# On the Iwasawa invariants of Artin representations

Aditya Karnataki  Chennai Mathematical Institute, H1, SIPCOT IT Park,
Kelambakkam, Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>and  Anwesh
Ray  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>
###### Abstract.

We study Iwasawa invariants associated to Selmer groups of Artin
representations, and criteria for the vanishing of the associated algebraic
Iwasawa invariants. The conditions obtained can be used to study natural
distribution questions in this context.

###### Key words and phrases:

Iwasawa theory, Artin representations, Euler characteristic, distribution
questions

###### 2020 Mathematics Subject Classification:

11R23, 11R34, 11F80 (Primary)

##  1\. Introduction

Let $p$ be an odd prime number and $\mathbb{Z}_{p}$ denote the ring of
$p$-adic integers. Let $K$ be a number field and fix an algebraic closure
$\bar{K}$ of $K$. A $\mathbb{Z}_{p}$-extension of $K$ is an infinite Galois
extension $K_{\infty}/K$ such that the Galois group
$\operatorname{Gal}(K_{\infty}/K)$ is isomorphic to $\mathbb{Z}_{p}$ as a
topological group. Let $K_{n}\subset K_{\infty}$ be the extension of $K$ for
which $\operatorname{Gal}(K_{n}/K)\simeq\mathbb{Z}/p^{n}\mathbb{Z}$, and
$h_{k}(K_{n})$ denote the cardinality of the $p$-primary part of the class
number of $K_{n}$. Writing $h_{p}(K_{n})=p^{e_{n}}$, Iwasawa [Iwa59] showed
that for all large enough values of $n$,

$e_{n}=p^{n}\mu+n\lambda+\nu,$

where $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{Z}$ are
invariants that depend on the extension $K_{\infty}/K$ and not on $n$. The
results of Iwasawa motivated Mazur [Maz72] to study the growth properties of
the $p$-primary Selmer groups of $p$-ordinary abelian varieties in
$\mathbb{Z}_{p}$-extensions. Mazur’s results were later extended to very
general classes of ordinary Galois representations by Greenberg, cf. [G+89].

Another class of representations that are natural to consider are Artin
representations. Let $K/\mathbb{Q}$ be a finite and totally imaginary Galois
extension with Galois group $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K/\mathbb{Q})$, and let
$\rho\mathrel{\mathop{\mathchar
58\relax}}\Delta\rightarrow\operatorname{GL}_{d}(\bar{\mathbb{Q}})$ be an
irreducible Artin representation. Let $p$ be an odd prime number and let
$\sigma_{p}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$ be an embedding
and via $\sigma_{p}$ we view $\rho$ as a representation
$\rho\mathrel{\mathop{\mathchar
58\relax}}\Delta\rightarrow\operatorname{GL}_{d}(\bar{\mathbb{Q}}_{p})$. Let
$v$ denote an archimedian prime of $K$, and set $K_{v}$ to denote the $v$-adic
completion of $K$. We shall identify $\operatorname{Gal}(K_{v}/\mathbb{R})$
with a subgroup $\Delta_{v}$ of $\Delta$. Set $d^{+}$ to denote the
multiplicity of the trivial character in $\rho_{|\Delta_{v}}$ and observe that
this number is well defined and independent of the choice of the archimedian
prime $v$. When $K$ is totally real, we find that $d^{+}=d$. Let
$\mathfrak{p}$ be a prime of $K$ that lies above $p$, and let
$\Delta_{\mathfrak{p}}\subset\Delta$ be the decomposition group
$\mathfrak{p}$. Following Greenberg and Vatsal [GV20], we make the following
assumptions.

######  Assumption 1.1.

With respect to notation above, assume that

  1. (1)

$p$ does not divide $[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$,

  2. (2)

$d^{+}=1$,

  3. (3)

there exists a $1$-dimensional representation $\epsilon_{\mathfrak{p}}$ of
$\Delta_{\mathfrak{p}}$ that occurs with multiplicity $1$ in
$\rho_{|\Delta_{\mathfrak{p}}}$.

In the special case $d=2$ and $d^{+}=1$, such representations are expected to
arise from Hecke eigenforms of weight $1$ on $\Gamma_{1}(N)$, where $N$ is the
Artin conductor of $\rho$. The conjecture has been settled in various special
cases, cf. [BDSBT01, Tay03] and references therein. The choice of the
character $\epsilon_{\mathfrak{p}}$ plays a role in the definition of the
Selmer groups associated to $\rho$. Take $\mathcal{K}$ be the completion of
$K$ at $\mathfrak{p}$. There is a natural isomorphism
$\operatorname{Gal}(\mathcal{K}/\mathbb{Q}_{p})\simeq\Delta_{\mathfrak{p}}$,
and set $\epsilon$ to denote the composite

$\operatorname{Gal}(\mathcal{K}/\mathbb{Q}_{p})\xrightarrow{\sim}\Delta_{\mathfrak{p}}\xrightarrow{\epsilon_{\mathfrak{p}}}\bar{\mathbb{Q}}^{\times}.$

Let $\chi$ denote the character associated to $\rho$ and $\mathbb{Q}(\chi)$
denote the field generated by the values of $\chi$. Let $\mathcal{F}$ be the
field generated by $\mathbb{Q}_{p}$, the values of $\chi$ and the values of
$\epsilon_{\mathfrak{p}}$. We regard $\rho$ as a representation a vector space
$V$ defined over $\mathcal{F}$ and note that $\dim_{\mathcal{F}}V=d$. Let
$V^{\epsilon_{\mathfrak{p}}}$ be the the maximal $\mathcal{F}$-subspace of $V$
on which $\Delta_{\mathfrak{p}}$ acts by $\epsilon_{\mathfrak{p}}$. By
hypothesis, $\dim_{\mathcal{F}}V^{\epsilon_{\mathfrak{p}}}=1$. Let
$\mathcal{O}$ be the valuation ring in $\mathcal{F}$ and $\varpi$ be its
uniformizer. We assume that $\rho$ is irreducible, and therefore, there is a
Galois stable $\mathcal{O}$-lattice $T\subset V$. This lattice is uniquely
determined up to scaling by a constant. We consider the divisible Galois
module $D\mathrel{\mathop{\mathchar 58\relax}}=V/T$ and let
$D^{\epsilon_{\mathfrak{p}}}$ be the image of $V^{\epsilon_{\mathfrak{p}}}$ in
$D$. Let $\mathbb{Q}_{\infty}$ denote the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. Let
$\mathbb{Q}_{n}\subset\mathbb{Q}_{\infty}$ be the subextension such that
$\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})\simeq\mathbb{Z}/p^{n}\mathbb{Z}$.
We set

$S_{\chi,\epsilon}(\mathbb{Q}_{n})\mathrel{\mathop{\mathchar
58\relax}}=\ker\left\\{H^{1}(\mathbb{Q}_{n},D)\longrightarrow\prod_{v\nmid
p}H^{1}(\mathbb{Q}_{n,v}^{\operatorname{nr}},D)\times
H^{1}(\mathbb{Q}_{n,\eta_{p}}^{\operatorname{nr}},D/D^{\epsilon_{\mathfrak{p}}})\right\\},$

where $\eta_{p}$ is the unique prime that lies above $p$.

######  Theorem 1.2 (Greenberg and Vatsal).

Suppose that the Assumption 1.1 holds. Then, the Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{n})$ is finite for all $n\in\mathbb{Z}_{\geq
0}$.

###### Proof.

The above result is [GV20, Proposition 3.1]. ∎

######  Definition 1.3.

Define the Selmer group over $\mathbb{Q}_{\infty}$ to denote the direct limit
with respect to restriction maps

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{n}S_{\chi,\epsilon}(\mathbb{Q}_{n}).$

Greenberg and Vatsal show that the Selmer group is cofinitely generated and
cotorsion over the Iwasawa algebra, which is a formal power series ring over
$\mathcal{O}$ in $1$-variable. Leveraging the results of Greenberg and Vatsal,
we study the Euler characteristic formula associated to these Selmer groups
and utilize it to study explicit conditions for the vanishing of Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$. For instance, we are able to prove
that there is an explicit relationship between the vanishing of the Selmer
group and the _$p$ -rationality_ of $K$ (in the sense of [MD90]).

######  Theorem 1.4 (Theorem 3.11).

Assume that

  1. (i)

the conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$,

  3. (iii)

$K$ is $p$-rational,

  4. (iv)

$p$ does not divide the class number of $K$.

Then, $S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$.

These criteria are illustrated in two special cases, namely that of
$2$-dimensional irreducible Artin representations (cf. Theorem 3.12) and
$3$-dimensional icosahedral Artin representations of $A_{5}$ (cf. Theorem
3.13). The Theorem 3.9 gives a more refined criterion that is equivalent to
the vanishing of $S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$. The relationship
between $p$-rationality and the Iwasawa invariants of number fields has been
studied by Hajir and Maire, cf. [HM19].

For simplicity, we then specialize our discussion to $2$-dimensional Artin
representation of dihedral type. Let $L$ is an imaginary quadratic field and
$\zeta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{L}\rightarrow\bar{\mathbb{Q}}^{\times}$ is a
character and
$\rho=\operatorname{Ind}_{L}^{\mathbb{Q}}\zeta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{Q}})$
be the associated Artin representation. Set $\zeta^{\prime}$ to be conjugate
character defined by setting $\zeta^{\prime}(x)=\zeta(cxc^{-1})$, where $c$
denotes complex conjugation. Assume that $\zeta^{\prime}=\zeta^{-1}$, thus
$\rho$ is dihedral type. Let $S(\rho)$ be the set of primes $p$ such that

  1. (1)

$p$ is odd and $p\nmid[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$,

  2. (2)

$p$ splits in $L$, $p\mathcal{O}_{L}=\pi\pi^{*}$, and $\pi$ is inert in $K/L$.

It is easy to see that if $\mathfrak{p}|p$ is a prime of $K$, then, the
conditions of Assumption of 1.1 are satisfied. Let $T(\rho)$ be the set of
primes $p\in S(\rho)$ such that the Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ vanishes for all of the primes
$\mathfrak{p}$ that lie above $p$. Set
$T^{\prime}(\rho)\mathrel{\mathop{\mathchar 58\relax}}=S(\rho)\backslash
T(\rho)$. In section 4, we apply our results to study the following natural
question.

######  Question 1.5.

What can be said about the densities of the sets of primes in $T(\rho)$ and
$T^{\prime}(\rho)$?

A conjecture of Gras predicts that any number field $K$ is $p$-rational at all
but finitely many primes $p$. This has conjectural implications to the
vanishing of Selmer groups for all but finitely many primes a compatible
family of Artin representations. It the case when $K/\mathbb{Q}$ is an
$S_{3}$-extension, we prove certain unconditional results by leveraging a
result of Maire and Rougnant [MR20] on the $p$-rationality of
$S_{3}$-extensions of $\mathbb{Q}$.

######  Theorem 1.6 (Theorem 4.10).

Let $K/\mathbb{Q}$ be an imaginary $S_{3}$ extension and $\rho$ be a
$2$-dimensional Artin representation that factors through
$\operatorname{Gal}(K/\mathbb{Q})$. Then,

$\\#\\{p\leq x\mid p\in T(\rho)\\}\geq c\log x.$

The result above is certainly weaker than what one is led to conjecture,
however, it does well to illustrate the effectiveness of the Theorem 3.11.
Throughout, we motivate our results by drawing upon analogues from the
classical Iwasawa theory of class groups.

###  1.1. Organization

Including the introduction, the article consists of four sections. In section
2 we review the classical Iwasawa theory of class groups and Artin
representations. We review results of Federer and Gross [FG80] which gives an
explicit relationship between $p$-adic regulators and Iwasawa invariants. We
end this section by reviewing some results of Greenberg and Vatsal [GV20] on
the Iwasawa theory of Artin representations. In the section 3, we discuss the
notion of the Euler characteristic of a cofinitely generated and cotorsion
module over the Iwasawa algebra. In section 4, we formulate and study some
natural distribution questions, which serve to illustrate our results.

##  2\. Iwasawa theory of class groups and Artin representations

###  2.1. Classical Iwasawa theory and Iwasawa invariants

We review the classical Iwasawa theory over the cyclotomic
$\mathbb{Z}_{p}$-extension of a number fields. The reader may refer to [Was97]
for a more comprehensive treatment. Throughout this section, we shall set $K$
to denote a number field and $p$ an odd prime number. Fix an algebraic closure
$\bar{K}$ of $K$. All algebraic extensions considered in this article shall
implicitly be assumed to be in $\bar{K}$. Let $\operatorname{Cl}(K)$ denote
the class group of $K$ and $\operatorname{Cl}_{p}(K)$ its $p$-primary part.

For $n\geq 1$, let $K(\mu_{p^{n}})$ be the number field extension of $K$
obtained by adjoining the $p^{n}$-th roots of unity to $K$. Denote by
$K(\mu_{p^{\infty}})$ the union of all extensions $K(\mu_{p^{n}})$. There is a
unique $\mathbb{Z}_{p}$-extension $K_{\infty}/K$ which is contained in
$K(\mu_{p^{\infty}})$, which is referred to as the _cyclotomic
$\mathbb{Z}_{p}$-extension_ of $K$. For $n\geq 1$, set $K_{n}/K$ to be the
sub-extension of $K_{\infty}/K$, for which $\operatorname{Gal}(K_{n}/K)$ is
isomorphic to $\mathbb{Z}/p^{n}\mathbb{Z}$. Setting
$K_{0}\mathrel{\mathop{\mathchar 58\relax}}=K$, refer to $K_{n}$ as the _$n$
-th layer_. Taking $\Gamma_{n}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K_{\infty}/K_{n})$ we identify
$\operatorname{Gal}(K_{n}/K)$ with $\Gamma/\Gamma_{n}$. For
$n\in\mathbb{Z}_{\geq 0}$, denote by $H_{p}(K_{n})$ the $p$-Hilbert class
field of $K_{n}$. In other words, $H_{p}(K_{n})$ is the maximal unramified
abelian $p$-extension of $K_{n}$ in $\bar{K}$. Set $X_{n}$ to denote the
Galois group $\operatorname{Gal}(H_{p}(K_{n})/K_{n})$, and identify $X_{n}$
with the maximal $p$-primary quotient of $\operatorname{Cl}(K_{n})$. Thus,
$[H_{p}(K_{n})\mathrel{\mathop{\mathchar 58\relax}}K_{n}]$ is equal to
$\\#\operatorname{Cl}_{p}(K_{n})$. Since the primes above $p$ are totally
ramified in $K_{\infty}$, it follows that $H_{p}(K_{n})\cap K_{\infty}=K_{n}$,
and thus, there are surjective maps $X_{m}\rightarrow X_{n}$ for all $m\geq
n$. Taking $X_{\infty}$ to be the inverse limit $\varprojlim_{n}X_{n}$, we
find that $X_{\infty}$ is both a $\mathbb{Z}_{p}$-module as well as a module
over $\Gamma$. On the other hand, letting $H_{p}(K_{\infty})$ denote the
maximal unramified abelian pro-$p$ extension of $K_{\infty}$, we identify
$X_{\infty}$ with the Galois group
$\operatorname{Gal}(H_{p}(K_{\infty})/K_{\infty})$.

On order to better study the algebraic structure of $X_{\infty}$, it proves
fruitful to view $X_{\infty}$ as a module over a completed group ring known as
the Iwasawa algebra. This algebra is defined as follows

$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Z}_{p}\llbracket\Gamma\rrbracket=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma_{n}].$

Choosing a topological generator $\gamma\in\Gamma$, we identify $\Lambda$ with
the formal power series ring $\mathbb{Z}_{p}\llbracket T\rrbracket$, by
setting $T\mathrel{\mathop{\mathchar 58\relax}}=(\gamma-1)$. As a
$\Lambda$-module, $X_{\infty}$ is finitely generated and torsion [Was97,
chapter 13].

Let $\mathcal{O}$ be a valuation ring with residue characteristic $p$, and let
$\varpi$ be a uniformizer of $\mathcal{O}$. Then, the Iwasawa algebra over
$\mathcal{O}$ is defined by extending coefficients to $\mathcal{O}$, as
follows $\Lambda_{\mathcal{O}}\mathrel{\mathop{\mathchar
58\relax}}=\Lambda\otimes_{\mathbb{Z}_{p}}\mathcal{O}$. The Iwasawa algebra
$\Lambda_{\mathcal{O}}$ is a local ring with maximal ideal
$\mathfrak{m}=(\varpi,T)$. A polynomial $f(T)\in\mathcal{O}\llbracket
T\rrbracket$ is said to be _distinguished_ if it is a monic polynomial whose
non-leading coefficients are divisible by $\varpi$. The Weierstrass
preparation theorem states that any power series $f(T)$ decomposes into a
product

$f(T)=\varpi^{\mu}\times g(T)\times u(T),$

where $\mu\in\mathbb{Z}_{\geq 0}$, $g(T)$ is a distinguised polynomial and
$u(T)$ is a unit in $\Lambda_{\mathcal{O}}$. The $\mu$-invariant of $f(T)$ is
the power of $\varpi$ above, and the $\lambda$-invariant is the degree of the
distinguished polynomial $g(T)$. The prime ideals of height $1$ are the
principal ideals $(\varpi)$ and $(g(T))$, where $g(T)$ is an irreducible
distinguished polynomial.

Given a finitely generated and torsion $\Lambda_{\mathcal{O}}$-module $M$,
there is a homomorphism of $\Lambda_{\mathcal{O}}$-modules

(2.1) $M\longrightarrow\left(\bigoplus_{i=1}^{s}\frac{\mathcal{O}\llbracket
T\rrbracket}{(\varpi^{\mu_{i}})}\right)\oplus\left(\bigoplus_{j=1}^{t}\frac{\mathcal{O}\llbracket
T\rrbracket}{(f_{i}(T)^{\lambda_{i}})}\right),$

with finite kernel and cokernel. Here, $\mu_{i},\lambda_{j}\geq 0$, and
$f_{j}(T)$ are irreducible distinguished polynomials. For further details, we
refer to [Was97, Theorem 3.12].

######  Definition 2.1.

The $\mu$-invariant $\mu_{p}(M)$ is the sum of the entries
$\sum_{i=1}^{s}\mu_{i}$ if $s>0$, and is set to be $0$ if $s=0$. On the other
hand, the $\lambda$-invariant $\lambda_{p}(M)$ is defined to be
$\sum_{i=1}^{s}\lambda_{i}\operatorname{deg}(f_{i})$ if $s>0$, and defined to
be $0$ if $t=0$. The characteristic element $f_{M}(T)$ is the product

$f_{M}(T)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{i}\varpi^{\mu_{i}}\times\prod_{j}f_{j}(T)^{\lambda_{j}}.$

We remark that the $\mu$-invariant and $\lambda$-invariant of $f_{M}(T)$ are
$\mu_{p}(M)$ and $\lambda_{p}(M)$ respectively.

######  Proposition 2.2.

Suppose that $M$ is a finitely generated and torsion
$\Lambda_{\mathcal{O}}$-module. Then, the following assertions hold,

  1. (1)

$\mu_{p}(M)=0$ if and only if $M$ is finitely generated as an
$\mathcal{O}$-module. In this case, $\lambda_{p}(M)$ is the $\mathcal{O}$-rank
of $M$.

  2. (2)

Letting $r_{p}(M)$ denote the order of vanishing of $f_{M}$ at $0$, we have
that

$\lambda_{p}(M)\geq r_{p}(M).$

  3. (3)

Write $f_{M}(T)$ as a power series

$f_{M}(T)=a_{r}T^{r}+a_{r+1}T^{r+1}+\dots+a_{\lambda}T^{\lambda},$

where $r=r_{p}(M)$ and $\lambda=\lambda_{p}(M)$. The $\mu$-invariant
$\mu_{p}(M)=0$ if and only if there is a coefficient $a_{i}$ not divisible by
$\varpi$.

  4. (4)

We have that $\varpi^{\mu}||a_{\lambda}$ and that $\varpi^{\mu+1}|a_{i}$ for
all $i<\lambda$.

###### Proof.

The results are easy observations that follow from the structural homomorphism
(2.1) and the definition of the Iwasawa invariants. ∎

######  Remark 2.3.

Let $\mathcal{O}^{\prime}$ be a valuation ring that is a finite extension of
$\mathcal{O}$, and $e$ be its ramification index. Setting
$M_{\mathcal{O}^{\prime}}\mathrel{\mathop{\mathchar
58\relax}}=M\otimes_{\mathcal{O}}\mathcal{O}^{\prime}$ and regard
$M_{\mathcal{O}^{\prime}}$ as a module over $\Lambda_{\mathcal{O}^{\prime}}$.
Then, it is easy to see that

$\mu_{p}(M_{\mathcal{O}^{\prime}})=e\mu_{p}(M)\text{ and
}\lambda_{p}(M_{\mathcal{O}^{\prime}})=\lambda_{p}(M).$

We denote the $\mu$-invariant (resp. $\lambda$-invariant) of $X_{\infty}$ by
$\mu_{p}(K)$ (resp. $\lambda_{p}(K)$). In this setting,
$\mathcal{O}\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{Z}_{p}$. Iwasawa
proved that for all large enough values of $n$, there is an invariant
$\nu_{p}(K)\in\mathbb{Z}$ for which

$\log_{p}\left(\\#\operatorname{Cl}_{p}(K_{n})\right)=p^{n}\mu_{p}(K)+n\lambda_{p}(K)+\nu_{p}(K),$

cf. [Was97, Theorem 13.13]. Moreover, Iwasawa conjectured that $\mu_{p}(K)=0$
for all number fields $K$, cf. [Iwa73]. For abelian extensions $K/\mathbb{Q}$
the conjecture has been proven by Ferrero and Washington [FW79].

###  2.2. The leading coefficient of the characteristic series

Let $K$ be a CM field with totally real subfield $K^{+}$. The Galois group
$\operatorname{Gal}(K/K^{+})$ acts on $X_{\infty}$. Let $\tau$ be the
generator of $\operatorname{Gal}(K/K^{+})$ and set

$X_{\infty}^{-}\mathrel{\mathop{\mathchar 58\relax}}=\\{x\in
X_{\infty}\mid\tau(x)=-x\\}.$

Then, $X_{\infty}^{-}$ is a $\Lambda$-module whose $\mu$-invariant (resp.
$\lambda$-invariant) is denoted $\mu_{p}^{-}(K)$ (resp. $\lambda_{p}^{-}(K)$).
Let $f_{K}^{-}(T)$ be the characteristic element associated to
$X_{\infty}^{-}$. Take $I$ to be the set of primes of $K^{+}$ that lie above
$p$ and split in $K$, and set $r_{p,K}\mathrel{\mathop{\mathchar
58\relax}}=\\#I$. Let $S$ be the set of places of $K$ dividing $p$ and
$\infty$. Let $U=U_{K}$ be the group of $S$-units of $K^{*}$ and $M=M_{K}$ be
the free abelian group of divisors at $S$. Given a
$\operatorname{Gal}(K/K^{+})$-module $N$, let

$N^{-}\mathrel{\mathop{\mathchar 58\relax}}=\\{n\in N\mid\tau(n)=-n\\}.$

Let $R$ be a ring, then we set $RN^{-}$ to denote the extension of scalars
$N^{-}\otimes R$. It is easy to see that both $U^{-}$ and $M^{-}$ are free
abelian group of rank $r_{p,K}$.

The map

$\phi\mathrel{\mathop{\mathchar 58\relax}}U^{-}\rightarrow M^{-}$

is defined by setting

$\phi(x)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{v|p}\operatorname{ord}_{p}\left(\operatorname{Norm}_{K_{v}/\mathbb{Q}_{p}}x\right).$

The induced map

$\phi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}U^{-}\rightarrow\mathbb{Q}M^{-}$

is an isomorphism (cf. [FG80, Proposition 1.4]). The inverse of $\phi$ can be
described as follows. For each prime $v\in I$, we choose a prime
$\widetilde{v}|v$ of $K$. Note that $\mathbb{Q}M^{-}$ has a basis

$\mathcal{B}=\\{\widetilde{v}-\tau(\widetilde{v})\mid v\in I\\}.$

Let $\mathfrak{p}$ be the prime ideal associated to $v$, and $h$ be a positive
integer such that $\mathfrak{p}^{h}=(\alpha)$ is principal. Both $\alpha$ and
$\tau(\alpha)$ are elements of $U$, and $\alpha/\tau(\alpha)\in U^{-}$.
Setting $f_{v}$ to denote the residue class degree of $v$, take

$\phi^{-1}\left(\widetilde{v}-\tau(\widetilde{v})\right)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{hf_{v}}\otimes\left(\alpha/\tau(\alpha)\right).$

Then, $\phi^{-1}$ is the inverse to $\phi$.

Define the homomorphism

$\lambda\mathrel{\mathop{\mathchar
58\relax}}U^{-}\rightarrow\mathbb{Q}_{p}M^{-}$

by setting

$\lambda(y)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{v|p}\log_{p}\left(N_{K_{v}/\mathbb{Q}_{p}}(y)\right).$

Composing $\phi^{-1}$ with $\lambda$, we obtain an endomorphism

$\lambda\phi^{-1}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}_{p}M^{-}\rightarrow\mathbb{Q}_{p}M^{-}.$

######  Definition 2.4.

With notation as above, define the regulator $\operatorname{Reg}_{p}(K)$ as
follows

$\operatorname{Reg}_{p}(K)\mathrel{\mathop{\mathchar
58\relax}}=\det\left(\lambda\phi^{-1}\mid\mathbb{Q}_{p}M^{-}\right),$

where it is understood that
$\operatorname{Reg}_{p}(K)\mathrel{\mathop{\mathchar 58\relax}}=1$ when
$r_{p,K}=0$.

######  Theorem 2.5 (Iwasawa, Greenberg).

With respect to notation above, $f_{K}^{-}(T)$ is divisible by $T^{r_{p,K}}$.

###### Proof.

We refer to works of Iwasawa [Iwa73] and Greenberg [Gre73] for the proof of
the statement. ∎

As a consequence of Theorem 2.5, one may write

$f_{K}^{-}(T)=a_{r}T^{r}+a_{r+1}T^{r+2}+\dots+a_{\lambda}T^{\lambda},$

where $r=r_{p,K}$ and $\lambda\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{p}^{-}(K)$. We note that as a consequence,

$\lambda_{p}^{-}(K)\geq r_{p,K}.$

For $a,b\in\mathbb{Q}_{p}$, we write $a\sim b$ to mean that $a=ub$ for some
$u\in\mathbb{Z}_{p}^{\times}$. Let $h_{K}$ (resp. $h_{K}^{-}$) denote the
class number (resp. number of elements in the minus part of the class group)
of $K$. Let $w_{K(\mu_{p})}$ be the number of roots of unity contained in
$K(\mu_{p})$.

######  Theorem 2.6 (Federer-Gross).

With respect to notation above, the following are equivalent

  1. (1)

$\operatorname{Reg}_{p}(K)\neq 0$,

  2. (2)

$a_{r}\neq 0$.

Assuming that these equivalent conditions hold, we have that

$a_{r}\sim\frac{h_{K}^{-}\left(\prod_{v\in
I}f_{v}\right)\operatorname{Reg}_{p}(K)}{w_{K(\mu_{p})}^{r_{p,K}}}.$

###### Proof.

The above result is [FG80, Proposition 3.9]. ∎

Suppose that $p\nmid f_{v}$ for all $v\in I$, and $p\nmid h_{K}$. Then, the
map $\phi^{-1}$ is defined over $\mathbb{Z}_{p}$ and it is easy to see that
$\operatorname{Reg}_{p}(K)$ is divisible by $p^{r_{p,K}}$. The normalized
regulator is defined as follows

$\mathcal{R}_{p}(K)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\operatorname{Reg}_{p}(K)}{p^{r_{p,K}}}.$

######  Corollary 2.7.

Let $K$ be a CM field for which

  1. (i)

$\mathcal{R}_{p}(K)\neq 0$,

  2. (ii)

$w_{K(\mu_{p})}\sim p$,

  3. (iii)

$p\nmid f_{v}$ for all $v\in I$,

  4. (iv)

$p\nmid h_{K}$.

Then, the following are equivalent

  1. (1)

$\mu_{p}^{-}(K)=0$ and $\lambda_{p}^{-}(K)=r_{p,K}$,

  2. (2)

$a_{r}$ is a unit in $\mathbb{Z}_{p}$,

  3. (3)

$p\nmid\mathcal{R}_{p}(K)$.

###### Proof.

With respect to above notation, write $f_{K}^{-}(T)=T^{r_{p,K}}g_{K}^{-}(T)$,
where $a_{r}=g_{K}^{-}(0)$. Then, $a_{r}$ is a unit in $\mathbb{Z}_{p}$ if and
only if $g_{K}^{-}(T)$ is a unit in $\Lambda$. This implies that

$\mu_{p}^{-}(K)=0\text{ and }\lambda_{p}^{-}(K)=r_{p,K}.$

Conversely, if

$\mu_{p}^{-}(K)=0\text{ and }\lambda_{p}^{-}(K)=r_{p,K},$

then the factorization $f_{K}^{-}(T)=T^{r_{p,K}}g_{K}^{-}(T)$ implies that
$g_{K}^{-}(T)$ must be a unit in $\Lambda$. Thus, we find that (1) and (2) are
equivalent. It follows from our assumptions that
$a_{r}\sim\mathcal{R}_{p}(K)$. Therefore, $a_{r}$ is a unit in
$\mathbb{Z}_{p}$ if and only if $p\nmid\mathcal{R}_{p}(K)$. This shows that
the conditions (2) and (3) are equivalent. ∎

When $K$ is an imaginary quadratic field, $X_{\infty}=X_{\infty}^{-}$ and thus
$\mu_{p}(K)=\mu_{p}^{-}(K)$ and $\lambda_{p}(K)=\lambda_{p}^{-}(K)$. In this
setting, we find that

$r_{p,K}\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}&0\text{ if
}p\text{ splits in }K;\\\ &1\text{ if }p\text{ is inert or ramified in }K.\\\
\end{cases}$

Note that $\mu_{p}(K)=0$ by the aforementioned result of Ferrero and
Washington [FW79].

######  Corollary 2.8.

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number. Then,
the following assertions hold.

  1. (1)

Suppose that $p$ splits in $K$. Then, $\lambda_{p}(K)=0$ if and only if
$p\nmid h_{K}$.

  2. (2)

Suppose that $p$ is inert in $K$. Then, $\lambda_{p}(K)=1$ if and only if
$p\nmid h_{K}$ and $p\nmid\mathcal{R}_{p}(K/\mathbb{Q})$.

###### Proof.

The result is a special case of Corollary 2.7. ∎

Let $K$ be an imaginary quadratic field and $p$ be an odd prime which is inert
in $K$. In this case $\lambda_{p}(K)\geq 1$ and the Corollary 2.8 asserts that
$\lambda_{p}(K)=1$ if and only if $\mathcal{R}_{p}(K/\mathbb{Q})$ is a unit in
$\mathbb{Z}_{p}$. The analysis of this condition leads to the statement of
Gold’s criterion.

######  Theorem 2.9 (Gold’s criterion).

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number which
splits in $K$. Assume that $p\nmid h_{K}$. Let $\mathfrak{p}|p$ be a prime
ideal, and $r$ be a positive integer not divisible by $p$, such that
$\mathfrak{p}^{r}$ is principal. Let $\alpha\in\mathcal{O}_{K}$ be a generator
of $\mathfrak{p}^{r}$. Setting $\bar{\mathfrak{p}}$ to denote the complex
conjugate of $\mathfrak{p}$, the following conditions are equivalent

  1. (1)

$\lambda_{p}(K)>1$,

  2. (2)

$\alpha^{p-1}\equiv 1\left(\mod{\bar{\mathfrak{p}}^{2}}\right)$.

###### Proof.

The result is a consequence of [Gol74, Theorems 3 and 4], and can also be seen
to follow from Corollary 2.8, cf. [San93, proof of Proposition 2.1]. ∎

###  2.3. Artin representations

We briefly discuss the results of Greenberg and Vatsal [GV20] on the Iwasawa
theory of Selmer groups associated to Artin representations. Let
$K/\mathbb{Q}$ be a finite Galois extension with Galois group
$\Delta\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K/\mathbb{Q})$. Fix an irreducible Artin
representation of dimension $d>1$

$\rho\mathrel{\mathop{\mathchar
58\relax}}\Delta\rightarrow\operatorname{GL}_{d}(\bar{\mathbb{Q}})$

and

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{n}S_{\chi,\epsilon}(\mathbb{Q}_{n})$

the Selmer group over the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$. Note that $S_{\chi,\epsilon}(\mathbb{Q}_{n})$ (resp.
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$) is an
$\mathcal{O}[\Gamma/\Gamma_{n}]$-module (resp.
$\Lambda_{\mathcal{O}}$-module).

It is easy to see that the restriction map

$H^{1}(\mathbb{Q}_{n},D)\rightarrow H^{1}(\mathbb{Q}_{\infty},D)^{\Gamma_{n}}$

induces a map between Selmer groups

$\iota_{n}\mathrel{\mathop{\mathchar
58\relax}}S_{\chi,\epsilon}(\mathbb{Q}_{n})\rightarrow
S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma_{n}}.$

The following control theorem shows that this map is injective with cokernel
which is independent of $n$.

######  Theorem 2.10 (Greenberg and Vatsal – Control theorem).

Suppose that $\chi$ is nontrivial, then, $\iota_{n}$ fits into a short exact
sequence of $\Gamma/\Gamma_{n}$-modules

$0\rightarrow S_{\chi,\epsilon}(\mathbb{Q}_{n})\rightarrow
S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma_{n}}\rightarrow
H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})\rightarrow 0.$

###### Proof.

The above result is [GV20, Proposition 4.1]. ∎

We note that
$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})\simeq(\mathcal{F}/\mathcal{O})^{t}$,
where $t$ is the multiplicity of the trivial representation of
$\Delta_{\mathfrak{p}}$ in $V/V^{\epsilon_{\mathfrak{p}}}$. Here, the action
of $\Gamma/\Gamma_{n}$ on
$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})$ is trivial.

Let $\mathcal{D}$ be a discrete $p$-primary $\Lambda_{\mathcal{O}}$-module. We
take $\mathcal{D}^{\vee}$ to be the Pontryagin dual defined as follows

$\mathcal{D}^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\mathcal{D},\mathbb{Q}_{p}/\mathbb{Z}_{p}\right).$

Say that $\mathcal{D}$ is cofinitely generated (resp. cotorsion) as a
$\Lambda_{\mathcal{O}}$-module, if $\mathcal{D}^{\vee}$ is finitely generated
(resp. torsion) as a $\Lambda_{\mathcal{O}}$-module. We set
$X_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ be the Pontryagin dual of
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$.

######  Theorem 2.11 (Greenberg-Vatsal).

With respect to notation above $X_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ is a
finitely generated and torsion $\Lambda_{\mathcal{O}}$-module that contains no
non-trivial finite $\Lambda_{\mathcal{O}}$-submodules.

###### Proof.

We refer to [GV20, Propositions 4.5 and 4.7] for the proof of the above
result. ∎

Set $\mu_{\chi,\epsilon}$ and $\lambda_{\chi,\epsilon}$ to denote the $\mu$
and $\lambda$-invariants of $S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\vee}$
respectively. The characteristic series of
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\vee}$ is denoted
$f_{\chi,\epsilon}(T)$.

##  3\. The Euler characteristic

In this section, we introduce the notion of the Euler characteristic
associated to a cofinitely generated and cotorsion module over
$\Lambda_{\mathcal{O}}$, and introduce conditions for it to be well defined.
For the Selmer group associated to an Artin representation, the Euler
characteristic is shown to equal the cardinality of
$S_{\chi,\epsilon}(\mathbb{Q})$. The results presented in this section have
consequences to the study of the Iwasawa $\mu$ and $\lambda$-invariants
associated to an Artin representation.

###  3.1. Definition and properties of the Euler characteristic

Let $\mathcal{D}$ be a cofinitely generated and cotorsion module over
$\Lambda_{\mathcal{O}}$. Since $\Gamma$ is pro-cyclic, it has cohomological
dimension $1$, we find that $H^{i}(\Gamma,\mathcal{D})=0$ for $i\geq 2$. Also
note that $H^{1}(\Gamma,\mathcal{D})$ is identified with the module of
coinvariants $\mathcal{D}_{\Gamma}$. This invariant encodes information about
the valuation of leading constant term of the characteristic series. For an
Artin representation that satisfies the conditions of Assumption 1.1 we
discuss conditions for the Euler characteristic to be well defined, and give
an explicit formula for it. The Euler characteristic formula in this context
can be viewed as an analogue of the result of Gross and Federer.

######  Proposition 3.1.

Let $\mathcal{D}$ be a cofinitely generated and cotorsion
$\Lambda_{\mathcal{O}}$-module, then,

  1. (1)

$\operatorname{corank}_{\mathbb{Z}_{p}}H^{0}(\Gamma,\mathcal{D})=\operatorname{corank}_{\mathbb{Z}_{p}}H^{1}(\Gamma,\mathcal{D})$.

  2. (2)

The module $H^{0}(\Gamma,\mathcal{D})$ is finite if and only if
$H^{1}(\Gamma,\mathcal{D})$ is finite.

###### Proof.

Since $\mathcal{D}$ is cofinitely generated as a
$\Lambda_{\mathcal{O}}$-module, it follows that both $\mathcal{D}^{\Gamma}$
and $\mathcal{D}_{\Gamma}$ are cofinitely generated as a
$\mathbb{Z}_{p}$-module. Part (1) follows from [How02, Theorem 1.1]. Since
$\mathcal{D}$ is cofinitely generated as a $\Lambda$-module, it follows that
$\mathcal{D}^{\Gamma}$ and $\mathcal{D}_{\Gamma}$ are both cofinitely
generated as $\mathbb{Z}_{p}$-modules. Part (2) follows from (1). ∎

######  Definition 3.2.

Let $\mathcal{D}$ be a cofinitely generated and cotorsion
$\Lambda_{\mathcal{O}}$-module. Then, we say that the Euler characteristic of
$\mathcal{D}$ is well defined if $H^{0}(\Gamma,\mathcal{D})$ (or equivalently
$H^{1}(\Gamma,\mathcal{D})$) is finite. Assuming that the characteristic of
$\mathcal{D}$ is well defined, the Euler characteristic is defined as follows

$\chi(\Gamma,\mathcal{D})\mathrel{\mathop{\mathchar 58\relax}}=\prod_{i\geq
0}\\#H^{i}(\Gamma,\mathcal{D})^{(-1)^{i}}=\frac{\\#H^{0}(\Gamma,\mathcal{D})}{\\#H^{1}(\Gamma,\mathcal{D})}.$

Recall from Definition 2.1 that $f_{\mathcal{D}^{\vee}}(T)$ is the
characteristic element of $\mathcal{D}^{\vee}$ and that
$\mu_{p}(\mathcal{D}^{\vee})$ (resp. $\lambda_{p}(\mathcal{D}^{\vee})$) is the
$\mu$-invariant (resp. $\lambda$-invariant). For the ease of notation, we set

$f_{\mathcal{D}}(T)\mathrel{\mathop{\mathchar
58\relax}}=f_{\mathcal{D}^{\vee}}(T),\mu_{p}(\mathcal{D})\mathrel{\mathop{\mathchar
58\relax}}=\mu_{p}(\mathcal{D}^{\vee})\text{ and
}\lambda_{p}(\mathcal{D})\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{p}(\mathcal{D}^{\vee}).$

From the expansion

$f_{\mathcal{D}}(T)=\sum_{i=0}^{\lambda}a_{i}T^{i},$

we note that $\lambda=\lambda_{p}(\mathcal{D})$. Moreover,
$\mu_{p}(\mathcal{D})=0$ if and only if $\varpi\nmid a_{i}$ for at least one
coefficient $a_{i}$. In particular, we have the have that $\varpi\nmid
f_{\mathcal{D}}(0)$ if and only if
$\mu_{p}(\mathcal{D})=\lambda_{p}(\mathcal{D})=0$. Fix an absolute value
$|\cdot|_{\varpi}$ on $\mathcal{O}$ normalized by setting

$|\varpi|_{\varpi}\mathrel{\mathop{\mathchar
58\relax}}=[\mathcal{O}\mathrel{\mathop{\mathchar
58\relax}}\varpi\mathcal{O}]^{-1}.$

######  Proposition 3.3.

Let $\mathcal{D}$ be a cofinitely generated and cotorsion $\Lambda$-module.
Then, with respect to the above notation, the following conditions are
equivalent

  1. (1)

$f_{\mathcal{D}}(0)\neq 0$,

  2. (2)

the Euler characteristic $\chi(\Gamma,\mathcal{D})$ is well defined.

Moreover, if the above equivalent conditions are satisfied, then,

$\chi(\Gamma,\mathcal{D})=|f_{\mathcal{D}}(0)|_{\varpi}^{-1}.$

###### Proof.

It is easy to see that if $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are
cofinitely generated $\Lambda_{\mathcal{O}}$-modules that are pseudo-
isomorphic, then $\mathcal{D}^{\Gamma}$ is finite if and only if
$(\mathcal{D}^{\prime})^{\Gamma}$ is finite. Therefore,
$\chi(\Gamma,\mathcal{D})$ is well defined if and only if
$\chi(\Gamma,\mathcal{D}^{\prime})$ is well defined. The characteristic series
is determined up to pseudo-isomorphism, and therefore,
$f_{\mathcal{D}}(T)=f_{\mathcal{D}^{\prime}}(T)$. We assume without loss of
generality that

$\mathcal{D}^{\vee}=\left(\bigoplus_{i=1}^{s}\frac{\mathcal{O}\llbracket
T\rrbracket}{(\varpi^{\mu_{i}})}\right)\oplus\left(\bigoplus_{j=1}^{t}\frac{\mathcal{O}\llbracket
T\rrbracket}{(f_{i}(T)^{\lambda_{i}})}\right).$

We identify $\left(\mathcal{D}^{\Gamma}\right)^{\vee}$ with

$\left(\mathcal{D}^{\vee}\right)_{\Gamma}=\mathcal{D}^{\vee}/T\mathcal{D}^{\vee}\simeq\left(\bigoplus_{i=1}^{s}\mathcal{O}/(\varpi^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\mathcal{O}/(f_{i}(0)^{\lambda_{i}})\right).$

Since

$f_{\mathcal{D}}(0)=\prod_{i}\varpi^{\mu_{i}}\times\prod_{j}f_{j}(0)^{\lambda_{j}},$

we deduce that

$\\#H^{0}(\Gamma,\mathcal{D})=\\#H^{1}(\Gamma,\mathcal{D}^{\vee})=\\#\left(\mathcal{D}^{\vee}\right)_{\Gamma}$

is finite if and only if $f_{\mathcal{D}}(0)$ is non-zero. Therefore, the
Euler characteristic is well defined if and only if $f_{\mathcal{D}}(0)$ is
non-zero.

Assuming that $f_{\mathcal{D}}(0)\neq 0$, we calculate the Euler
characteristic. First, we note that the previous argument implies that

$\\#H^{0}(\Gamma,\mathcal{D})=|f_{\mathcal{D}}(0)|_{\varpi}^{-1}.$

On the other hand, we find that

$\\#H^{1}(\Gamma,D)=\\#H^{0}(\Gamma,D^{\vee})=\\#\left(D^{\vee}\right)^{\Gamma}.$

Identify $(\mathcal{D}^{\vee})^{\Gamma}$ with the kernel of the multiplication
by $T$ endomorphism of $\mathcal{D}^{\vee}$. Since $f_{\mathcal{D}}(0)$ is
assumed to be non-zero, it follows that none of the terms $f_{j}(T)$ are
divisible by $T$. It follows from this that the multiplication by $T$ map is
injective and hence, $(\mathcal{D}^{\vee})^{\Gamma}=0$. Thus it has been shown
that

$\chi(\Gamma,D)=|f_{\mathcal{D}}(0)|_{\varpi}^{-1}.$

∎

######  Corollary 3.4.

Suppose that the Euler characteristic of $\mathcal{D}$ is well defined. Then
the following conditions are equivalent

  1. (1)

$\chi(\Gamma,\mathcal{D})=1$,

  2. (2)

$\mu_{p}(\mathcal{D})=0$ and $\lambda_{p}(\mathcal{D})=0$.

###### Proof.

It is easy to see that $\mu_{p}(\mathcal{D})=0$ and
$\lambda_{p}(\mathcal{D})=0$ if and only if $f_{\mathcal{D}}(T)$ is a unit in
$\Lambda_{\mathcal{O}}$. Thus, the condition (2) is equivalent to the
condition that $\varpi\nmid f_{\mathcal{D}}(0)$. According to Proposition 3.3,

$\chi(\Gamma,\mathcal{D})=|f_{\mathcal{D}}(0)|_{\varpi}^{-1}.$

Therefore $\chi(\Gamma,\mathcal{D})=1$ if and only if $f_{\mathcal{D}}(0)$ is
a unit in $\mathcal{O}$. ∎

###  3.2. Calculating the Euler characteristic

Let $\rho$ be an Artin representation with character $\chi$ and set $\epsilon$
to be the chosen character with respect to which the Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ is defined. We shall assume
throughout that the Assumption 1.1 is satisfied.

######  Proposition 3.5.

With respect to above notation, the following conditions are equivalent

  1. (1)

the Euler characteristic $\chi(\Gamma,S_{\chi,\epsilon}(\mathbb{Q}_{\infty}))$
is defined,

  2. (2)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

###### Proof.

The Euler characteristic is defined if and only if
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma}$ is finite. By the control
theorem, we have the short exact sequence

$0\rightarrow S_{\chi,\epsilon}(\mathbb{Q})\rightarrow
S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma}\rightarrow
H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})\rightarrow 0,$

where $H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})$ is a cofree
$\mathcal{O}$-module. According to Theorem 1.2, the Selmer group
$S_{\chi,\epsilon}(\mathbb{Q})$ is finite. Therefore, the Euler characteristic
is well defined if and only if
$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$. ∎

######  Theorem 3.6.

Let $\rho$ be an Artin representation for which

  1. (1)

the conditions of Assumption 1.1 are satisfied.

  2. (2)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

Then, the Euler characteristic is given by

$\chi(\Gamma,S_{\chi,\epsilon}(\mathbb{Q}_{\infty}))=\\#S_{\chi,\epsilon}(\mathbb{Q}).$

###### Proof.

According to Proposition 3.5, the Euler characteristic is well defined. We are
to calculate the orders of the finite abelian groups
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma}$ and
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})_{\Gamma}$. According to Theorem 2.11,
$X_{\chi,\epsilon}(\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\vee}$ does not contain any
nontrivial finite $\Lambda_{\mathcal{O}}$-submodules. Since the Euler
characteristic is well defined,
$X_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma}$ is finite, and hence is
zero. In other words, $S_{\chi,\epsilon}(\mathbb{Q}_{\infty})_{\Gamma}=0$. On
the other hand, it follows from Theorem 2.10 that

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})^{\Gamma}=S_{\chi,\epsilon}(\mathbb{Q}).$

Therefore, we find that
$\chi(\Gamma,S_{\chi,\epsilon}(\mathbb{Q}_{\infty}))=\\#S_{\chi,\epsilon}(\mathbb{Q})$.
∎

######  Corollary 3.7.

Let $\rho$ be an Artin representation and $\epsilon\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\mathcal{K}/\mathbb{Q}_{p})\rightarrow\mathcal{O}^{\times}$
be a character for which the following assumptions are satisfied.

  1. (i)

The conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

Then, the following conditions are equivalent

  1. (1)

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$,

  2. (2)

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ is finite,

  3. (3)

$\mu_{\chi,\epsilon}=\lambda_{\chi,\epsilon}=0$,

  4. (4)

$\chi(\Gamma,S_{\chi,\epsilon}(\mathbb{Q}_{\infty}))=1$,

  5. (5)

$S_{\chi,\epsilon}(\mathbb{Q})=0$.

###### Proof.

It is clear that (1) implies (2). On the other hand, by Theorem 2.11,
$X_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ contains no nontrivial finite
$\Lambda_{\mathcal{O}}$-submodules. Hence, if
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ is finite, it must be equal to $0$.
Therefore, (1) and (2) are equivalent.

We note that a $\Lambda_{\mathcal{O}}$-module $M$ is finite if and only if it
is pseudo-isomorphic to the trivial module. On the other hand, $M\sim 0$ if
and only if the $\mu$ and $\lambda$-invariants of $M$ are $0$. Therefore (2)
and (3) are equivalent.

The equivalence of (3) and (4) follows from Corollary 3.4, and the equivalence
of (4) and (5) follows from Proposition 3.3. ∎

###  3.3. The structure of the Selmer group $S_{\chi,\epsilon}(\mathbb{Q})$

In this subsection, we analyze the structure of
$S_{\chi,\epsilon}(\mathbb{Q})$ and establish conditions for the vanishing of
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$. We set $d(\chi)$ to denote $d$ and
write $d(\chi)=d^{+}(\chi)+d^{-}(\chi)$, reflecting the action of the
archimedian places on the representation. At each prime $\mathfrak{p}|p$, let
$\mathcal{U}_{\mathfrak{p}}$ denote the principal units at $\mathfrak{p}$, and
set $\mathcal{U}_{p}$ to denote the product
$\prod_{\mathfrak{p}|p}\mathcal{U}_{\mathfrak{p}}$. The decomposition group
$\Delta_{\mathfrak{p}}$ naturally acts on $\mathcal{U}_{\mathfrak{p}}$, and
$\mathcal{U}_{p}$ is identified with the induced representation
$\operatorname{Ind}_{\Delta_{\mathfrak{p}}}^{\Delta}\mathcal{U}_{\mathfrak{p}}$.
Let $U_{K}$ be the group of units of $\mathcal{O}_{K}$ that are principal
units at all primes $\mathfrak{p}|p$. The diagonal inclusion map

$U_{K}\hookrightarrow\mathcal{U}_{p}$

is a $\Delta$-equivariant map, and induces a $\mathbb{Z}_{p}$-linear map

$\lambda_{p}\mathrel{\mathop{\mathchar
58\relax}}U_{K}\otimes\mathbb{Z}_{p}\rightarrow\mathcal{U}_{p}.$

We recall that the character $\epsilon$ arises from a choice of prime
$\mathfrak{p}|p$ and character
$\epsilon_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}\Delta_{\mathfrak{p}}\rightarrow\mathcal{O}^{\times}$. Let
$\mathfrak{p}^{\prime}$ be any other prime above $p$, and write
$\mathfrak{p}^{\prime}=\delta(\mathfrak{p})$ for $\delta\in\Delta$. Then
conjugation by $\delta$ gives an isomorphism
$c_{\delta}\mathrel{\mathop{\mathchar
58\relax}}\Delta_{\mathfrak{p}^{\prime}}\xrightarrow{\sim}\Delta_{\mathfrak{p}}$.
We set $\epsilon_{\mathfrak{p}^{\prime}}$ to denote the composite

$\Delta_{\mathfrak{p}^{\prime}}\xrightarrow{c_{\delta}}\Delta_{\mathfrak{p}}\xrightarrow{\epsilon_{\mathfrak{p}}}\mathcal{O}^{\times}.$

Thus for each prime $\mathfrak{p}|p$, we have made a choice of character
$\epsilon_{p}$ of $\Delta_{\mathfrak{p}}$. For $\mathfrak{p}|p$, we have a
decomposition of $\mathcal{O}[\Delta_{\mathfrak{p}}]$-modules,

$\mathcal{U}_{\mathfrak{p},\mathcal{O}}=\mathcal{U}_{\mathfrak{p},\mathcal{O}}^{\epsilon_{\mathfrak{p}}}\times\mathcal{V}_{\mathfrak{p},\mathcal{O}},$

where the action of $\Delta_{\mathfrak{p}}$ on
$\mathcal{U}_{\mathfrak{p},\mathcal{O}}^{\epsilon_{\mathfrak{p}}}$ is via
$\epsilon_{\mathfrak{p}}$. Use the following notation

$\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}|p}\mathcal{U}_{\mathfrak{p},\mathcal{O}}^{\epsilon_{\mathfrak{p}}},\text{
and
}\mathcal{V}_{p,\mathcal{O}}=\prod_{\mathfrak{p}|p}\mathcal{V}_{\mathfrak{p},\mathcal{O}}.$

Then, we find that

$\mathcal{U}_{p,\mathcal{O}}=\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\times\mathcal{V}_{p,\mathcal{O}}.$

The above decomposition is that of $\mathcal{O}[\Delta]$-modules. In the same
way, $\bar{U}_{p,\mathcal{O}}$ decomposes into a product

$\bar{U}_{p,\mathcal{O}}=\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\times\bar{V}_{p,\mathcal{O}}.$

Let $\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$ (resp.
$\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$) denote the
$\chi$-isotypic component of
$\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)$ (resp.
$\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)$). Greenberg and Vatsal
show that $\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$ has
finite index in
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$, provided the
assertions of Assumption 1.1 are satisfied.

######  Proposition 3.8.

With respect to notation above, suppose that the Assumption 1.1 is satisfied.
Then, the following assertions hold

  1. (1)

$\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$ has finite index in
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$,

  2. (2)

there is a short exact sequence of $\mathcal{O}$-modules

$0\rightarrow H^{1}_{\operatorname{nr}}(\mathbb{Q},D)\rightarrow
S_{\chi,\epsilon}(\mathbb{Q})\rightarrow\mathrm{Hom}_{\mathcal{O}[\Delta]}\left(\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}/\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi},D\right)\rightarrow
0.$

Here, $H^{1}_{\operatorname{nr}}(\mathbb{Q},D)$ is the subgroup of
$H^{1}(\mathbb{Q},D)$ consisting of cohomology classes that are unramified at
all primes.

###### Proof.

The result follows from the proof of [GV20, Proposition 3.1]. ∎

######  Theorem 3.9.

Assume that

  1. (i)

the conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

Then, the following conditions are equivalent

  1. (1)

$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$ and
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}=\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$.

  2. (2)

$\mu_{\chi,\epsilon}=\lambda_{\chi,\epsilon}=0$.

  3. (3)

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$,

###### Proof.

The result follows as a direct consequence of Corollary 3.7 and Proposition
3.8. ∎

######  Lemma 3.10.

With respect to notation above, assume that $p\nmid\\#\operatorname{Cl}(K)$.
Then, we find that $H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$.

###### Proof.

Since $p\nmid\\#\Delta$, we find that
$H^{1}(\Delta,D^{\operatorname{G}_{K}})=0$, and thus from the inflation-
restriction sequence, $H^{1}(\mathbb{Q},D)$ injects into
$\operatorname{Hom}\left(\operatorname{G}_{K}^{\operatorname{ab}},D\right)^{\Delta}$.
Let $H_{K}$ be the Hilbert class field of $K$. We find that
$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)$ injects into
$\operatorname{Hom}(\operatorname{Gal}(H_{K}/K),D)^{\Delta}$. Since
$p\nmid\\#\operatorname{Cl}(K)$, we find that

$\operatorname{Hom}(\operatorname{Gal}(H_{K}/K),D)=0,$

and hence, $H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$. ∎

######  Theorem 3.11.

Assume that

  1. (i)

the conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$,

  3. (iii)

$K$ is $p$-rational,

  4. (iv)

$p\nmid\\#\operatorname{Cl}(K)$.

Then, $S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$.

###### Proof.

Since $p\nmid\\#\operatorname{Cl}(K)$, it follows from Lemma 3.10 that
$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$. Let $G_{p}$ be the maximal
pro-$p$ extension of $K$ which is unramified outside $p$. Since
$p\nmid\\#\operatorname{Cl}(K)$, we find that $G_{p}^{\operatorname{ab}}$ is
isomorphic to $\mathcal{U}_{p}/\bar{U}$. On the other hand, $K$ is
$p$-rational if and only if the Leopoldt conjecture is true at $p$ and there
is no $p$ torsion in $G_{p}^{\operatorname{ab}}$ (cf. [MR20] or [MD90]). This
implies that

$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}/\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}=0.$

The results then follow from Theorem 3.9. ∎

###  3.4. Special cases

In this subsection, we consider certain special cases. First, we consider some
$2$-dimensional Artin representations of dihedral type. Let $(L,\zeta)$ be a
pair, where $L$ is an imaginary quadratic extension of $\mathbb{Q}$ and
$\zeta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{L}\rightarrow\bar{\mathbb{Q}}^{\times}$ is a
character. Consider the $2$-dimensional Artin representation

$\rho=\rho_{(L,\zeta)}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Ind}_{\operatorname{G}_{L}}^{\operatorname{G}_{\mathbb{Q}}}\zeta.$

When restricted to $L$, the representation $\rho$ decomposes into a direct sum
of characters

$\rho_{|\operatorname{G}_{L}}=\left({\begin{array}[]{cc}\zeta&\\\
&\zeta^{\prime}\\\ \end{array}}\right).$

Here, $\zeta^{\prime}$ is given as follows

$\zeta^{\prime}(x)=\zeta(cxc^{-1}),$

where $c$ denotes the complex conjugation. We remark that the representation
$\rho$ is a self-dual representation of dihedral type if and only if
$\zeta^{\prime}=\zeta^{-1}$. Let $p$ be an odd prime number which splits in
$L$ as a product $p\mathcal{O}_{L}=\mathfrak{p}\mathfrak{p}^{*}$. We set
$\epsilon_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}=\zeta_{|\operatorname{G}_{\mathfrak{p}}}$ and note that
$\epsilon_{\mathfrak{p}^{*}}=\zeta^{\prime}_{|\operatorname{G}_{\mathfrak{p}}}$.
The field $K=L(\zeta,\zeta^{\prime})$ is the extension of $L$ generated by
$\zeta$ and $\zeta^{\prime}$. Choose an extension $\mathcal{F}/\mathbb{Q}_{p}$
containing $\mathbb{Q}_{p}(\chi,\epsilon_{\mathfrak{p}})$ and set
$\mathcal{O}$ to denote its valuation ring, and $D$ the associated
$\mathcal{O}$-divisible module. With respect to such a choice, we let
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ be the associated the Selmer group
over the cyclotomic $\mathbb{Z}_{p}$-extension.

######  Theorem 3.12.

With respect to notation above, assume that the following conditions hold

  1. (i)

$\rho$ is irreducible,

  2. (ii)

the order of $\zeta$ is coprime to $p$,

  3. (iii)

$\epsilon_{\mathfrak{p}}$ is nontrivial and
$\epsilon_{\mathfrak{p}}\neq\epsilon_{\mathfrak{p}^{*}}$.

Then, the following conditions are equivalent

  1. (1)

$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$ and
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}=\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$.

  2. (2)

$\mu_{\chi,\epsilon}=\lambda_{\chi,\epsilon}=0$.

  3. (3)

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$.

If $p\nmid h_{K}$ and $K$ is $p$-rational, then, the above conditions are
satisfied.

###### Proof.

The result follows from Theorem 3.9 once we show that

  1. (i)

the conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

Consider the conditions for Assumption 1.1 to hold.

  1. (1)

The field $K$ is the extension of $L$ cut out by $\zeta$ and $\zeta^{\prime}$.
Since the order of $\zeta$ is assumed to be coprime to $p$, the same holds for
$\zeta^{\prime}$. Since $p$ is odd, it follows that
$p\nmid[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$.

  2. (2)

Since $K$ is an imaginary quadratic field complex conjugation acts via the
matrix $\left({\begin{array}[]{cc}&1\\\ 1&\\\ \end{array}}\right)$, whose
eigenvalues are $1$ and $-1$ respectively. Therefore, $d^{+}=d^{-}=1$.

  3. (3)

The third condition follows from the assumption that $\epsilon_{\mathfrak{p}}$
is nontrivial and $\epsilon_{\mathfrak{p}}\neq\epsilon_{\bar{\mathfrak{p}}}$.

Finally, we note that since $\epsilon_{\bar{\mathfrak{p}}}$ is nontrivial, and
hence $H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

By the proof of Theorem 3.11, we note that
$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$ and
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}=\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$
if $p$ is such that $p\nmid h_{K}$ and $K$ is $p$-rational, then, the above
conditions are satisfied. ∎

Next, we consider the group of rotational symmetries of the icosahedron. This
group is simply $A_{5}$. An orthogonal set is a set of $6$ points, so that
three pairwise orthogonal lines can be drawn between pairs of them. The
midpoints of the $30$ edges of an icosahedron can be partitioned into $5$
orthogonal sets such that these sets are permuted by $A_{5}$. Let $g\in A_{5}$
be an element that corresponds to rotation of the icosahedron with axis
$(x,y,z)$ and angle $\theta$. Then the corresponding matrix is

$\left({\begin{array}[]{ccc}\cos\theta+(1-\cos\theta)x^{2}&(1-\cos\theta)xy-z\sin\theta&(1-\cos\theta)xz+y\sin\theta\\\
(1-\cos\theta)xy+z\sin\theta&\cos\theta+(1-\cos\theta)x^{2}&(1-\cos\theta)yz-x\sin\theta\\\
(1-\cos\theta)xz-y\sin\theta&(1-\cos\theta)yz+x\sin\theta&\cos\theta+(1-\cos\theta)z^{2}.\end{array}}\right)$

This gives a representation

$r\mathrel{\mathop{\mathchar
58\relax}}A_{5}\rightarrow\operatorname{GL}_{3}(\bar{\mathbb{Q}}).$

Let $K/\mathbb{Q}$ be a Galois extension with Galois group
$\operatorname{Gal}(K/\mathbb{Q})$ isomorphic to $A_{5}$, and $\varrho$ be the
composite

$\varrho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{Gal}(K/\mathbb{Q})\xrightarrow{\sim}A_{5}\xrightarrow{r}\operatorname{GL}_{3}(\bar{\mathbb{Q}}),$

where the initial map is the quotient map. Any $5$-cycle corresponds to a
rotation by an angle of $\frac{2\pi}{5}$, and therefore has eigenvalues
$e^{\frac{2\pi i}{5}},e^{-\frac{2\pi i}{5}},1$. Let $g\in A_{5}$ be a
$5$-cycle and $D$ be the group generated by $g$, and
$L\mathrel{\mathop{\mathchar 58\relax}}=K^{D}$ be the field fixed by $D$. Let
$p\geq 7$ be a prime which is split in $L$ and is inert in $K/L$. Then, the
restriction to the decomposition group at $p$ is of the form
$\operatorname{diag}\left(\alpha_{p},\alpha_{p}^{-1},1\right)$, where
$\alpha_{p}$ is an unramified character of order $5$. Let
$\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\mathcal{O}^{\times}$ be an
even character which is ramified at $p$, and set
$\rho\mathrel{\mathop{\mathchar 58\relax}}=\varrho\otimes\psi$.

######  Theorem 3.13.

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

  1. (1)

$H^{1}_{\operatorname{nr}}(\mathbb{Q},D)=0$ and
$\left(\mathcal{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}=\left(\bar{U}_{p,\mathcal{O}}^{[\epsilon]}\right)^{\chi}$.

  2. (2)

$\mu_{\chi,\epsilon}=\lambda_{\chi,\epsilon}=0$.

  3. (3)

$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})=0$.

If $p\nmid h_{K}$ and $K$ is $p$-rational, then, the above conditions are
satisfied.

###### Proof.

As in the proof of Theorem 3.12, the result follows from Theorem 3.9 once we
show that

  1. (i)

the conditions of Assumption 1.1 are satisfied.

  2. (ii)

$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$.

Let us verify the conditions of the Assumption 1.1.

  1. (1)

Since it is assumed that $p\geq 7$, it follows that $p\nmid\\#A_{5}$, and
hence, $p\nmid[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$.

  2. (2)

Let $v$ be an archidean prime, and $\sigma_{v}$ be a generator of the
decomposition group at $v$. Then,
$\rho(\sigma_{v})=\varrho(\sigma_{v})\psi(\sigma_{v})=\varrho(\sigma_{v})$.
Since $\varrho$ only gives rise to rotational symmetries and
$\varrho(\sigma_{v})$ has order $2$, it must correspond to the rotation with
$\theta=\pi$. This means that it is conjugate to
$\operatorname{diag}(1,-1,-1)$, and hence, $d^{+}=1$.

  3. (3)

For the third condition, it has been arranged that
$\varrho_{|\Delta_{\mathfrak{p}}}=\operatorname{diag}(\alpha_{p},\alpha_{p}^{-1},1)$,
where $\alpha_{p}$ is an unramified character of order $5$, and hence,
$\epsilon_{p}\mathrel{\mathop{\mathchar 58\relax}}=\alpha_{p}\psi$ is a
ramified character which occurs with multiplicity $1$.

Finally, we note that $H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{p}})=0$
since the characters $\alpha_{p}^{-1}\psi$ and $\psi$ are nontrivial. This is
because $\psi$ is ramified at $p$, while $\alpha_{p}$ is not. ∎

##  4\. Distribution questions for Artin representations

###  4.1. Iwasawa theory of class groups

In this section, we introduce and study some natural distribution questions
for Iwasawa invariants of class groups. These discussions serve to motivate
similar questions for Artin representations, which we discuss in the next
subsection. Given a number field $K$ and a prime number $p$, the Iwasawa
invariants $\mu_{p}(K)$ and $\lambda_{p}(K)$ are natural invariants to
consider associated with the growth of $p$-primary parts of class groups in
the cyclotomic extension of $K$. One considers the following question.

######  Question 4.1.

Given an imaginary quadratic $K/\mathbb{Q}$, how does $\mu_{p}(K)$ and
$\lambda_{p}(K)$ vary as $p\rightarrow\infty$? In other words, for a given
pair $(\mu,\lambda)$, what can be said about the upper and lower densities of
the set of primes $p$ for which

$\lambda_{p}(K)=\lambda.$

We call the above set $\mathcal{F}_{\lambda}$, and its upper (resp. lower)
density $\bar{\mathfrak{d}}_{\lambda}$ (resp.
$\underline{\mathfrak{d}}_{\lambda}$). We note that if $p$ is inert or
ramified in $K$ and $p\nmid h_{K}$, then, $h_{p}(K_{n})=0$ for all $n$. In
particular, $\lambda_{p}(K)=0$. This implies in particular that
$\underline{\mathfrak{d}}_{0}\geq\frac{1}{2}$. On the other hand, if $p$
splits in $K$, then, $r_{p,K}\geq 1$ and thus, $\lambda_{p}(K)\geq 1$. This
implies that $\mathfrak{d}_{0}=\frac{1}{2}$. Let $p$ be a prime which splits
in $K$ for which $p\nmid h_{K}$. The former condition is satisfied by
$\frac{1}{2}$ of the primes and the latter condition is satisfied for all but
finitely many primes. The following result is a corollary to Gold’s criterion.

######  Corollary 4.2.

Let $p$ be an odd prime number which splits into
$\mathfrak{p}\mathfrak{p}^{*}$ in $\mathcal{O}_{K}$. Suppose that $r>1$ is an
integer not divisible by $p$ and such that $\mathfrak{p}^{r}=(\alpha)$. Then,
the following conditions are equivalent

  1. (1)

$\lambda_{p}(K)>1$,

  2. (2)

$\operatorname{Tr}(\alpha)^{p-1}\equiv 1\mod{p^{2}}$.

Indeed we find that for each number $a_{0}\in[1,p-1]$, there is precisely one
congruence class $a$ modulo-$p^{2}$ such that is $a_{0}$ mod-$p$ and satisfies
$a^{p-1}\equiv 1\mod{p^{2}}$. Therefore, the probability that an integer
$a\in\mathbb{Z}/p^{2}\mathbb{Z}$ satisfies the congruence $a^{p-1}\equiv
1\mod{p^{2}}$ is $\frac{p-1}{p^{2}}=\frac{1}{p}-\frac{1}{p^{2}}$. Let
$N_{K}(X)$ be the number of primes $p\leq X$ such that $p$ is split in $K$ and
$\lambda_{p}(K)>1$. Thus, the heuristic suggests that

$N_{K}(X)\sim\sum_{p\in\Omega^{\prime}(X)}\left(\frac{1}{p}-\frac{1}{p^{2}}\right)\sim\log\log
X,$

where $\Omega^{\prime}(X)$ is the set of primes $p\leq X$ that split in $K$.
This leads us to the following expectation.

######  Conjecture 4.3.

Let $M_{p}(X)$ be the number of primes $\leq X$ that split in $K$ for which
$\lambda_{p}(K)>1$, then, $M_{p}(X)=\frac{1}{2}\log\log X+O(1)$.

In particular the set of primes $p$ for which $\lambda_{p}(K)>1$ is infinite
of density $0$, and thus, the conjecture in particular predicts that
$\mathfrak{d}_{1}=\frac{1}{2}$. Horie [Hor87] has proven the infinitude of
primes $p$ such that $\lambda_{p}(K)>1$ and Jochnowitz [Joc94] on the other
hand the infinitude of primes $p$ for which $\lambda_{p}(K)=1$.

On the other hand, one can fix a prime and study the variation of
$\lambda_{p}(K)$ as $K$ ranges over all imaginary quadratic fields. We
separate this problem into two cases, namely that of imaginary quadratic
fields in which $p$ is inert, and those in which $p$ splits. For the primes
$p$ that are inert in $K$ Ellenberg, Jain and Venkatesh [EJV11] make the
following prediction based on random matrix heuristics.

######  Conjecture 4.4 (Ellenberg, Jain, Venkatesh).

Amongst all imaginary quadratic fields $K$ in which $p$ is inert, the
proportion for which $\lambda_{p}(K)=r$ is equal to

$p^{-r}\prod_{t>r}\left(1-p^{-t}\right).$

We note that $\lambda_{p}(K)=0$ if and only if $p\nmid h_{K}$. The probability
that $p\nmid h_{K}$ is, according to the Cohen-Lenstra heuristic, predicted to
be equal to

$\prod_{t>0}\left(1-p^{-t}\right).$

###  4.2. Artin representations

Given an Artin representation $\rho$, we are interested in understanding the
variation of $\mu$ and $\lambda$-invariants as $p$ ranges over all prime
numbers. We specialize our discussion to odd $2$-dimensional Artin
representations of dihedral type

$\rho=\operatorname{Ind}_{\operatorname{G}_{L}}^{\operatorname{G}_{\mathbb{Q}}}\zeta,$

where $L$ is an imaginary quadratic field and $\zeta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{L}\rightarrow\bar{\mathbb{Q}}^{\times}$ is a
character. Let $\zeta^{\prime}$ be the character defined by
$\zeta^{\prime}(\sigma)=\zeta(c\sigma c^{-1})$, where
$c\in\operatorname{G}_{\mathbb{Q}}$ denotes the complex conjugation. Here,
$L/\mathbb{Q}$ is an imaginary quadratic field. Since it is assumed that
$\rho$ is of dihedral type, it follows that $\zeta^{\prime}=\zeta^{-1}$ and
the extension $K$ is simply the extension of $L$ that is fixed by the kernel
of $\zeta$. Furthermore, assume that $2\nmid[K\mathrel{\mathop{\mathchar
58\relax}}L]$. Let $S(\rho)$ be the set of primes $p$ such that

  1. (1)

$p$ is odd and $p\nmid[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$,

  2. (2)

$p$ splits in $L$, $p\mathcal{O}_{L}=\pi\pi^{*}$, and $\pi$ is inert in $K/L$.

For $p\in S(\rho)$, it is then clear from the assumptions that the Assumption
1.1 holds, and that
$H^{0}(\Delta_{\mathfrak{p}},D/D^{\epsilon_{\mathfrak{p}}})=0$ for any of the
primes $\mathfrak{p}|p$ of $K$ and the character
$\epsilon_{\mathfrak{p}}=\zeta_{|\Delta_{\mathfrak{p}}}$. Let $\epsilon$ be
the character associated to this choice of $\mathfrak{p}$ and
$\epsilon_{\mathfrak{p}}$.

######  Definition 4.5.

Let $T(\rho)$ be the set of primes $p\in S(\rho)$ such that the Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})$ vanishes for all of the primes
$\mathfrak{p}$ that lie above $p$. Set
$T^{\prime}(\rho)\mathrel{\mathop{\mathchar 58\relax}}=S(\rho)\backslash
T(\rho)$.

We note that the Theorem 3.11 implies that for $p\in S(\rho)$ such that $K$ is
$p$-rational, then, $p\in T(\rho)$.

Let us recall some heuristics on the $p$-rationality condition. The
$\mathbb{Z}_{p}$-rank of $\mathcal{U}_{p}$ is $n=[K\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}]=r_{1}+2r_{2}$. On the other hand, the rank of $U_{K}$ is
$k\mathrel{\mathop{\mathchar 58\relax}}=r_{1}+r_{2}-1$. The probabilty that a
random matrix with $n$ columns and $k$ rows does not have full
$\mathbb{Z}_{p}$-rank is

$\operatorname{Pr}_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=1-\frac{\prod_{i=0}^{k-1}\left(p^{n}-p^{i}\right)}{p^{n}}\leq\frac{1}{p^{n-k+1}}+\frac{1}{p^{n-k+2}}+\dots+\frac{1}{p^{n}}.$

Therefore, the expected number of primes $p$ at which $K$ is not $p$-rational
is finite, since according to this heuristic,

$\sum_{p}\operatorname{Pr}_{k,n}\leq\sum_{p}\left(\frac{1}{p^{r_{2}+2}}+\frac{1}{p^{r_{2}+3}}+\dots+\frac{1}{p^{n}}\right)=\zeta(r_{2}+2)+\zeta(r_{2}+3)+\dots+\zeta(n)<\infty.$

This is indeed a conjecture due to Gras.

######  Conjecture 4.6 (Gras).

Let $K$ be a number field. Then for all large values of $p$, $K$ is
$p$-rational.

This leads us to make the following conjecture.

######  Conjecture 4.7.

With respect to above notation, the set of primes $T^{\prime}(\rho)$ is
finite. Thus, there are only finitely many pairs $(\mathfrak{p},\epsilon)$
where $\mathfrak{p}$ is a prime above $p\in S(\rho)$ and
$\epsilon\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{L_{\mathfrak{p}}}\rightarrow\bar{\mathbb{Q}}_{p}^{\times}$
is a character, such that the associated Selmer group
$S_{\chi,\epsilon}(\mathbb{Q}_{\infty})\neq 0$.

There is a relationship between Gras’ conjecture and the generalized abc-
conjecture, stated below, cf. [Voj98].

######  Conjecture 4.8 (Generalized abc-conecture).

Let $K$ be a number field and $I$ be an ideal in $\mathcal{O}_{K}$, the
radical of $I$ is defined as follows

$\operatorname{Rad}(I)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}|p}N(\mathfrak{p}),$

where the product is over all prime ideals $\mathfrak{p}$ dividing $I$, and
$N(\mathfrak{p})\mathrel{\mathop{\mathchar
58\relax}}=\\#\left(\mathcal{O}_{K}/\mathfrak{p}\mathcal{O}_{K}\right)$ is the
norm of $\mathfrak{p}$. The generalized abc conjecture predicts that for any
$\epsilon>0$, there exists a constant $C_{K,\epsilon}>0$ such that

$\prod_{v}\operatorname{max}\\{|a|_{v},|b|_{v},|c|_{v}\\}\leq
C_{K,\epsilon}\left(\operatorname{Rad}(abc)\right)^{1+\epsilon},$

holds for all non-zero $a,b,c\in\mathcal{O}_{K}$ such that $a+b=c$.

Let us recall a recent result of Maire and Rougnant, cf. [MR20, Theorem A].

######  Theorem 4.9 (Maire-Rougnant).

Let $K/\mathbb{Q}$ be an imaginary $S_{3}$ extension. Then, the generalized
abc-conjecture for $K$ implies that there is a constant $c>0$ such that

$\\#\\{p\leq x\mid K\text{ is }p\text{-rational}\\}\geq c\log x.$

The above result has implication to the vanishing of Iwasawa modules (and
invariants).

######  Theorem 4.10.

Let $K/\mathbb{Q}$ be an imaginary $S_{3}$ extension and $\rho$ be a
$2$-dimensional Artin representation that factors through
$\operatorname{Gal}(K/\mathbb{Q})$. Then,

$\\#\\{p\leq x\mid p\in T(\rho)\\}\geq c\log x.$

###### Proof.

Theorem 3.11 implies that for $p\in S(\rho)$ such that $K$ is $p$-rational,
then, $p\in T(\rho)$. The result thus follows. ∎

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