}$ be the group of $p^{n}$-th
roots of unity contained in $\bar{F}$ and
$\mu_{p^{\infty}}\mathrel{\mathop{\mathchar 58\relax}}=\cup_{n\geq
1}\mu_{p^{n}}$. Let $F(\mu_{p^{n}})$ (resp. $F(\mu_{p^{\infty}})$) be the
field extension of $F$ generated by $\mu_{p^{n}}$ (resp. $\mu_{p^{\infty}}$).
The cyclotomic $\mathbb{Z}_{p}$-extension of $F$ is the unique
$\mathbb{Z}_{p}$-extension of $F$ contained in $F(\mu_{p^{\infty}})$, and is
denoted by $F_{\operatorname{cyc}}$. Let $K$ be a finite extension of
$\mathbb{Q}_{p}$ and $\mathcal{O}$ be its valuation ring. Given an integral
Galois representation, $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$,
let $\mathbf{T}_{\rho}$ be the underlying $\mathcal{O}$-lattice on which
$\operatorname{G}_{F,S}$ acts by $\rho$ and set
$\mathbf{A}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}\left(K/\mathcal{O}\right)$.

######  Definition 2.2.

With respect to notation above, the fine Selmer group over
$F_{\operatorname{cyc}}$ associated with $\rho$ is
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$.

Thus [SW18, Lemma 3.2] implies that the definition of
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ does not
depend on the choice of $S$.

Set $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(F_{\operatorname{cyc}}/F)$ and let

$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{O}\llbracket\Gamma\rrbracket=\varprojlim_{n}\mathcal{O}[\Gamma/\Gamma^{p^{n}}]$

be the associated Iwasawa algebra. Given a finitely generated $\Lambda$-module
$\mathfrak{M}$, there is a map of $\Lambda$-modules

$\mathfrak{M}\longrightarrow\Lambda^{\alpha}\oplus\left(\bigoplus_{i=1}^{s}\Lambda/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(f_{j}(T))\right)$

with finite kernel and cokernel. Here,
$\alpha=\operatorname{rank}_{\Lambda}\mathfrak{M}$ and each power series
$f_{j}(T)$ is a _distinguished polynomial_. In other words,
$f_{j}(T)\in\Lambda$ is a monic polynomial all of whose non-leading
coefficients are non-units in $\mathcal{O}$. The $\mu$-invariant is the sum
$\sum_{i}\mu_{i}$. Given an integral Galois representation

$\rho\mathrel{\mathop{\mathchar
58\relax}}G_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O}),$

set
$\mu^{\operatorname{fn}}(\rho)=\mu^{\operatorname{fn}}(\rho/F_{\operatorname{cyc}})$
to be the $\mu$-invariant of $Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$, the
Pontryagin dual of the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$. The
following is Conjecture A in [CS05].

######  Conjecture 2.3 (Coates-Sujatha).

Let $F$ be a number field, $E$ be an elliptic curve over $F$, and $S$ be a
finite set of primes of $F$ containing $S_{p}$, all archimedean primes and all
primes at which $E$ has bad reduction. If

$\rho\mathrel{\mathop{\mathchar
58\relax}}G_{F,S}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$

is the $p$-adic Galois representation arising from $E$, then
$\mu^{\operatorname{fn}}(\rho)=0$.

The following result provides evidence for the above conjecture.

######  Theorem 2.4 (Coates-Sujatha).

Let $F$ be a number field, $S$ be a finite set of primes of $F$ containing
$S_{p}$ and all archimedean primes and

$\rho\mathrel{\mathop{\mathchar
58\relax}}G_{F,S}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$

be the $p$-adic Galois representation associated to an elliptic curve
$E_{/F}$. Suppose that there is a number field $L/F$ such that

  1. (1)

$L\subset F(E[p^{\infty}])$,

  2. (2)

the extension $F(E[p^{\infty}])/L$ is pro-$p$.

Then, $\mu^{\operatorname{fn}}(\rho)$ is equal to $0$ if and only if the
Iwasawa’s $\mu=0$ conjecture holds for the cyclotomic
$\mathbb{Z}_{p}$-extension $L_{\operatorname{cyc}}/L$.

###### Proof.

This result is [CS05, Corollary 3.5]. ∎

######  Theorem 2.5.

Let $F$ be an abelian extension of $\mathbb{Q}$, and

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

be the $p$-adic Galois representation associated to an elliptic curve
$E_{/\mathbb{Q}}$. Assume that the residual representation $\bar{\rho}$ is
reducible, i.e., there are characters

$\varphi_{1},\varphi_{2}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathbb{F}_{p}^{\times}$

such that $\bar{\rho}=\left({\begin{array}[]{cc}\varphi_{1}&\ast\\\
0&\varphi_{2}\\\ \end{array}}\right)$. Let $S$ be a finite set of primes of
$F$ containing $S_{p}$, all archimedean primes and all primes of bad reduction
of $E_{/F}$. Let $\rho_{F,S}$ be $\rho|_{G_{F,S}}$ i.e. $\rho_{F,S}$ is the
representation $\rho$ when viewed as a representation $G_{F,S}$. Then,
$\mu^{\operatorname{fn}}(\rho_{F,S}/F_{\operatorname{cyc}})$ is equal to $0$.

###### Proof.

Let $\mathbb{Q}(\varphi_{i})$ be the extension of $\mathbb{Q}$ which is fixed
by $\operatorname{ker}\varphi_{i}$. Note that $\mathbb{Q}(\varphi_{i})$ is an
abelian extension of $\mathbb{Q}$. Let $L$ be the composite of the fields $F$,
$\mathbb{Q}(\varphi_{1})$ and $\mathbb{Q}(\varphi_{2})$. Since it is a
composite of abelian extensions of $\mathbb{Q}$, it follows that
$L/\mathbb{Q}$ is abelian. By the result of Ferrero and Washington [FW79], the
Iwasawa $\mu$-invariant vanishes for any abelian extension. Note that by
construction, the extension $F\left(E[p^{\infty}]\right)/L$ is pro-$p$. Since
the conditions of Theorem 2.4 are met, the result follows. ∎

We remark that the representation $\bar{\rho}$ in the statement of Theorem 2.5
is reducible if $E$ admits a rational $p$-isogeny. In particular, if
$E(\mathbb{Q})[p]\neq 0$, then the residual representation is of the form
$\bar{\rho}=\left({\begin{array}[]{cc}1&\ast\\\ 0&\bar{\chi}_{p}\\\
\end{array}}\right)$, where $\bar{\chi}_{p}$ is the mod-$p$ cyclotomic
character. Although the $\mu=0$ conjecture for fine Selmer groups was
originally stated in [CS05] for the $2$-dimensional Galois representations
associated to elliptic curves, the following generalized conjecture is
expected to hold (cf. [LS18, Conjecture A, p.74]):

######  Conjecture 2.6.

Let $p$ be a prime number, $F$ be a number field and $\mathcal{O}$ be the ring
of integers of a finite extension of $\mathbb{Q}_{p}$. Let $S$ be a finite set
of primes of $F$ containing the primes that lie above $p$ and
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be an integral Galois representation. Then, $\mu^{\operatorname{fn}}(\rho)$ is
equal to $0$.

##  3\. A Criterion for $\mu=0$ for the fine Selmer group of a Galois
representation

In this section, we shall discuss various criteria for the vanishing of the
$\mu$-invariant of the fine Selmer group associated to a Galois
representation. As in the previous section, let $p$ be an odd prime, $F$ be a
number field and $S$ be a finite set of primes containing all archimedean
primes and the primes above $p$. Let $\mathcal{O}$ be the valuation ring of a
finite extension $K/\mathbb{Q}_{p}$ with uniformizer $\varpi$ and let
$\mathbb{F}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{O}/\varpi$ be its
residue field. Fix an integral Galois representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
on an $\mathcal{O}$-lattice $\mathbf{T}_{\rho}$. Let $\bar{\rho}$ denote the
residual representation $\rho\mod{\varpi}$ and set
$\mathbf{V}_{\bar{\rho}}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}\mathbb{F}$ to
be the underlying vector space on which $\operatorname{G}_{F,S}$ acts via
$\bar{\rho}$. Setting
$\mathbf{V}_{\rho}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}K$, recall that
$\mathbf{A}(\rho)=\mathbf{V}_{\rho}/\mathbf{T}_{\rho}$. We identify
$\mathbf{V}_{\bar{\rho}}$ with $\mathbf{A}(\rho)[\varpi]$. Denote by
$\rho^{\wedge}$ the dual of the representation $\rho$ and by $\chi_{p}$ be the
$p$-adic cyclotomic character. We set $\rho^{*}$ to denote the representation
$\rho^{\wedge}\otimes\chi_{p}$. Recall that $\bar{\chi}_{p}$ is the mod-$p$
cyclotomic character, and $\bar{\rho}^{\wedge}$ is the dual representation to
$\bar{\rho}$. Identify $\bar{\rho}^{*}$ with
$\bar{\rho}^{\wedge}\otimes\bar{\chi}_{p}$. We note that
$\mathbf{T}_{\rho^{*}}$ is isomorphic to the twist
$\mathbf{T}_{{\rho}^{\wedge}}\otimes_{\mathcal{O}}\mathcal{O}(\chi_{p})$.
Therefore, to summarize, $\rho$ and $\rho^{*}$ are integral representations
acting on the $\mathcal{O}$-modules $\mathbf{T}_{\rho}$ and
$\mathbf{T}_{\rho^{*}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}\left(\mathbf{T}_{\rho},\mathcal{O}(\chi_{p})\right)$,
respectively. The representations $\bar{\rho}$ and $\bar{\rho}^{*}$ act on the
$\mathbb{F}$-vector spaces $\mathbf{V}_{\bar{\rho}}$ and
$\mathbf{V}_{\bar{\rho}^{*}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}\left(\mathbf{V}_{\bar{\rho}},\mathbb{F}(\bar{\chi}_{p})\right)$,
respectively. We also note that $\mathbf{A}(\rho)$ is a $p$-divisible
$K/\mathcal{O}$-module which is identified with
$\mathbf{T}_{\rho}\otimes_{\mathcal{O}}K/\mathcal{O}$.

######  Conjecture 3.1 (Weak Leopoldt Conjecture for $\rho$).

With respect to notation above,
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho))=0$.

We refer the reader to [PR00, Appendix B] for a list of cases where the Weak
Leopoldt Conjecture is known.

We prove a criterion which shows that the vanishing of
$\mu^{\operatorname{fn}}(\rho)$ depends only on the residual representation
$\bar{\rho}$. In the special case when $\rho$ is the Galois representation on
the $p$-adic Tate module of an elliptic curve, a brief sketch of the proof is
given in [Suj10, Proposition 4.6].

######  Theorem 3.2.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation such that the weak Leopoldt conjecture holds for
$\rho^{*}$. Then, the following are equivalent.

  1. (1)

The fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is
cotorsion over $\Lambda$ and $\mu^{\operatorname{fn}}(\rho)=0$.

  2. (2)

The cohomology group
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{{\bar{\rho}}^{*}})$ is equal
to $0$.

The proof of the above theorem is provided later in this section, following
Lemma 3.5. The vanishing of
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{{\bar{\rho}}^{*}})$ is studied
in greater detail in this article. Given a $\operatorname{G}_{F,S}$-module
$M$, the Iwasawa cohomology group
$H^{i}_{\operatorname{Iw}}\left(F_{S}/F_{\operatorname{cyc}},M\right)$ is
defined to be the inverse limit
$\varprojlim_{n}H^{i}\left(F_{S}/F_{n},M\right)$ with respect to corestriction
maps. Given a $\Lambda$-module $M$, we denote the Pontryagin dual by
$M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$.
Recall that the fine Selmer group of $\mathbf{A}(\rho)$ over
$F_{\operatorname{cyc}}$ is
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$, and
$Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})^{\vee}$.

######  Theorem 3.3.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation such that the weak Leopoldt conjecture holds for
$\rho^{*}$. Then,
$H^{1}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho^{*}))^{\vee}$ contains no
non-zero finite $\Lambda$-submodule (i.e., $\Lambda$-submodule having finite
cardinality).

###### Proof.

The result is well known for Galois representations which are ordinary at $p$
(see [Gre89, Proposition 5]) and the general case follows from [OV02, Theorem
4.7] specialized to the cyclotomic $\mathbb{Z}_{p}$-extension
$F_{\operatorname{cyc}}$. ∎

######  Corollary 3.4.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation and assume that the weak Leopoldt conjecture holds
for $\rho^{*}$. Then, the following are equivalent:

  1. (1)

$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})$ is finite,

  2. (2)

$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})=0$.

###### Proof.

For ease of notation, denote $\mathbf{A}(\rho^{*})$ by $\mathbf{A}$ throughout
the proof. From the long exact sequence in cohomology associated with

$0\rightarrow\mathbf{A}[\varpi]\rightarrow\mathbf{A}\xrightarrow{\varpi}\mathbf{A}\rightarrow
0,$

we have

$\dots\rightarrow
H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A})^{\vee}\rightarrow
H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])^{\vee}\rightarrow
H^{1}(F_{S}/F_{\operatorname{cyc}},\mathbf{A})^{\vee}.$

From our hypothesis on Weak Leopoldt conjecture for $\rho^{*}$, we obtain that
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])^{\vee}$ is a
$\Lambda$-submodule of
$H^{1}(F_{S}/F_{\operatorname{cyc}},\mathbf{A})^{\vee}$. By Theorem 3.3,
$H^{1}(F_{S}/F_{\operatorname{cyc}},\mathbf{A})^{\vee}$ does not contain any
non-zero finite $\Lambda$-submodule which proves the corollary. ∎

From the Poitou-Tate sequence associated to $\mathbf{A}(\rho)$, we obtain the
following exact sequences (see [PR00, Section 1.3.1] for more details):

(3.1) $\begin{split}0&\rightarrow
H^{0}\left(F_{\operatorname{cyc}},\mathbf{A}(\rho)\right)\rightarrow\bigoplus_{v\in
S}K_{v}^{0}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\rightarrow
H^{2}_{\operatorname{Iw}}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})^{\vee}\rightarrow\\\
&R_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\rightarrow 0,\\\
0&\rightarrow
R_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\rightarrow
H^{1}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho))\rightarrow\bigoplus_{v\in
S}K^{1}_{v}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\rightarrow\\\
&H_{\operatorname{Iw}}^{1}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})^{\vee}\rightarrow
H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho)).\end{split}$

######  Lemma 3.5.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation. The following statements are equivalent

  1. (1)

$Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is $\Lambda$-torsion with
$\mu=0$,

  2. (2)

$H^{2}_{\operatorname{Iw}}\left(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}}\right)$
is $\Lambda$-torsion with $\mu=0$.

###### Proof.

From the Poitou-Tate sequence, we get the following exact sequence

$0\rightarrow Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\rightarrow
H^{2}_{\operatorname{Iw}}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})\rightarrow\bigoplus_{v\in
S}K_{v}^{0}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})^{\vee}.$

Every non-archimedean prime $v$ is finitely decomposed in
$F_{\operatorname{cyc}}$, hence, it is not hard to see that
$K_{v}^{0}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})^{\vee}$ is finitely
generated as a $\mathbb{Z}_{p}$-module. As a $\Lambda$-module,
$K_{v}^{0}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})^{\vee}$ is thus clearly
torsion with $\mu$-invariant equal to $0$. The result is seen to follow from
this. ∎

We now give a proof of Theorem 3.2. Denote the mod-$p$ Iwasawa algebra by
$\Omega\mathrel{\mathop{\mathchar 58\relax}}=\Lambda/\varpi\Lambda$.

###### Proof of Theorem 3.2.

For ease of notation, denote $\mathbf{A}({\rho^{*}})$ by $\mathbf{A}$
throughout the proof. Identify $\mathbf{A}[\varpi]$ with
$\mathbf{V}_{\bar{\rho}^{*}}$. Note that
$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})$
is finitely generated as a $\Lambda$-module. This follows easily from the
Poitou-Tate sequence (3.1) together with the proof of Lemma 3.5. According to
the Lemma 3.5, condition (1) is equivalent to
$H^{2}_{\operatorname{Iw}}\left(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}}\right)$
being a finitely generated $\mathcal{O}$-module. From the short exact sequence

${0\rightarrow\mathbf{T}_{\rho^{*}}\xrightarrow{\varpi}\mathbf{T}_{\rho^{*}}\rightarrow\mathbf{A}[\varpi]\rightarrow
0},$

we obtain

$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})\xrightarrow{\varpi}H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})\rightarrow
H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])\rightarrow
0.$

The isomorphism

$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])\simeq\frac{H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})}{\varpi
H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},{\mathbf{T}_{\rho^{*}}})}$

shows that (1) is equivalent to the finiteness of
$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])$.

We show that (2) is also equivalent to the finiteness of
$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])$.
Given a finitely generated $\Omega$-module $M$, we set $E^{j}(M)$ to denote
the Ext-group $\operatorname{Ext}^{j}_{\Omega}(M,\Omega)$. Setting
$E^{i,j}\mathrel{\mathop{\mathchar
58\relax}}=E^{j}\left(H^{i}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])^{\vee}\right)$,
we find that the Iwasawa cohomology group
$H_{\operatorname{Iw}}^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])$
is related to the cohomology groups
$H^{i}\left(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi]\right)^{\vee}$ via
Jannsen’s spectral sequence [Jan13, Corollary 13]. The spectral sequence
states that

$E^{i,j}\Rightarrow
H_{\operatorname{Iw}}^{i+j}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi]).$

Letting $W_{i}$ to be the cohomology group
$H^{i}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])^{\vee}$, note that for
$j>0$, the adjoint Iwasawa module $E^{j}(W_{i})$ is $\Omega$-torsion. Recall
that condition (2) is the requirement that $W_{2}=0$. According to Corollary
3.4, $W_{2}=0$ if and only if $W_{2}$ is finite. On the other hand, it is
clear that $W_{2}$ is finite if and only if
$E^{2,0}=\operatorname{Hom}_{\Omega}(W_{2},\Omega)$ is $0$. Therefore,
$W_{2}=0$ (i.e., (2) holds) if and only if $E^{2,0}=0$. From the above
spectral sequence, it follows that (2) is equivalent to the finiteness of
$H^{2}_{\operatorname{Iw}}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}[\varpi])$.
∎

######  Lemma 3.6.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation. With notation as above, suppose that
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})=0$. Then, the
weak Leopoldt conjecture is true for $\rho^{*}$ and
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is
cotorsion over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$.

###### Proof.

Consider the short exact sequence of Galois modules

$0\rightarrow\mathbf{V}_{\bar{\rho}^{*}}\rightarrow\mathbf{A}(\rho^{*})\xrightarrow{\varpi}\mathbf{A}(\rho^{*})\rightarrow
0,$

from which we find that
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})$ surjects
onto $H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho^{*}))[\varpi]$. From
the assumption
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})=0$, it
follows that
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho^{*}))[\varpi]=0$. The
vanishing of
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho^{*}))[\varpi]$ implies
that $H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{A}(\rho^{*}))=0$. Thus, it
follows that the weak Leopoldt conjecture holds for $\rho^{*}$. It follows
from Theorem 3.2 that
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is
cotorsion over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$. ∎

The following criterion will be used in establishing new cases of the $\mu=0$
conjecture for the fine Selmer group.

######  Theorem 3.7.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}G_{F,S}\to\operatorname{GL}_{n}(\mathcal{O})$ be a Galois
representation as above. Suppose that
$H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})=0$. Then,
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is
cotorsion over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$. Furthermore,
the weak Leopoldt conjecture holds for $\rho^{*}$.

###### Proof.

Since the $p$-cohomological dimension of
$\Gamma=\operatorname{Gal}(F_{\operatorname{\mathrm{cyc}}}/F)$ is $1$, by the
Hochschild-Serre spectral sequence, the restriction map

$H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})\rightarrow
H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})^{\Gamma}$

is surjective. Hence, if $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})=0$, then,

$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})^{\Gamma}=0.$

Since $\Gamma$ is pro-$p$, this implies that
$H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{\bar{\rho}^{*}})=0$, and the
result follows from Lemma 3.6. ∎

In the remainder of this section, we outline sufficient conditions for the
vanishing of $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})$. We shall illustrate
these conditions for Galois representations that arise from elliptic curves.
In section 5, it is shown that these conditions are satisfied for a number of
examples of _dihedral Galois representations_ of interest. Given $i\geq 0$, we
let ${}^{i}_{S}(\mathbf{V}_{\bar{\rho}})$ be defined as follows

${}^{i}_{S}(\mathbf{V}_{\bar{\rho}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{i}(F_{S}/F,\mathbf{V}_{\bar{\rho}})\longrightarrow\bigoplus_{v\in
S}H^{i}(F_{v},\mathbf{V}_{\bar{\rho}})\right\\}.$

Set $L$ to be the Galois extension of $F$ _cut out_ by $\bar{\rho}$. In other
words, it is the extension of $F$ fixed by the kernel of $\bar{\rho}$. Set
$H_{L}$ to be the mod-$p$ class group $H_{L}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Cl}(L)\otimes\mathbb{F}_{p}$. Denote by
$H_{L}^{\prime}$ the maximal quotient of $H_{L}$ such that the primes of $L$
that lie above $S$ are split in the corresponding subextension of the Hilbert
class field of $L$. Note that $H_{L}^{\prime}$ is a stable with respect to the
natural action of $G=\operatorname{Gal}(L/F)$ on $H_{L}$. Given a prime $v$ of
$F$, set $\operatorname{G}_{v}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{F}_{v}/F_{v})$, which is viewed as a
subgroup of $\operatorname{Gal}(\bar{F}/F)$ after fixing an embedding
$\bar{F}\to\bar{F_{v}}$. Set $\rho_{|v}$ (resp. $\bar{\rho}_{|v}$) to denote
the restriction of $\rho$ (resp. $\bar{\rho}$) to $\operatorname{G}_{v}$.

######  Theorem 3.8.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a Galois representation and $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathbb{F})$
be the associated residual representation. With respect to notation above,
assume that the following conditions are satisfied:

  1. (1)

$H^{1}(G,\mathbf{V}_{\bar{\rho}})=0$,

  2. (2)

$\operatorname{Hom}_{G}(H_{L}^{\prime},\mathbf{V}_{\bar{\rho}})=0$.

  3. (3)

At each prime $v\in S$, the local representation $\bar{\rho}_{|v}$ does not
have the trivial $1$-dimensional representation as a subrepresentation.

Then, $\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ is
cotorsion over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$. Furthermore,
the weak Leopoldt conjecture is true for $\rho^{*}$.

###### Proof.

According to Theorem 3.7, if $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})=0$,
then the assertions follow. We show that
$H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})=0$, thus proving the result. Note
that $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})$ fits into an exact sequence

$0\rightarrow{}^{2}_{S}(\mathbf{V}_{\bar{\rho}^{*}})\rightarrow
H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})\rightarrow\bigoplus_{v\in
S}H^{2}(F_{v},\mathbf{V}_{\bar{\rho}^{*}}).$

By Global duality of -groups [NSW08, Theorem 8.6.7],

${}^{2}_{S}(\mathbf{V}_{\bar{\rho}^{*}})\simeq{}^{1}_{S}\left(\mathbf{V}_{\bar{\rho}}\right)^{\vee}.$

By local Tate duality [Ser13, pp. 91-92],
$H^{2}(F_{v},\mathbf{V}_{\bar{\rho}^{*}})\simeq
H^{0}(F_{v},\mathbf{V}_{\bar{\rho}})^{\vee}$, and hence, is trivial since we
are assuming that the trivial representation does not occur as a
subrepresentation of $\bar{\rho}_{|v}$.

The hypothesis $H^{1}(G,\mathbf{V}_{\bar{\rho}})=0$, together with the
inflation-restriction sequence applied to the extensions $F\subset L\subset
F_{S}$, implies that ${}^{1}_{S}\left(\mathbf{V}_{\bar{\rho}}\right)$ injects
into $\operatorname{Hom}_{G}(H_{L}^{\prime},\mathbf{V}_{\bar{\rho}})$. Since
it is assumed that
$\operatorname{Hom}_{G}(H_{L}^{\prime},\mathbf{V}_{\bar{\rho}})=0$, it follows
that ${}^{1}_{S}\left(\mathbf{V}_{\bar{\rho}}\right)=0$, and hence
${}^{2}_{S}(\mathbf{V}_{\bar{\rho}^{*}})=0$ as well. Thus, putting everything
together, we have shown that $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})$
$=0$. ∎

Let $E$ be an elliptic curve defined over a number field $F$ and let $p$ be an
odd prime number. Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$
be the Galois representation on the $p$-adic Tate module of $E$ and let
$\bar{\rho}$ be its residual representation. Let $G$ denote the Galois group
$\operatorname{Gal}(L/F)$. For a prime $v$ of $F$, denote by $F_{v}$ the
completion of $F$ at $v$ and by $\mathbb{F}_{v}$ the residue field of $F_{v}$.

######  Theorem 3.9.

Let $E_{/F}$ be an elliptic curve. Let $S$ be the set of primes of $F$ that
consists of primes $S_{p}$, all archimedean primes and the primes at which $E$
has bad reduction. Let $\rho\mathrel{\mathop{\mathchar
58\relax}}G_{F,S}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$ be the
$p$-adic Galois representation attached to $E$. With respect to above
notation, assume that the following conditions are satisfied

  1. (1)

the residual representation $\bar{\rho}$ is irreducible,

  2. (2)

$\operatorname{Hom}_{G}\left(H_{L}^{\prime},E[p]\right)=0$,

  3. (3)

for every prime $v\in S$, we assume that $E(\mathbb{F}_{v})[p]=0$.

Then, the following assertions hold:

  1. (a)

the weak Leopoldt conjecture is true for $\rho$,

  2. (b)

$\mathcal{R}_{p^{\infty}}(E[p^{\infty}]/F_{\operatorname{cyc}})$ is cotorsion
over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$.

###### Proof.

We verify that the conditions of Theorem 3.8. Clearly, the conditions (2) and
(3) are satisfied. Note that $\mathbf{V}_{\bar{\rho}}$ is identified with
$E[p]$. That condition (1) is satisfied follows from [PS21, Lemma 2.2]. This
requires the assumption that $\bar{\rho}$ is irreducible. Hence, Theorem 3.8
implies that the weak Leopoldt conjecture is true for $\rho^{*}$ and
$\mathcal{R}_{p^{\infty}}(E[p^{\infty}]/F_{\operatorname{cyc}})$ is cotorsion
over $\Lambda$ with $\mu^{\operatorname{fn}}(\rho)=0$. As $\rho$ is the
$p$-adic Galois representation attached to an elliptic curve,
$\rho\simeq\rho^{*}$. This finishes the proof of the theorem. ∎

We note here that for elliptic curves $E_{/\mathbb{Q}}$ with Mordell Weil rank
$0$ and the Tate-Shafarevich group $\Sh(E/\mathbb{Q})[p]=0$, the upper bound
given by [PS21, Theorem 4.2] provides sufficient conditions for the vanishing
of $\operatorname{Hom}_{G}\left(H_{L},E[p]\right)$. To illustrate this,
consider the following example.

######  Example 3.10.

Let $E$ be the elliptic curve over $F=\mathbb{Q}$ given by 11a2 and
Weierstrass equation $y^{2}+y=x^{3}-x^{2}-7820x-263580$ and set $p=7$.
According to the data provided by the L-functions and Modular forms database
[LMF22],

  1. (1)

$\operatorname{rank}E(\mathbb{Q})=0$,

  2. (2)

$E$ has conductor $11$ and thus, has good reduction at $7$,

  3. (3)

$E$ has split multiplicative reduction at $11$, thus,
$\\#E(\mathbb{F}_{11})=10$,

  4. (4)

$\\#E(\mathbb{F}_{7})=8-a_{7}(E)=10$,

  5. (5)

the mod-$7$ representation is irreducible,

  6. (6)

the Tamagawa product is $1$,

  7. (7)

$\Sh(E/\mathbb{Q})=0$.

Set $S=\\{7,11,\infty\\}$. The conditions (1) and (3) are satisfied. In order
to see that condition (2) is satisfied, it suffices to show that the stronger
vanishing condition $\operatorname{Hom}_{G}\left(H_{L},E[p]\right)=0$. We
refer to the notation in [PS21, Theorem 4.2]. Since the Mordell-Weil rank of
$E$ is $0$ and $\Sh(E/\mathbb{Q})=0$, it follows that the mod-$p$ Selmer group
defined in _loc.cit._ is $0$. Furthermore, since $7\nmid\\#E(\mathbb{F}_{11})$
the set $\mathcal{I}$ in _loc.cit._ is empty. Thus, by the upper bound of
[PS21, Theorem 4.2], we find that
$\operatorname{Hom}_{G}\left(H_{L},E[p]\right)=0$, and in particular, (2) is
also satisfied for $E_{/\mathbb{Q}}$ at $p=7$.

We note here that our condition
$\operatorname{Hom}_{G}\left(H_{L}^{\prime},E[p]\right)=0$ is a weaker
condition than $\operatorname{Hom}_{G}\left(H_{L},E[p]\right)=0$. Since
explicit computation with class groups of fields generated by torsion points
of elliptic curves is difficult, we do not provide an example to illustrate
this.

##  4\. The Adjoint representation of a modular form and $\mu=0$ for the fine
Selmer group

In this section, we prove results for the vanishing of the $\mu$-invariant of
the fine Selmer group for the first Tate-twist of the _adjoint Galois
representation_ associated with a newform. Here, by newform, we mean a new
cuspidal eigenform. We will use this convention throughout the article. When
the newform has weight $2$ and trivial nebentypus, the _symmetric square
representation_ is a direct summand of this Tate-twist, and hence, in this
case, we effectively obtain results for the symmetric square representation as
well. Moreover, given a rational elliptic curve $E$ of squarefree conductor,
we prove results for the vanishing of the $\mu$-invariant of the fine Selmer
group for $\operatorname{Ad}^{0}(\rho_{E,p})$. Here $\rho_{E,p}$ is the
$p$-adic Galois representation attached to $E$ and
$\operatorname{Ad}^{0}(\rho_{E,p})$ is the subrepresentation of the adjoint
representation of $\rho_{E,p}$ consisting of matrices with trace $0$.

The study of these adjoint Galois representations is of special significance
in deformation theory. We shall introduce ideas from deformation theory to
study the vanishing of the $\mu$-invariant of the fine Selmer group. Our
methods also extend to Artin representations and _neat_ representations (in
the sense of Mazur) and we prove results for the vanishing of the
$\mu$-invariants of the fine Selmer groups of their first Tate-twists.

For a number field $K$, we will be studying continuous mod $p$ representations
of $\operatorname{Gal}(\overline{K}/K)$ which are unramified outside a finite
set of primes of $K$ and their lifts to characteristic $0$. Our focus will be
on $\mu$-invariants of fine Selmer groups of such representations. Of
particular significance are the Galois representations arising from modular
eigenforms.

###  4.1. Adjoints of Galois representations

The notion of _unobstructedness_ arises naturally in the study of Galois
deformation theory. It was initially introduced by Mazur in [Maz89] and is
recalled below. Let $\mathbb{F}$ be a finite field of characteristic $p$, $K$
be a number field and $S$ be a finite set of primes of $K$ containing all
primes dividing $p$ and all archimedean primes. Let
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{K,S}\to\operatorname{GL}_{n}(\mathbb{F})$ be a continuous
representation. Recall, from § 2, that $K_{S}$ is the maximal extension of $K$
unramified at all primes of $K$ lying outside $S$ and $\operatorname{G}_{K,S}$
is the corresponding Galois group $\operatorname{Gal}(K_{S}/F)$.

Let $\mathcal{O}$ be the ring of integers of a finite extension of
$\mathbb{Q}_{p}$ with residue field $\mathbb{F}$. Let
$\operatorname{CNL}_{\mathcal{O}}$ be the category of complete local
noetherian $\mathcal{O}$-algebras $R$ with residue field isomorphic to
$\mathbb{F}$. Given $R\in\operatorname{CNL}_{\mathcal{O}}$, let $\mathfrak{m}$
be the maximal ideal of $R$ and fix a residual isomorphism of
$\mathcal{O}$-algebras $R/\mathfrak{m}\xrightarrow{\sim}\mathbb{F}$. As $R$ is
noetherian, there exist $\alpha_{1},\cdots,\alpha_{m}\in R$ such that
$\mathfrak{m}=(\alpha_{1},\cdots,\alpha_{m})$. Since $R$ is a complete local
$\mathcal{O}$-algebra, there exists a morphism $\phi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{O}\llbracket X_{1},\cdots,X_{m}\rrbracket\to R$ which sends
$X_{i}$ to $\alpha_{i}$ for all $1\leq i\leq m$ (see [Eis95, Theorem 7.16
(a)]). Note that the map $\mathcal{O}\to R/\mathfrak{m}$ induced by $\phi$ is
surjective and $\alpha_{1},\cdots,\alpha_{m}$ generate $\mathfrak{m}$.
Therefore, it follows, from [Eis95, Theorem 7.16 (b)], that $\phi$ is
surjective. Hence, we get a presentation for $R$ of the form

$R\simeq\frac{\mathcal{O}\llbracket
X_{1},\dots,X_{m}\rrbracket}{\left(f_{1},\dots,f_{n}\right)}.$

An $R$-lift of $\bar{\rho}$ is a Galois representation
$\rho_{R}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(R)$ such that
$\bar{\rho}=\rho_{R}\mod{\mathfrak{m}}$. The subgroup of matrices
$A\in\operatorname{GL}_{n}(R)$ that reduce to the identity modulo the maximal
ideal of $R$ is denoted by $\widehat{\operatorname{GL}}_{n}(R)$. Two $R$-lifts
$\rho_{R}$ and $\rho_{R}^{\prime}$ of $\bar{\rho}$ are _strictly equivalent_
if $\rho_{R}=A\rho_{R}^{\prime}A^{-1}$ for some matrix
$A\in\widehat{\operatorname{GL}}_{n}(R)$. An _$R$ -deformation_ of
$\bar{\rho}$ is a strict equivalence class of $R$-lifts.

Given a Galois representation $\rho_{R}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(R)$, set
$\operatorname{Ad}\rho_{R}$ to be the adjoint representation of $\rho_{R}$. As
an $R$-module, $\operatorname{Ad}\rho_{R}$ consists of all $n\times n$
matrices with entries in $R$. The Galois action on $\operatorname{Ad}\rho_{R}$
is defined by the adjoint action

$\sigma\cdot v=\rho_{R}(\sigma)v\rho_{R}(\sigma)^{-1},$

where, $\sigma\in\operatorname{G}_{K,S}$ and $v\in\operatorname{Ad}\rho_{R}$.
Thus, we may view $\operatorname{Ad}\rho_{R}$ as a Galois representation

$\operatorname{Ad}\rho_{R}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n^{2}}(R).$

In particular, specializing this construction to the case $R=\mathbb{F}$ and
$\rho_{R}=\bar{\rho}$, we get the adjoint representation of $\bar{\rho}$:

$\operatorname{Ad}\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n^{2}}(\mathbb{F}).$

######  Definition 4.1.

A representation $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathbb{F})$
is said to be _unobstructed_ if
$H^{2}(K_{S}/K,\operatorname{Ad}\bar{\rho})=0$. Following this definition, we
say that $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
is _unobstructed_ if the corresponding residual representation
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathbb{F})$
(obtained by composing $\rho$ with the natural surjective map
$\mathcal{O}\to\mathbb{F}$) is unobstructed.

Associated to $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{K,S}\to\operatorname{GL}_{n}(\mathbb{F})$, consider a _functor of
deformations_

$\operatorname{Def}_{\bar{\rho}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{CNL}_{\mathcal{O}}\rightarrow\operatorname{Sets},$

which takes $R\in\operatorname{CNL}_{\mathcal{O}}$ to the set
$\operatorname{Def}_{\bar{\rho}}(R)$ consisting of $R$-deformations of
$\bar{\rho}$ considered as a representation of the group
$\operatorname{G}_{K,S}$. When $\bar{\rho}$ is absolutely irreducible, there
is a _universal Galois representation_ unramified away from $S$ (see [Maz89,
Proposition 1]), which we denote by

$\rho^{\operatorname{univ}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}\left(R_{\bar{\rho}}\right).$

Here, $R_{\bar{\rho}}$ is the _universal deformation ring_ associated to
$\bar{\rho}$ and the group $\operatorname{G}_{K,S}$ (the set $S$ being
suppressed in our notation). It is universal in the sense that for any
$R$-deformation $\rho_{R}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(R)$ of
$\bar{\rho}$, there exists a unique homomorphism of complete noetherian local
$\mathcal{O}$-algebras $\psi\mathrel{\mathop{\mathchar
58\relax}}R_{\bar{\rho}}\rightarrow R$ such that the following diagram
commutes

${\text{GL}_{n}(R_{\bar{\rho}})}$${\operatorname{G}_{K,S}}$${\text{GL}_{n}(R),}$$\scriptstyle{\rho^{\operatorname{univ}}}$$\scriptstyle{\rho_{R}}$

where the vertical map is induced by $\psi$.

In general, finding the explicit structure of the universal deformation ring
$R_{\bar{\rho}}$ is a very difficult problem. However, if $\bar{\rho}$ is
unobstructed, then we know that

$R_{\bar{\rho}}\simeq\mathcal{O}\llbracket X_{1},\cdots,X_{d}\rrbracket$

where $d=\dim_{\mathbb{F}}(H^{1}(G_{K,S},\operatorname{Ad}\bar{\rho}))$ (see
[Maz89, Proposition 2]). Indeed, [Maz89, Section 1.6] implies that
$R_{\bar{\rho}}$ is a quotient of $\mathcal{O}\llbracket
X_{1},\cdots,X_{d}\rrbracket$. Since $\bar{\rho}$ is unobstructed, [Maz89,
Proposition 2] implies that the Krull dimension of $R_{\bar{\rho}}$ is at
least $d+1$. This allows us to conclude that $R_{\bar{\rho}}$ is isomorphic to
$\mathcal{O}\llbracket X_{1},\cdots,X_{d}\rrbracket$. In particular,
specializing to the case $K=\mathbb{Q}$ and $n=2$, we get:

######  Proposition 4.2.

Suppose that $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathbb{F})$
is absolutely irreducible and unobstructed. Then:

  1. (1)

If $\bar{\rho}$ is odd, then $R_{\bar{\rho}}$ is isomorphic (as a complete
noetherian local $\mathcal{O}$-algebra) to the formal power series ring
$\mathcal{O}\llbracket X_{1},X_{2},X_{3}\rrbracket$.

  2. (2)

If $\bar{\rho}$ is even, then $R_{\bar{\rho}}$ is isomorphic (as a complete
noetherian local $\mathcal{O}$-algebra) to the formal power series ring
$\mathcal{O}\llbracket X\rrbracket$.

###### Proof.

If $\bar{\rho}$ is odd, then the global Euler characteristic formula implies
that
$\dim_{\mathbb{F}}(H^{1}(G_{\mathbb{Q},S},\operatorname{Ad}\bar{\rho}))=3$. If
$\bar{\rho}$ is even, then the global Euler characteristic formula implies
that
$\dim_{\mathbb{F}}(H^{1}(G_{\mathbb{Q},S},\operatorname{Ad}\bar{\rho}))=1$.
The result now follows from [Maz89, Proposition 2] and the explanations given
above. ∎

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathcal{O})$ be a
representation. As we are assuming that $p$ is odd, we get a direct sum
decomposition

$\operatorname{Ad}\rho=\operatorname{Ad}^{0}\rho\oplus\mathcal{O}$

as $G_{\mathbb{Q},S}$-representations with summands
$\operatorname{Ad}^{0}\rho$ and $\mathcal{O}$ corresponding to trace $0$
matrices and scalars, respectively. We have a similar decomposition for
$\operatorname{Ad}\bar{\rho}$ as well. Recall that we denoted the $p$-adic
cyclotomic character by $\chi_{p}$. So we have

$\operatorname{Ad}\rho(1)=\operatorname{Ad}^{0}\rho(1)\oplus\mathcal{O}(\chi_{p})=\operatorname{Sym}^{2}(\rho)\otimes\det(\rho)^{-1}\chi_{p}\oplus\mathcal{O}(\chi_{p}).$

Thus, if $\det(\rho)=\chi_{p}$, then $\operatorname{Sym}^{2}(\rho)$ is a
subrepresentation of $\operatorname{Ad}\rho(1)$. We obtain the following
criterion for the vanishing of the $\mu$-invariant of the fine Selmer group of
the first Tate-twist of the adjoint representation (without assuming
$\dim(\rho)=2$):

######  Theorem 4.3.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be a continuous Galois representation. If $\rho$ is _unobstructed_ , then the
following assertions hold:

  1. (1)

the weak Leopoldt conjecture is true for $\operatorname{Ad}\rho$.

  2. (2)

the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\operatorname{Ad}\rho(1))/K_{\operatorname{cyc}})$
associated to $\operatorname{Ad}\rho(1)$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\operatorname{Ad}\rho(1))=0$.

  3. (3)

If $n=2$ and $r=\operatorname{Sym}^{2}(\rho)\otimes(\det(\rho))^{-1}\chi_{p}$,
then the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(r)/K_{\operatorname{cyc}})$ associated to
$r$ is cotorsion over $\Lambda$ with $\mu^{\operatorname{fn}}(r)=0$.

###### Proof.

Observe that
$\left(\operatorname{Ad}\rho(1)\right)^{*}=\operatorname{Ad}\rho$.
Unobstructedness of $\rho$ implies that
$H^{2}(K_{S}/K,\operatorname{Ad}\bar{\rho})=0$. Hence, the assertions (1) and
(2) above follow directly from Theorem 3.7. For (3), consider the case when
$n=2$ and $r=\operatorname{Sym}^{2}(\rho)\otimes(\det(\rho))^{-1}\chi_{p}$. In
this case, we find that $r^{*}=\operatorname{Ad}^{0}\rho$. Hence,
$H^{2}(K_{S}/K,\mathbf{V}_{\bar{r}^{*}})$ is a direct summand of
$H^{2}(K_{S}/K,\operatorname{Ad}\bar{\rho})$, and therefore is equal to $0$.
The assertion in this case follows once again from Theorem 3.7. ∎

The unobstructedness of modular Galois representations $\rho$ is studied in
greater detail in [Wes04]. Before recalling the main result of [Wes04], we
briefly describe its setup. Let $f$ be a newform (i.e. a new cuspidal
eigenform) of weight $k\geq 2$ on $\Gamma_{1}(N)$, where $N\in\mathbb{Z}_{\geq
1}$. Let $S$ be a finite set of primes of $\mathbb{Q}$ containing the primes
dividing $N$ and $\infty$. Denote by $F$ the field of Fourier coefficients of
$f$ and let $\mathfrak{p}$ be a prime of $\mathcal{O}_{F}$. Set $\mathcal{O}$
to be the completion of $\mathcal{O}_{F}$ at $\mathfrak{p}$. Fix a uniformizer
$\varpi$ of $\mathcal{O}$ and set $\mathbb{F}$ to denote the residue field
$\mathcal{O}/\varpi$. Let $p$ be the prime of $\mathbb{Q}$ lying below
$\mathfrak{p}$ and let $S_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}=S\cup\\{p\\}$. Let

$\rho_{f,\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S_{\mathfrak{p}}}\rightarrow\operatorname{GL}_{2}(\mathcal{O})$

be an integral Galois representation associated to $f$ and the prime
$\mathfrak{p}$ by the construction of Eichler-Shimura and Deligne. Let
$\bar{\rho}_{f,\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S_{\mathfrak{p}}}\rightarrow\operatorname{GL}_{2}(\mathbb{F})$
be the residual representation obtained by reducing $\rho_{f,\mathfrak{p}}$
modulo $\varpi$. Note that the construction of Eichler-Shimura and Deligne
gives a Galois representation over the fraction field of $\mathcal{O}$ and the
Galois representation $\rho_{f,\mathfrak{p}}$ arises from a choice of a Galois
stable $\mathcal{O}$-lattice $\mathbf{T}$ in it. When the residual
representation is absolutely irreducible, there is a unique choice of
$\mathbf{T}$. In all the examples that we consider, the residual
representation will be absolutely irreducible. We will mostly be working with
this setup in the rest of the section.

Denote by $\operatorname{Obs}(f)$ the set of primes $\mathfrak{p}$ of
$\mathcal{O}_{F}$ at which $\rho_{f,\mathfrak{p}}$ is _obstructed_ (i.e., not
unobstructed). An important notion in this context is that of a _congruence
prime_.

######  Definition 4.4.

Let $d$ be a divisor of $N$. We say that $\mathfrak{p}$ is a _congruence prime
of level $d$_ for $f$ if there is a newform $f^{\prime}$ such that

  * •

$f^{\prime}$ has weight $k$ and level $\Gamma_{1}(d)$;

  * •

$f^{\prime}$ is _not_ Galois conjugate to $f$;

  * •

$\bar{\rho}_{f^{\prime},\bar{\mathfrak{p}}}\simeq\bar{\rho}_{f,\bar{\mathfrak{p}}}$
for some prime $\bar{\mathfrak{p}}$ of $\overline{\mathbb{Q}}$ above
$\mathfrak{p}$.

Denote by $\operatorname{Cong}(f)$ the set of all congruence primes of $f$ (as
$d$ ranges through all divisors of $N$).

Since there are only finitely many newforms whose level is a divisor of $N$,
it is easy to see, by comparing their Hecke eigenvalues which are not equal,
that $\operatorname{Cong}(f)$ is finite. Let $\varphi$ denote Euler’s totient
function.

The following explicit result is due to Weston and has been subsequently
generalized in various directions.

######  Theorem 4.5 (Weston).

Let $f$ be a Hecke newform of weight $k\geq 2$ on $\Gamma_{1}(N)$ and let $M$
be the conductor of the nebentypus of $f$. Consider two cases.

  1. (1)

First, consider the case $k>2$. Then, for all but finitely many primes
$\mathfrak{p}$ of $\mathcal{O}_{F}$, the Galois representation
$\rho_{f,\mathfrak{p}}$ is unobstructed. Furthermore, if $N$ is assumed to be
squarefree, we have that

$\operatorname{Obs}(f)\subseteq\left\\{\mathfrak{p}|p;p\leq
k+1\right\\}\cup\left\\{\mathfrak{p}|p;p|\left(N\varphi(N)\prod_{\ell|(N/M)}(\ell+1)\right)\right\\}\cup\operatorname{Cong}(f).$

  2. (2)

Consider the case $k=2$. Then, the set $\operatorname{Obs}(f)$ of obstructed
primes $\mathfrak{p}$ has Dirichlet density zero.

###### Proof.

The first part is obtained by combining [Wes04, Theorem 5.4] with [Wes05,
Theorem 1]. The second part follows directly from [Wes04, Theorem 5.5] (see
also [Maz97, Corollary 2] when $f$ is a non-CM modular newform having weight
$2$, trivial nebentypus and rational Hecke eigenvalues). ∎

Combining the above theorem with Theorem 4.3, one obtains an immediate
application towards the vanishing of the $\mu$-invariant of the fine Selmer
group.

######  Theorem 4.6.

Let $f$ be a Hecke newform of weight $k\geq 2$ on $\Gamma_{1}(N)$ and let $M$
be the conductor of the nebentypus $\epsilon$ of $f$. Let $F$ be the number
field generated by its Hecke eigenvalues and let $\mathcal{O}_{F}$ be the ring
of integers of $F$. Consider two cases.

  1. (1)

If $k>2$, then, for all but finitely many primes $\mathfrak{p}$ of
$\mathcal{O}_{F}$, the following assertions hold:

    1. (a)

the weak Leopoldt conjecture holds for
$\operatorname{Ad}\rho_{f,\mathfrak{p}}$.

    2. (b)

Let $r$ denote any one of the representations
$\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p}$
or $\operatorname{Ad}\rho_{f,\mathfrak{p}}(1)$. Then the fine Selmer group
associated to $r$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(r)=0$.

Furthermore, if $N$ is squarefree, then the above assertions hold for all
primes outside the finite set

$\operatorname{Obs}(f)\subseteq\left\\{\mathfrak{p}|p;p\leq
k+1\right\\}\cup\left\\{\mathfrak{p}|p;p|\left(N\varphi(N)\prod_{\ell|(N/M)}(\ell+1)\right)\right\\}\cup\operatorname{Cong}(f).$

  2. (2)

Consider the case $k=2$. Then, for a set of primes $\mathfrak{p}$ of
$\mathcal{O}_{F}$ of Dirichlet density one, the following assertions hold:

    1. (a)

the weak Leopoldt conjecture is true for
$\operatorname{Ad}\rho_{f,\mathfrak{p}}$.

    2. (b)

For
$r\in\\{\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes\epsilon^{-1},\operatorname{Ad}\rho_{f,\mathfrak{p}}(1)\\}$,
the fine Selmer group associated to $r$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(r)=0$.

###### Proof.

The result follows directly from Theorems 4.3 and 4.5. ∎

######  Remark 4.7.

If $f$ is a newform of weight $2$ with trivial nebentypus, then, under the
notation established above, Theorem 4.6 implies that for a set of primes
$\mathfrak{p}$ of $\mathcal{O}_{F}$ of Dirichlet density one, the fine Selmer
group associated to the symmetric square representation
$\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})$ is co-torsion over $\Lambda$
with
$\mu^{\operatorname{fn}}(\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}}))=0$.

######  Remark 4.8.

For Galois representations associated to elliptic curves of rank $0$ (not the
adjoint or symmetric square representation, but the representation itself),
similar results can be proved for a set of primes $\mathfrak{p}$ of Dirichlet
density $1$, see [Wut07]. On the other hand, for modular forms of weight
$k>2$, Theorem 4.6 proves that the $\mu$-invariant of the first Tate twist of
the adjoint representation vanishes for all but finitely many primes, which is
stronger than showing that it vanishes for a set of primes of Dirichlet
density $1$. Moreover, the set of primes outside which the $\mu$-invariant is
known to vanish, is made explicit in the squarefree level case by the above
result. Also, in the weight $2$ case, when $f$ coincides with an abelian
variety of $\operatorname{GL}_{2}$-type, there is no assumption made on the
rank of this abelian variety.

######  Remark 4.9.

Hatley [Hat16] has generalized [Wes05, Theorem 1] to modular newforms of
arbitrary level. Thus, combining [Hat16, Theorem 3.6] with Theorem 3.7, one
can explicitly describe a finite set of primes of $\mathcal{O}_{F}$ outside of
which
$\mu^{\operatorname{fn}}({\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p}})$
and $\mu^{\operatorname{fn}}(\operatorname{Ad}\rho_{f,\mathfrak{p}}(1))$
vanish even when the level of $f$ is not squarefree. Since the description of
this set is tedious, we will not give it here. We refer the reader to [Hat16,
Theorem 3.6] for more details.

Specializing to the case when $N=1$, $S=\\{\infty\\}$ and $f$ is the unique
normalized cusp form of level $1$ and weight $k\in\\{12,16,18,20,22,26\\}$, we
get, for every prime $p$, the $p$-adic Galois representation

$\rho_{f,p}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},\\{p,\infty\\}}\to\operatorname{GL}_{2}(\mathbb{Z}_{p})$

attached to $f$. Let

$\bar{\rho}_{f,p}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},\\{p,\infty\\}}\to\operatorname{GL}_{2}(\mathbb{F}_{p})$

be the reduction of $\rho$ modulo $p$.

Combining [Wes04, Theorem 2] with Theorem 4.3, we get:

######  Theorem 4.10.

Let $f$ be the unique normalized cusp form of level $1$ and weight
$k\in\\{12,16,18,20,22,26\\}$. If $p>k+1$ is a prime such that
$\bar{\rho}_{f,p}$ is absolutely irreducible, then

  1. (1)

the weak Leopoldt conjecture is true for $\operatorname{Ad}\rho_{f,p}$.

  2. (2)

For
$r\in\\{\operatorname{Sym}^{2}(\rho_{f,p})(2-k),\operatorname{Ad}\rho_{f,p}(1)\\}$,
the fine Selmer group associated to $r$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(r)=0$.

###### Proof.

In the setting of the theorem above, Weston obtains [Wes04, Theorem 2] by
proving that $\rho_{f,p}$ is unobstructed i.e.
$H^{2}(\mathbb{Q}_{\\{p,\infty\\}}/\mathbb{Q},\operatorname{Ad}(\bar{\rho}_{f,p}))=0$
(see the proof of [Wes04, Theorem 5.6] for more details). Combining this with
Theorem 4.3 proves the theorem. Note that, in the cases at hand,
$\det(\rho_{f,p})=\chi_{p}^{k-1}$. Hence, we have
$\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p}=\operatorname{Sym}^{2}(\rho_{f,p})(2-k)$.
∎

If $f$ is one of the newforms considered in Theorem 4.10, then the set of
primes $p$ such that $\bar{\rho}_{f,p}$ is _not_ absolutely irreducible is
given in [Wes04, Section 5.4]. In particular if $f=\Delta$, then the
hypotheses of Theorem 4.10 are satisfied by primes $p\geq 17$ and $p\neq 691$.

Weston’s work on unobstructedness of modular Galois representations has been
generalized in several directions by various authors. In the setting of
Hilbert modular forms, such results have been obtained by Gamzon [Gam16].
Using his main result, we get:

######  Theorem 4.11.

Let $K$ be a totally real field, $f$ be a Hilbert modular newform over $K$ and
$S$ be a finite set of primes of $K$ containing all prime divisors of the
level of $f$ and all archimedean primes of $K$. Let $F$ be the number field
generated by its Hecke eigenvalues and let $\mathcal{O}_{F}$ be the ring of
integers of $F$. For a prime $\mathfrak{p}$ of $\mathcal{O}_{F}$, let $p$ be
the rational prime lying below $\mathfrak{p}$, $S_{\mathfrak{p}}$ be
$S\cup\\{\text{primes of }K\text{ lying above }p\\}$ and

$\rho_{f,\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}G_{K,S_{\mathfrak{p}}}\to\operatorname{GL}_{2}(\mathcal{O}_{F,\mathfrak{p}})$

be the corresponding Galois representation attached to $f$. Suppose the
following hypotheses hold:

  1. (1)

$f$ has no CM,

  2. (2)

$f$ is not a twist of a base change of a Hilbert newform over a proper
subfield $E$ of $K$,

  3. (3)

All weights of $f$ are greater than $2$.

Then, for all but finitely many primes $\mathfrak{p}$ of $\mathcal{O}_{F}$,
the following assertions hold:

  1. (a)

the weak Leopoldt conjecture is true for
$\operatorname{Ad}^{0}\rho_{f,\mathfrak{p}}$.

  2. (b)

The fine Selmer group associated to
$\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p}$
is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p})=0$.

###### Proof.

Note that, under the hypotheses of the theorem above, [Gam16, Theorem 1.1]
implies that
$H^{2}(K_{S_{\mathfrak{p}}}/K,\operatorname{Ad}^{0}\bar{\rho}_{f,\mathfrak{p}})=0$
for all but finitely many primes $\mathfrak{p}$ of $\mathcal{O}_{F}$. We note
here that
$\left(\operatorname{Sym}^{2}(\rho_{f,\mathfrak{p}})\otimes(\det(\rho_{f,\mathfrak{p}}))^{-1}\chi_{p}\right)^{*}$
is identified with $\operatorname{Ad}^{0}\rho_{f,\mathfrak{p}}$. Theorem 3.7
gives the result. ∎

The analogous problem for $\operatorname{GSp}_{4}$-representations is studied
by Broshi, Mullath, Sorensen and Weston [BMSW20]. On the other hand, Guiraud
in [Gui20] has established a related generalization (in the spirit of Gamzon)
in the setting of regular algebraic conjugate self-dual cuspidal (RACSDC)
automorphic representations (see [Gui20, Theorem 1.2] for more details). We
expect that Theorem 4.6 can be suitably generalized to such settings.

We end this subsection by proving an analogue of Theorem 4.6 for the vanishing
of $\mu$-invariant of $\operatorname{Ad}^{0}\rho$, where $\rho$ is the Galois
representation associated to an elliptic curve.

######  Theorem 4.12.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with squarefree conductor $N$
and let $S$ be the set of primes dividing $N$ and $\infty$. For a prime $p$,
let

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

be the $p$-adic Galois representation attached to $E$. Then, for infinitely
many primes $p$, the following assertions hold:

  1. (1)

the weak Leopoldt conjecture holds for
$\operatorname{Ad}^{0}\rho_{E,p}(1)=\operatorname{Sym}^{2}(\rho_{E,p})$.

  2. (2)

The fine Selmer group associated to $\operatorname{Ad}^{0}\rho_{E,p}$ is
cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\operatorname{Ad}^{0}\rho_{E,p})=0$.

###### Proof.

Recall that
$(\operatorname{Ad}^{0}\rho_{E,p})^{*}=\operatorname{Ad}^{0}\rho_{E,p}(1)$.
So, by Theorem 3.7, it suffices to prove that for infinitely many primes p, we
have
$H^{2}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p}(1))=0$.
Using Poitou-Tate duality, we get the following exact sequence (see [Was97,
Proposition 10]):

(4.1) $0\to
H^{0}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p})\to\oplus_{\ell\in
S}H^{0}(G_{\ell},\operatorname{Ad}^{0}\bar{\rho}_{E,p})\oplus
H^{0}(G_{p},\operatorname{Ad}^{0}\bar{\rho}_{E,p})\to\\\
H^{2}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p}(1))^{*}\to{}^{1}_{S_{p}}(\operatorname{Ad}^{0}\bar{\rho}_{E,p})\to
0.$

Hence, we conclude that
$H^{2}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p}(1))=0$
if the following conditions hold:

  1. (1)

$H^{0}(G_{\ell},{\operatorname{Ad}^{0}\bar{\rho}_{E,p}})=0$ for all $\ell\in
S$,

  2. (2)

$H^{0}(G_{p},{\operatorname{Ad}^{0}\bar{\rho}_{E,p}})=0$,

  3. (3)

${}^{1}_{S_{p}}(\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$.

Suppose $\bar{\rho}_{E,p}$ is absolutely irreducible. Hence, it follows that
$H^{0}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$.
Note that, in [Wes05], Weston defines a suitable Selmer group
$H^{1}_{\emptyset}(G_{\mathbb{Q}},\mathbf{A}(\operatorname{Ad}^{0}\rho_{E,p}))$
of $\mathbf{A}(\operatorname{Ad}^{0}\rho_{E,p})$ (see [Wes05, p. 204]). This
Selmer group is also extensively studied in [DFG05]. Since
$H^{0}(\mathbb{Q}_{S_{p}}/\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$,
we get, using the proof of [Wes05, Lemma 7] and [Wes05, Equation 2.3], that
${}^{1}_{S_{p}}(\operatorname{Ad}^{0}\bar{\rho}_{E,p})$ is a subgroup of
$H^{1}_{\emptyset}(G_{\mathbb{Q}},\mathbf{A}(\operatorname{Ad}^{0}\rho_{E,p}))$.
Let $f_{E}$ be the weight $2$ modular form associated to $E$. From [DFG05,
Theorem 3.7], we get that the length of
$H^{1}_{\emptyset}(G_{\mathbb{Q}},\mathbf{A}(\operatorname{Ad}^{0}\rho_{E,p}))$
is the $p$-valuation of the congruence ideal $\eta_{f_{E}}^{\emptyset}$
defined in [DFG05, Section 1.7.3]. As $\eta_{f_{E}}^{\emptyset}$ is an ideal
of $\mathbb{Z}$, we conclude that
$H^{1}_{\emptyset}(G_{\mathbb{Q}},\mathbf{A}(\operatorname{Ad}^{0}\rho_{E,p}))=0$
for all but finitely many primes $p$ (see the proofs of [Wes04, Theorem 5.4,
Theorem 5.6] for more details). Note that $\bar{\rho}_{E,p}$ is absolutely
irreducible for all but finitely many primes $p$. Hence, we conclude that
${}^{1}_{S_{p}}(\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$ for all but finitely
many primes $p$.

Now $E$ is a rational elliptic curve of conductor $N$. So it follows, from the
modularity of rational elliptic curves, that there exists a newform $f_{E}$ of
level $\Gamma_{0}(N)$ and weight $2$ with rational Fourier coefficients such
that $\rho_{E,p}=\rho_{f_{E},p}$ for all primes $p$.

Now, in addition to irreducibility of $\bar{\rho}_{E,p}$, suppose $p>N^{2}$
and $H^{0}(G_{\ell},\operatorname{Ad}^{0}\bar{\rho}_{E,p})\neq 0$ for some
$\ell\in S$. As $N$ is squarefree and $f_{E}$ has trivial nebentypus, it
follows that $\bar{\rho}_{E,p}$ is reducible, semi-simple and hence,
unramified at $\ell$ (see the proof of [Wes05, Lemma 11] for more details).
So, we conclude, from [Edi97, (B) of p. 221], that $p$ is a congruence prime
of level dividing $\dfrac{N}{\ell}$ (see Definition 4.4 for the definition of
congruence primes). Since $\bar{\rho}_{E,p}$ is absolutely irreducible for all
but finitely many primes $p$ and there are only finitely many congruence
primes, we conclude that for all but finitely many primes $p$,
$H^{0}(G_{\ell},\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$ for all $\ell\in S$.

Note that if $p$ is a supersingular prime of $E$, then the restriction of
$\bar{\rho}_{E,p}$ to $G_{p}$ is absolutely irreducible and hence,
$H^{0}(G_{p},\operatorname{Ad}^{0}\bar{\rho}_{E,p})=0$. A celebrated theorem
of Elkies ([Elk87, Theorem 2]) implies that $E$ has infinitely many
supersingular primes. Combining this with all the analysis given above proves
the theorem. ∎

###  4.2. Artin representations

Note that, in Theorem 4.5, the weight of the modular newforms is always
assumed to be greater than $1$. Therefore, it does not shed any light on the
unobstructedness of Artin representations as they arise only in the setting of
weight $1$ modular forms. Moreover, an Artin representation can be considered
as a $p$-adic representation for every prime $p$ (as we will see below). So it
is easy to formulate the question studied by Weston [Wes04] for Artin
representations of arbitrary dimensions. This question is studied by Böckle,
Guiraud, Kalyanswamy and Khare in [BGKK18] in more generality. In particular,
they focus on the vanishing of $H^{2}$ for arbitrary Artin representations
rather than restricting to the case of adjoint of Artin representations. We
will now briefly describe their setup following [BGKK18, Section 6.1].

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{n}(\mathbb{C})$
be a non-trivial irreducible Artin representation (which is not necessarily
odd). So it is a continuous, irreducible representation with finite non-
trivial image. Let $E$ be the finite extension of $\mathbb{Q}$ fixed by
$\ker(\rho)$. Let $G=\operatorname{Gal}(E/\mathbb{Q})$ and $h_{E}$ be the
class number of $E$. Let $S$ be the set of primes of $\mathbb{Q}$ consisting
of primes which are ramified in $E$ and $\infty$.

As $\operatorname{Im}(\rho)$ is finite, there exists a number field $F$ such
that, under a suitable basis,
$\operatorname{Im}(\rho)\subset\operatorname{GL}_{n}(F)$. Let
$\mathcal{O}_{F}$ be the ring of integers of $F$ and $\mathfrak{p}$ be a prime
of $\mathcal{O}_{F}$. Let $F_{\mathfrak{p}}$ be the completion of $F$ at
$\mathfrak{p}$, $\mathcal{O}$ be its ring of integers and $\mathbb{F}$ be its
residue field. Let $p$ be the rational prime lying below $\mathfrak{p}$ and
let $S_{\mathfrak{p}}\mathrel{\mathop{\mathchar 58\relax}}=S\cup\\{p\\}$.
Then, under a suitable basis, the representation
$\rho_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S_{\mathfrak{p}}}\to\operatorname{GL}_{n}(F_{\mathfrak{p}})$,
obtained by composing $\rho\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S_{\mathfrak{p}}}\to\operatorname{GL}_{n}(F)$ with the
map $F\to F_{\mathfrak{p}}$ induced by completion, takes values in
$\operatorname{GL}_{n}(\mathcal{O})$.

Recall that we have fixed $p$ to be an odd prime. Moreover, assume that
$p\nmid|G|.h_{E}$, $p$ is unramified in $E$ and
$p\geq\text{max}\\{\ell^{|G|}\mid\ell\in S\\}$. Then, the residual
representation $\bar{\rho}_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S_{\mathfrak{p}}}\to\mathrm{GL}_{n}(\mathbb{F})$
obtained by reducing $\rho_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S_{\mathfrak{p}}}\to\operatorname{GL}_{n}(\mathcal{O})$
modulo the maximal ideal of $\mathcal{O}$ is absolutely irreducible (see
[BGKK18, Section 6.1] for more details). Thus it is natural to ask whether
$H^{2}(\mathbb{Q}_{S_{\mathfrak{p}}}/\mathbb{Q},\bar{\rho}_{\mathfrak{p}})=0$
for all but finitely many such primes $\mathfrak{p}$. This question is studied
in [BGKK18].

To be precise, in the setup described above, they prove:

######  Theorem 4.13 (Böckle-Guiraud-Kalyanswamy-Khare).

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_{n}(\mathbb{C})$
be a non-trivial irreducible Artin representation and let $F$ be a number
field over which $\rho$ is defined as above. Let $\mathcal{O}_{F}$ be the ring
of integers of $F$. Let $S$ be the set of primes of $\mathbb{Q}$ consisting of
primes at which $\rho$ is ramified and $\infty$. If
$H^{0}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),\rho)=0$, then
$H^{2}(\mathbb{Q}_{S_{\mathfrak{p}}}/\mathbb{Q},\bar{\rho}_{\mathfrak{p}})=0$
for all but finitely many primes $\mathfrak{p}$ of $\mathcal{O}_{F}$.

###### Proof.

This is a part of Proposition $6.6$ of [BGKK18]. ∎

Combining this theorem with Theorem 3.7, we get:

######  Theorem 4.14.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_{n}(\mathbb{C})$
be a non-trivial irreducible Artin representation and let $F$ be a number
field over which $\rho$ is defined as above. Let $\mathcal{O}_{F}$ be the ring
of integers of $F$. Suppose
$H^{0}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),\rho)=0$. Then, for all but
finitely many primes $\mathfrak{p}$ of $\mathcal{O}_{F}$, the following
assertions hold:

  1. (a)

the weak Leopoldt conjecture is true for $\rho_{\mathfrak{p}}$.

  2. (b)

The fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho_{\mathfrak{p}}^{*})/\mathbb{Q}_{\operatorname{cyc}})$
associated to $\rho_{\mathfrak{p}}^{*}$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho_{\mathfrak{p}}^{*})=0$.

######  Remark 4.15.

Note that, in [BGKK18, Proposition 6.6], it is also proved that if

$0<\dim(H^{0}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),\rho))<\dim(\rho)$

and Heuristic $6.5$ of [BGKK18] holds, then
$H^{2}(\mathbb{Q}_{S_{\mathfrak{p}}}/\mathbb{Q},\bar{\rho}_{\mathfrak{p}})=0$
for all but finitely many primes $\mathfrak{p}$ of $\mathcal{O}_{F}$.
Combining this result with Theorem 3.7, we conclude that
$\mu^{\operatorname{fn}}(\rho_{\mathfrak{p}}^{*})=0$ for all but finitely many
primes $\mathfrak{p}$ of $\mathcal{O}_{F}$ in these cases as well.

Note that the same question for Artin representations of arbitrary number
fields is analyzed in Section $6.2$ of [BGKK18].

###  4.3. Neat Representations

Let $K$ be a number field, $\mathbb{F}$ be a finite field of characteristic
$p$ and $S$ be a finite set of primes of $K$ containing all primes of $K$
dividing $p$ and all archimedean primes. Let
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{K,S}\to\operatorname{GL}_{n}(\mathbb{F})$ be a continuous,
absolutely irreducible representation. Let $\mathcal{O}$ be the ring of
integers of a finite extension of $\mathbb{Q}_{p}$ with residue field
$\mathbb{F}$. Fix a uniformizer $\varpi$ of $\mathcal{O}$.

Suppose $\bar{\rho}$ is unobstructed and let
$d=\dim_{\mathbb{F}}(H^{1}(K_{S}/K,\operatorname{Ad}\bar{\rho}))$. Then, from
[Maz89, Proposition 2], it follows that the residual representation
$\bar{\rho}$ lifts to the universal representation unramified away from $S$

$\rho^{\operatorname{univ}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}\left(\mathcal{O}\llbracket
X_{1},\cdots,X_{d}\rrbracket\right),$

which we view as a family of Galois representations. Indeed, given any
$d$-tuple $a=(a_{1},\cdots,a_{d})\in\mathcal{O}^{d}$, the $\mathcal{O}$-valued
homomorphism $\varphi_{a}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{O}\llbracket
X_{1},\cdots,X_{d}\rrbracket\rightarrow\mathcal{O}$ sending $X_{i}$ to $\varpi
a_{i}$ gives rise to a Galois representation $\rho_{a}$ given by the composite

(4.2) $\rho_{a}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\xrightarrow{\rho^{\operatorname{univ}}}\operatorname{GL}_{n}\left(\mathcal{O}\llbracket
X_{1},\cdots,X_{d}\rrbracket\right)\xrightarrow{\varphi_{a}^{*}}\operatorname{GL}_{n}(\mathcal{O}),$

where the second homomorphism is induced by $\varphi_{a}$. Since $\bar{\rho}$
is unobstructed, it follows from Theorem 4.3 that the weak Leopoldt conjecture
holds for $\operatorname{Ad}\rho_{a}$, and moreover,
$\mu^{\operatorname{fn}}(\operatorname{Ad}\rho_{a}(1))=0$. Thus, we get a
family of Galois representations parametrized by $\mathcal{O}^{d}$ for which
the $\mu$-invariant of the fine Selmer group of the adjoint representation
vanishes.

Recall that if $K=\mathbb{Q}$ and $n=2$, then we get that $d=3$ if
$\bar{\rho}$ is odd and $d=1$ if $\bar{\rho}$ is even (see Proposition 4.2).

In [Maz89], Mazur introduced the notion of _neatness_ and constructed odd
Galois representations of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
that are unobstructed and unramified away from a single prime $p$. Böckle
[B9̈9] gave a similar construction for _even_ , unobstructed Galois
representations of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. We
relate the notion of neatness to vanishing of the $\mu$-invariant of fine
Selmer groups.

Fix an absolutely irreducible residual representation
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathbb{F})$
as above, let $L/K$ be the splitting field cut out by $\bar{\rho}$. Denote by
$S^{\prime}$ the set of primes of $L$ that lie above the non-archimedean
primes of $S$ and set $G$ to be the Galois group $G=\operatorname{Gal}(L/K)$.
For a prime $w\in S^{\prime}$, denote by $L_{w}$ the completion of $L$ at $w$.
Denote the subgroups of $L^{\times}$ and $L_{w}^{\times}$ consisting of of
$p$-th roots of unity by $\mu_{p}(L)$ and $\mu_{p}(L_{w})$, respectively.
Denote the ring of integers of $L$ and $L_{w}$ by $\mathcal{O}_{L}$ and
$\mathcal{O}_{L_{w}}$, respectively.

An $\mathbb{F}_{p}[G]$-module $M$ is said to be _coprime_ to
$\operatorname{Ad}\bar{\rho}$ if the tensor product
$M\otimes_{\mathbb{F}_{p}}(\operatorname{Ad}\bar{\rho})^{\vee}$ does not
contain the identity representation.

######  Definition 4.16.

We define three Galois modules $M_{1}$, $M_{2}$ and $M_{3}$ associated to $L$
as follows:

  1. (1)

$M_{1}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{coker}\left\\{\mu_{p}(L)\rightarrow\bigoplus_{w\in
S^{\prime}}\mu_{p}(L_{w})\right\\}$;

  2. (2)

$M_{2}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{\mathcal{O}_{L}^{*}/(\mathcal{O}_{L}^{*})^{p}\rightarrow\bigoplus_{w\in
S^{\prime}}\mathcal{O}_{L_{w}}^{*}/(\mathcal{O}_{L_{w}}^{*})^{p}\right\\}$;

  3. (3)

$M_{3}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Cl}(L)\otimes\mathbb{F}_{p}$, where
$\operatorname{Cl}(L)$ is the class group of $L$.

######  Definition 4.17 (Mazur).

With notation as above, $\bar{\rho}$ is said to be _neat_ if $p$ does not
divide the cardinality of $\operatorname{Im}(\bar{\rho})$ and $M_{1}$, $M_{2}$
and $M_{3}$ are coprime to $\operatorname{Ad}\bar{\rho}$.

See [Maz89, Section 1.12] for more details.

######  Theorem 4.18 (Mazur).

Let $\mathbb{F}$ be a finite field of characteristic $p$, $K$ be a number
field and $S$ be a set of primes of $K$ containing all primes above $p$ and
all archimedean primes. Let $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathbb{F})$
be a _neat_ residual representation. Then $\bar{\rho}$ is _unobstructed_.

###### Proof.

The reader is referred to the proof of [Maz89, Proposition 7]. ∎

######  Corollary 4.19.

Let $\bar{\rho}$ be a neat Galois representation as in Theorem 4.18, let
$d=\dim_{\mathbb{F}}(H^{1}(K_{S}/K,\operatorname{Ad}\bar{\rho}))$ and let
$\rho_{a}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$
be the $\mathcal{O}$-deformation of $\bar{\rho}$ associated to
$a=(a_{1},\cdots,a_{d})\in\mathcal{O}^{d}$ as above (see (4.2) for the
definition of $\rho_{a}$). Then, the following assertions hold:

  1. (1)

the weak Leopoldt conjecture is true for $\operatorname{Ad}\rho_{a}$,

  2. (2)

the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\operatorname{Ad}\rho_{a}(1))/K_{\operatorname{cyc}})$
associated to $\operatorname{Ad}\rho_{a}(1)$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\operatorname{Ad}\rho_{a}(1))=0$. If $n=2$, then the
same assertion holds for
$\operatorname{Sym}^{2}(\rho_{a})\otimes(\det(\rho_{a}))^{-1}\chi_{p}$.

###### Proof.

The result is a direct consequence of Theorem 4.18 and Theorem 4.3. ∎

In section 1.13 of _loc.cit._ , Mazur constructs examples of $2$-dimensional
neat, odd Galois representations associated to certain $S_{3}$-extensions of
$L/\mathbb{Q}$ ( which means $d=3$). The examples give a family of odd
representations $\bar{\rho}$ for which the above Corollary applies. More
specifically, let $p$ be a prime number which can be represented as
$27+4a^{3}$, where $a$ is an integer. Let $L$ be the splitting field of
$f(x)\mathrel{\mathop{\mathchar 58\relax}}=x^{3}+ax+1$, the Galois group
$\operatorname{Gal}(L/\mathbb{Q})$ is isomorphic to $S_{3}$. The discriminant
of $f(x)$ is $-p$ and therefore, $p$ is the only prime that is ramified in
$L$. There is a natural inclusion of $S_{3}$ into
$\operatorname{GL}_{2}(\mathbb{F}_{p})$, via which we obtain a Galois
representation $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$.
We refer to _loc. cit._ for further details. Such Galois representations are
referred to as _special $S_{3}$-representations_ since they are unobstructed,
i.e.,
$H^{2}(\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}},\operatorname{Ad}\bar{\rho})=0$.
Since $\bar{\rho}$ is an $S_{3}$-representation, it is easy to see that
$\bar{\rho}$ is a direct summand of $\operatorname{Ad}\bar{\rho}$. Hence, it
follows that
$H^{2}(\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}},{\bar{\rho}})=0$. Note
that $\bar{\rho}$ is self-dual i.e. $\bar{\rho}^{\vee}=\bar{\rho}$. Therefore,
it follows from Theorem 3.7 that for any representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$
that lifts $\bar{\rho}$, the following assertions hold:

  1. (1)

the weak Leopoldt conjecture is true for $\rho$,

  2. (2)

the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho(1))/\mathbb{Q}_{\operatorname{cyc}})$
associated to $\rho(1)$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho(1))=0$.

Similarly in [B9̈9, Section 3.1], Böckle constructs examples of
$2$-dimensional neat, even Galois representations associated to certain
totally real $S_{3}$-extensions of $L/\mathbb{Q}$ (which means $d=1$). The
examples give a family of even representations $\bar{\rho}$ for which the
above Corollary applies. To be precise, let $p$ be a prime number which can be
represented as $4a^{6}-27$, where $a\geq 2$ is an integer and $L$ be the
splitting field of $f(x)\mathrel{\mathop{\mathchar 58\relax}}=x^{3}-a^{2}x-1$.
So the Galois group $\operatorname{Gal}(L/\mathbb{Q})$ is isomorphic to
$S_{3}$. The discriminant of $f(x)$ is $p$ and therefore, $p$ is the only
prime that is ramified in $L$. Using the arguments of the previous paragraph,
we obtain a Galois representation $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$.
In [B9̈9, Theorem 3.1], Böckle proves that $\bar{\rho}$ is neat and hence,
unobstructed if the following hypotheses are satisfied:

  1. (1)

$p$ satisfies the Ankeney-Artin-Chowla conjecture,

  2. (2)

$\left(u\left(\dfrac{3}{2a^{2}}\right)-1\right)(2a^{3}+9)/6-\left(u\left(\dfrac{3}{3-2a^{3}}\right)-1\right)\not\equiv\dfrac{1}{243}(4a^{6}-27)(9/4+a^{3})\pmod{(4a^{6}-27)^{2}}$,
where for $x\in\mathbb{Z}_{p}$, $u(x)\mathrel{\mathop{\mathchar
58\relax}}=x/x^{p}$.

It is verified in _loc.cit._ that these hypotheses are satisfied for all $108$
primes $p$ of the form $4a^{6}-27$ with $2\leq a\leq 1000$ (see [B9̈9, Section
3.1] for more details). As noted above, $\bar{\rho}$ is a direct summand of
$\operatorname{Ad}\bar{\rho}$ in this case as well. Therefore, we conclude,
using the same arguments as above, that for any representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},\\{p,\infty\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$
that lifts $\bar{\rho}$, the following assertions hold:

  1. (1)

the weak Leopoldt conjecture is true for $\rho$.

  2. (2)

The fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho(1))/\mathbb{Q}_{\operatorname{cyc}})$
associated to $\rho(1)$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho(1))=0$.

##  5\. Residually dihedral representations

In this section, we study residually dihedral Galois representations that
arise from modular forms to illustrate Theorem 3.8.

To be precise, let $p$ be an odd prime, $\mathbb{F}$ be a finite field of
characteristic $p$, $S$ be a finite set of primes of $\mathbb{Q}$ containing
$p$ and $\infty$ and $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ be a
continuous representation. We assume that $\bar{\rho}$ is dihedral, which is
to say that there exists a quadratic extension $K$ of $\mathbb{Q}$ and a
character $\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K,S^{\prime}}\rightarrow\mathbb{F}^{\times}$,
where $S^{\prime}$ is the set of primes of $K$ that lie above the primes in
$S$, such that $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathbb{F})$
is isomorphic to the induced representation
$\operatorname{Ind}_{\operatorname{G}_{K,S^{\prime}}}^{\operatorname{G}_{\mathbb{Q},S}}(\psi)$.
For the ease of notation, we will denote this representation by
$\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$.

Let $\mathcal{O}^{\prime}$ be the ring of integers of a finite extension of
$\mathbb{Q}_{p}$ with residue field $\mathbb{F}$ and fix a uniformizer
$\varpi$ of $\mathcal{O}^{\prime}$. We say that a representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathcal{O}^{\prime})$ is
_residually dihedral_ if the residual representation
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ obtained by
reducing $\rho$ modulo $\varpi$ is dihedral as described above. Note that a
residually dihedral representation $\rho$ is not necessarily dihedral.

Now suppose $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ is dihedral
which means
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$
for some character $\psi\mathrel{\mathop{\mathchar
58\relax}}G_{K,S^{\prime}}\to\mathbb{F}^{\times}$. Then the projective image
of $\bar{\rho}$ (i.e. the image of $\bar{\rho}$ under the natural surjective
map $\operatorname{GL}_{2}(\mathbb{F})\to\operatorname{PGL}_{2}(\mathbb{F})$
obtained by going modulo scalars) is either a dihedral group or a cyclic
group. Moreover, $\bar{\rho}$ is absolutely irreducible if and only if the
projective image of $\bar{\rho}$ is a dihedral group.

In addition to being dihedral, now assume $\bar{\rho}$ is also odd and
absolutely irreducible. Combining the work of Khare and Wintenberger on
Serre’s conjecture ([KW09, Theorem 9.1]) and work of Kisin ([Kis09, Theorem
0.1]), we see that $\bar{\rho}$ lifts to a $p$-adic modular Galois
representation $\rho_{f,\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathcal{O})$
without increasing the set $S$ of ramified primes for the representation.
There is however, always a natural choice of lift
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(W(\mathbb{F}))$
(where $W(\mathbb{F})$ is the ring of Witt vectors of $\mathbb{F}$), letting
$\tilde{\psi}$ denote the Teichmüller lift of $\psi$, we see that
$\rho=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\tilde{\psi})$
is a lift of $\bar{\rho}$.

In the setting of dihedral representations $\bar{\rho}$ as above, we will
provide explicit criteria for the conditions in Theorem 3.8 to be satisfied.
As a result, we obtain that if $\bar{\rho}$ satisfies these criteria, then for
all characteristic $0$ lifts $\rho$ of $\bar{\rho}$, and in particular for the
lifts arising from modular newforms, the weak Leopoldt conjecture holds for
$\rho^{*}$ and $\mu^{\operatorname{fn}}(\rho)=0$ (see Theorem 5.8). Note that
lifts of such representations arising from CM modular forms have been studied
in [BN18].

######  Lemma 5.1.

Let $\mathcal{O}$ be the ring of integers of a finite extension of
$\mathbb{Q}_{p}$ with residue field $\mathbb{F}$. Let
$\rho\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathcal{O})$ be a Galois
representation such that its residual representation
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$
is dihedral as above. Let $L$ be the extension of $\mathbb{Q}$ cut out by
$\bar{\rho}$ and let $G=\operatorname{Gal}(L/\mathbb{Q})$. Then
$H^{1}(G,\mathbf{V}_{\bar{\rho}})=0$ (i.e. condition (1) in Theorem 3.8 is
satisfied).

###### Proof.

Observe that the order of $G$ is coprime to $p$. Therefore, it follows that
$H^{1}(G,\mathbf{V}_{\bar{\rho}})=0$. ∎

Let
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$
and let $L$ be the field extension of $\mathbb{Q}$ which is fixed by the
kernel of $\bar{\rho}$ as above. Note that $K\subset L$. Let $K_{1}$ and
$L_{1}$ be the mod-$p$ Hilbert class fields of $K$ and $L$, respectively. In
other words, $\operatorname{Gal}(K_{1}/K)$ and $\operatorname{Gal}(L_{1}/L)$
are identified with the mod-$p$ quotients of the class groups of $K$ and $L$,
respectively. Now $S$ is a finite set of primes of $\mathbb{Q}$ containing
$p$, $\infty$ and the primes $\ell\neq p$ at which $\bar{\rho}$ is ramified.
So $S$ contains the primes that ramify in $L$. Set $K_{1,S}$ (resp. $L_{1,S}$)
to be the maximal subextension of $K_{1}$ (resp. $L_{1}$) in which the primes
of $K$ (resp. $L$) above $S$ split completely. Thus,
$\operatorname{Gal}(K_{1,S}/K)$ (resp. $\operatorname{Gal}(L_{1,S}/L)$) is
identified with the maximal quotient of
$\operatorname{Cl}(K)/p\operatorname{Cl}(K)$ (resp.
$\operatorname{Cl}(L)/p\operatorname{Cl}(L)$) such that the primes of $K$
(resp. $L$) above $S$ split completely in the corresponding subextension. The
fields defined fit into the following diagram:

$K$$L$$K_{1,S}$$L_{1,S}$$K_{1}$$L_{1}$

Note that all fields in the diagram are Galois over $\mathbb{Q}$ and that
$G=\operatorname{Gal}(L/\mathbb{Q})$ acts naturally on
$H_{L}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(L_{1}/L)$ and
$H_{L,S}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(L_{1,S}/L)$.
Here, the action is described as follows. Given $v\in H_{L}$ and $\sigma\in
G$, pick a lift $\tilde{\sigma}$ of $\sigma$ to
$\operatorname{Gal}(L_{1}/\mathbb{Q})$. Since $H_{L}$ is a normal subgroup of
$\operatorname{Gal}(L_{1}/\mathbb{Q})$ it follows that
$\tilde{\sigma}v\tilde{\sigma}^{-1}$ belongs to $H_{L}$. Since $H_{L}$ is
abelian, $\tilde{\sigma}v\tilde{\sigma}^{-1}$ is independent of the choice of
lift $\tilde{\sigma}$ of $\sigma$. The action is defined by setting
$\sigma\cdot v\mathrel{\mathop{\mathchar
58\relax}}=\tilde{\sigma}v\tilde{\sigma}^{-1}$ for any choice of lift
$\tilde{\sigma}$ of $\sigma$. The module $H_{L,S}$ is a $G$-stable quotient of
$H_{L}$, i.e., a quotient by a $G$-stable submodule.

######  Lemma 5.2.

Suppose that
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ is a dihedral
representation and let $L$, $K_{1,S}$ and $L_{1,S}$ be the fields defined as
above. Assume that $K_{1,S}\cdot L=L_{1,S}$. Let
$G=\operatorname{Gal}(L/\mathbb{Q})$ and $M$ be an irreducible
$\mathbb{F}_{p}[G]$-module such that $\operatorname{dim}M>1$. Then, with
respect to notation above,

$\operatorname{Hom}_{G}(H_{L,S},M)=0.$

###### Proof.

Since it is assumed that $K_{1,S}\cdot L=L_{1,S}$, it follows that the
$G$-action on $H_{L,S}$ factors through an action of the quotient
$\operatorname{Gal}(K/\mathbb{Q})$. Indeed, both $K_{1,S}$ and $L$ are abelian
extensions of $K$ and hence, so is $L_{1,S}=L.K_{1,S}$. Therefore,
$\operatorname{Gal}(L/K)$ acts trivially on $\operatorname{Gal}(L_{1,S}/L)$.
As $\operatorname{Gal}(K/\mathbb{Q})$ is abelian, it follows that any
irreducible representation of $\operatorname{Gal}(K/\mathbb{Q})$ is one-
dimensional. The order of $G$ is coprime to $p$ and hence, it follows that
$H_{L,S}$ decomposes into $1$-dimensional representations of $G$ that factor
through the action of $\operatorname{Gal}(K/\mathbb{Q})$. Combining this with
the assumption that $M$ is an irreducible $G$-module with
$\operatorname{dim}M>1$, we get that $\operatorname{Hom}_{G}(H_{L,S},M)=0$. ∎

######  Lemma 5.3.

Assume that $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ is dihedral
i.e.
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$,
and that its projective image is a dihedral group. Let $L$, $K_{1,S}$ and
$L_{1,S}$ be the fields defined as above and let
$G=\operatorname{Gal}(L/\mathbb{Q})$. Moreover, assume that the condition
$K_{1,S}\cdot L=L_{1,S}$ of Lemma 5.2 is satisfied. Then,

$\operatorname{Hom}_{G}(H_{L,S},V_{\bar{\rho}})=0,$

i.e., condition (2) of Theorem 3.8 is satisfied for $\bar{\rho}$.

###### Proof.

Since the projective image of $\bar{\rho}$ is dihedral, it follows that
${\bar{\rho}}$ is an irreducible representation of $G$ of dimension $2$.
Hence, the lemma follows directly from Lemma 5.2. ∎

We shall provide some examples in which the condition $K_{1,S}\cdot L=L_{1,S}$
is satisfied. First, we obtain sufficient conditions for condition (3) in
Theorem 3.8 to hold. Let $\nu$ be a generator of
$\operatorname{Gal}(K/\mathbb{Q})$ and $\psi^{\nu}$ be defined as follows:

$\psi^{\nu}(x)\mathrel{\mathop{\mathchar
58\relax}}=\psi(\tilde{\nu}x\tilde{\nu}^{-1}),$

where $\tilde{\nu}$ is a lift of $\nu$ in $G_{\mathbb{Q},S}$. Note that in the
above formula, $\psi(\tilde{\nu}x\tilde{\nu}^{-1})$ is independent of the
choice of the lift $\tilde{\nu}$ of $\nu$.

######  Lemma 5.4.

Assume that
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$.
Then the projective image of $\bar{\rho}$ is a dihedral group if and only if
$\psi\neq\psi^{\nu}$. Moreover, the image of $\bar{\rho}$ is itself a non-
abelian dihedral group if and only if $\psi^{-1}=\psi^{\nu}$ and
$\psi^{\nu}\neq\psi$.

###### Proof.

Recall that the projective image of
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$
is dihedral if and only if $\bar{\rho}$ is irreducible.

Let $V$ be the vector space underlying $\bar{\rho}$. As
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$,
we get, after choosing a suitable basis of $V$, that
$\bar{\rho}(g)=\begin{pmatrix}\psi(g)&0\\\ 0&\psi^{\nu}(g)\end{pmatrix}$ for
all $g\in G_{K}$ and $\bar{\rho}(\tilde{\nu})=\begin{pmatrix}0&1\\\
a&0\end{pmatrix}$ for some $a\in\mathbb{F}.$ Now
$G_{\mathbb{Q}}=G_{K}\cup\tilde{\nu}G_{K}$. Therefore, $\bar{\rho}$ is
reducible if and only if there exists a non-zero $v\in V$ which is a common
eigenvector for $\bar{\rho}(\tilde{\nu})$ and $\bar{\rho}(g)$ for all $g\in
G_{K}$. If $\psi=\psi^{\nu}$, then such an eigenvector clearly exists and
hence, $\bar{\rho}$ is reducible.

If $\psi\neq\psi^{\nu}$, then there exists a $g_{0}\in G_{K}$ such that
$\psi(g_{0})\neq\psi^{\nu}(g_{0})$. So if a non-zero vector $v\in V$ is an
eigenvector for $\bar{\rho}(g_{0})$, then either
$\bar{\rho}(g_{0})v=\psi(g_{0})v$ or $\bar{\rho}(g_{0})v=\psi^{\nu}(g_{0})v$.
Let $w=\bar{\rho}(\tilde{\nu})v$. Then $w\neq 0$. If
$\bar{\rho}(g_{0})v=\psi(g_{0})v$, then
$\bar{\rho}(g_{0})w=\psi^{\nu}(g_{0})w$. As $\psi(g_{0})\neq\psi^{\nu}(g_{0})$
and $w\neq 0$, it follows that $w$ is not a scalar multiple of $v$. This means
that $v$ is not an eigenvector of $\bar{\rho}(\tilde{\nu})$. If
$\bar{\rho}(g_{0})v=\psi^{\nu}(g_{0})v$, then we similarly conclude that $v$
is not an eigenvector of $\bar{\rho}(\tilde{\nu})$. Therefore, if
$\psi\neq\psi^{\nu}$, then $\bar{\rho}$ is irreducible. This finishes the
proof of the first part of the lemma.

Note that the image of $\bar{\rho}$ is a non-abelian dihedral group if and
only if $\bar{\rho}(G_{K})$ is a cyclic group of order $>2$ and
$\bar{\rho}(\tilde{\nu}g\tilde{\nu}^{-1})=\bar{\rho}(g^{-1})$ for all $g\in
G_{K}$. Further, $\bar{\rho}(\tilde{\nu}g\tilde{\nu}^{-1})=\bar{\rho}(g^{-1})$
for all $g\in G_{K}$ if and only if $\psi^{\nu}=\psi^{-1}$. Moreover, if this
holds, then $\bar{\rho}(G_{K})$ is cyclic. If $\psi^{\nu}=\psi^{-1}$, then
$\bar{\rho}(G_{K})$ is cyclic of order $>2$ if and only if
$\psi^{-1}\neq\psi$. Therefore, we conclude that the image of $\bar{\rho}$ is
a non-abelian dihedral group if and only if $\psi^{\nu}=\psi^{-1}$ and
$\psi^{\nu}\neq\psi$. ∎

######  Lemma 5.5.

Assume that
$\bar{\rho}=\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$.
Let $\ell$ be a prime and $L$ be the extension of $\mathbb{Q}$ cut out by
$\ker(\bar{\rho})$. Suppose that the trivial representation is a
subrepresentation of $\bar{\rho}_{|\ell}$. Let $w\mid\ell$ be a choice of
prime of $K$ that lies above $\ell$. Then, at least one of the following
conditions hold:

  1. (1)

$\ell$ splits in $K$ and either $\psi_{|w}=1$ or $\psi^{\nu}_{|w}=1$,

  2. (2)

$\ell$ is either inert or ramified in $K$, and the unique prime $w$ of $K$
lying above $\ell$ splits completely in $L$.

###### Proof.

Assume that $\ell$ splits in $K$. Then, we have that
$\bar{\rho}_{|w}=\left({\begin{array}[]{cc}\psi_{|w}&0\\\ 0&\psi^{\nu}_{|w}\\\
\end{array}}\right)$, and the result follows in this case.

Now suppose $\ell$ is either inert or ramified in $K$. Then $w$ is the unique
prime of $K$ lying above $\ell$ and we can choose $\tilde{\nu}\in G_{\ell}$.
As the trivial representation is a subrepresentation of $\bar{\rho}_{|\ell}$,
it follows that either $\psi_{|w}=1$ or $\psi^{\nu}_{|w}=1$.

As $\tilde{\nu}\in G_{\ell}$ and $G_{w}$ is normal in $G_{\ell}$, it follows
that $\tilde{\nu}G_{w}\tilde{\nu}^{-1}=G_{w}$. Therefore, it follows that if
$\psi_{|w}=1$, then $\psi^{\nu}_{|w}=1$ and vice versa. This finishes the
proof of the Lemma.

∎

Combining Lemmas 5.1, 5.3 and 5.5, we obtain the following result.

######  Theorem 5.6.

Let $\mathcal{O}$ be the ring of integers in a finite extension of
$\mathbb{Q}_{p}$ with residue field $\mathbb{F}$ and $S$ be a set of primes of
$\mathbb{Q}$ containing $p$ and $\infty$. Consider a Galois representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathcal{O})$
and let $\bar{\rho}$ be its residual representation. Assume that the following
assertions hold:

  1. (1)

For some quadratic field extension $K/\mathbb{Q}$ and a character
$\psi\mathrel{\mathop{\mathchar
58\relax}}G_{K,S^{\prime}}\rightarrow\mathbb{F}^{\times}$ (where $S^{\prime}$
is the set of primes of $K$ lying above primes in $S$), the residual
representation $\bar{\rho}$ is isomorphic to the induced representation
$\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi)$.
In addition, we assume that the projective image of $\bar{\rho}$ is a non-
abelian dihedral group.

  2. (2)

With respect to above notation, we assume that $K_{1,S}\cdot L=L_{1,S}$ (cf.
Lemma 5.3).

  3. (3)

At each prime $\ell\in S$, the local representation $\bar{\rho}_{|\ell}$ does
not have the trivial representation as a subrepresentation.

Then, the following assertions hold:

  1. (a)

the weak Leopoldt conjecture is true for $\rho^{*}$,

  2. (b)

the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/\mathbb{Q}_{\operatorname{cyc}})$
associated to $\rho$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho)=0$.

###### Proof.

Note that condition (3) of Theorem 3.8 is satisfied by assumption (3) above.
It follows from Lemmas 5.1 and 5.3 that the remaining conditions of Theorem
3.8 are satisfied, and thus the result follows. ∎

We reiterate that Lemma 5.5 gives sufficient conditions for condition (3) of
Theorem 5.6 to hold.

Note that the above theorem involves conditions on the residual representation
$\bar{\rho}$ and the set of primes $S$. Via purely Galois theoretic
computations, we construct a class of examples of residual representations
$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathbb{F})$
satisfying the above conditions.

Note that the condition $K_{1,S}\cdot L=L_{1,S}$ is satisfied if $K_{1}\cdot
L=L_{1}$. Examples satisfying this condition are constructed in [DW20, section
5.2] (see [DW20, Table 1]). We will adapt the approach of [DW20, Section 5.2]
to our context to obtain a class of examples satisfying the conditions of
Theorem 5.6. We will now briefly outline our strategy to get examples.

Let $K$ be an imaginary quadratic extension of $\mathbb{Q}$ and $p$ be an odd
prime. Assume that there is an odd prime $q\neq p$ such that $q$ divides the
class number of $K$. Let $N$ be the maximal extension of $K$ contained in the
Hilbert class field of $K$ with odd degree of extension
$[N\mathrel{\mathop{\mathchar 58\relax}}K]$. By class field theory, $N$ is a
Galois extension of $K$ and $\operatorname{Gal}(N/K)$ is identified with the
odd part of $\operatorname{Cl}(K)$ i.e. with
$\operatorname{Cl}(K)/\operatorname{Cl}_{2}(K)$, where
$\operatorname{Cl}_{2}(K)$ is the $2$-Sylow subgroup of
$\operatorname{Cl}(K)$. Note that $\operatorname{Cl}_{2}(K)$ is stable under
the action of $\operatorname{Gal}(K/\mathbb{Q})$ on $\operatorname{Cl}(K)$.
Therefore, using class field theory, we conclude that $N$ is Galois over
$\mathbb{Q}$.

Let $\nu$ be a lift of the generator of $\operatorname{Gal}(K/\mathbb{Q})$ in
$G_{0}=\operatorname{Gal}(N/\mathbb{Q})$ with order $2$. Since
$|\operatorname{Gal}(N/K)|$ is odd, such a lift does exist. Let
$H_{0}=\operatorname{Gal}(N/K)$.

######  Lemma 5.7.

Under the notation and set-up established above, $\nu g\nu^{-1}=g^{-1}$ for
all $g\in H_{0}$.

###### Proof.

As $H_{0}$ is normal in $G_{0}$, $\nu H_{0}\nu^{-1}=H_{0}$. Let $H_{1}=\\{h\in
H_{0}\mid\nu h\nu^{-1}=h\\}$ and $H_{2}=\\{h\in H_{0}\mid\nu
h\nu^{-1}=h^{-1}\\}$. So $H_{1}$ and $H_{2}$ are subgroups of $H_{0}$ and
moreover, they are normal in $G_{0}$. As $H_{0}$ has odd order, we have
$H_{1}\cap H_{2}=\\{1\\}$. We now claim that $H_{0}=H_{1}H_{2}$.

Let $g\in H_{0}$ and suppose $\nu g\nu^{-1}=g^{\prime}\in H_{0}$. Let
$g_{1}=gg^{\prime}$ and $g_{2}=g(g^{\prime})^{-1}$. So $g^{2}=g_{1}g_{2}$. As
$\nu^{2}=1$, we get that $\nu g_{1}\nu^{-1}=g_{1}$ and $\nu
g_{2}\nu^{-1}=g_{2}^{-1}$. So $g_{1}\in H_{1}$ and $g_{2}\in H_{2}$. Thus
$g^{2}\in H_{1}H_{2}$ for all $g\in H_{0}$. As $H_{0}$ is abelian and has odd
order, every element of $H_{0}$ is a square and hence, $H_{0}=H_{1}H_{2}$.

Suppose $|H_{1}|=k>1$. Note that $G^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=G_{0}/H_{2}$ is an abelian group of order $2|H_{1}|=2k$. So if
$N^{\prime}\subset N_{0}$ is the fixed field of $H_{2}$, then $N^{\prime}$ is
an abelian extension of $\mathbb{Q}$ with
$\operatorname{Gal}(N^{\prime}/\mathbb{Q})=G^{\prime}$. Note that $N^{\prime}$
is an unramified extension of $K$. Therefore, a prime $\ell$ of $\mathbb{Q}$
is ramified in $N^{\prime}$ if and only if it is ramified in $K$. As $K$ is
quadratic over $\mathbb{Q}$, the order of the inertia group at any rational
prime $\ell$ in $\operatorname{Gal}(N^{\prime}/\mathbb{Q})$ is at most $2$.

As $k$ is odd and $G^{\prime}$ is abelian, $G^{\prime}$ contains a unique
subgroup $H^{\prime}$ of order $2$. Thus the subfield of $N^{\prime}$ fixed by
$H^{\prime}$ is an unramified abelian extension of $\mathbb{Q}$ of degree
$k>1$. This gives us a contradiction. Hence, $|H_{1}|=1$ which means
$H_{0}=H_{2}$ proving the lemma. ∎

Let $L^{\prime}$ be a cyclic, unramified extension of $K$ with
$[L^{\prime}\mathrel{\mathop{\mathchar 58\relax}}K]=n>1$, $n$ odd (i.e.
$L^{\prime}\subset N$) and $p\nmid n$. Note that such an extension exists as
we are assuming that the class number of $K$ is divisible by an odd prime
$q\neq p$. From Lemma 5.7, we conclude that $L^{\prime}$ is Galois over
$\mathbb{Q}$, with $\operatorname{Gal}(L^{\prime}/\mathbb{Q})=D_{n}$, the
dihedral group with $2n$ elements. As $n$ is odd,
$\operatorname{Gal}(L^{\prime}/\mathbb{Q})$ is non-abelian.

######  Theorem 5.8.

Let $\mathbb{F}$ be a field of characteristic $p$. Let $K$ be an imaginary
quadratic field with class number divisible by an odd prime different from
$p$. Let $L^{\prime}$ be a cyclic extension of $K$ contained in the Hilbert
class field of $K$ with $[L^{\prime}\mathrel{\mathop{\mathchar
58\relax}}K]=n>1$, $n$ odd and $p\nmid n$. Let $\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L^{\prime}/K)\rightarrow\mathbb{F}^{\times}$ and
$\eta\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q}}\to\mathbb{F}^{\times}$ be non-trivial characters. Set
$\bar{\rho}$ to denote the induced representation
$\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi\eta)$
of $G_{\mathbb{Q}}$. Let $L$ be the extension of $\mathbb{Q}$ cut out by
$\bar{\rho}$. Let $S$ be a set of prime numbers containing $p$, $\infty$ and
the primes which are ramified in $L$. Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathcal{O})$
be a lift of $\bar{\rho}$. Assume that the following conditions are satisfied:

  1. (1)

If $\ell\in S$, $\ell$ is split in $K$ and $w$ is a fixed place of $K$ lying
above $\ell$, then $|\psi(G_{w})|\neq|\eta(G_{\ell})|$,

  2. (2)

If $\ell$ is either inert or ramified in $K$ and $w$ is the unique place of
$K$ lying above $\ell$, then $\eta_{|\ell}$ is non-trivial and is not the
quadratic character of $G_{\ell}$ corresponding to $K_{w}$,

  3. (3)

$L_{1,S}=L\cdot K_{1,S}$.

Then, the following assertions hold:

  1. (a)

the weak Leopoldt conjecture is true for $\rho^{*}$,

  2. (b)

the fine Selmer group
$\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/\mathbb{Q}_{\operatorname{cyc}})$
associated to $\rho$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho)=0$.

###### Proof.

Note that, under the hypotheses of the theorem above, conditions (1) and (2)
of Theorem 5.6 hold. Therefore, by Theorem 5.6, it suffices to check that _no_
condition of Lemma 5.5 is satisfied. Let $\ell\in S$ and fix a place $w$ of
$K$ lying above $\ell$.

Suppose $\ell$ is split in $K$. Now condition (1) implies that
$\psi\eta_{|w}\neq 1$. As $\operatorname{Gal}(L^{\prime}/\mathbb{Q})$ is a
non-abelian dihedral group, it follows that $\psi^{\nu}=\psi^{-1}$. As $\eta$
is a character of $G_{\mathbb{Q}}$, we get that $\eta^{\nu}=\eta$. Therefore,
condition (1) implies that $(\psi\eta)^{\nu}_{|w}=\psi^{-1}\eta_{|w}\neq 1$.
Thus condition (1) of Lemma 5.5 is not satisfied.

Now suppose $\ell$ is either inert or ramified in $K$. As $L^{\prime}$ is an
unramified extension of $K$ and $K_{w}$ is a quadratic extension of
$\mathbb{Q}_{\ell}$, it follows that the image of $G_{\ell}$ in
$\operatorname{Gal}(L^{\prime}/\mathbb{Q})$ is abelian. As
$\operatorname{Gal}(L^{\prime}/\mathbb{Q})=D_{n}$ with $n$ odd, it follows
that the image of $G_{\ell}$ in $\operatorname{Gal}(L^{\prime}/\mathbb{Q})$ is
an abelian group of order $2$. Hence, $w$ is totally split in $L^{\prime}/K$.
Now as $\eta_{|\ell}$ is not the quadratic character corresponding to $K_{w}$
and $\eta_{|\ell}\neq 1$, it follows that $\eta_{|w}\neq 1$ and hence, $w$ is
not completely split in $L/K$. Therefore, condition (2) of Lemma 5.5 is not
satisfied. This finishes the proof of the theorem. ∎

Examples: Let $p=3$ and $K=\mathbb{Q}(\sqrt{-D})$ with

$D\in\\{239,971,1259,2243,2699,2843\\}.$

We verify, using LMFDB [LMF22], that the class number of $K$ is $15$ (see
Imaginary quadratic number fields with class number 15). Let $L^{\prime}$ be
the unramified, abelian extension of $K$ of degree $5$. Let $\eta$ be the mod
$3$ cyclotomic character of $\bar{\chi}_{3}$ of $G_{\mathbb{Q}}$. So
$L=L^{\prime}(\mu_{3})$. We verify, from LMFDB [LMF22], that $L_{1}=L\cdot
K_{1}$ which means condition (3) of Theorem 5.8 is satisfied for all finite
set $S$ of primes of $\mathbb{Q}$. Let

$\mathcal{S}_{1}=\\{\ell\text{ prime }\mid\ell\equiv 1\pmod{3}\text{ and
}\ell\text{ is split in }K\text{ but not totally split in }L^{\prime}\\}$

and

$\mathcal{S}_{2}=\\{\ell\text{ prime }\mid\ell\equiv-1\pmod{3}\text{ and
}\ell\text{ is not inert in }K\\}.$

Let $\mathcal{S}=\mathcal{S}_{1}\cup\mathcal{S}_{2}$. Note that, by Chebotarev
density theorem, $\mathcal{S}$ is an infinite set with Dirichlet density
$9/20$. Let $S^{\prime}$ be a finite subset of $\mathcal{S}$ and let
$S=S^{\prime}\cup\\{3,D,\infty\\}$. Then $S$ satisfies conditions (1) and (2)
of Theorem 5.8. Let $\mathbb{F}$ be the finite field with $81$ elements and
let $\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L/K)\rightarrow\mathbb{F}^{\times}$ be a non-
trivial character. Let $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{F})$ be the induced
representation
$\operatorname{Ind}_{\operatorname{G}_{K}}^{\operatorname{G}_{\mathbb{Q}}}(\psi\bar{\chi}_{3})$
of $G_{\mathbb{Q},S}$. If $\rho\mathrel{\mathop{\mathchar
58\relax}}G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathcal{O})$ is a lift of
$\bar{\rho}$, then by Theorem 5.8, we conclude that

  1. (a)

the weak Leopoldt conjecture is true for $\rho^{*}$,

  2. (b)

the fine Selmer group
$\mathcal{R}_{3^{\infty}}(\mathbf{A}(\rho)/{\mathbb{Q}_{\operatorname{cyc}}})$
associated to $\rho$ is cotorsion over $\Lambda$ with
$\mu^{\operatorname{fn}}(\rho)=0$.

Moreover, observe that $\bar{\rho}$ is self-dual i.e.
$\bar{\rho}^{\vee}=\bar{\rho}$. Hence, we conclude that the weak Leopoldt
conjecture is true for $\rho(1)$.

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