For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+\delta_{F,p}$ variables in his innovative paper published in 1978, where $\delta_{F,p}$ denotes the Leopoldt defect for $F$ and $p$. In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of $F$ at $p$), and construct a $p$-adic Artin $L$-function of $d+1+\delta_{F,p}$ variables, which interpolates critical values of the Artin $L$-function associated to a $p$-unramified Artin representation of the absolute Galois group $G_F$.
Our construction
is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case.

§ INTRODUCTION

Let $F$ be a CM field of degree $2{d}$ over $\mathbb{Q}$ and $F^+$
its maximal totally real subfield. The ring of integers of $F$ is denoted as $\mathfrak{r}_F$. We fix a prime number $p$, an algebraic closure $\overline{\mathbb{Q}}$ of the rational number field $\mathbb{Q}$, an embedding $\iota_\infty \colon \overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$ of $\overline{\mathbb{Q}}$ into the complex number field $\mathbb{C}$, and
an isomorphism $\boldsymbol{\iota}\colon \mathbb{C}\xrightarrow{\, \sim \,}\mathbb{C}_p$ between $\mathbb{C}$ and the completion $\mathbb{C}_p$ of a fixed algebraic closure $\overline{\mathbb{Q}}_p$ of the $p$-adic number field $\mathbb{Q}_p$. Set $\iota_p:=\boldsymbol{\iota}\circ \iota_\infty \colon \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$.
Suppose that the prime ideal $(p)$ of $\mathbb{Q}$ generated by $p$ is unramified in $F^+$, and that every prime ideal of $F^+$ lying above $(p)$ is completely decomposed in the quadratic extension $F /F^+$. We fix a $p$-ordinary CM type $\Sigma_{F}$ of $F$, whose choice amounts to choosing one of $\mathfrak{P}$ or $\mathfrak{P}^c$ for each prime $\mathfrak{p}$ of $F^+$ lying above $(p)$ decomposed as $\mathfrak{p}\mathfrak{r}_F=\mathfrak{P}\mathfrak{P}^c$;
see Section <ref> for its precise definition.
Let $F_{\max}$ be the composition of all $\mathbb{Z}_p$-extensions of $F$ (in $\overline{\mathbb{Q}}$) and $\Gamma_{F,\mathrm{max}}$ the Galois group of $F_{\max}/F$. Due to global class field theory, we see that $\Gamma_{F,\mathrm{max}}$ is a free $\mathbb{Z}_p$-module of rank
equal to or greater than $d+1$; the equality holds if and only if the Leopoldt conjecture for $F$ and $p$ is true.

Now let $\widehat{\mathbb{Z}}^\mathrm{ur}_p$ be the ring of integers of the $p$-adic completion $\widehat{\mathbb{Q}}^\mathrm{ur}_p$ of the maximal unramified extension $\mathbb{Q}_p^\mathrm{ur}$ of $\mathbb{Q}_p$, and let $\widehat{\mathcal{O}}^\mathrm{ur}$ denote a complete discrete valuation ring which is finite over
$\widehat{\mathbb{Z}}^\mathrm{ur}_p$ and contained in the ring of integers $\mathcal{O}_{\mathbb{C}_p}$ of $\mathbb{C}_p$.
Then, under certain assumptions, Katz [25] has constructed a $p$-adic Hecke $L$-function over $F$ as an element of the multi-variable Iwasawa algebra $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$. The $p$-adic $L$-function which Katz has constructed interpolates the values at $0$ of the Hecke $L$-functions for various algebraic Hecke characters with appropriate infinity type and conductor dividing a power of $p$.
Later Hida–Tilouine [22] has generalised the construction of Katz, and relaxed constraints on the conductors of algebraic Hecke characters appearing in the interpolation property. The results of Katz and Hida–Tilouine are summarised as follows.

[Katz, Hida–Tilouine]
Let $\psi$ be an algebraic Hecke character of finite order on $F$ such that $F_\psi = (\overline{\mathbb{Q}})^{\mathrm{Ker} (\psi^{\mathrm{gal}})}$ is at most tamely ramified at every prime of $F$ lying above $(p)$.
Let us fix a totally imaginary element $\delta$ of $F$ satisfying the conditions
($1_\delta$)—($3_\delta$) introduced in Section <ref>.
We further assume that the ideal  $(2\delta)$ generated by $2\delta$ is relatively prime to the conductor of $\psi$.
Then there exists a unique element $L_{p, \Sigma_F}(\psi )$ of $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$ satisfying

\begin{multline*}
\frac{\eta^{\mathrm{gal}}(L_{p, \Sigma_{F}}(\psi))}
{C_{\mathrm{CM},p,F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}} =
(\mathfrak{r}_{F}^\times : \mathfrak{r}_{{F}^+}^\times )
\frac{(-1)^{w_\eta d}\,  i^{\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert }}
{\sqrt{|D_{F^+}|} 2^d (2\delta)_{\Sigma_F}^{\mathsf{r}_\eta }}
\prod_{v \in \Sigma_{F,p}} \mathrm{Eul}((\psi\eta)_v,0)
\frac{L ((\psi\eta)_\infty, 0) L(\psi\eta,0)}{\Omega_{\mathrm{CM},\infty,F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}}, \\
\mathrm{Eul}((\psi\eta)_v,0)=
\begin{cases}
L((\psi \eta)_{v^c},0)^{-1}L((\psi\eta)_v^\vee,1)^{-1} & \text{if $\psi\eta$ is unramified at $v$}, \\
\epsilon((\psi \eta)_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)^{-1} & \text{if $\psi\eta$ is ramified at $v$},
\end{cases}
\end{multline*}

for each algebraic Hecke character $\eta$ satisfying the following conditions:

* the Galois character $\eta^{\mathrm{gal}}\colon G_F\longrightarrow \overline{\mathbb{Q}}^\times_p$
associated to $\eta$ factors through $\Gamma_{F,\mathrm{max}}$;

* the infinity type of $\eta$ satisfies both $-w_\eta-r_{\eta,\sigma}\leq -1$ and $r_{\eta,\sigma}\geq 0$ for every $\sigma\in \Sigma_F$.

Here $D_{F^+}$ denotes the absolute discriminant of $F^+$, $\mathsf{t}$ denotes the diagonal element $\sum_{\sigma \in \Sigma_{F}} \sigma$ in $\mathbb{Z} [\Sigma_{F}]$, and $(w_\eta,\mathsf{r}_\eta)\in \mathbb{Z}\times \mathbb{Z}^{\Sigma_F}$ denotes the infinity type of the algebraic Hecke character $\eta$ defined as $\eta\bigl((x_\sigma)_{\sigma\in \Sigma_F}\bigr)=\prod_{\sigma\in \Sigma_F}x_\sigma^{w_\eta+r_{\eta,\sigma}}\overline{x}_\sigma^{-r_{\eta,\sigma}}$. See Remark <ref> for details on multi-index notation used here.
The symbols $\epsilon((\psi\eta)_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)$ and $L((\psi\eta)_\infty,s)$ respectively denote Deligne's local constant and the archimedean local factor (or the gamma factor) of $L(\psi\eta,s)$; see Remark <ref> and Remark <ref> for details.
The definition of the modified complex CM period $\Omega_{\mathrm{CM},\infty,F} \in \mathbb{C}^{\times,\Sigma_F}$ (resp. the $p$-adic CM preiod $C_{\mathrm{CM},p} \in
(\widehat{\mathcal{O}}^\mathrm{ur})^{\times, \Sigma_F}$) will be reviewed in Section <ref>; see Definition <ref> for details.

Let  $ M_{\Sigma_F}$  denote the maximal abelian pro-$p$ extension of $F_{\max}$ which is unramified outside the primes above $\Sigma_{F,p}$. Let $F_\psi$ denote the finite extension of $F$ corresponding to the kernel of the finite Galois character $\psi^\mathrm{gal}\colon G_F\rightarrow \overline{\mathbb{Q}}^\times$ associated to $\psi$. Set $\mathcal{O}=\mathbb{Z}_p[\mathrm{Im}\,\psi^\mathrm{gal}]$. Now assume that $F_\mathrm{max}$ and $F_\psi$ are linearly disjoint over $F$.
Note that the $\psi$-part $\mathrm{Gal}( M_{\Sigma_F}F_\psi/ F_{\max} F_\psi)_\psi$
of the Galois group $\mathrm{Gal}( M_{\Sigma_F}F_\psi/ F_{\max} F_\psi)\otimes_{\mathbb{Z}_p}\mathcal{O}$ is a finitely generated torsion
${\mathcal{O}}[[\Gamma_{F,\mathrm{max}}]]$-module as explained <cit.>, and thus its characteristic ideal is defined as a nontrivial ideal of $\mathcal{O}[[\Gamma_{F,\max}]]$. Then the (multi-variable) Iwasawa main conjecture for CM fields is formulated as follows.

Let $\psi$ be an algebraic Hecke character of finite order defined on $F$ such that $F_\psi$ is at most tamely ramified at every prime of $F$ lying above $(p)$.Then we have

\mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]}(\mathrm{Gal}( M_{\Sigma_F}F_\psi / F_{\max} F_\psi)_\psi \widehat{\otimes}_{\mathcal{O}} \widehat{\mathcal{O}}^\mathrm{ur} ) = (L_{p, \Sigma_F}(\psi ))

as an equality of ideals of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$.

Under certain technical assumptions, Ming-Lun Hsieh has proved the Iwasawa main conjecture for $\psi$.

Let $\psi$ be an algebraic Hecke character of finite order on $F$ such that $F_\psi$ is at most tamely ramified at every prime of $F$ lying above $(p)$.
Assume that the Leopoldt conjecture is true for $F$ and $p$. We further assume the following four conditions

* the prime $p$ divides neither $3$, the relative class number $h^-_{F}$ of $F/F^+$, nor the order of $\psi$;

* the character $\psi$ is anticyclotomic;

* the local Galois character $\psi^{\mathrm{gal}} \vert_{ G_{F_v}} $
is unramified and non-trivial for every finite place $v$ of $F$
lying above $(p)$;

* the global Galois character $\psi^{\mathrm{gal}} \vert_{G_{F(\sqrt{p^\ast})}}$ is non-trivial
with $p^\ast =(-1)^{\frac{p-1}{2}} p$.

Here $\psi^{\mathrm{gal}}\colon G_F \longrightarrow \overline{\mathbb{Q}}^\times_p$
is the Galois character associated to $\psi$ via global class field theory, as before.
Then the Iwasawa main conjecture for $\psi$ holds true.

The result above is Corollary 2 of [24]. In fact, Hsieh verified, under several assumptions, the inclusion

\mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]}(\mathrm{Gal}( M_{\Sigma_F}/F_\psi F_{\max} )_\psi \otimes_{\mathcal{O}} \widehat{\mathcal{O}}^\mathrm{ur} ) \supset (L_{p, \Sigma_F}(\psi ))

in <cit.> predicted by the Iwasawa main conjecture for $\psi$ when the Leopoldt conjecture for $F$ and $p$ holds true, adopting the method of Eisenstein congruences on the unitary group $\mathrm{GU}(2,1)$.
Note that the Iwasawa algebra $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$ is isomorphic to the ring of formal power series in $d+1$ variables under the Leopoldt conjecture for $F$ and $p$. Meanwhile, Hida obtains, under other technical assumptions, an equality between the algebraic characteristic ideal and the analytic ideal <cit.> in the anticyclotomic Iwasawa algebra, which is a $d$-variable quotient of the Iwasawa algebra $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$. A standard specialization argument combined with these two results implies the desired equality in $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$.
When the CM field $F$ under consideration is a composite of a totally real field and an imaginary quadratic field,
Hsieh has another result <cit.> on the Iwasawa main conjecture for $\psi$ by using Rubin's equality of the
two-variable Iwasawa main conjecture for an imaginary quadratic field <cit.> in place of the anticyclotomic main conjecture.

In the present article, we construct a $p$-adic Artin $L$-function $L_{p, \Sigma_F}(\rho )$ associated to an Artin representation $\rho$ on $F$ unramified at every prime ideal lying above $(p)$.
Since the Galois character $\psi^{\mathrm{gal}}$
associated to an algebraic Hecke character $\psi$ of finite order is an Artin representation of degree $1$, Theorems A and Theorem B below can be respectively regarded as extensions of Theorems 1 and Theorem 2.

[a part of Theorem <ref>]
Let $\rho$ be an Artin representation of $G_F$ unramified at every prime ideal of $F$ lying above $(p)$, and suppose that
the field $F_\rho $ corresponding to the kernel of $\rho$ is also a CM field.
Assume that the Iwasawa main conjecture for $\psi'$ is true for any intermediate field $F'$ of the extension $F_\rho /F$ and for any
algebraic Hecke character $\psi'$ of finite order on $F'$, whose associated Galois character $\psi^{\prime,\mathrm{gal}}$ factors through $\mathrm{Gal}(F_\rho/F')$.
Then there exists a unique element $L_{p, \Sigma_F }(\rho )$ of $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\mathrm{max}}]]$ satisfying

\begin{multline} \label{equation:interpolation_property_TheoremA}
\!\!\!\!\!\! \frac{\eta^\mathrm{gal} (L_{p,\Sigma_F} (M(\rho )))}{\bigl(C^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}_{\mathrm{CM}, p,F} \bigr)^{r(\rho )}} =
\left(
\frac{(-1)^{w_\eta d}\, i^{\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert}}
{2^d (2\delta)_{\Sigma_F}^{\!\!\mathsf{r}_\eta}}
\right)^{r(\rho )} \!\!\!\!
\prod_{v\in \Sigma_{F,p}} \! \!\!\mathrm{Eul} ((\rho\otimes \eta)_v,0)
\dfrac{L((\rho\otimes \eta)_\infty,0)L(\rho \otimes \eta, 0)}{\bigl(\Omega^{w_\eta\mathsf{t}+2\mathsf{r}_\eta}_{\mathrm{CM}, \infty, F} \bigr)^{r(\rho )}}\\
\mathrm{Eul}((\rho\otimes \eta)_v,0)=
\begin{cases}
L((\rho\otimes \eta)_{v^c},0)^{-1}L((\rho\otimes \eta)_v^\vee, 1)^{-1} & \text{if $\eta$ is unramified at $v$}, \\
\epsilon((\rho \otimes \eta)_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)^{-1} & \text{ if $\eta$ is ramified at $v$},
\end{cases}
\end{multline}

for each algebraic Hecke character $\eta$ satisfying the conditions (i) and (ii) appearing in Theorem 1. Here $L(\rho\otimes \eta,s)$ denotes the Hasse–Weil $L$-function associated to the pure motive $M(\rho)\otimes M(\eta)$, and $\epsilon((\rho\otimes \eta)_v,\mathbf{e}_{F,v}^{(2\delta)}, dx_v)$ is Deligne's local constant at $v\in \Sigma_{F,p}$; see (<ref>) and (<ref>) for precise definitions.

[a part of Theorem <ref>]
Let $\rho$ be an Artin representation of $G_F$, which is unramified at
every prime ideal of $F$ lying above $p$, and suppose that the field $F_\rho $ corresponding to the kernel of $\rho$ is also a CM field.
Assume that the Iwasawa main conjecture for $\psi '$ is true for any intermediate field $F'$ of the extension $F_\rho /F$ and for any
algebraic Hecke character $\psi'$ of finite order on $F'$.
Then we have

\mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]}(\mathrm{Gal}( M_{\Sigma_F}/F_\rho F_{\max} )_\rho \widehat{\otimes}_{\mathcal{O}}\widehat{\mathcal{O}}^\mathrm{ur}  )= (L_{p, \Sigma_F }(M(\rho) ))

as an equality of ideals of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$.

Let us briefly explain our strategy to prove Theorems A and B.
One of the key ingredients is Brauer's induction theorem <cit.> appearing in representation theory of finite groups.
Indeed, it provides us a virtual decomposition
$\rho= \sum^s_{i=1} a_i \mathrm{Ind}^G_{G_{F_i}} \psi_i^\mathrm{gal} $ of
the Artin representation $\rho$ under consideration, where, for each $i$, $F_i$ is an intermediate field of $F_\rho /F$, $G_{F_i}$ is the absolute Galois group of $F_i$, and $\psi_i^\mathrm{gal}\colon \mathrm{Gal}(F_\rho/F_i)\rightarrow \overline{\mathbb{Q}}^\times$ is a Galois character of finite order.
Then we define our $p$-adic $L$-function
$ L_{p,\Sigma_F}(\rho )$
as the product
\prod_{i=1}^s
\mathrm{pr}_i (L_{p,\Sigma_F}(\psi_i))^{a_i}
up to multiples of minor constants,
where $\psi_i$ is an algebraic Hecke character on $F_i$ corresponding to $\psi^\mathrm{gal}$ via global class field theory, and  $\mathrm{pr}_i$ is the composition
$\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F_i,\mathrm{max}}]]\twoheadrightarrow  \widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(F_iF_{\max}/F_i)]] \hookrightarrow \widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$.
Note that the decomposition of $\rho$ as above due to Brauer induction principle is just a virtual one, and the integers $a_i$ can be negative.
Hence $L_{p,\Sigma_F}(\rho )$ is constructed in the field of fractions of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$, and we do not know a priori whether it is contained in $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$ or not; that is, the integrality of the $p$-adic Artin $L$-function is not clear in this stage. Meanwhile, we can prove that, on the algebraic side, the characteristic ideal has a quite similar decomposition

\begin{multline}\label{equation:descent_argument_algebraic_side_introduction}
\mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]}(\mathrm{Gal}( M_{\Sigma_F}/F_\rho F_{\max} )_\rho  \widehat{\otimes}_\mathcal{O}\widehat{\mathcal{O}}^\mathrm{ur} ) \\
= \displaystyle \prod_{i=1}^s
\mathrm{pr}_i \bigl( \mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F_i,\mathrm{max}}]]}(\mathrm{Gal}( M_{\Sigma_F}/F_i F_{\max} )_{\psi_i}  \widehat{\otimes}_{\mathcal{O}}\widehat{\mathcal{O}}^\mathrm{ur}) \bigr)^{a_i}.
\end{multline}

Although the product in the right-hand side of (<ref>) is just a fractional ideal of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$ and it is not clear whether it is an integral ideal of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$, the left-hand side of (<ref>) is indeed an integral ideal of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$ by definition. Therefore, admitting the Iwasawa main conjecture for every algebraic Hecke character $\psi_i$ appearing in the virtual decomposition of $\rho$,
we may conclude that the principal ideal
$( L_{p,\Sigma_F}(\rho ))$ must be an integral ideal of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$. In order to check that our $p$-adic Artin $L$-function $L_{p,\Sigma_F}(M(\rho) )$  thus constructed satisfies the interpolation property (<ref>), we need to put together the interpolation formula for each $L_{p,\Sigma_{F_i}}(\psi_i)$ into the desired form
(<ref>). This hard task will be done in Section <ref>. We will close the Introduction by giving several comments concerning the present article.

§.§.§ Remarks related to preceding results

The prototype of this work is Ralph Greenberg's study of a $p$-adic Artin $L$-function [13, 16] over a totally real number field $K$.
He has constructed a one-variable $p$-adic $L$-function associated to a totally even Artin representation $\rho$ of $G_K$. The construction based upon the Brauer induction principle has appeared in [13], and we owe the main strategy of the proof to [13]. There are, however, quite a lot of difficulties which did not matter in the case over a totally real field.
We shall explain technical difficulties appearing in our generalisation of Greenberg's work to the case over a CM field.

Firstly, the Brauer induction argument in Greenberg's case is covered by one-variable objects, but in the present work,
the number of variables changes drastically throughout the Brauer induction argument. Here let us admit Leopoldt's conjecture for simplicity.
Then $p$-adic Hecke $L$-functions $L_p(\psi_i)$ appearing in the construction of the $p$-adic Artin $L$-function are of one variable when $F_i$ is totally real. Contrastingly in our CM situation,
$p$-adic Hecke $L$-functions $L_{p,\Sigma_{F_i}}(\psi_i)$ are of $[F_i :\mathbb{Q}]+1$ variables for each $i$. Thus, in order to take the product of $p$-adic Hecke $L$-functions, we need to specialise each $L_{p,\Sigma_{F_i}}(\psi_i)$ into the $(d+1)$-variable one via $\mathrm{pr}_i$, but the characteristic ideals appearing in the right-hand side of (<ref>) do not behave compatibly in general with respect to such specialisation procedures.
To overcome this difficulty, we adopt techniques developed in our previous work [19]. In particular, inductive descent arguments developed there play crucial roles in the verification of the key decomposition formula (<ref>) on the algebraic side.

Secondly, the proof of the desired interpolation property (<ref>) becomes much more complicated than that in [13]. Indeed, only $L$-values concern the interpolation properties of the $p$-adic Artin $L$-functions for totally real fields, and thus one readily verifies it just by using inductivity of the Artin $L$-functions.
However, there are so many factors appearing in the interpolation properties of the $p$-adic Artin $L$-functions for CM fields; for example, the gamma factor $L((\rho\otimes \eta)_\infty,0)$, the modified complex CM periods $\Omega_{\mathrm{CM},\infty,F}$, the $p$-adic CM periods $C_{\mathrm{CM},p,F}$, the modified $p$-Euler factor $\mathrm{Eul}((\rho\otimes \eta)_v,0)$ and so on. All of these factors do not appear in [13], and the matching of some factors requires much harder arguments.
Amongst them, the matching of Deligne's local constants requires quite delicate arguments and is not straightforward at all. To establish its matching, we formulate and verify in Appendix <ref> an extension of classical Davenport–Hasse relation <cit.> for Gauss sums to the case where the conductor of the multiplicative character is a power of a prime number (Theorem <ref>), which seems to be a result of independent interest and will play important roles in the construction of $p$-adic Artin $L$-functions for more general motives in future works.

§.§.§ Notation

We mainly use the fraktur $\mathfrak{r}$ for the ring of integers
of an algebraic number field (which is often regarded
as the base field of a certain motive);
the calligraphic $\mathcal{O}$
is kept to denote the ring of integers for a $p$-adic field
(which is often regarded as the coefficient field of the $p$-adic
realisation of a certain motive).
The absolute norm of a fractional ideal $\mathfrak{a}$
of an algebraic number field is denoted by $\mathcal{N}\mathfrak{a}$.
We fix an algebraic closure $\overline{\mathbb{Q}}$ of the rational number field $\mathbb{Q}$ and
regard all algebraic number fields (that is, all finite extensions of $\mathbb{Q}$) as subfields of $\overline{\mathbb{Q}}$.

For an algebraic number field $\mathsf{K}$, let  $\mathbb{A}_\mathsf{K}$
(resp. $\mathbb{A}_\mathsf{K}^\times$) denote the ring of adèles
(resp. the group of idèles) of $\mathsf{K}$. The finite part
(resp. the archimedean part) of the ring of adèles
$\mathbb{A}_\mathsf{K}$ is denoted by $\mathbb{A}_{\mathsf{K},\mathrm{fin}}$
(resp. $\mathbb{A}_\mathsf{K}^\infty$).

We shall fix the notion of the standard additive character throughout this article.
For each finite prime $v$ of $K$, we define $\mathbf{e}_{\mathsf{K},v}\colon  \mathsf{K}_{v} \longrightarrow \mathbb{C}^{\times}$ to be

\begin{align} \label{eq:standard_additive}
\mathbf{e}_{\mathsf{K},v} (x )=\exp(-2\pi \sqrt{-1} \mathrm{Tr}_{\mathsf{K}/\mathbb{Q}}(\tilde{x}))
\end{align}

where $\tilde{x}$ denotes an arbitrary element of $\bigcup_{n=1}^{\infty} \mathfrak{P}_v^{-n}$
(regarded as a $\mathfrak{r}_\mathsf{K}$-submodule of $\mathsf{K}$)
such that $\tilde{x}-x$ is contained in the completion of the ring of integers of $\mathsf{K}$ at $v$.

Let $\mathbb{C}_p$ be the $p$-adic completion of the fixed algebraic closure $\overline{\mathbb{Q}}_p$ of $\mathbb{Q}_p$ and $\mathcal{O}_{\mathbb{C}_p}$ its
ring of integers. For the ring of integers $\mathcal{O}$ of a finite extension of $\mathbb{Q}_p$,
we use the symbol ${\widehat{\mathcal{O}}}^\mathrm{ur}$  for the discrete valuation ring ${\mathcal{O}} \widehat{\mathbb{Z}}_p^{\mathrm{ur}}$, where $\widehat{\mathbb{Z}}_p^{\mathrm{ur}}$ denotes the ring of integers of the $p$-adic completion of the maximal unramified extension of $\mathbb{Q}_p$.

In this article we adopt the geometric normalisation
of global class field theory. Specifically, for a finite abelian extension $\mathsf{L/K}$ of algebraic number fields, the reciprocity map $(-,\mathsf{L/K}) \colon \mathbb{A}^\times_\mathsf{K} \rightarrow \mathrm{Gal}(\mathsf{L/K})$
is normalised so that it sends a uniformiser $\varpi_{\mathfrak{p}}$ of a prime ideal $\mathfrak{p}$ relatively prime to the conductor of the extension $\mathsf{L/K}$
to the geometric Frobenius element
$\mathrm{Frob}_{\mathfrak{p}}$ in $\mathrm{Gal}(\mathsf{L/K})$. In other words,
$a^{(\varpi_{\mathfrak{p}}, \mathsf{L/K})^{-1}}\equiv a^{q_{\mathfrak{p}}} \, (\text{mod}\, \mathfrak{p})$
holds for each $a$ in $\mathfrak{r}_\mathsf{K}$ where $q_{\mathfrak{p}}$ denotes
the cardinality of the residue field $\mathfrak{r}_\mathsf{K}/\mathfrak{p}$.

If $\mathsf{K}$ is an algebraic number field, the absolute Galois group
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathsf{K})$ is denoted by
$G_\mathsf{K}$. For a place $v$ of $\mathsf{K}$, the decomposition group
and the inertia group at $v$ are denoted by $D_v$ and $I_v$, respectively.
For a (possibly infinite) abelian Galois extension $\mathsf{L/K}$
of $\mathsf{K}$ and the ring of integer $\mathcal{O}$ of
a finite extension of $\mathbb{Q}_p$, we define
$\widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(\mathsf{L/K})]]^\sharp$ as the
free $\widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(\mathsf{L/K})]]$-module of rank one
on which $G_\mathsf{K}$ acts via the universal tautological character

\begin{align*}
G_\mathsf{K} \rightarrow
\widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(\mathsf{L/K})]]^\times ; \ g \mapsto
\end{align*}

We finally remark that, as for Hodge–Tate $p$-adic representations,
the Hodge–Tate weights are normalised so that
the $p$-adic cyclotomic character $\chi_{p,\mathrm{cyc}}$ is of Hodge–Tate weight $-1$.

§ REVIEW OF $P$-ADIC HECKE $L$-FUNCTIONS FOR CM FIELDS

In this section, we review theory of $p$-adic Hecke $L$-functions for CM
fields developed by Katz [25], Hida and Tilouine

§.§ General settings

Let $F$ be a CM field of degree $2d$ and $F^+$ the maximal totally real subfield of $F$. We use the symbol $c$ for the complex conjugation of $F$, that is, a unique generator of the Galois group of $F/F^+$.
The composition of all $\mathbb{Z}_p$-extensions of $F$ in $\overline{\mathbb{Q}}$ is denoted as $F_\mathrm{max}$, and the Galois group of $F_{\mathrm{max}}/F$ is denoted as $\Gamma_{F,\mathrm{max}}$. Then $\Gamma_{F,\mathrm{max}}$ is a free $\mathbb{Z}_p$-module of finite rank, and one readily observes, by using global class field theory, that the $\mathbb{Z}_p$-rank of $\Gamma_{F,\mathrm{max}}$ is equal to or greater than $d+1$, which is exactly equal to $d+1$ if and only if the Leopoldt conjecture for $F$ and $p$ is true.

We choose and fix an odd prime number $p$.
We impose the following two assumptions on $F$ and $p$:

(unr$_{F,p}$) the field $F^+$ is absolutely unramified
at $p$,

(ord$_{F,p}$) all places of $F^+$ lying above
$p$ split completely in $F$.

By virtue of the ordinarity condition (ord$_{F,p}$), there
exists a $p$-ordinary CM type $\Sigma_{F}$ of
$F$ (also called a $p$-adic CM type in literature),
which is defined as a subset of the set $I_{F}$ of all embeddings of
$F$ into $\overline{\mathbb{Q}}$ satisfying the following two conditions:

– we have $I_{F} =\Sigma_{F} \sqcup \Sigma_F^c$ (disjoint union) where
$\Sigma_{F}^c=\{ \sigma\circ c \in I_F \mid \sigma \in \Sigma _{F}\}$;

– we have $\{ \text{places of $F$ lying
above $p$} \} = \Sigma_{F,p} \sqcup \Sigma^c_{F,p}$
(disjoint union) where $\Sigma_{F,p}$ denotes the set of places of $F$ induced by embeddings $\iota_p \circ \sigma$ for
all $\sigma$ in $\Sigma_{F}$, and $\Sigma^c_{F,p}$
is defined as $\Sigma^c_{F,p}:=\{\mathfrak{P}^c \mid \mathfrak{P}\in \Sigma_{F,p}\}$.

We take a $p$-ordinary CM type $\Sigma_{F}$ of $F$ and fix it once and for all.

§.§ Period relations of CM periods

In this subsection, we recall the definition of
the complex and $p$-adic CM periods, which
appear in the interpolation formulae of the $p$-adic Hecke $L$-functions
for CM fields. For this purpose, we first construct a Hilbert–Blumenthal
abelian variety equipped with complex multiplication by
$\mathfrak{r}_{F}$ as follows.

Consider a diagonal embedding $\mathfrak{r}_F\hookrightarrow \mathbb{C}^{\Sigma_{F}}\, ; x\mapsto (\sigma(x))_{\sigma \in \Sigma_F}$ of $\mathfrak{r}_F$ into $\mathbb{C}^{\Sigma_F}$ with respect to the fixed $p$-ordinary CM-type $\Sigma_{F}$.
The image $\Sigma_{F}(\mathfrak{r}_{F})$  of this embedding then forms a
$\mathbb{Z}$-lattice of $\mathbb{C}^{\Sigma_{F}}$, and thus we can define a complex torus $\mathbb{C}^{\Sigma_{F}}/\Sigma_F (\mathfrak{r}_F)$.
In order to regard
this complex torus as an abelian variety, we equip it with a polarisation by choosing any element $\delta$ of
$F$ satisfying the following three conditions:

* $\delta$ is relatively prime to $p$;

* $\delta$ is purely imaginary, that is,
its complex conjugate $\delta^c$ coincides with $-\delta$;

* the imaginary part $-i\sigma(\delta)$ of
$\sigma(\delta)$ is positive for all $\sigma$ in
$\Sigma_{F}$. Here $i$ denotes the imaginary unit and
we regard $\sigma(\delta)$ as an element of $\mathbb{C}$
via the fixed embedding $\iota_\infty\colon \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}$.

Using $\delta$, let us define an alternating pairing $\langle \  , \
\rangle_\delta:=(uv^c-u^cv)/2\delta$ on $\mathfrak{r}_{F}$.
The existence of such an alternating pairing implies that there is an abelian variety $X_{\Sigma_F}(\mathfrak{r}_{F})_{/\mathbb{C}}$ over $\mathbb{C}$
whose $\mathbb{C}$-valued points are identified with the complex torus $\mathbb{C}^{\Sigma_{F}}/\Sigma_F (\mathfrak{r}_F)$. The details are as follows.
Let $\mathfrak{c}$ denote a fractional ideal of $F^+$ defined as $\mathfrak{c}=\mathrm{Tr}_{F/F^+}(\mathfrak{d}_{F^+}/2\delta)^{-1}$. Here $\mathfrak{d}_{{F}^+}$ denotes the absolute different of $F^+$. Then $\mathfrak{c}$ is relatively prime to $p$ due to the assumptions (unr$_{F,p}$) and $(1_\delta)$.
Then there exists an abelian variety denoted by
$X_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathbb{C}}\otimes_{\mathfrak{r}_{F^+}} \mathfrak{c}$ over $\mathbb{C}$
whose $\mathbb{C}$-valued points are identified with the complex torus $\mathbb{C}^{\Sigma_{F}}/\Sigma_F (\mathfrak{c} \mathfrak{r}_F)$,
and the alternating pairing $\langle \  , \
\rangle_\delta$ induces a $\mathfrak{c}$-polarisation $\lambda_\delta \colon
X_{\Sigma_{F}}(\mathfrak{r}_{F})^t_{/\mathbb{C}} \xrightarrow{\sim}
X_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathbb{C}}\otimes_{\mathfrak{r}_{F^+}} \mathfrak{c}$ where
$X_{\Sigma_{F}}(\mathfrak{r}_{F})^t_{/\mathbb{C}}$ denotes the dual abelian variety of $X_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathbb{C}}$.
Note that $2\delta$ is chosen so that its image under  $F \hookrightarrow F_\mathfrak{P}\xrightarrow{\, \sim \,} F_\mathfrak{p}^+$ generates the absolute different $\mathfrak{d}_{F^+_\mathfrak{p}}$ of $F^+_{\mathfrak{p}}$ for each place $\mathfrak{p}$ of $F^+$ lying above $p$ and a unique $\mathfrak{P}\in \Sigma_{F,p}$ satisfying $\mathfrak{P}\mid \mathfrak{p}$  (see <cit.> for details). The condition (unr$_{F,p}$) then implies that the image of $2\delta$ in $F_\mathfrak{P}$ (or  $F_\mathfrak{p}^+$) is a unit.

Next we endow the $\mathfrak{c}$-polarised abelian variety
$(X_{\Sigma_{F}}(\mathfrak{r}_F)_{/\mathbb{C}}, \lambda_\delta)$
with a $\Gamma_{00}(p^\infty)$-level structure $i_\delta \colon (\mathfrak{d}_{F^+}^{-1}\otimes_\mathbb{Z} \mathbb{G}_m)[p^\infty]\hookrightarrow X(\mathfrak{r}_F)_{/\mathbb{C}}$ induced by a composite map

\begin{align*}
(\mathfrak{d}_{F^+}^{-1} &\otimes_\mathbb{Z} \mathbb{G}_m)[p^\infty](\mathbb{C})
\cong
\prod_{\mathfrak{p} \mid p\mathfrak{r}_{F^+} } \mathfrak{d}_{F^+_\mathfrak{p}}^{-1}\mathfrak{p}^{-\infty}  / \mathfrak{r}_{F^+_{\mathfrak{p}}}  \overset{\sim}{\longrightarrow}
\prod_{\mathfrak{P}\in \Sigma_{F,p}} \mathfrak{d}_{F_{\mathfrak{P}}}^{-1}\mathfrak{P}^{-\infty} / \mathfrak{r}_{F_{\mathfrak{P}}}
\\
&\xrightarrow{\times 2\delta}
\prod_{\mathfrak{P} \in \Sigma_{F,p}}
\mathfrak{P}^{-\infty}/\mathfrak{r}_{F_{\mathfrak{P}}}
\overset{\sim}{\longrightarrow} \prod_{\mathfrak{P} \in \Sigma_{F,p}}
\mathfrak{P}^{-\infty}/\mathfrak{r}_{F}
\hookrightarrow \mathbb{C}^{\Sigma_F}/\Sigma_{F} (\mathfrak{r}_F )=X_{\Sigma_F}(\mathfrak{r}_F)(\mathbb{C})
\end{align*}

(see <cit.>). The triple
\lambda_\delta, i_\delta)$ then has a model
$\mathcal{X}_{\Sigma_F}(\mathfrak{r}_{F}) $ over the valuation
ring $\mathcal{W}=\overline{\mathbb{Q}}\cap \widehat{\mathbb{Z}}_p^{\mathrm{ur}}$
due to the theory of complex multiplication combined with
Serre and Tate's criterion for good reduction and

By construction, there exists a canonical complex uniformisation
$\Pi \colon \mathbb{C}^{\Sigma_F} \twoheadrightarrow X(\mathfrak{r}_F)(\mathbb{C})$ defined as the quotient
with respect to the lattice $\Sigma_F(\mathfrak{r}_F)$. It induces an isomorphism

\begin{align*}
\Pi^\ast \colon \mathrm{Fil}^1H^1_{\mathrm{dR}}(X_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathbb{C}})
\xrightarrow{\, \sim \, } \bigoplus_{\sigma \in \Sigma_F} \mathbb{C}\,  {\rm d}u_\sigma,
\end{align*}

where $(u_\sigma)_{\sigma \in \Sigma_F}$ denotes the coordinate of $\mathbb{C}^{\Sigma_F}$. Set  $\omega_{\mathrm{trans}}(\mathfrak{r}_F)=
(\Pi^{\ast})^{-1} \left( \sum_{\sigma \in \Sigma_F} {\rm d}u_\sigma \right)$.

On the other hand, the $p$-part of the $\Gamma_{00}(p^\infty)$-level structure induces an isomorphism
$\hat{i}_\delta \colon ((\mathfrak{d}_{F^+}^{-1}\otimes_\mathbb{Z} \mathbb{G}_m)_{/\widehat{\mathcal{O}}^{\mathrm{ur}}})^{\wedge} \xrightarrow{\sim} (\mathcal{X}_{\Sigma_F}(\mathfrak{r}_{F})
between the formal completions along the identity sections over $\widehat{\mathcal{O}}^{\mathrm{ur}}$ where
_{/\widehat{\mathcal{O}}^{\mathrm{ur}}}$ is the base extension of $ \mathcal{X}_{\Sigma_F}(\mathfrak{r}_{F})$  to $\widehat{\mathcal{O}}^{\mathrm{ur}}$. Note that $((\mathfrak{d}_{F^+}^{-1}\otimes_\mathbb{Z} \mathbb{G}_m)_{/\widehat{\mathcal{O}}^{\mathrm{ur}}})^{\wedge}$ is decomposed into $\prod_{\sigma\in \Sigma_F}{\mathbb{G}_m^\wedge}_{/\widehat{\mathcal{O}}^{\mathrm{ur}}}$ corresponding to a natural isomorphism $\mathfrak{d}_{F^+}^{-1}\otimes_\mathbb{Z}\widehat{\mathcal{O}}^\mathrm{ur} \xrightarrow{\, \sim \,} \prod_{\sigma\in \Sigma_F} \widehat{\mathcal{O}}^\mathrm{ur} \, ; x\otimes 1 \mapsto (\iota_p\circ \sigma(x))_{\sigma\in \Sigma_F}$.
Thus, pulling back by $\hat{i}_\delta$, we obtain an isomorphism

\begin{align*}
\hat{i}_\delta^\ast \colon \mathrm{Fil}^1H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\widehat{\mathcal{O}}^{\mathrm{ur}}})
\xrightarrow{\, \sim \, }
\bigoplus_{\sigma \in \Sigma_{F}} \widehat{\mathcal{O}}^{\mathrm{ur}}\dfrac{{\rm d}T_\sigma}{T_\sigma}
\end{align*}

where $T_\sigma$ denotes the formal parameter of the component ${\mathbb{G}_m^\wedge}_{/\widehat{\mathcal{O}}^\mathrm{ur}}$ corresponding to $\sigma\in \Sigma_F$. Now set  $\omega_{\mathrm{can}}(\mathfrak{r}_F)$ to be $\omega_{\mathrm{can}}(\mathfrak{r}_F)=
(\hat{i}_\delta^{\ast})^{-1} \left(
\sum_{\sigma \in \Sigma_F} \dfrac{{\rm d}T_\sigma}{T_\sigma}\right)$.

Let us choose and fix a basis $\omega$ of $\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathcal{W}} )$, which is
a free $\mathfrak{r}_{F^+} \otimes_{\mathbb{Z}}\mathcal{W}$-module of rank one.

* We define a complex CM period $C_{\mathrm{CM}, \infty ,F}
\Sigma_{F}}\in \bigl(
\mathfrak{r}_{F^+}\otimes_{\mathbb{Z}}\mathbb{C}\bigr)^\times$ to be a constant satisfying

\begin{align*}
\omega = C_{\mathrm{CM}, \infty ,F} \, \omega_{\mathrm{trans}}(\mathfrak{r}_F)
\end{align*}

as an equation in
$\mathrm{Fil}^1H^1_{\mathrm{dR}}(X_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\mathbb{C}})$. We also define a modified complex CM period $\Omega_{\mathrm{CM},\infty,F}=(\Omega_{\mathrm{CM},\infty,F,\sigma})_{\sigma\in \Sigma_F}$ by setting $\Omega_{\mathrm{CM},\infty, F,\sigma}:=(2\pi i)^{-1}C_{\mathrm{CM},\infty,F,\sigma}$ for each $\sigma\in \Sigma_F$.

* We define a $p$-adic CM period $C_{\mathrm{CM}, p ,F} =\bigl(C_{\mathrm{CM},p,F,\sigma}\bigr)_{\sigma\in
\Sigma_{F}}\in \bigl(
\mathfrak{r}_{F^+}\otimes_{\mathbb{Z}}\widehat{\mathcal{O}}^{\mathrm{ur}}\bigr)^\times$
to be a constant satisfying

\begin{align*}
\omega = C_{\mathrm{CM}, p, F} \,  \omega_{\mathrm{can}}(\mathfrak{r}_F)
\end{align*}

as an equation in $\mathrm{Fil}^1H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma_{F}}(\mathfrak{r}_{F})_{/\widehat{\mathcal{O}}^{\mathrm{ur}}})$.

Both of the complex and  $p$-adic periods do depend on the choice of an
$\mathfrak{r}_{F^+}  \otimes_{\mathbb{Z}}\mathcal{W} $-basis
$\omega$ of $\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma_{F}}(\mathfrak{r}_F)_{/\mathcal{W}})$,
but the “ratio of them” is independent of $\omega$;
namely, when $\omega$ is replaced by $a\omega$ for $a\in
\bigl(\mathfrak{r}_{F^+} \otimes_\mathbb{Z} \mathcal{W}\bigr)^\times$,
the resulting (complex and $p$-adic) periods are both multiplied by
the same constant $a$.

Later we need to compare (complex and $p$-adic) CM periods for various CM fields. Let $F'$ be a CM field which is absolutely unramified at $p$ and contains $F$. Then one readily checks that $\Sigma_{F'}=\{\sigma'\in I_{F'} \mid \sigma'\vert_{F}\in \Sigma_F\}$ is indeed a $p$-ordinary CM type of $F'$. We say that $\Sigma_{F'}$ is induced from $\Sigma_F$. Now let us choose and fix a basis $\omega$ of $\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma}(\mathfrak{r}_{F})_{/\mathcal{W}})$.
Since $\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma}(\mathfrak{r}_{F'})_{/\mathcal{W}})$ is canonically identified with
$\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma}(\mathfrak{r}_{F})_{/\mathcal{W}} ) \otimes_{\mathfrak{r}_{F}}\mathfrak{r}_{F'}$,
we can choose $\omega \otimes_{\mathfrak{r}_{F}}\mathfrak{r}_{F'}$ as a basis of $\mathrm{Fil}^1 H^1_{\mathrm{dR}}(\mathcal{X}_{\Sigma}(\mathfrak{r}_{F'})_{/\mathcal{W}} )$.  This observation implies the following period relation, which plays an important role in the proof of Theorem <ref>.

Let $F$, $F'$, $\omega$ and $\omega\otimes_{\mathfrak{r}_F}\mathfrak{r}_{F'}$ be as above. For each $?\in \{\infty, p\}$, consider the CM periods $C_{\mathrm{CM}, ? ,F}=\bigl(C_{\mathrm{CM}, ? ,F, \sigma}\bigr)_{\sigma \in \Sigma_F} $ for $F$ and $C_{\mathrm{CM}, ? ,F'}=\bigl(C_{\mathrm{CM}, ? ,F', \sigma'}\bigr)_{\sigma'\in \Sigma_{F'}}$ for $F'$ defined with respect to $\omega$ and $\omega\otimes_{\mathfrak{r}_F}\mathfrak{r}_{F'}$, as in Definition $\ref{definition:complex_CM_period&p-adic_CM_period}$. Then we have $C_{\mathrm{CM}, ? ,F', \sigma'}  = C_{\mathrm{CM}, ? ,F, \sigma' \vert_F}$ for every $\sigma'\in \Sigma_{F'}$. A similar statement holds true for the modified CM periods  $\Omega_{\mathrm{CM},\infty,F}$ and $\Omega_{\mathrm{CM},\infty,F'}$.

§.§ $p$-adic Hecke $L$-functions for CM fields

Before stating the interpolation formula of $p$-adic $L$-functions for CM fields, we here make a remark concerning purity of infinity types of algebraic Hecke characters. Let $\eta =(\eta_v)_v \colon \mathbb{A}_F^\times/F^\times \rightarrow \overline{\mathbb{Q}}^\times$ be an algebraic Hecke character of a CM field $F$, and $\Sigma_F$ a $p$-ordinary CM type of $F$. Then there exist an integer $w_\eta\in \mathbb{Z}$ and an integer-valued vector $\mathsf{r}_\eta=(r_{\eta,\sigma})_{\sigma\in \Sigma_F}\in \mathbb{Z}^{\Sigma_F}$ satisfying

\begin{align*}
\eta_\infty(x_\infty) &=\prod_{\sigma \in \Sigma_F} x_\sigma^{w_\eta+r_{\eta,\sigma}} \overline{x}_\sigma^{-r_{\eta,\sigma}} \qquad \text{for } x_\infty=(x_\sigma)_{\sigma\in \Sigma_F} \in F\otimes_{\mathbb{Q}}\mathbb{R}\cong \mathbb{C}^{\Sigma_{F}}
\end{align*}

where $\bar{\cdot}$ denotes the complex conjugation in $\mathbb{C}$. We refer to the pair $(w_\eta, \mathsf{r}_\eta) \in \mathbb{Z}\times \mathbb{Z}^{\Sigma_F}$ as the infinity type of $\eta$.

To lighten the notation, we use the following multi-index notation in interpolation formula of $p$-adic $L$-functions. Let $\mathsf{a}=(a_\sigma)_{\sigma\in \Sigma_F}\in \Sigma_F$ be an integer-valued vector. Then we put $\lvert \mathsf{a}\rvert:=\sum_{\sigma \in \Sigma_F} a_\sigma$, and for a $\mathbb{C}$-valued or $\widehat{\mathcal{O}}^\mathrm{ur}$-valued vector $\mathsf{w}=(w_\sigma)_{\sigma\in \Sigma_F}$, let $\mathsf{w}^{\mathsf{a}}$ denote the product $\prod_{\sigma\in \Sigma_F} w_\sigma^{a_\sigma}$. Furthermore for each $x\in F$, we write $(x)_{\Sigma_F}^\mathsf{a}:=\prod_{\sigma\in \Sigma_F}\sigma(x)^{a_\sigma}$. Finally let $\mathsf{t}\in \Sigma_F$ denote a particular vector whose components are all equal to $1$.

Let $\mathfrak{C}$ be an integral ideal of $F$ relatively prime to $p$. Then $\mathfrak{C}$ is decomposed as $\mathfrak{C}=\mathfrak{FF}_c\mathfrak{I}$ where $\mathfrak{F}$, $\mathfrak{F}_c$ and $\mathfrak{I}$ are integral ideals of $F$ such that

* both $\mathfrak{F}$ and $\mathfrak{F}_c$ are the products of prime ideals which split completely over $F^+$;

* $\mathfrak{I}$ is the products of prime ideals which inert or ramify over $F^+$;

* $\mathfrak{F}$ and $\mathfrak{F}_c$ are relatively prime and satisfy $\mathfrak{F}_c^c\supset \mathfrak{F}$.

We choose and fix such a decomposition of $\mathfrak{C}$, which is not unique in general. We write $F_{\mathfrak{C}p^\infty}$ the ray class field modulo $\mathfrak{C}p^\infty$ of $F$. Let $L(\eta,s)=\sum_{(0)\neq \mathfrak{A}\subset \mathfrak{r}_F} \eta(\mathfrak{A})\mathcal{N}\mathfrak{A}^{-s}$ denote the complex Hecke $L$-function of $\eta$,  and write $L^\mathfrak{C}(\eta,s)=\prod_{\mathfrak{L}\mid \mathfrak{C}}(1-\eta(\mathfrak{L})\mathcal{N}\mathfrak{L}^{-s})L(\eta,s)$ for the Hecke $L$-function whose Euler factors at all places dividing $\mathfrak{C}$ are depleted. Here we use the same symbol $\eta$ for the ideal character associated to $\eta$ by abuse of notation. Furthermore let $L_\infty(\eta,s):=L_\infty(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))$ denote the archimedean factors (gamma factors) of $L(\eta,s)$, for which we will discuss later in Remark <ref>.   Now we are ready to introduce the results of Katz, Hida and Tilouine.

Let $p$ be an odd prime number, ${F}$ a CM field of degree $2d$ and $F^+$ the maximal totally real subfield of $F$. Assume that
${F}$, ${F}^+$ and $p$ satisfy both  (unr$_{F,p}$) and (ord$_{F,p}$). Choose a purely imaginary element $\delta$ of ${F}$ so that it satisfies all the conditions $(1_\delta)$, $(2_\delta)$ and $(3_\delta)$ stated at the beginning of Section $\ref{ssc:period_relation}$. Let $\mathfrak{C}$ be an integral ideal of ${F}$ relatively prime to $p$, and fix a decomposition $\mathfrak{C}=\mathfrak{FF}_c\mathfrak{I}$ as above. We assume that the principal ideal $(2\delta)$ is relatively prime to $\mathfrak{C}$.

Then, for each $p$-ordinary CM type $\Sigma_F$ of $F$, there exists a unique element
$L_{p,\Sigma_{F}}^{\mathrm{KHT}}({F})$ in the Iwasawa algebra
$\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}({F}_{\mathfrak{C}p^\infty}/F)]]$ satisfying

\begin{multline} \label{eq:p-adic_Hecke}
\frac{\eta^{\mathrm{gal}}(L_{p, \Sigma_{F}}^{\mathrm{KHT}}({F}))}
{C_{\mathrm{CM},p,F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}} =
(\mathfrak{r}_{F}^\times : \mathfrak{r}_{{F}^+}^\times )
\frac{(-1)^{w_\eta d}\,  i^{\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert }}
{\sqrt{|D_{F^+}|} 2^d (2\delta)_{\Sigma_F}^{\mathsf{r}_\eta }}
\prod_{v \in \Sigma_{F,p}} \mathrm{Eul}_v(\eta,0)
\frac{L (\eta_\infty, 0) L^{\mathfrak{C}}(\eta,0)}{\Omega_{\mathrm{CM},\infty,F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}}, \\
\mathrm{Eul}_v(\eta,0)=
\begin{cases}
L(\eta_{v^c},0)^{-1}\left(1-\dfrac{1}{\eta_v(\varpi_v)\mathcal{N}v}\right) & \text{if $\eta$ is unramified at $v$}, \\
\epsilon(\eta_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)^{-1} & \text{ if $\eta$ is ramified at $v$},
\end{cases}
\end{multline}

for every algebraic Hecke character $\eta=(\eta_v)_v \colon \mathbb{A}_F^\times/F^\times \rightarrow \overline{\mathbb{Q}}^\times$ with conductor dividing $\mathfrak{C}p^{\infty}$ satisfying

* the conductor of $\eta$ is divisible by all prime factors
of $\mathfrak{F}$;

* the infinity type $(w_\eta,\mathsf{r}_\eta)\in \mathbb{Z}\times \mathbb{Z}^{\Sigma_F}$ of $\eta$ satisfies both $-w_\eta-r_{\eta,\sigma}\leq -1$ and $r_{\eta,\sigma}\geq 0$ for every $\sigma\in \Sigma_F$.

Here $\epsilon (\eta_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)$ denotes Deligne's local constant; see Remark $\ref{rem:Hecke_epsilon}$ for details.

For each finite place $v$ of $F$, we define an additive character $\mathbf{e}^{(2\delta)}_{F,v} \colon F_v\rightarrow \mathbb{C}^\times$ on $F_v$ by

\begin{align*}
\mathbf{e}_{F,v}^{(2\delta)} \colon F_v \rightarrow \mathbb{C}^\times \,; x \mapsto \mathbf{e}_{F,v}\bigl((2\delta)_v^{-1}x\bigr),
\end{align*}

where $\mathbf{e}_{F,v}$ denotes the standard additive character $\mathbf{e}_{F,v}$ introduced in (<ref>). Let $dx_v$ denote a Haar measure on $F_v$ normalised so that the volume of the ring of integers of $F_v$ is equal to $1$. In this remark, we here calculate Deligne's local constant $\epsilon(\eta_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)^{-1}$ appearing in the interpolation formula (<ref>) of $L^{\mathrm{KHT}}_{p,\Sigma_F}(F)$.  The functional equation of the local constant <cit.> and the behaviour under unramified twist <cit.> imply the equality

\begin{align*}
\epsilon(\eta_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)^{-1}=\epsilon(\eta_v^{-1} \omega_{1,v}, \mathbf{e}_{F,v}^{(-2\delta)},dx_v)=\epsilon(\eta_v^{-1},\mathbf{e}^{(-2\delta)}_{F,v},dx_v) \omega_{1,v}(\varpi_v)^{e(\eta_v)},
\end{align*}

where $\omega_{1,v}\colon F_v^\times \rightarrow \mathbb{C}^\times$ is the norm character and $e(\eta_v)$ is the exponent of the conductor of $\eta_v$. Here note that the Haar measure $dx_v$ is self-dual since $F_v$ is unramified over $\mathbb{Q}_p$. By <cit.>, the local constant $\epsilon(\eta_v,\mathbf{e}^{(2\delta)}_{F,v},d_v)^{-1}$ is thus explicitly described as

\begin{align*}
\epsilon(\eta_v, &\mathbf{e}^{(2\delta)}_{F,v}, dx_{v})^{-1} = \epsilon(\eta_v^{-1}, \mathbf{e}^{(-2\delta)}_{F,v}, dx_{v})  \mathcal{N}v^{-e(\eta_v)} \\
&=\mathcal{N}v^{-e(\eta_v)}\int_{\mathfrak{P}_v^{-e(\eta_v)}}  \eta_v(x) \mathbf{e}^{(-2\delta)}_{F,v}(x) \, dx_v \\
&=\mathcal{N}v^{-e(\eta_v)}\int_{\mathfrak{r}_{F_v}^\times} \eta_v(\varpi_v^{-e(\eta_v)}x) \mathbf{e}_{F,v}(\varpi_v^{-e(\eta_v)}(-2\delta)_v^{-1}x) \, \bigl( \mathcal{N}v^{e(\eta_v)}dx_{v}\bigr) \\
&=\sum_{x\in (\mathfrak{r}_{F,v}/\mathfrak{P}_v^{e(\eta_v)})^\times} \eta_v(\varpi_v^{-e(\eta_v)}x)\mathbf{e}_{F,v}(\varpi_v^{-e(\eta_v)}(-2\delta)_v^{-1}x) \int_{1+\mathfrak{P}_v^{e(\eta_v)}}dx_v \\
&=\bigl(\mathcal{N}v\;  \eta_v(\varpi_v)\bigr)^{-e(\eta_v)}\sum_{x\in (\mathfrak{r}_{F_v}/\mathfrak{P}_v^{e(\eta_v)})^\times} \eta_v(x) \exp(2\pi i\, \mathrm{Tr}_{F_v/\mathbb{Q}_p} (\varpi_v^{-e(\eta_v)}(2\delta)_v^{-1}x)),
\end{align*}

where $\mathfrak{P}_v$ denotes the maximal ideal of $F_v$. This implies that $\prod_{v\in \Sigma_{F,v}}\epsilon(\eta_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)^{-1}$ coincides with  $\mathrm{Local}(\eta,v\text{'s})$ introduced by Katz <cit.> and $W_p(\eta)$ introduced by Hida and Tilouine <cit.>.

On the Hodge realisation of the pure motive $H_{\mathrm{B},\sigma}(M(\eta)_{/F})_\mathbb{C}$ with respect to $\sigma\in \Sigma_F$, the real-valued points of Deligne's torus $\mathbb{S}_m(\mathbb{R})=\mathbb{C}^\times$ acts via $z\mapsto z^{w_\eta+r_{\eta,\sigma}}\bar{z}^{-r_{\eta,\sigma}}$; this implies that the Hodge type of $H_{\mathrm{B},\sigma}(M(\eta)_{/F})_\mathbb{C}$ is $(-w_\eta-r_{\eta,\sigma}, r_{\eta,\sigma})$.[We adopt the convention in <cit.> in the present article.] Hence, under the assumption (ii) in Theorem <ref>, the archimedean factor $L(\eta_\infty, s):=\prod_{\sigma\in \Sigma_F}L(\eta_\sigma,s)$ of $L(\eta,s)$ is described as

\begin{align} \label{eq:arch_L_eta}
L(\eta_\infty,s)= \prod_{\sigma \in \Sigma_F} \Gamma_\mathbb{C}(s+w_\eta+r_{\eta,\sigma})=\prod_{\sigma\in \Sigma_F} 2\cdot (2\pi)^{-s-w_\eta-r_{\eta,\sigma}}\Gamma(s+w_\eta+r_{\eta,\sigma})
\end{align}

where $\Gamma(s)$ is the gamma function and $\Gamma_\mathbb{C}(s)$ denotes $2\cdot (2\pi)^{-s}\Gamma(s)$. We thus have

\begin{align*}
\dfrac{i^{\lvert -w_\eta\mathsf{t}-\mathsf{r}_\eta\rvert}L(\eta_\infty,0)}{2^d (2\delta)^{\mathsf{r}_\eta}_{\Sigma_F} \Omega_{\mathrm{CM},\infty}^{w_\eta\mathsf{t}+2\mathsf{r}_\eta}}&= \prod_{\sigma \in \Sigma_F} \dfrac{i^{-w_\eta\mathsf{t}-r_{\eta,\sigma}} 2\cdot (2\pi)^{-w_\eta-r_{\eta,\sigma}}\Gamma(w_\eta+r_{\eta,\sigma})}{2\cdot \sigma(2\delta)^{r_{\eta,\sigma}} (2\pi i)^{-w_\eta-2r_{\eta,\sigma}}C_{\mathrm{CM},\infty,F,\sigma}^{w_\eta+2r_{\eta,\sigma}}} \\
&=\prod_{\sigma\in \Sigma_F} \dfrac{(2\pi)^{r_{\eta,\sigma}}\Gamma(w_\eta+r_{\eta,\sigma})}{(-i\sigma(2\delta))^{r_{\eta,\sigma}}C_{\mathrm{CM},\infty,F}^{w_\eta+2r_{\eta,\sigma}}}=\dfrac{(2\pi)^{\lvert \mathsf{r}_\eta\rvert}\prod_{\sigma\in \Sigma_F} \Gamma(w_\eta+r_{\eta,\sigma})}{\mathrm{Im}(2\delta)^{\mathsf{r}_\eta}C_{\mathrm{CM},\infty,F}^{w\mathsf{t}+2\mathsf{r}_\eta}}.
\end{align*}

The last term is exactly what appeas in <cit.> and <cit.>. Note that $i^{\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert}L(\eta_\infty,0)$ is regarded as the modified $\infty$-Euler factor $L^{(i)}_\infty(\mathrm{Res}_{F/\mathbb{Q}}\,M(\eta))$ introduced in <cit.>, and $\Omega_{\mathrm{CM},\infty,F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}$ amounts to the modified period $\Omega^{(i)}(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))$ defined in <cit.>; indeed, the algebraicity of the right-hand side of (<ref>) <cit.> implies that Deligne's period $C^+(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))$ should coincide up to non-zero multiples of algebraic numbers with $(2\pi i)^{\lvert -\mathsf{r}_\eta\rvert} C_{\mathrm{CM}, \infty, F}^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}$, and $\tau(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))$ loc. cit. is equal to $\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert$. Furthermore one readily observes that $p$ is ordinary for the motive  $\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta)$ in the sense of <cit.> or <cit.>, and $\prod_{v\in \Sigma_{F,p}}\mathrm{Eul}_v(\eta, s)$ is nothing but (the quotient of) the modified $p$-Euler factor $L^{(i)}_p(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))/L_p(\mathrm{Res}_{F/\mathbb{Q}}\, M(\eta))$ introduced in <cit.> (see <cit.>).

The $p$-adic $L$-function $L_{p, \Sigma_{F}}^{\mathrm{KHT}}({F})$ does depend on the choice of $\delta$ satisfying
the conditions $(1_\delta)$, $(2_\delta)$ and $(3_\delta)$, but we can explicitly describe effects on the interpolation formula  when we replace $\delta$
by another element $\delta'$ satisfying the polarisation conditions $(1_\delta)$, $(2_\delta)$ and $(3_\delta)$.
In particular, the principal ideal of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}({F}_{\mathfrak{C}p^\infty}/F)]]$ generated by
$L_{p, \Sigma_{F}}^{\mathrm{KHT}}({F})$ does not change after such a replacement of $\delta$. Refer, for example, to <cit.>.
We also remark that the assumption (unr$_{F^+}$) is not required
in Katz, Hida and Tilouin's original construction of $L_{p,\Sigma_{F}}^{\mathrm{KHT}}(F)$.

§ CONSTRUCTING THE $P$-ADIC ARTIN $L$-FUNCTION IN THE FIELD OF FRACTIONS OF THE IWASAWA ALGEBRA

In this section, we will construct a $p$-adic Artin $L$-function as an element of the field of fractions of the Iwasawa algebra.
In the next section, we will show that this $p$-adic $L$-function is contained in the Iwasawa algebra under several assumptions.

§.§ $L$-factors and local $\epsilon$-factors

For a CM field $F$, let $\eta=(\eta_v)_v \colon \mathbb{A}_F^\times/F^\times\rightarrow \mathbb{C}^\times$ be an algebraic Hecke character and  $\rho\colon G_F \rightarrow {\rm Aut}_{\overline{\mathbb{Q}}}(V_\rho)$ an Artin representation. As is well known, the image of $\eta$ is contained in the multiplicative group $E^\times$ of a number field $E$. By extending $E$ if necessary, we may assume that $\rho$ is also defined over $E$. Then $\eta$ corresponds to a unique continuous character $\eta^\mathrm{gal}\colon G_F\rightarrow E_p^\times$ via the global Artin reciprocity map $\mathrm{rec}_F\colon \mathbb{A}_F^\times/F^\times \rightarrow G_F^\mathrm{ab}$,  where $E_p$ is an appropriate finite extension of $\mathbb{Q}_p$ containing $\iota_p(E)$.  For each finite place $v$ of $F$, let $W_{F_v}\subset G_{F_v}$ (resp. ${}^\prime W_{F_v}$) be the Weil group (resp. the Weil–Deligne group) of $F_v$. Following Fontaine's recipe proposed in <cit.>, we associate with $V_\rho\otimes_{E_p} \eta^\mathrm{gal}$ a Weil–Deligne representation $W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})$ of ${}^\prime W_{F_v}$. It is defined over $\mathcal{E}$, the completion of the maximal unramified extension of $E_p$ if $v$ lies above $p$ and $E_p$ otherwise.
Set $W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})_\mathbb{C}:=W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})\otimes_{\mathcal{E},\boldsymbol{\iota}^{-1}}\mathbb{C}$.
One readily observes by construction that $W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})_\mathbb{C}\cong V_\rho \otimes_{\mathbb{C}} W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}$ is a complex vector space of dimension $r(\rho)$ on which $\gamma\in W_{F_v}$ acts as $\iota_\infty \bigl(\rho(\gamma) \eta_v(\mathrm{rec}_{F,v}^{-1}(\gamma^{\mathrm{ab}}))\bigr)$, where $\eta_v$ is the $v$-component of the Hecke character $\eta=(\eta_v)_v$, $\gamma^\mathrm{ab}$ is the image of $\gamma$ in the abelianisation $W_{F_v}^\mathrm{ab}$ of $W_{F_v}$, and  $\mathrm{rec}_{F,v}\colon F_v^\times \xrightarrow{\, \sim \,} W_{F_v}^\mathrm{ab}$ denotes the local Artin reciprocity map at $v$.

Let $v$ be a finite place of $F$. Under these settings, the $L$-factor and the local $\epsilon$-factor of the pure motive $M(\rho)\otimes M(\eta)$ at $v$ are defined as follows:

\begin{align}
L((\rho\otimes \eta)_v,s) &:=L(W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes \eta^\mathrm{gal})_\mathbb{C},s) \label{equation:L_factor}\\
\det\bigl(1-\mathrm{Frob}_v \,  \mathcal{N}v^{-s} \mid \bigl(W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})_\mathbb{C}\bigr)^{I_v,N=0}\bigr)^{-1},  \nonumber \\
\epsilon((\rho\otimes \eta)_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)&:=\epsilon(W\widehat{D}_{\mathrm{pst},v}(V_\rho\otimes_{E_p} \eta^\mathrm{gal})_\mathbb{C},\mathbf{e}_{F,v}^{(2\delta)},dx_{v}). \label{equation:local_constant}
\end{align}

The right-hand side of (<ref>) denotes Deligne's local constant defined in <cit.> with respect to the additive character $\mathbf{e}^{(2\delta)}_{F,v}$ and the normalised Haar measure $dx_v$ of $F_v$; see Remark <ref> for details on $\mathbf{e}^{(2\delta)}_{F,v}$ and $dx_v$. In particular, if both $\eta$ and $\rho$ are unramified at $v$, we have $L((\rho\otimes \eta)_v, s)=\det(1-\eta_v(\varpi_v)\rho(\mathrm{Frob}_v)\mathcal{N}v^{-s})^{-1}$ where $\varpi_v$ denotes a uniformiser of $F_v$ and $\epsilon((\rho\otimes \eta)_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)$ is equal to $1$.
We set $L(\rho\otimes \eta,s):=\prod_{v \colon \text{finite places of $F$}}L((\rho\otimes \eta)_v,s)$.

Concerning the archimedean local factor, let $(w_\eta, \mathsf{r}_\eta)\in \mathbb{Z}\times \mathbb{Z}^{\Sigma_F}$ be the infinity type of $\eta$ as in Section <ref>. Then the Hodge realisation $H_{\mathrm{B},\sigma}(M(\rho)\otimes_EM(\eta))_\mathbb{C}$ of $M(\rho) \otimes_E M(\eta)$ is an $r(\rho)$-dimensional $\mathbb{C}$-vector space of Hodge type $(-w_\eta-r_{\eta,\sigma},r_{\eta,\sigma})$, and thus the archimedean local $L$-factor of $M(\rho)\otimes M(\eta)$ is described as

\begin{align} \label{eq:arch_L_rho}
L((\rho\otimes \eta)_\infty,s)=\prod_{\sigma\in \Sigma_F}L((\rho\otimes \eta)_\sigma,s)=\prod_{\sigma\in \Sigma_F} \Gamma_\mathbb{C}(s+w_\eta+r_{\eta,\sigma})^{r(\rho)}.
\end{align}

§.§ Gluing the $p$-adic Hecke $L$-functions

Now let us state the main result of this section (refer to Remark <ref> for multi-index notation used in the statement).

Let $p$ be an odd prime number, ${F}$ a CM field of degree $2d$ and $F^+$ the maximal totally real subfield of $F$. Assume that
${F}$, ${F}^+$ and $p$ satisfy both  (unr$_{F,p}$) and (ord$_{F,p}$).
Choose a purely imaginary element $\delta$ of ${F}$ so that it satisfies all the conditions $(1_\delta)$, $(2_\delta)$ and $(3_\delta)$ stated at the beginning of Section $\ref{ssc:period_relation}$.

In addition, let $\rho\colon G_F\rightarrow {\rm Aut}_E(V_\rho)$ be an Artin representation unramified at any primes lying above $p$, and assume that the field $F_\rho$ corresponding to $\mathrm{ker}(\rho)$ is also a CM field.
Then, for each $p$-ordinary CM type $\Sigma_F$ of $F$, there exists an element $L_{p,\Sigma_F} (M(\rho ))$ of $\mathrm{Frac} (\widehat{\mathcal{O}}^{\mathrm{ur}}[[
\Gamma_{F,\mathrm{max}}]])$ satisfying

\begin{multline} \label{eq:interpolation_Lp_Artin}
\frac{\eta^\mathrm{gal} (L_{p,\Sigma_F} (M(\rho )))}{\bigl(C^{w_\eta \mathsf{t}+2\mathsf{r}_\eta}_{\mathrm{CM}, p,F} \bigr)^{r(\rho )}} =
\left(
\frac{(-1)^{w_\eta d}\, i^{\lvert -w_\eta \mathsf{t}-\mathsf{r}_\eta\rvert}}
{2^d (2\delta)_{\Sigma_F}^{\mathsf{r}_\eta}}
\right)^{r(\rho )}
\prod_{v\in \Sigma_{F,p}}\mathrm{Eul}_v (\rho\otimes \eta,0)
\dfrac{L_\infty(\rho\otimes \eta,0)L(\rho \otimes \eta, 0)}{\bigl(\Omega^{w_\eta\mathsf{t}+2\mathsf{r}_\eta}_{\mathrm{CM}, \infty, F} \bigr)^{r(\rho )}}\\
\mathrm{Eul}_v(\rho\otimes \eta,0)=
\begin{cases}
L((\rho\otimes \eta)_{v^c},0)^{-1}L((\rho\otimes \eta)_v^\vee, 1)^{-1} & \text{if $\eta$ is unramified at $v$}, \\
\epsilon((\rho \otimes \eta)_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)^{-1} & \text{ if $\eta$ is ramified at $v$},
\end{cases}
\end{multline}

for any algebraic Hecke character $\eta \colon \mathbb{A}_F^\times/F^\times\rightarrow \mathbb{C}^\times$ such that

* the associated Galois character $\eta^{\mathrm{gal}}$ factors through $\Gamma_{F,\mathrm{max}};$

* the infinity type $(w_\eta,\mathsf{r}_\eta)\in \mathbb{Z}\times \mathbb{Z}^{\Sigma_F}$ of $\eta$ satisfies both $-w_\eta-r_{\eta,\sigma}\leq -1$ and $r_{\eta,\sigma}\geq 0$ for every $\sigma\in \Sigma_F$.

Here we use the symbol $L_{p,\Sigma_F}(M(\rho))$ to express that it is indeed the $p$-adic $L$-function of the Artin motive $M(\rho)$. Note that the interpolation formula (<ref>) complies the generalformulation of Coates and Perrin-Riou [5, 4]; see also Remark <ref> and <ref>.

Set $G=\mathrm{Gal}(F_\rho/F)=G_F/\ker(\rho)$. Then, by Brauer's induction theorem <cit.>, the Artin representation $\rho$ is decomposed (as a virtual representation of $G$) into

\begin{equation}\label{equation:brauer_induction}
\rho= \sum^s_{j=1} a_j \mathrm{Ind}^G_{G_j} \psi_j,
\end{equation}

where $G_j:=\mathrm{Gal}(F_\rho/F_j)$ is a subgroup of $G$, $\psi_j$ is an abelian character of $G_j$ and  $a_j$ is an integer for each $j$. Note that each $F_j$ is a CM field because it is an intermediate field of the extension $F_\rho/F$ of CM fields; see <cit.> for details. Furthermore since the dimensions of the both sides of (<ref>) obviously coincide, we obtain an identity

\begin{align}\label{eq:brauer_induction_dim}
r(\rho)=\sum_{j=1}^s a_j (G:G_j) \dim \psi_j =\sum_{j=1}^s a_j [F_j:F].
\end{align}

For each $1\leq j\leq s$, the  $p$-ordinary CM type $\Sigma_F$ induces a unique $p$-ordinary CM type $\Sigma_{F_j}:=\{\tilde{\sigma}\in I_{F_j} \mid \tilde{\sigma}\rvert_F \in \Sigma_F\}$ of $F_j$. Now let $\mathfrak{C}_j$ denote the conductor of $\psi_j$, which is relatively prime to $p$ by construction. Fix a decomposition $\mathfrak{C}_j=\mathfrak{F}_j\mathfrak{F}_{j,c}\mathfrak{I}_j$ of $\mathfrak{C}_j$ as is explained right before Theorem <ref>.
Then, by Theorem <ref>, the $p$-adic $L$-function $L^{\mathrm{KHT}}_{p,\Sigma_{F_j}}(F_j)$ for the CM field $F_j$ and the $p$-ordinary CM type $\Sigma_{F_j}$ uniquely exists for each $j$ as an element of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{j,\mathfrak{C}_ip^\infty}/F_j)]]$. It is characterised by the interpolation property (note that $[F_j:\mathbb{Q}]=2d[F_j:F]$):

\begin{multline}\label{equation:katz1}
\frac{\xi^{\mathrm{gal}}(L^{\mathrm{KHT}}_{p,\Sigma_{F_j}}(F_j))}
{C_{\mathrm{CM},p,F_j }^{w_\xi \mathsf{t}_j+2\mathsf{r}_\xi}} =
\dfrac{(\mathfrak{r}_{F_j}^\times : \mathfrak{r}_{F_j^+}^\times )
}{\sqrt{\lvert D_{F_j^+}\rvert}}
\frac{(-1)^{w_\xi d [F_j :F]} \, i^{\lvert -w_\xi \mathsf{t}_j-\mathsf{r}_\xi\rvert}}{2^{d[F_j:F]}(2\delta)_{\Sigma_{F_j}}^{\mathsf{r}_{\xi}}}
\prod_{\tilde{v}\in \Sigma_{F_j,p}}\mathrm{Eul}_{\tilde{v}}(\xi,0)\frac{L(\xi_\infty,0)L^{\mathfrak{C}_j}(\xi,0)}{\Omega_{\mathrm{CM},\infty ,F_j }^{w_\xi\mathsf{t}_j+2\mathsf{r}_\xi}}
\end{multline}

for any algebraic Hecke character $\xi \colon \mathbb{A}_{F_j}^\times/F_j^\times\rightarrow \overline{\mathbb{Q}}^\times$ satisfying both (i) and (ii) in Theorem <ref>. Here $\mathsf{t}_j$ denotes an integer-valued vector $(1,1,\ldots,1)\in \mathbb{Z}^{\Sigma_{F_j}}$. We emphasise that, in (<ref>), we choose the auxiliary element for polarisation over $F_j$ appearing in (<ref>) to be the same as $\delta$ fixed in the assumption of Theorem <ref> for any $j$. Indeed, one may readily check that $\delta$ also satisfies ($1_\delta$), ($2_\delta$) and ($3_\delta$) for $F_j$.

Now consider the $\psi_j$-twisting map

\begin{align*}
\mathrm{Tw}_{\psi_j} \colon \widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{j,\mathfrak{C}_ip^\infty}/F_j)]]\rightarrow \widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{j,\mathfrak{C}_jp^\infty}/F_j)]] \,; \, g\mapsto \psi_j(g) g
\end{align*}

and let us define $L_{p,\Sigma_{F_j}}(\psi_j) \in \widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F_j ,\mathrm{max}}]]$ to be the image of

\begin{equation}\label{equation:katz2}
\mathrm{Tw}_{\psi_j}\left( \tfrac
{\sqrt{\lvert D_{F_j^+}\rvert}}
\mathfrak{r}^\times_{F_j^+})}
L_{p,\Sigma_{F_j}}^{\mathrm{KHT}}(F_j)\right) \in \widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{j,\mathfrak{C}_ip^\infty}/F_j)]]
\end{equation}

under the natural surjection $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{j,\mathfrak{C}_j p^\infty}/F_j)]] \twoheadrightarrow
\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F_j ,\mathrm{max}}]]$.
Let us consider the composite map

\begin{align*}
\xymatrix{\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F_j ,\mathrm{max}}]] \ar@{->>}[r]^(.4){\mathrm{pr}_j} & \widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_j F_{\mathrm{max}}/F_j)]] \ar@{^(->}[r]^(.6){\mathrm{incl}_j} & \widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\mathrm{max}}]]}
\end{align*}

where $\mathrm{pr}_j$ denotes the ring homomorphism induced by the quotient map of the Galois group
$\Gamma_{F_j ,\mathrm{max}} =\mathrm{Gal}(F_{j,\mathrm{max}}/F_j) \twoheadrightarrow  \mathrm{Gal}(F_j F_{\mathrm{max}}/F_j)$ and
$\mathrm{incl}_j$ denotes the ring homomorphism induced by the inclusion
$  \mathrm{Gal}(F_j F_{\mathrm{max}}/F_j) \subset \mathrm{Gal}(F_{\mathrm{max}}/F)=\Gamma_{F, \mathrm{max}}$.
We set

\begin{equation}\label{equation:katz3}
L_{p,\Sigma_F}(M(\rho ))=
\displaystyle \prod_{j=1}^s
\mathrm{incl}_j\circ \mathrm{pr}_j( L_{p,\Sigma_{F_j}}(\psi_j))^{a_j},
\end{equation}

which is a priori an element of the field of fractions of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\mathrm{max}}]]$. In the rest of the proof, we assemble the interpolation formulae of the $p$-adic $L$-functions appearing in the right hand side of (<ref>), and deduce the desired interpolation property of $L_{p,\Sigma_F}(M(\rho))$.

[$\blacktriangleright$ Infinity types of $\psi_j\eta_j$]
Let $\eta\colon \mathbb{A}_F^\times/F^\times \rightarrow \mathbb{C}^\times$ be an algebraic Hecke character satisfying the conditions (a) and (b) in Theorem <ref>. The restriction $\eta^{\mathrm{gal}}\vert_{G_{F_j}}$ of $\eta^{\mathrm{gal}}$ then corresponds to $\eta_j:=\eta\circ \mathrm{Norm}_{F_j/F}$ via global class field theory. By abuse of notation, we use the same symbol $\psi_j$ for a finite Hecke character corresponding to $\psi_j$ via global class field theory. Since the Norm map $\mathrm{Norm}_{F_j/F}$ induces

\begin{align*}
\mathbb{A}_{F_j}^{\infty,\times}=\mathbb{C}^{\Sigma_{F_j}}\rightarrow  \mathbb{A}_{F}^{\infty,\times}=\mathbb{C}^{\Sigma_{F}}\,; \, (x_\tau)_{\tau\in \Sigma_{F_j}} \mapsto \left(\prod_{\tau\vert_F=\sigma} x_\tau\right)_{\sigma\in \Sigma_F},
\end{align*}

one readily observes that the infinity type $(w_{\eta_j}, \mathsf{r}_{\eta_j})\in \mathbb{Z}\times \mathbb{Z}^{\Sigma_{F_j}}$ of $\psi_j\eta_j$ satisfies equalities $w_{\eta_j}=w_\eta$ and  $r_{\eta_j,\tau}=r_{\eta,\tau\vert_F}$ for each $\tau\in \Sigma_{F_j}$. From these formulae, we see that $\psi_j\eta_j$ satisfies the condition (ii) of Theorem <ref> for $F_j$.
Furthermore since $\mathfrak{C}_j$ is the conductor of $\psi_j$ and $\eta_j$ is ramified only at places lying above $p$, the condition (i) of Theorem <ref> is always fulfilled for $\psi_j\eta_j$. By (<ref>), (<ref>) and (<ref>), we have

\begin{align}\label{equation:specialization_p-adicL}
&\dfrac{\eta^{\mathrm{gal}} (L_{p,\Sigma_F}(M(\rho )))}
{\prod_{j=1}^s \bigl( C_{\mathrm{CM},p,F_i }^{w_{\eta}\mathsf{t}_j+2\mathsf{r}_{\eta_j}}\bigr)^{a_j}} = \prod_{j=1}^s \left(\dfrac{\eta^{\mathrm{gal}}\vert_{\mathrm{Gal}(\overline{F}/F_j)}\psi_j(L_{p,\Sigma_{F_j}}^{\mathrm{KHT}}(F_j))}{\Omega_{\mathrm{CM},p,F_j}^{w_{\eta}\mathsf{t}_j+2\mathsf{r}_{\eta_j}}}\right)^{a_j}
\\
&\qquad =  \prod_{j=1}^s
\left(\frac{(-1)^{w_\eta d [F_j :F]} \, i^{\lvert -w_\eta \mathsf{t}_j-\mathsf{r}_{\eta_j}\rvert}}{2^{d[F_j:F]}(2\delta)_{\Sigma_{F_j}}^{\mathsf{r}_{\eta_j}}}
\prod_{\tilde{v}\in \Sigma_{F_j,p}}A_{\tilde{v}}(\psi_j\eta_j,0)\frac{L((\psi_j\eta_j)_\infty,0)L(\psi_j\eta_j,0)}{\Omega_{\mathrm{CM},\infty ,F_j }^{w_\eta\mathsf{t}_j+2\mathsf{r}_{\eta_j}}}\right)^{a_j}, \nonumber
\end{align}

where we have substituted $w_{\eta_j}=w_\eta$. Note that, since $\psi_j\eta_j$ is ramified at each finite place $\widetilde{v}$ dividing $\mathfrak{C}_j$ by definition, the Euler factor of $L(\psi_j\eta_j,s)$ at such $\widetilde{v}$ is equal to $1$, and thus we have replaced the $L$-value $L^{\mathfrak{C}_j}(\eta_j\psi_j,0)$ by $L(\eta_j\psi_j,0)$ in (<ref>).  We shall modify each term of (<ref>) and show that (<ref>) is equivalent to the desired interpolation formula (<ref>).

[$\blacktriangleright$ $L$-values]
For each $j=1,2,\ldots,s$,  Mackey's decomposition theorem <cit.> provides a decomposition of $\mathrm{Ind}^G_{G_j}\, \psi_j$ appearing in  (<ref>) as $G_{F_v}$-representations

\begin{align} \label{equation:Mackey}
\mathrm{Ind}^G_{G_j}\, \psi_j \vert_{G_{F_v}} =\bigoplus_{[g]\in G_{F_j}\backslash G_F/G_{F_v}} \mathrm{Ind}^{G_{F_v}}_{G_{F_j}^g\cap G_{F_v}} \, \psi_j^g
\end{align}

where $G_{F_j}^g:=g^{-1}G_{F_j}g$ and $\psi_j^g$ is defined as $\psi^g_j(x)=\psi_j(g xg^{-1})$ for $x\in G_{F_j}^g\cap G_{F_v}$.
Combining (<ref>) with the Tensor Product Theorem <cit.>, we obtain an isomorphism of (virtual) representations of ${}^\prime W_{F_v}$:

\begin{align} \label{eq:Brauer_L}
W\widehat{D}_{\mathrm{pst},v}(\rho\otimes \eta^\mathrm{gal})_\mathbb{C} &= \sum_{j=1}^s a_j \sum_{[g]\in G_{F_j}\backslash G_F/G_{F_v}}  \bigl(\mathrm{Ind}^{W_{F_v}}_{G_{F_j}^g\cap W_{F_v}} \, \psi_j^g \bigr)\otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C} \\
&= \sum_{j=1}^s a_j\sum_{[g]\in G_{F_j}\backslash G_F/G_{F_v}}  \mathrm{Ind}^{W_{F_v}}_{G_{F_j}^g \cap W_{F_v}} \,\left( \psi_j^g \otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{G_{F_j}^g\cap W_{F_v}} \right).\nonumber
\end{align}

Now let $\tilde{v}_0$ denote a unique place of $F_j$ fixed by $G_{F_v}$ (we supress $j$ by abuse of notation). Set $W_{F_{j,\tilde{v}_0^g}}:=G_{F_j}\cap W_{F_v}^{g^{-1}}$, which is indeed regarded as the Weil group of $F_{j,\tilde{v}_0^g}$. If we regard $\psi_j^g\otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})\vert_{G_{F_j}^g}$ as a representation of $W_{F_{j,\tilde{v}_0^g}}$ via  $W_{F_{j,\tilde{v}_0^g}} \xrightarrow{\, \sim \,} G_{F_j}^g\cap W_{F_v}\, ; \gamma \mapsto g^{-1}\gamma g$, it is isomorphic to $\psi_j\otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{W_{F_{j,\tilde{v}_0^g}}}$. We thus obtain

\begin{align}
\prod_{j=1}^s &\prod_{\tilde{v}\mid v} L((\psi_j\eta_j)_w,s)^{a_j} =\prod_{j=1}^s \prod_{\tilde{v}\mid v} L(\psi_j\otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{W_{F_{j,\tilde{v}}}},s)^{a_j}  \label{eq:Brauer_L_factor}\\
&=\prod_{j=1}^s\prod_{[g]\in G_{F_j}\backslash G_F/G_{F_v}} L( \psi_j^g \otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{G_{F_j}^g\cap W_{F_v}},s)^{a_j} \nonumber\\
&=\prod_{j=1}^s\prod_{[g]\in G_{F_j}\backslash G_F/G_{F_v}} L\left(  \mathrm{Ind}^{W_{F_v}}_{G_{F_j}^g \cap W_{F_v}} \,\left(\psi_j^g \otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{G_{F_j}^g\cap W_{F_v}}\right),s\right)^{a_j} \nonumber\\
&=L\left(\sum_{j=1}^s a_j\sum_{[g]\in G_{F_j}\backslash G_F/G_{F_v}}  \mathrm{Ind}^{W_{F_v}}_{G_{F_j}^g \cap W_{F_v}} \,\left( \psi_j^g \otimes W\widehat{D}_{\mathrm{pst},v}(\eta^\mathrm{gal})_\mathbb{C}\vert_{G_{F_j}^g\cap W_{F_v}} \right) ,s\right)  \nonumber \\
&=L((\rho\otimes \eta)_v,0) \nonumber
\end{align}

where $\prod_{\tilde{v}\mid v}$ denotes the product over all the finite places $\tilde{v}$ of $F_j$ lying above $v$. The third equality follows due to inductivity of $L$-fuctors (see <cit.>) and the last equality follows from (<ref>). By taking the product over all the finite places of $F$ and substituting $s=0$, we obtain the desired equality $\prod_{j=1}^s L(\psi_j\eta,0)^{a_j}=L(\rho\otimes \eta,0)$.

[$\blacktriangleright$ Archimedean local $L$-factors]
By (<ref>), the product of archimedean $L$-factors in the left-hand side of (<ref>) is rewritten as

\begin{align*}
\prod_{j=1}^s L((\psi_j\eta_j)_\infty,0)^{a_j} &=\prod_{j=1}^s \prod_{\tau\in \Sigma_{F_j}} \Gamma_\mathbb{C}(w_\eta+r_{\eta_j,\tau})^{a_j}=\prod_{j=1}^s \prod_{\sigma\in \Sigma_F}\prod_{\substack{\tau\in \Sigma_{F_j} \\ \tau\vert_F=\sigma}} \Gamma_\mathbb{C}(w_\eta+r_{\eta_j,\tau})^{a_j}.
\end{align*}

Since the number of $\tau\in\Sigma_{F_j}$ satisfying $\tau\vert_F=\sigma$ is equal to $[F_j:F]$ for any $\sigma \in \Sigma_F$, we have

\begin{align*}
\prod_{j=1}^s L((\psi_j\eta_j)_\infty,0)^{a_j} &=\prod_{j=1}^s \prod_{\sigma\in \Sigma_F}\prod_{\substack{\tau\in \Sigma_{F_j} \\ \tau\vert_F=\sigma}} \Gamma_\mathbb{C}(w_\eta+r_{\eta,\tau\vert_F})^{a_j}=\prod_{\sigma\in \Sigma_F}\Gamma_\mathbb{C}(w_\eta+r_{\eta,\sigma})^{\sum_{j=1}^s a_j[Fj\colon:F]},
\end{align*}

which coincides with $L((\rho\otimes \eta)_\infty,0)=\prod_{\sigma\in \Sigma_F}\Gamma_\mathbb{C}(w_\eta+r_{\eta,\sigma})^{r(\rho)}$ by (<ref>) and (<ref>).

[$\blacktriangleright$ Periods]
Similarly to the computation of the archimedean local $L$-factors, we have

\begin{align*}
\prod_{j=1}^s \bigl( C_{\mathrm{CM},p,F_j }^{w_{\eta}\mathsf{t}_j+2\mathsf{r}_{\eta_j}}\bigr)^{a_j}
&= \prod_{j=1}^s \prod_{\tau \in \Sigma_{F_j}} \bigl(C_{\mathrm{CM},p,F_j ,\tau }^{w_{\eta} +2{r}_{\eta_j,\tau}}  \bigr)^{a_j}=
\prod_{j=1}^s \prod_{\sigma \in \Sigma_F} \prod_{\substack{ \tau \in \Sigma_{F_j}\\   \tau \vert_F =\sigma }} \bigl(C_{\mathrm{CM},p,F_j ,\tau }^{w_{\eta} +2{r}_{\eta_j,\tau}}  \bigr)^{a_j} \\
&= \prod_{j=1}^s \prod_{\sigma \in \Sigma_F} \prod_{\substack{ \tau \in \Sigma_{F_j}\\  \tau \vert_F =\sigma }} \bigl(C_{\mathrm{CM},p,F ,\tau\vert_F }^{w_{\eta} +2{r}_{\eta,\tau\vert_F}}  \bigr)^{a_j}=\prod_{\sigma\in \Sigma_F} \bigl(C_{\mathrm{CM},p,F,\sigma}^{w_\eta+2r_{\eta,\sigma}}\bigr)^{\sum_{j=1}^s a_j[F_j:F]},
\end{align*}

which is equal to $\bigl(C_{\mathrm{CM},p,F}^{w_\eta\mathsf{t}+2\mathsf{r}_\eta}\bigr)^{r(\rho)}$ due to (<ref>). Here the third equality follows from Lemma <ref>. The equality $\prod \bigl(\Omega_{\mathrm{CM},\infty,F_j}^{w_\eta\mathsf{t}_j+2\mathsf{r}_{\eta_j}}\bigr)^{a_j}=\bigl(\Omega_{\mathrm{CM},\infty,F}^{w_\eta\mathsf{t}+2\mathsf{r}_{\eta}}\bigr)^{r(\rho)}$ is also obtained by a very similar calculation.

[$\blacktriangleright$ Modified $\infty$-factors and ther coefficients]
Likewise we can calculate as

\begin{align*}
\prod_{j=1}^s \left(\dfrac{(-1)^{w_\eta d[F_j:F]}\, i^{\lvert -w_\eta\mathsf{t}_j-\mathsf{r}_{\eta_j}\rvert}}{2^{d[F_j:F]}(2\delta)_{\Sigma_{F_j}}^{\mathsf{r}_{\eta_j}}}\right)^{a_j} &=\prod_{j=1}^s  \left( \dfrac{(-1)^{w_\eta d[F_j:F]}}{2^{d[F_j:F]}}  \prod_{\sigma\in \Sigma_F} \prod_{\substack{\tau\in \Sigma_{F_j} \\ \tau\vert_F=\sigma}} \dfrac{i^{ -w_\eta-r_{\eta,\tau\vert_F}}}{\tau(2\delta)^{r_{\eta,\tau\vert_F}}}\right)^{a_j} \\
&=\prod_{j=1}^s \left( \dfrac{(-1)^{w_\eta d[F_j:F]}}{2^{d[F_j:F]}}  \prod_{\sigma\in \Sigma_F}  \dfrac{i^{ (-w_\eta-r_{\eta,\sigma})[F_j:F]}}{\tau(2\delta)^{r_{\eta,\sigma}[F_j:F]}}\right)^{a_j} \\
&= \left( \dfrac{(-1)^{w_\eta d}\, i^{ \lvert -w_\eta \mathsf{t}-\mathsf{r}_{\eta}\rvert}}{2^{d}(2\delta)_{\Sigma_F}^{\mathsf{r}_\eta}}   \right)^{\sum_{j=1}^s a_j[F_j:F]} \\
&=\left( \dfrac{(-1)^{w_\eta d}\, i^{ \lvert -w_\eta \mathsf{t}-\mathsf{r}_{\eta}\rvert}}{2^{d}(2\delta)_{\Sigma_F}^{\mathsf{r}_\eta}}   \right)^{r(\rho)},
\end{align*}

where the third equality follows from the definition of our multi-index notation and the last equality follows from (<ref>).

[$\blacktriangleright$ Modified $p$-Euler factors at unramified places]

Let $v\in \Sigma_{F,p}$ be a $p$-adic place of $F$ at which $\eta$ is unramified. Then, since $\mathrm{Eul}_{\tilde{v}}(\psi_j\eta_j,0)=L_{\tilde{v}^c}(\psi_j\eta_j,0)^{-1}L_{\tilde{v}}((\psi_j\eta_j)^{-1},1)^{-1}$ for every $\tilde{v}\in \Sigma_{F_j,p}$ lying above $\tilde{v}$, we have

\begin{align*}
\prod_{j=1}^s \prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} \mathrm{Eul}_{\tilde{v}}(\psi_j\eta_j,0)^{a_j}&= \prod_{j=1}^s \prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} L((\psi_j\eta_j)_{\tilde{v}^c},0)^{-a_j}L((\psi_j\eta_j)_{\tilde{v}}^{-1},1)^{-a_j}\\
&= L ((\rho\otimes \eta)_{v^c},0)^{-1} L ((\rho^\vee \otimes \eta^{-1})_v,1)^{-1} \\
&=L ((\rho\otimes \eta)_{v^c},0)^{-1} L ((\rho \otimes \eta)_v^\vee ,1)^{-1}=\mathrm{Eul}_v(\rho\otimes \eta, 0)
\end{align*}

due to (<ref>); note that (<ref>) implies the equality $\rho^{\vee}=\sum_{j=1}^s a_j \mathrm{Ind}^{G}_{G_j} \psi_j^{-1}$.

[$\blacktriangleright$ Modified $p$-Euler factors at ramified places]
Let $v\in \Sigma_{F,p}$ be a $p$-adic place of $F$ at which $\eta$ is ramified.
In this situation, we have $\mathrm{Eul}_v(\rho\otimes \eta,0)=\epsilon((\rho\otimes \eta)_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v)$ and $\mathrm{Eul}_{\tilde{v}}(\rho\otimes \eta,0)=\epsilon((\psi_j\eta_j)_{\tilde{v}},\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}},dx_{\tilde{v}})$ for each $\tilde{v}\in \Sigma_{F_j,p}$ lying above $v$. We shall verify the equality among the local constants

\begin{align} \label{eq:Brauer_induction_epsilon}
\prod_{j=1}^s\prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}}  \epsilon((\psi_j\eta_j)_{\tilde{v}},\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}},dx_{\tilde{v}})^{a_j}=\epsilon((\rho\otimes \eta)_v,\mathbf{e}^{(2\delta)}_{F,v},dx_v),
\end{align}

which implies the desired equality

\begin{align} \label{eq:Brauer_induction_A}
\prod_{j=1}^s\prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} \mathrm{Eul}_{\tilde{v}}(\psi_j\eta_j,0) =\mathrm{Eul}_v(\rho\otimes \eta,0).
\end{align}

Now let us prove (<ref>). Note that $\psi_j$  is unramified at each $\tilde{v}\in \Sigma_{F_j,p}$ since $F/F_j$ is unramified at $\tilde{v}$ by construction.  Hence, if we let $\mathfrak{P}_{\tilde{v}}^{e(\eta_{j,\tilde{v}})}$ denotes the conductor of the $\tilde{v}$ component of $\eta_j=\eta\circ \mathrm{Norm}_{F_j/F}$, the left-hand side of (<ref>) is calculated as

\begin{align}
\prod_{j=1}^s &\prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} \epsilon ((\psi_j\eta_j)_{\tilde{v}}, \mathbf{e}_{F_j,\tilde{v}}^{(2\delta)},dx_{\tilde{v}})^{a_j} =
\prod_{j=1}^s \prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}}\epsilon ((\eta_j)_{\tilde{v}}, \mathbf{e}_{F_j,\tilde{v}}^{(2\delta)},dx_{\tilde{v}})^{a_j} \psi_{j,\tilde{v}}(\varpi_v^{e(\eta_{j,\tilde{v}})})^{a_j} \label{equation:epsilonfactor_into_two_terms} \\
&=\prod_{j=1}^s\prod_{\substack{\tilde{v} \in \Sigma_{F_j , p} \\ \tilde{v} \mid v}}
\psi_{j,\tilde{v}}(\varpi_v^{e(\eta_{j,\tilde{v}})})^{a_j}
\left(\int_{\mathfrak{P}_{\tilde{v}}^{-e(\eta_{j,\tilde{v}})}} \eta_{j,\tilde{v}}^{-1}(x)\mathbf{e}_{F,v}^{(2\delta)}(x)dx_{\tilde{v}}\right)^{a_j}
\nonumber \\
&=\prod_{j=1}^s\prod_{\substack{\tilde{v} \in \Sigma_{F_j , p} \\ \tilde{v} \mid v}}  \psi_{j,\tilde{v}}(\varpi_v^{e(\eta_{j,\tilde{v}})})^{a_j}
\left(\int_{\mathfrak{r}_{F_{j,\tilde{v}}}} \eta_{j,\tilde{v}}^{-1}(\varpi_v^{-e(\eta_{j,\tilde{v}})}x)\mathbf{e}_{F,v}^{(2\delta)}(\varpi_v^{-e(\eta_{j,\tilde{v}})}x)d(\varpi_v^{-e(\eta_{j,\tilde{v}})}x_{\tilde{v}})\right)^{a_j} \nonumber \\
&=\prod_{j=1}^s \prod_{\substack{\tilde{v} \in \Sigma_{F_j , p} \\ \tilde{v} \mid v}} \psi_{j,\tilde{v}}(\varpi_v^{e(\eta_{j,\tilde{v}})})^{a_j} \eta_{j,\tilde{v}}(\varpi_v^{e(\eta_{j,\tilde{v}})})^{a_j}
\left(\sum_{x\in (\mathfrak{r}_{F_j}/\mathfrak{P}_{\tilde{v}}^{e(\eta_{j,\tilde{v}})})^\times}
\eta_{j,\tilde{v}}^{-1}(x)\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}}(\varpi_v^{-e(\eta_{j,\tilde{v}})}x)\right)^{a_j}. \nonumber
\end{align}

Here $\varpi_v$ is a uniformiser of $F_v$ and we also regard it as a uniformiser of $F_{j,\tilde{v}}$ due to unramifiedness of $F_{j,\tilde{v}}/F_v$. The first equality follows from <cit.>, and the second equality follows from <cit.>
Note that $e(\eta_{j,\tilde{v}})$ coincides with $e(\eta_v)$, the exponent of the conductor  of $\eta_v$, since the norm map $\mathrm{Norm}_{F_{j,\tilde{v}}/F_v}$ induces a surjection $1+\mathfrak{P}_{\tilde{v}}^k\twoheadrightarrow 1+\mathfrak{P}_v^k$
for every natural number $k$ (see <cit.>; recall again that $F_{j,\tilde{v}}/F_v$ is unramified).

We may calculate the product concerning $\eta_{j,\tilde{v}}$'s further by using a generalisation of Hasse and Davenport's relation for characters with prime power conductor (Theorem <ref>), whose proof shall be given in Appendix <ref>. Indeed, we may apply Theorem <ref> by setting

\begin{align*}
s&=f_{\tilde{v}\mid v}, &  \mathbb{F}_q &= \mathfrak{r}_{F} /\mathfrak{P}_v, & \mathbb{F}_{q^s}&=\mathfrak{r}_{F_j}/\mathfrak{P}_{\tilde{v}}, & \chi &= \eta_v \\
n&=e (\eta_v), & W_n (\mathbb{F}_q ) &= \mathfrak{r}_{F}/\mathfrak{P}_v^{e(\eta_v)}, &  W_n(\mathbb{F}_{q^s}) &=\mathfrak{r}_{F_j}/\mathfrak{P}_{\tilde{v}}^{e(\eta_v)}, & \psi_{n,\mathbb{F}_q} &=\mathbf{e}_{F,v}^{(2\delta)}(\varpi_v^{-e(\eta_v)} -)
\end{align*}

where $f_{\tilde{v}\vert v}$ denotes the inertia degree of $\tilde{v}\vert v$, which coincides with the extension degree of $F_{j,\tilde{v}}/F_v$. Note that $\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}}(\varpi_v^{-e(\eta_v)}-)=\mathbf{e}^{(2\delta)}_{F,v}(\varpi_v^{-e(\eta_v)}-)\circ \mathrm{Tr}_{F_{j,\tilde{v}}/F_v}$ holds since both $\varpi_v$ and $\delta$ are elements of $F_v$, and thus one may regard $\mathbf{e}_{F_j,\tilde{v}}^{(2\delta)}(\varpi_v^{-e(\eta_v)}-)$ as $\psi_{n,\mathbb{F}_q^s}$ appearing in Appendix <ref>. Applying Theorem <ref>, we have

\begin{align}%\label{equation:checked_by_generalized DH3}
& \prod_{j=1}^s \prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} \eta_{v}\circ \mathrm{Norm}_{F_{j,\tilde{v}}}(\varpi_v^{e(\eta_v)})^{a_j}
\left(
\sum_{x\in  (\mathfrak{r}_{F_j}/\mathfrak{P}_{\tilde{v}}^{e(\eta_v)})^\times}
\eta_{j,\tilde{v}}^{-1}(x)\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}}
\right) ^{ a_j} \label{equation:localepsilon1} \\
& =  \prod_{j=1}^s \prod_{\substack{\tilde{v} \in \Sigma_{F_j , p}
\\ \tilde{v}\mid v}} \eta_v(\varpi_v^{e(\eta_v)})^{a_jf_{\tilde{v}\vert v}} \nonumber
\\
&\hspace*{8em} \cdot
\left( (-1)^{e (\eta_v)(f_{\tilde{v}\vert v}-1)}  \left(\sum_{x\in (\mathfrak{r}_{F}/\mathfrak{P}_v^{e(\eta_v)})^\times}\eta_v^{-1}(x)\mathbf{e}^{(2\delta)}_{F,v}(\varpi_v^{-e(\eta_v)}x) \right)^{f_{\tilde{v}\vert v}}\right)^{ a_j} \nonumber \\
&\hspace{30em}  (\text{by Theorem~\ref{theorem:D-H_relation_overW_n}}) \nonumber \\
(-1)^{\sum_{j=1}^s a_i \sum_{\tilde{v}\mid v} e(\eta_v) (f_{\tilde{v}\mid v}-1)}
\nonumber \\
&\hspace*{8em} \cdot
\left(  \eta_v(\varpi_v^{e(\eta_v)})
\sum_{x\in (\mathfrak{r}_{F}/\mathfrak{P}_v^{e(\eta_v)})^\times}
\eta_v^{-1} (x)\mathbf{e}^{(2\delta)}_{F,v}(\varpi_v^{-e(\eta_v)}x)\right)^{\sum_{j=1}^s a_j\sum_{\tilde{v}\mid v} f_{\tilde{v}\mid v} } \nonumber \\
&=(-1)^{\sum_{j=1}^sa_j \sum_{\tilde{v}\mid v}e(\eta_v) (f_{\tilde{v}\mid v}-1)}\epsilon(\eta_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)^{r(\rho)}, \nonumber
\end{align}

where $\sum_{\tilde{v}\mid v}$ denotes the summation over $\tilde{v}\in \Sigma_{F_i,p}$ lying above $v$. The last equality follows from <cit.> and the identity $r(\rho) = \sum_{j=1}^s a_j[F_j:F]=\sum_{j=1}^s \sum_{\tilde{v}\mid v} a_j f_{\tilde{v}\mid v}$,
which follows from the unramifiedness of $F_{j,\tilde{v}}/F_v$. Although there appears an unexpected signature $(-1)^{\sum_{j=1}^s \sum_{\tilde{v}\mid v}e(\eta_v)f_{\tilde{v}\mid v}}$, it will be finally cancelled out with the signature appearing in the computation of the product concerning $\psi_j$'s.

Next, let us calculate the product concerning $\psi_j$'s in (<ref>). For $j=1,2,\ldots,s$, let $\tilde{v}_0$ be a unique place of $F_j$ fixed by $G_{F_v}$. Then $g\mapsto \tilde{v}_0^g$ induces a bijection between $G_{F_j}\backslash G_F/G_{F_v}$ and the set of all the places of $F_j$ lying above $v$ (see, for example, <cit.>), and we can regard $G_{F_{j,\tilde{v}_0^g}}:=G_{F_j}\cap G_{F_v}^{g^{-1}}$ as the decomposition subgroup of $G_{F_j}$ at $\tilde{v}_0^g$. Since $F_{j,\tilde{v}_0^g}/F_v$ is an unramified extension of degree $f_{\tilde{v}_0^g \mid v}$, we see that $g\mathrm{Frob}_v^{f_{\tilde{v}_0^g\mid v}}g^{-1}\in G_{F_{j,\tilde{v}_0^g}}$ is indeed a lift of the geometric Frobenius $\mathrm{Frob}_{\tilde{v}_0^g}$ at $\tilde{v}_0^g$.
Now consider the induced representation $\mathrm{Ind}_{G_{F_j}^g \cap G_{F_v}}^{G_{F_v}} \psi_j^g :=\mathrm{Hom}_{\overline{\mathbb{Q}}[G_{F_j}^g\cap G_{F_v}]} (\overline{\mathbb{Q}}[G_{F_v}], \overline{\mathbb{Q}}(\psi_j^g))$ where $\psi_j^g$ is defined as $x\mapsto \psi_j(gxg^{-1})$. The action of $\mathrm{Frob}_v$ on $\mathrm{Ind}_{G_{F_j}^g \cap G_{F_v}}^{G_{F_v}} \psi_j^g$ is[We let $\gamma \in G_{F_v}$ act on $\mathrm{Ind}_{G_{F_j}^g \cap G_{F_v}}^{G_{F_v}} \psi_j^g$ by $(\gamma f)(x):=f(x\gamma)$ for $f\in \mathrm{Ind}_{G_{F_j}^g \cap G_{F_v}}^{G_{F_v}} \psi_j^g$ and $x\in \overline{\mathbb{Q}}[G_{F_v}]$.] represented by a square matrix

\begin{align*}
\begin{bmatrix}
&  &  & & \psi_j^g(\mathrm{Frob}_v^{f_{\tilde{v}_0^g\mid v}}) \\
1 &   &   & &  \\
&1  &  & & \\
&   & \ddots   & &   \\
&  &  & 1 & \\
\end{bmatrix}=
\begin{bmatrix}
&  &  & &  \psi_j(\mathrm{Frob}_{\tilde{v}_0^g})\\
1 &   &   & &  \\
& 1 &  & & \\
&   &  \ddots  & &   \\
&  &  &1  & \\
\end{bmatrix}
\end{align*}

of degree $f_{\tilde{v}_0^g\mid v}$ with respect to the basis $\{ \phi_{-k} \}_{0\leq k\leq  f_{\tilde{v}_0^g\mid v}-1}$, where $\phi_{-k}\colon \overline{\mathbb{Q}}[G_{F_v}]\rightarrow \overline{\mathbb{Q}}(\psi_j^g)$ denotes a $\overline{\mathbb{Q}}[G_{F_j}^g\cap G_{F_v}]$-equivariant homomorphism sending $\mathrm{Frob}_v^{-k}$ to $1$ and $\mathrm{Frob}_v^{-k'}$ to $0$ for $k'=0,1,\ldots,f_{\tilde{v}_0^g\mid v}-1$ with $k'\neq k$. The determinant of the above matrix is equal to $(-1)^{f_{\tilde{v}_0^g\mid v}-1}\psi_j(\mathrm{Frob}_{\tilde{v}_0^g})$, and we thus obtain the equality

\begin{align} \label{equation:Frob_action_induced}
\det \bigl(\mathrm{Frob}_v;\, \mathrm{Ind}_{G_{F_j}^g\cap G_{F_v}}^{G_{F_v}}\psi_j^g\bigr) &=(-1)^{f_{\tilde{v}_0^g\mid v}-1} \psi_j(\mathrm{Frob}_{\tilde{v}_0^g})=(-1)^{f_{\tilde{v}_0^g\mid v}-1} \psi_{j,\tilde{v}_0^g}(\varpi_v).
\end{align}

Combining (<ref>) with Mackey decomposition (<ref>), we may calculate as

\begin{align}\label{equation:localepsilon2}
\prod_{j=1}^s \prod_{\substack{\tilde{v} \in \Sigma_{F_i , p} \\ \tilde{v}\mid v}}  &\psi_{j,\tilde{v}} (\varpi_v^{e(\eta_v)})^{ a_i}
=\prod_{j=1}^s \prod_{[g]\in G_{F_j}\backslash G_F/G_{F_v}}  \psi_{j,\tilde{v}_0^g} (\varpi_v^{e(\eta_v)})^{ a_j}  \\
&=\prod_{j=1}^s \prod_{[g]\in G_{F_j}\backslash G_F/G_{F_v}} \left\{(-1)^{e(\eta_v)(f_{\tilde{v}_0^g\mid v}-1)} \det (\mathrm{Frob}_v^{e(\eta_v)};\, \mathrm{Ind}_{G_{F_j}^g\cap G_{F_v}}^{G_{F_v}}\psi_j^g)\right\}^{a_j} \nonumber\\
&\hspace*{27em} (\text{by \eqref{equation:Frob_action_induced}}) \nonumber \\
&= (-1)^{\sum_{j=1}^sa_j \sum_{\tilde{v}\mid v} e(\eta_v)(f_{\tilde{v}\mid v}-1)} \nonumber \\
&\hspace*{8em} \cdot  \det \left(\mathrm{Frob}_v^{e(\eta_v)};\, \sum_{j=1}^s a_j\bigoplus_{[g]\in G_{F_j}\backslash G_F/G_{F_v}}\mathrm{Ind}_{G_{F_j}^g\cap G_{F_v}}^{G_{F_v}}\psi_j^g\right) \nonumber\\
&= (-1)^{\sum_{j=1}^sa_j \sum_{\tilde{v}\mid v} e(\eta_v)(f_{\tilde{v}\mid v}-1)} \det\left(\mathrm{Frob}_v^{e(\eta_v)} \,;\, V_\rho\vert_{G_{F_v}}\right) \quad (\text{by \eqref{equation:brauer_induction} and \eqref{equation:Mackey}}). \nonumber
\end{align}

We here emphasise that the signature appearing in (<ref>) completely coincides with that in (<ref>).
Therefore, by putting (<ref>), (<ref>) and (<ref>), we finally obtain

\begin{align*}
\prod_{j=1}^s \prod_{\substack{\tilde{v}\in \Sigma_{F_j,p} \\ \tilde{v}\mid v}} \epsilon((\psi_j\eta_j)_{\tilde{v}},\mathbf{e}^{(2\delta)}_{F_j,\tilde{v}},dx_{\tilde{v}})^{a_j}&=\epsilon(\eta_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v)^{r(\rho)}\det\bigl(\mathrm{Frob}_v^{e(\eta_v)}\, ; \, V_\rho\bigr) \\
&=\epsilon((\rho\otimes \eta)_v,\mathbf{e}_{F,v}^{(2\delta)},dx_v),
\end{align*}

due to <cit.>.

We thus complete the proof of Theorem <ref>.

The interpolation formula (<ref>) uniquely characterises the $p$-adic Artin $L$-function $L_{p,\Sigma_F}(M(\rho))$ due to the $p$-adic identity theorem <cit.>; indeed we may readily extend <cit.> to elements of the field of fractions of the Iwasawa algebra $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\mathrm{max}}]]$.  In particular $L_{p,\Sigma_F}(M(\rho))$ does not depend on any Brauer decomposition (<ref>) of $\rho$ which we choose in the proof of Theorem <ref>, because the interpolation formula (<ref>) does not contain any data concerning a particular choice of the decomposition (<ref>).

The integrality conjecture below is regarded as a $p$-adic counterpart of Artin's conjecture for (complex) Artin $L$-functions.

The $p$-adic Artin  $L$-function $L_{p,\Sigma_F} (M(\rho )) \in \mathrm{Frac} (\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\mathrm{max}}]])$
constructed in Theorem <ref> is indeed an element of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\mathrm{max}}]]$.

§ THE INTEGRALITY OF THE $P$-ADIC ARTIN $L$-FUNCTION AND THE IWASAWA MAIN CONJECTURE

Let $\rho\colon \mathrm{Gal}(\overline{F}/F)\rightarrow {\rm Aut}_{\overline{\mathbb{Q}}}(V_\rho)$ be an Artin representation.
Choose a $G_{F}$-stable $\mathcal{O}$-lattice $T$ of the Artin representation $V_\rho$ and set $A =T \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$.
We denote by $S$ the set of primes of $F$ containing the primes dividing $\infty$, the primes dividing $(p)$ and
the ramified primes of $\eta$. For any algebraic extension $F'$ of $F$, we denote by $S_{F'}$ the set of prims of $F'$ over $S$.
When $F'$ is a finite extension of $F$, we define the Selmer group $\mathrm{Sel}_{A} (F') $ as follows:

\begin{equation}\label{equation:definitionSelmergp_rho}
\mathrm{Sel}_{A} (F') = \mathrm{Ker}
\left[
H^1 (F'_{S_{F'}} /F' ,A ) \longrightarrow
\underset{\substack{\lambda \in S_{F'} \\ \lambda \nmid p\infty}}{\prod}
H^1 (I_{\lambda} ,A)
\times \underset{\substack{\mathfrak{p}\in \Sigma^c_{p,F'}}}{\prod}
H^1 (I_{\mathfrak{p}} ,A )
\right]
\end{equation}

where we denote by $\Sigma^c_{p,F'}$ the set of primes of $K$ which are over the set of primes in $\Sigma^c_{p}$.
When $F'$ is an algebraic extension of $F$ of infinite degree, we define $\mathrm{Sel}_{A} (F') $ by taking the inductive limit of the Selmer groups
defined over intermediate extensions of finite degree over $F$.
Similarly, we define the strict Selmer group as follows:
When $F'$ is a finite extension of $F$, we define the strict Selmer group $\mathrm{Sel}^{\mathrm{str}}_{A} (F')$ as follows:

\begin{equation}
\mathrm{Sel}^{\mathrm{str}}_{A} (F') = \mathrm{Ker}
\left[
H^1 (F'_{S_{F'}}/F' ,A ) \longrightarrow
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
H^1 ({F'}_{\lambda}^{\mathrm{ur}} ,A)
\times \underset{\substack{\mathfrak{p}\in \Sigma^c_{p,F'}}}{\prod}
H^1 (D_{\mathfrak{p}} ,A )
\right] .
\end{equation}

By the same reason as above, $\mathrm{Sel}^{\mathrm{str}}_{A} (F') $ makes sense even when $F'$ is an algebraic extension of $F$ of infinite degree.

Let $F$ be a CM field and $\psi$ a Hecke character of finite order on $F$ whose conductor is prime to $(p)$.
Since the Galois representation $V_\psi$ associated to $\psi$ is of dimension $1$, we denote by $T_\psi$ the unique $G_F$-stable lattice
such that $T_\psi \otimes_{\mathbb{Z}_p} \mathbb{Q}_p $ is isomorphic to $V_\psi$. We set $A_\psi =T_\psi  \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$.
The Selmer group $\mathrm{Sel}_{A_{\psi }}(F_{\mathrm{max}})$ is defined in the same way as
Then the following equality of principal ideals of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal} (F_{\mathrm{max}} /F )]]$ should be true:

(L_{p,\Sigma_F}(\psi )) = \mathrm{char}_{\widehat{\mathcal{\mathcal{O}}}^{\mathrm{ur}}[[\mathrm{Gal} (F_{\mathrm{max}} /F )]]}
\bigl(\mathrm{Sel}_{A_{\psi }}(F_{\mathrm{max}})^\vee \widehat{\otimes}_{\mathcal{O}}\widehat{\mathcal{O}}^{\mathrm{ur}} \bigr)

where $L_{p,\Sigma_F}(\psi )$ is defined as in (<ref>).

Choose a $G_{F}$-stable $\mathcal{O}$-lattice $T$ of $V_\rho$ and set $A =T \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$.
Then the following equality of principal fractional ideals of $\widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(F_{\max}/F)]]$ should be true:

\begin{equation}\label{equation:IMC_for_Artinrep}
(L_{p ,\Sigma_F} (M(\rho ))) =
\mathrm{char}_{\widehat{\mathcal{O}}^\mathrm{ur}[[\mathrm{Gal}(F_{\max}/F)]]} \bigl(\mathrm{Sel}_A (F_{\mathrm{max}})^\vee
\widehat{\otimes}_{\mathcal{O}} \widehat{\mathcal{O}}^{\mathrm{ur}} \bigr)
\end{equation}

* The left-hand side of (<ref>) is independent of the choice of a lattice $T$ and
the right-hand side of (<ref>) depends on the choice of a lattice $T$ a priori.
In fact, we can check that the right-hand side of (<ref>) is independent of the choice of a lattice $T$ by
Lemma <ref>.

* By Lemma <ref>, the variant of Conjecture <ref> which is formulated by replacing
$\mathrm{Sel}_A (F_{\mathrm{max}})$ by $\mathrm{Sel}^\mathrm{str}_A (F_{\mathrm{max}})$ is equivalent to Conjecture <ref>.

* Note that Theorem <ref> constructs $L_{p,\Sigma_F} (M(\rho ))$ as an element of
$\mathrm{Frac} (\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\mathrm{max}}]])$.
Since the right-hand side of (<ref>) is integral ideal of
$\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\max}]]$, the validity of (<ref>)
implies that $(L_{p,\Sigma_F} (M(\rho )))$ is an integral ideal of

Let $M(\rho)$ be the Artin motive corresponding to an Artin representation $\eta$ of $G_F$ which is unramified at any prime of $F$ over $p$
such that $\mathrm{ker}(\rho )$ is a CM-extension of $F$. Assume that the CM Iwasawa main conjecture $($Conjecture $\ref{conjecture:iwasawamainconjecture})$ is true for any intermediate field $K$ of the extension
$F_{\mathrm{max}} /F$ which is finite over $F$.
Then Conjectures $\ref{conjecture:integrality}$ and $\ref{conjecture:iwasawamainconjectureartin}$ hold true.

Though we assume that the fixed prime number $p$ is odd throughout the present article, we would like to explain here the situation of the case $p=2$.
Indeed, the only issue concerning the prime $2$ is that, due to Hasse's unit index theorem <cit.>, the denominator of the modification factor $\frac{\sqrt{\lvert D_{F^+}\rvert}}{(\mathfrak{r}_{F}^\times :\mathfrak{r}_{F^+}^\times)}$ appearing in (<ref>) might be divisible by $2$ even if $F$ is assumed to be absolutely unramified at $(2)$. Therefore we can still verify by the same proof that the $2$-adic Artin $L$-function $L_{2,\Sigma_F}(M(\rho))$ constructed as above is an element of $\widehat{\mathcal{O}}^\mathrm{ur}[[\Gamma_{F,\max}]]\otimes_{\mathbb{Z}}\mathbb{Q}$ and the main conjecture (<ref>) holds true up to $\mu$-invariants.

Before proving the theorem, we prepare some notation and some lemmas.

As before, choose a $G_{F}$-stable $\mathcal{O}$-lattice $T$ of the Artin representation $V_\rho$ and set $A =T \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$.
Then the natural surjection

\mathrm{Sel}_{A} (F_{\mathrm{max}})^\vee \twoheadrightarrow \mathrm{Sel}^{\mathrm{str}}_{A} (F_{\mathrm{max}})^\vee

is a pseudo-isomorphism of $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-modules.
Also for any finite extension $F'$ of $F$,
the natural surjection

\begin{align*}
\mathrm{Sel}_{A} (F'F_{\mathrm{max}})^\vee &\twoheadrightarrow \mathrm{Sel}^{\mathrm{str}}_{A} (F'F_{\mathrm{max}})^\vee & \left( \mathrm{resp}.\
\mathrm{Sel}_{A} (F'_{\mathrm{max}})^\vee \twoheadrightarrow \mathrm{Sel}^{\mathrm{str}}_{A} (F'_{\mathrm{max}})^\vee
\right)
\end{align*}

is a pseudo-isomorphism of $\mathcal{O}[[\mathrm{Gal}(F'F_{\mathrm{max}} /F')]]$-modules $($resp. $\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]$-modules$)$.

To prove Lemma <ref>, we prepare some notation. We define the module $\mathrm{Loc}^i_M (K)$ to be

\begin{equation}
\mathrm{Loc}^i_M (K) =  \underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod}
\frac{ H^i (K_{\mathfrak{p}} ,M) }{H^i _{\mathrm{ur}}(K_{\mathfrak{p}} ,M)}\times
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
\frac{H^i (K_{\lambda} ,M)}{H^i_{\mathrm{ur}} (K_{\lambda} ,M)}
\end{equation}

for any integer $i$ such that $0\leq i \leq 2$, any intermediate field
$K$ of $F_\mathrm{max}/F$ and any subquotient $M$ of the $\mathrm{Gal}
(F_{S}/F)$-module $A$.
Similarly, we define the module $\mathrm{Loc}^{\mathrm{str}, i}_M (K)$ to be

\begin{equation}
\mathrm{Loc}^{\mathrm{str}, i}_M (K) =  \underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod}  H^i (K_{\mathfrak{p}} ,M) \times
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
\frac{H^i (K_{\lambda} ,M)}{H^i_{\mathrm{ur}} (K_{\lambda} ,M)}
\end{equation}

for any integer $i$ such that $0\leq i \leq 2$, any intermediate field
$K$ of $F_\mathrm{max}/F$ and any subquotient $M$ of the $\mathrm{Gal}
(F_{S}/F)$-module $A$.

Let us consider the following commutative diagram.

\begin{equation}
\begin{CD}
0 @>>> \mathrm{Sel}_A (F_{\mathrm{max}}) @>>> H^1 (F_S / F_{\mathrm{max}} ,A)  @>{\mathrm{loc}_A}>> \mathrm{Loc}^1_A (F_{\mathrm{max}})  \\
@. @A{f}AA @AA{g}A @AA{h}A\\
0 @>>> \mathrm{Sel}^{\mathrm{str}}_{A} (F_{\mathrm{max}}) @>>> H^1 (F_S / F_{\mathrm{max}} ,A)  @>>{\mathrm{loc}^{\mathrm{str}}_{A}}> \mathrm{Loc}^{\mathrm{str},1}_{A} (F_{\mathrm{max}}) .
\\
\end{CD}
\end{equation}

By construction, the left vertical map $f$ is injective. In order to verify the assertion for $\mathrm{Sel}_{A} (F_{\mathrm{max}})^\vee \twoheadrightarrow \mathrm{Sel}^{\mathrm{str}}_{A} (F_{\mathrm{max}})^\vee$, it suffices to show that $(\mathrm{Coker}\, f)^\vee$ is a pseudo-null $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
The right vertical map $h$ is surjective and we have

\mathrm{Ker}(h)
\cong \underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,F}}}{\prod} H^1_{\mathrm{unr}} (F_\mathfrak{p} ,A) \cong
\underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,F}}}{\prod} H^1 (F^{\mathrm{unr}}_\mathfrak{p} / F_\mathfrak{p} ,A^{I_{F_\mathfrak{p}}}).

Since the $\mathbb{Z}_p$-rank of the decomposition group $D_{\mathfrak{p}}$ is equal to or greater than $2$ due to <cit.>, so is the codimension of the support of $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module $(\mathrm{Ker}\, h)^\vee$; in other words, $(\mathrm{Ker}\, h)^\vee$ is a pseudonull $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
This implies that $(\mathrm{Coker}\, f)^\vee$ is also pseudonull, because it is a subquotient of $(\mathrm{Ker}\, h)^\vee$.
One readily verifies the other assertions by the same argument, also using <cit.>.

Let $F'$ be a finite extension of $F$ contained in $F_S$ which is a CM-field of degree $2d'$ over $\mathbb{Q}$.
Let $\tau : G_{F'} \longrightarrow V_{\tau}$ be an Artin representation of $G_{F'}$.
We choose a $G_{F'}$-stable lattice $T$ and we denote $V_{\tau}$ by $A$.
Let $F'_{\mathrm{max}}$ denote the composite of all $\mathbb{Z}_p$-extensions of $F'$.
Then the localization map $H^1 (F_S / F'_{\mathrm{max}} ,A) \overset{\mathrm{loc}^{\mathrm{str}}_A}{\longrightarrow} \mathrm{Loc}^{\mathrm{str},1}_A (F'_{\mathrm{max}}) $
is surjective.

Let $A^\ast = \mathrm{Hom}_{\mathbb{Z}_p} (T ,\mu_{p^\infty})$.

\overline{\mathrm{Loc}^{\mathrm{str},1}_A (K)} =  \underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod}  H^1 (K_{\mathfrak{p}} ,A) \times
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
\frac{H^1 (K_{\lambda} ,A)}{H^1_{\mathrm{ur}} (K_{\lambda} ,A)_{\mathrm{div}}}

for any finite extension $K$ of $F'$ where $H^1_{\mathrm{ur}} (K_{\lambda} ,A)_{\mathrm{div}}$ is the maximal divisible subgroup of
$H^1_{\mathrm{ur}} (K_{\lambda} ,A)$. We similarly define

\overline{\mathrm{Loc}^{\mathrm{str},1}_{A^\ast} (K)}
=  \underset{\substack{\mathfrak{p}\in \Sigma_{p ,K}}}{\prod}  H^1 (K_{\mathfrak{p}} ,A^\ast) \times
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
\frac{H^1 (K_{\lambda} ,A^\ast)}{H^1_{\mathrm{ur}} (K_{\lambda} ,A^\ast)_{\mathrm{div}}}

for any finite extension $K$ of $F'$ contained in $F_S$. We define a Bloch–Kato type Selmer group $\mathrm{Sel}^{\mathrm{BK}}_A (K)$
(resp. $\mathrm{Sel}^{\mathrm{BK}}_{A^\ast} (K)$) to be the kernel of the localization map
$H^1 (K_S / K ,A) \overset{\mathrm{loc}^\mathrm{str}_A}{\longrightarrow} \overline{\mathrm{Loc}^{\mathrm{str},1}_A (K)} $
(resp. $H^1 (K_S / K ,A^\ast) \overset{\mathrm{loc}^\mathrm{str}_{A^\ast}}{\longrightarrow} \overline{\mathrm{Loc}^{\mathrm{str},1}_{A^\ast} (K)}$).
For a finite Galois module $A^\ast [p^m]$,
we define $\overline{\mathrm{Loc}^{\mathrm{str},1}_{A^\ast [p^m]} (K)}$ by

\overline{\mathrm{Loc}^{\mathrm{str},1}_{A^\ast [p^m]} (K)}
=  \underset{\substack{\mathfrak{p}\in \Sigma_{p ,K}}}{\prod}  H^1 (K_{\mathfrak{p}} ,A^\ast [p^m]) \times
\underset{\substack{\lambda \in S_K \\  \lambda \nmid p\infty}}{\prod}
\frac{H^1 (K_{\lambda} ,A^\ast [p^m])}{\iota_m ^{-1} (H^1_{\mathrm{ur}} (K_{\lambda} ,A^\ast )_{\mathrm{div}})}

using the natural map $\iota_m : H^1 (K_{\lambda} ,A^\ast [p^m])
\longrightarrow H^1 (K_{\lambda} ,A^\ast )$ and the inverse image $\iota_m^{-1} $.
Then we define a Selmer group $\mathrm{Sel}^{\mathrm{BK}}_{A^\ast [p^m]} (K)$ with a finite coefficient $A^\ast [p^m]$
to be the kernel of the localization map
$H^1 (K_S / K ,A^\ast [p^m]) \overset{\mathrm{loc}^\mathrm{str}_A}{\longrightarrow} \overline{\mathrm{Loc}^{\mathrm{str},1}_{A^\ast [p^m]} (K)} $.
Then we have the following exact sequence by the global duality theorem:

\begin{equation}\label{equation:surjectiveity_localization}
\mathrm{Sel}^{\mathrm{BK}}_{A} (F'_{\mathrm{max}}) \longrightarrow
H^1 (F_S /F'_{\mathrm{max}},A) \overset{\overline{\mathrm{loc}^\mathrm{str}_A}}{\longrightarrow} \varinjlim_{K} \overline{\mathrm{Loc}^{\mathrm{str},1}_A (K)}
\longrightarrow  \left( \varprojlim_{K,m}  \mathrm{Sel}^{\mathrm{BK}}_{A^\ast [p^m]} (K) \right)^\vee
\end{equation}

For the last term of (<ref>), we have the following isomorphism of $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-modules:

\begin{equation}\label{equation:compactSel_discreteSel}
\left( \varprojlim_{K,m}  \mathrm{Sel}^{\mathrm{BK}}_{A^\ast [p^m]} (K) \right)^\vee
\cong \mathrm{Hom}_{\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]} ((\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}})^\vee )^\iota , \mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]])
\end{equation}

where $K$ runs through the set of finite extensions of $F'$ contained in $F'_{\mathrm{max}} $ and $(\ \ )^\iota$ denotes the twist of the structure of $\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]$ via
the involution $\iota : \mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]] \longrightarrow \mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]$ sending
$g \in \mathrm{Gal}(F'_{\mathrm{max}} /F')$ to $g^{-1}$.
The proof of (<ref>) goes in the same way as <cit.> by replacing the ring $\mathcal{R} /(H,p^n)$
by $\mathcal{O}/(\varpi^n) [\mathrm{Gal}(F'_{\mathrm{max}} /F')/U]$ where $\varpi$ is a uniformizer of $\mathcal{O}$ and $U$ is an open subgroup of
$\mathrm{Gal}(F'_{\mathrm{max}} /F')$. So we refer the reader to [32] for the proof of (<ref>).
Let us denote by $K_{\tau}$ the finite extension of $F'$ which corresponds to the kernel of the Artin representation $\tau$.
Since $F'_{\mathrm{max}}K_{\tau} (\mu_p) /F'_{\mathrm{max}}$ is a finite extension, the kernel of the restriction map

\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}}) \longrightarrow \mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}}K_{\tau} (\mu_p))

is a cofinitely generated $\mathcal{O}$-module. Hence $\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}})^\vee$ is a torsion $\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]$-module if
$\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}}K_{\tau} (\mu_p))^\vee$ is a torsion
$\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}}K_{\tau} (\mu_p) /F'K_{\tau} (\mu_p))]]$-module.
Since $F'_{\mathrm{max}}K_{\tau} (\mu_p)$ contains all $p$-power roots of unity, the action of $G_{F'_{\mathrm{max}}K_{\tau} (\mu_p)}$
on $A^\ast$ is trivial. By definition, we have

\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}}K_{\tau} (\mu_p))^\vee \cong  \mathrm{Gal}(L /F'_{\mathrm{max}}K_{\tau} (\mu_p) )

where $L$ denotes the maximal, abelian, pro-$p$-extension of $F'_{\mathrm{max}}K_{\tau} (\mu_p)$ in which all primes are unramified.
It is classically well-known that
$\mathrm{Gal}(L /F'_{\mathrm{max}}K_{\tau} (\mu_p)) $ is a torsion module over
$\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}}K_{\tau} (\mu_p) /F'_{\mathrm{max}}K_{\tau} (\mu_p))]]$.
Thus we have shown that $\mathrm{Sel}^{\mathrm{BK}}_{A^\ast } (F'_{\mathrm{max}})^\vee$ of (<ref>) is a torsion
$\mathcal{O}[[\mathrm{Gal}(F'_{\mathrm{max}} /F')]]$-module, which implies that the right-hand side of
(<ref>) is zero. Thus the map $\overline{\mathrm{loc}^\mathrm{str}_A}$ in (<ref>)
is surjective. By definition, we have a natural surjection $q: \varinjlim_{K} \overline{\mathrm{Loc}^{\mathrm{str},1}_A (K)}
\twoheadrightarrow \mathrm{Loc}^{\mathrm{str},1}_A (F'_{\mathrm{max}})$ and
${\mathrm{loc}^\mathrm{str}_A}$ is equal to the composite $q\circ \overline{\mathrm{loc}^\mathrm{str}_A}$. Thus ${\mathrm{loc}^\mathrm{str}_A}$ is also a surjection.
This completes the proof.

For any finitely generated torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module $X$, we define the
$\mu$-invariant $\mu (X)$ to be the length of the $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]_{(\varpi)}$-module
$X_{(\varpi)}$ where $\varpi$ is a uniformizer of $\mathcal{O}$ and $(\ \ )_{(\varpi)}$ denotes the localization at the prime ideal
of $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$ generated by $\varpi$.

Let $\rho' \colon G_{F} \longrightarrow V_{\rho '}$ be an Artin representation of $G_{F}$.
Let $T$ and $T'$ be $G_F$-stable lattices of $V_{\rho'}$. We denote by $A$ and $A'$ the discrete Galois representations
$T\otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$ and  $T' \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$
Then the $\mu$-invariant of $\mathrm{Sel}_A (F_\mathrm{max}) $ coincides with that of $\mathrm{Sel}_{A'} (F_\mathrm{max}) $.

For a number field $K$, a set of primes $S$ of $K$ and a discrete $G_K$-module $M$,
we define a module $\mathcyr{Sh}^i (K,S,M)$ which will be used in the proof of the above lemma as follows:

\mathcyr{Sh}^i (K,S,M) = \mathrm{Ker} \left[ H^i (K_S /K ,M) \longrightarrow \bigoplus_{w \in S} H^i (K_w ,M)\right]

where $i$ is $1$ or $2$.

By Lemma <ref>, it suffices to prove the anologous assertion for
the $\mu$-invariants of $\mathrm{Sel}^{\mathrm{str}}_A (F_\mathrm{max}) $ and
$\mathrm{Sel}^{\mathrm{str}}_{A'} (F_\mathrm{max}) $.
By replacing $T$ with an appropriate homothetic lattice, we may assume
without loss of generality that there is a $G_F$-stable surjection $A
\twoheadrightarrow A'$. Then we obtain the following fundamental diagram:

\begin{equation}\label{equation:fundamentaldiagram_mu}
\begin{CD}
0 @>>> \mathrm{Sel}^{\mathrm{str}}_A (F_{\mathrm{max}}) @>>> H^1 (F_S / F_{\mathrm{max}} ,A)  @>{\mathrm{loc}^\mathrm{str}_A}>> \mathrm{Loc}^{\mathrm{str},1}_A (F_{\mathrm{max}})  \\
@. @VVV @VV{f_A}V @VV{g_A}V \\
0 @>>> \mathrm{Sel}^{\mathrm{str}}_{A'} (F_{\mathrm{max}}) @>>> H^1 (F_S / F_{\mathrm{max}} ,A')  @>>{\mathrm{loc}^\mathrm{str}_{A'}}> \mathrm{Loc}^{\mathrm{str},1}_{A'} (F_{\mathrm{max}}) .
\\
\end{CD}
\end{equation}

The vertical maps $f_A$ and $g_A$ are induced from the surjection
$A\twoheadrightarrow A'$, and $f_A$ fits into the long exact sequence

\begin{align} \label{equation:f_A}
\begin{tikzcd}[ampersand replacement=\&]
\& H^0(F_S / F_{\mathrm{max}} ,A) \rar \ar[draw=none]{d}[name=X,anchor=center]{} \& H^0 (F_S /
F_{\mathrm{max}} ,A') \ar[rounded corners,
to path={
-- ([xshift=2ex]\tikztostart.east)
|- (X.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]{dll}{} \\
H^1 (F_S / F_{\mathrm{max}} ,C) \rar \& H^1 (F_S /
F_{\mathrm{max}} ,A) \ar{r}{f_A} \ar[draw=none]{d}[name=Y,anchor=center]{} \& H^1 (F_S / F_{\mathrm{max}} ,A') \ar[rounded corners,
to path={
-- ([xshift=2ex]\tikztostart.east)
|- (Y.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]{dll}{}\\
H^2 (F_S / F_{\mathrm{max}} ,C) \rar \& H^2 (F_S /
F_{\mathrm{max}} ,A), \&
\end{tikzcd}
\end{align}

where $C$ denotes a $G_F$-module of finite order defined as the kernel of $A \twoheadrightarrow A'$. The groups $ H^0 (F_S / F_{\mathrm{max}} ,A)$ and  $ H^0 (F_S / F_{\mathrm{max}} ,A')$ are cofinitely generated
$\mathcal{O}$-modules and their $\mathcal{O}$-coranks are both equal to $\mathrm{dim}_{\mathcal{K}}H^0 (F_S / F_{\mathrm{max}} ,V_{\rho'})$ where $\mathcal{K}$ is the
field of fractions of $\mathcal{O}$. The kernel of the natural map $H^0 (F_S / F_{\mathrm{max}} ,A) \longrightarrow  H^0 (F_S / F_{\mathrm{max}} ,A')$
is isomorphic to $H^0 (F_S / F_{\mathrm{max}} ,C)$, which is a finite group.
Hence the cokernel of the natural map $H^0 (F_S / F_{\mathrm{max}} ,A) \longrightarrow  H^0 (F_S / F_{\mathrm{max}} ,A')$ is also finite since
they are cofinitely generated $\mathcal{O}$-module of the same corank.
Thus, we have

\begin{equation}\label{equation:char_H^0}
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]} \, \mathrm{Coker}
\left[H^0 (F_S / F_{\mathrm{max}} ,A) \longrightarrow  H^0 (F_S / F_{\mathrm{max}} ,A')\right]^\vee =0 .
\end{equation}

Let us denote by $\mathcal{T}$ a free $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]^\sharp
\otimes_{\mathcal{O}} T $ with continuous
$\mathrm{Gal} (F_S / F)$-action where $ \mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]^\sharp $
denotes a free $ \mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]] $-module of rank $1$ on which $G_F$ acts via the tautological character
$G_F \twoheadrightarrow \mathrm{Gal}(F_{\mathrm{max}} /F) \hookrightarrow \mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]^\times$.
By Shapiro's lemma, we have

\begin{equation}\label{equation:shapiro_h^2_A}
H^2 (F_S / F_{\mathrm{max}} ,A) \cong H^2 (F_S / F , \mathcal{T}^\vee),
\end{equation}

$\mathcal{T}^\vee$ denotes the Pontrjagin dual of $\mathcal{T}$.
Let $x$ be any non-zero element of $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$.
Since $\mathcal{T}$ is a cofree $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module, the multiplication
by $x$ on $\mathcal{T}^\vee$ is surjective. Then the multiplication
by $x$ on $H^2 (F_S / F , \mathcal{T}^\vee)$ is surjective since the $p$-cohomological dimension of $\mathrm{Gal} (F_S / F )$ is equal to $2$, which
implies that $H^2 (F_S / F , \mathcal{T}^\vee)$ is a divisible $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
By taking the Pontrjagin dual, we see that $H^2 (F_S / F , \mathcal{T}^\vee)^\vee$ is a torsion-free $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
By (<ref>), $H^2 (F_S / F_{\mathrm{max}} ,A) ^\vee$ is a torsion-free $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
By definition, we have the exact sequence

\begin{equation}\label{equation:h^2_torsion}
\bigoplus_{w \in S_{\mathrm{max}}}
H^2 ( F_{\mathrm{max},w} ,A)^\vee \longrightarrow
H^2 (F_S / F_{\mathrm{max}} ,A)^\vee  \\
\longrightarrow
\mathcyr{Sh}^2 (F_{\mathrm{max}} ,S_{\mathrm{max}},A)^\vee
\longrightarrow 0 .
\end{equation}

Note that we have $
\displaystyle{\bigoplus_{w \in S_{\mathrm{max}}}
H^2 ( F_{\mathrm{max},w} ,A)^\vee =0 }
$ for the first module of (<ref>) since the cohomological dimension of
the absolute Galois group of $F_{\mathrm{max},w}$ is equal to $1$ for every
$w \in S_{\mathrm{max}}$ (see <cit.>).
We want to show that the third module $\mathcyr{Sh}^2 (F_{\mathrm{max}} ,S_{\mathrm{max}},A)^\vee$ of (<ref>) is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
By the global duality theorem, we can show that
$\mathcyr{Sh}^2 (F_{\mathrm{max}} ,S_{\mathrm{max}},A)^\vee$ of (<ref>) is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module
if and only if $\mathcyr{Sh}^1 (F_{\mathrm{max}} ,S_{\mathrm{max}},A^\ast)^\vee$ of (<ref>) is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module where $A^\ast$ is $\mathrm{Hom} (T ,\mu_{p^\infty})$ with $T=\varprojlim_m A[p^m] $. We refer the reader
to <cit.> for the proof of this fact.
Now we denote by $K_{\rho '}$ the finite extension of $F$ which corresponds to the kernel of the Artin representation $\rho'$.
Since $F_{\mathrm{max}}K_{\rho'} (\mu_p) /F_{\mathrm{max}}$ is a finite extension, the kernel of the restriction map

\mathcyr{Sh}^1 (F_{\mathrm{max}} ,S_{\mathrm{max}},A^\ast) \longrightarrow \mathcyr{Sh}^1 (F_{\mathrm{max}}K_{\rho '} (\mu_p) ,S_{\mathrm{max}},A^\ast)

is a cofinitely generated $\mathcal{O}$-module. Hence $\mathcyr{Sh}^1 (F_{\mathrm{max}} ,S_{\mathrm{max}},A^\ast)^\vee$ of
(<ref>) is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module if
$\mathcyr{Sh}^1 (F_{\mathrm{max}}K_{\rho '}(\mu_p) ,S_{F_{\mathrm{max}}K_{\rho '}(\mu_p)},A^\ast)^\vee$ is a torsion module over
$\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}}K_{\rho '} (\mu_p)  /FK_{\rho '} (\mu_p) )]]$.
Since $F_{\mathrm{max}}K_{\rho '}(\mu_p)$ contains all $p$-power roots of unity, the action of $G_{F_{\mathrm{max}}K_\rho (\mu_p)}$
on $A^\ast$ is trivial. By definition, we have

\mathcyr{Sh}^1 (F_{\mathrm{max}}K_{\rho '} (\mu_p) ,S_{F_{\mathrm{max}}K_{\rho '} (\mu_p)},\mathbb{Q}_p /\mathbb{Z}_p)^\vee \cong
\mathrm{Gal}(L_0 /F_{\mathrm{max}}K_{\rho '} (\mu_p) )

where $L_0$ denotes the maximal, abelian, pro-$p$-extension of $F_{\mathrm{max}}K_{\rho '} (\mu_p)$ in which all primes are unramified and the primes in
$S_{F_{\mathrm{max}}K_\rho (\mu_p)}$ split completely. It is classically well-known that
$\mathrm{Gal}(L_0 /F_{\mathrm{max}}K_{\rho '} (\mu_p) )$ is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}}K_{\rho '} (\mu_p) /F K_{\rho '} (\mu_p))]]$-module.
Thus we have shown that $\mathcyr{Sh}^2 (F_{\mathrm{max}} ,S_{\mathrm{max}},A)^\vee$ of (<ref>) is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module.
By the exact sequence (<ref>),
$H^2 (F_S / F_{\mathrm{max}} ,A)^\vee $ is a torsion $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module. By combining the fact that
$H^2 (F_S / F_{\mathrm{max}} ,A) ^\vee$ is a torsion-free $\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]$-module, we conclude that

\begin{equation}\label{equation:H^2=0}
H^2 (F_S / F_{\mathrm{max}} ,A)^\vee=0 .
\end{equation}

By (<ref>), (<ref>) and (<ref>), we have

\begin{equation}
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
(\mathrm{Ker}\,  f_A)^\vee \cong
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
H^1 (F_S / F_{\mathrm{max}} ,C)^\vee
\end{equation}

\begin{equation}
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\, (\mathrm{Coker}\, f_A )^\vee \cong
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
H^2 (F_S / F_{\mathrm{max}} ,C)^\vee .
\end{equation}

By similar reasons, we have
By (<ref>), (<ref>) and (<ref>), we have

\begin{equation}
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\, (\mathrm{Ker}\, g_A)^\vee \cong
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
\mathrm{Loc}^{\mathrm{str},1}_C (F_{\mathrm{max}}) ^\vee
\end{equation}

\begin{equation}
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
(\mathrm{Coker} \, g_A )^\vee \cong
\mathrm{Char}_{\mathcal{O}[[\mathrm{Gal}(F_{\mathrm{max}} /F)]]}\,
\mathrm{Loc}^{\mathrm{str},2}_C (F_{\mathrm{max}})^\vee .
\end{equation}

Since $\mathrm{loc}_{A}$ and $\mathrm{loc}_{A'}$ are surjective by applying Lemma <ref>
to the case $F'=F$ and $\tau =\rho'$, we have the following equality
by applying the snake lemma to the above fundamental diagram (<ref>) localized at $(\varpi)$:

\begin{multline}\label{equation:comparison_of_mu-invariants}
\mu (\mathrm{Sel}^{\mathrm{str}}_A (F_\mathrm{max}))-\mu (\mathrm{Sel}^{\mathrm{str}}_{A'} (F_\mathrm{max}))
\\ =
\mu (H^1 (F_S / F_{\mathrm{max}} ,C)) - \mu (H^2 (F_S / F_{\mathrm{max}} ,C))
- \mu (\mathrm{Loc}^{\mathrm{str},1}_C (F_{\mathrm{max}})) + \mu (\mathrm{Loc}^{\mathrm{str},2}_C (F_{\mathrm{max}})).
\end{multline}

Let us calculate $\mu (H^1 (F_S / F_{\mathrm{max}} ,C)) - \mu (H^2 (F_S / F_{\mathrm{max}} ,C)) $.
Let $K$ be an intermediate field of the extension $F_{\mathrm{max}} /F$ which is finite over $F$.
The global Euler–Poincaré characteristic formula of the Galois cohomology shows that

\begin{equation}\label{equation:local_behaviour1}
\frac{\# H^1 (F_S / K ,C)}{\# H^0 (F_S / K ,C) \# H^2 (F_S / K ,C)} =
\underset{\substack{v:\text{prime of $K$} \\  v \vert \infty}}{\prod}  \# H^0 (K_v , C) = \# C ^{\frac{[K: \mathbb{Q}]}{2}} .
\end{equation}

Since the local Euler–Poincaré characteristic formula of the Galois cohomology says that the local Euler–Poincaré characteristic
of the case of the absolute Galois group of an $\ell$-adic field acting on a module of $p$-power order with $\ell \nmid p$ is zero,
we have

\begin{equation}\label{equation:local_behaviour2a}
\frac{\# \mathrm{Loc}^{\mathrm{str},1}_{C} (K)}{\# \mathrm{Loc}^{\mathrm{str},0}_{C} (K) \# \mathrm{Loc}^{\mathrm{str},2}_{C} (K)} =
\underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod}   \frac{\#  H^1 (K_{\mathfrak{p}},C)}
{\#  H^0 (K_{\mathfrak{p}} ,C) \#  H^2 (K_{\mathfrak{p}},C)} .
\end{equation}

By the local Euler–Poincaré characteristic
of the case of the absolute Galois group of an $\ell$-adic field acting on a module of $p$-power order, we have

\begin{equation}\label{equation:local_behaviour2b}
\underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod}   \frac{\#  H^1 (K_{\mathfrak{p}},C)}
{\#  H^0 (K_{\mathfrak{p}} ,C) \#  H^2 (K_{\mathfrak{p}},C)}= \underset{\substack{\mathfrak{p}\in \Sigma^c_{p ,K}}}{\prod} \# C^{[K_{\mathfrak{p}}:\mathbb{Q}_p]}
= \# C ^{\frac{[K: \mathbb{Q}]}{2}}.
\end{equation}

By (<ref>), (<ref>) and (<ref>),
the ratio of
$\frac{\# H^1 (F_S / K ,C)}{ \# H^2 (F_S / K ,C)}$ and $\frac{\# \mathrm{Loc}^{\mathrm{str},1}_{C} (K)}{ \# \mathrm{Loc}^{\mathrm{str},2}_{C} (K)}$ is bounded
when $K$ runs through
the set of intermediate fields of the extension $F_{\mathrm{max}} /F$ which are finite over $F$.
Thus the right-hand side of (<ref>) is zero and this completes the proof of the lemma.

On the analytic side, we recall that we defined $L_p(M(\rho ))$ as in (<ref>).
On the algebraic side, we recall that we chose a $G_{F}$-stable $\mathcal{O}$-lattice $T$ of the Artin representation $V_\rho$ and set $A =T \otimes_{\mathbb{Z}_p} \mathbb{Q}_p /\mathbb{Z}_p$.
Let $F_i$ and $\psi_i$ be as in the Brauer decomposition given in (<ref>).
We claim that

\begin{equation}\label{equation:Selmer_by_Brauer_induction}
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F )]]}\mathrm{Sel}_{A}(F_\mathrm{max})^\vee
\prod^s_{i=1} \mathrm{incl}_i \left(
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} (F_i F_\mathrm{max} /F_i)]]}\mathrm{Sel}_{A_{\psi_i}}(F_i F_\mathrm{max})^\vee
\right)^{a_i}
\end{equation}

By Shapiro's lemma, the right-hand side of (<ref>) is equal to

\prod^s_{i=1} \left(
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F)]]}\mathrm{Sel}_{\mathrm{Ind}^F _{F_i}A_{\psi_i}}( F_\mathrm{max})^\vee
\right)^{a_i}.

Hence it suffices to check that we have

\begin{equation}\label{equation:Selmer_by_Brauer_induction2}
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F )]]}\mathrm{Sel}_{A}(F_\mathrm{max})^\vee
\\ =
\prod^s_{i=1} \left(
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F)]]}\mathrm{Sel}_{\mathrm{Ind}^F _{F_i}A_{\psi_i}}( F_\mathrm{max})^\vee
\right)^{a_i}.
\end{equation}

We note that some of $a_i$ can be negative. Without loss of generality, we may assume that there exists a positive integer $s' \leq s$ such that
$a_1 ,\ldots , a_{s'}$ are positive and $a_{s'+1} ,\ldots , a_{s}$ are negative.
Then the equality (<ref>) is equivalent to the following equality:

\begin{multline}\label{equation:Selmer_by_Brauer_induction3}
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F )]]}\mathrm{Sel}_{A \oplus \bigoplus^{s}_{i=s'+1} \mathrm{Ind}^F _{F_i}A_{\psi_i}^{\oplus (-a_i)}}(F_\mathrm{max})^\vee
\\ =
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} ( F_\mathrm{max} /F)]]}\mathrm{Sel}_{ \bigoplus^{s'}_{i=1} \mathrm{Ind}^F _{F_i}A_{\psi_i}^{\oplus a_i}}( F_\mathrm{max})^\vee .
\end{multline}

Let $T $ be a $G_F$-stable lattices of $V_\rho$ such that $A=T\otimes \mathbb{Q}_p /\mathbb{Z}_p$.
For each $i$, we have a $G_{F_i}$-stable lattice $T_{\psi_i}$ of $V_{\psi_i}$ such that $A_{\psi_i}=T_{\psi_i} $.
Let $V_{\rho'} = V_\rho \oplus \bigoplus^{s}_{i=s'+1} \mathrm{Ind}^F _{F_i}V_{\psi_i}^{\oplus (-a_i)} $.
Through the isomorphism $V_{\rho' } \cong \bigoplus^{s'}_{i=1} \mathrm{Ind}^F _{F_i}V_{\psi_i}^{\oplus a_i}$,
T \oplus \bigoplus^{s}_{i=s'+1} \mathrm{Ind}^F _{F_i}T_{\psi_i}^{\oplus (-a_i)}
and $
T \oplus \bigoplus^{s'}_{i=1} \mathrm{Ind}^F _{F_i}T_{\psi_i}^{\oplus a_i}
are both $G_F$-stable lattices of $V_{\rho'}$. Hence (<ref>) is true modulo $\mu$-invariant.
By applying Lemma <ref> to $V_\rho$, we also see that the equality is true without the ambiguity of the $\mu$-ivvariant.
Thus we proved that (<ref>)
is true, so is (<ref>).
For each $i$, our $p$-adic $L$-function $L_{p,\Sigma_F}(\psi_i)$ defined in (<ref>) satisfies the Iwasawa main conjecture

(L_{p,\Sigma_{F_i}}(\psi_i)) = \mathrm{char}_{\widehat{\mathcal{O}}^{\mathrm{ur}} [[\mathrm{Gal} (F_{i,\mathrm{max}} /F_i)]]}
\mathrm{Sel}_{A_{\psi_i}}(F_{i ,\mathrm{max}})^\vee \otimes_{\mathcal{O}} \widehat{\mathcal{O}}^{\mathrm{ur}}

by the assumption of the validity of
the CM Iwasawa main conjecture (Conjecture <ref> for $K=F_i$ and $\psi =\psi_i$).
By the definition of $L_{p,\Sigma_F}(M(\rho ))$ given in (<ref>), we have

\begin{equation}\label{equation:equality using L_p(M(rho))}
(L_{p,\Sigma_F}(M(\rho ))) = \prod^s_{i=1} \mathrm{incl}_i \circ \mathrm{pr}_i
\left(
\mathrm{char}_{\widehat{\mathcal{O}}^{\mathrm{ur}} [[\mathrm{Gal} (F_{i,\mathrm{max}} /F_i)]]}
\mathrm{Sel}_{A_{\psi_i}}(F_{i ,\mathrm{max}})^\vee
\right)^{a_i}
\otimes_{\mathcal{O}} \widehat{\mathcal{O}}^{\mathrm{ur}}
\end{equation}

in $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\mathrm{Gal}(F_{i,\mathrm{max}}/F_i)]]$.
Thus we need to prove the descent property

\begin{equation}\label{equation:selmer_descent}
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} (F_i F_\mathrm{max} /F_i)]]}\mathrm{Sel}_{A_{\psi_i}}(F_i F_\mathrm{max})^\vee
\mathrm{pr}_i \left( \mathrm{char}_{\mathcal{O}[[\mathrm{Gal} (F_{i,\mathrm{max}} /F_i)]]} \mathrm{Sel}_{A_{\psi_i}}(F_{i ,\mathrm{max}})^\vee
\right)
\end{equation}

for each $i$.
In fact, if (<ref>) holds true,
we obtain the main equality (<ref>) of Conjecture <ref>
by combining with (<ref>) and (<ref>).

In the rest of the proof, we need to show (<ref>). In place of proving the descent property directly for
the Selmer group, we will show the descent property for the strict Selmer group.
Recall that the Pontrjagin duals of
$\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i}}(F_i F_\mathrm{max}) \hookrightarrow \mathrm{Sel}_{A_{\psi_i}}(F_i F_\mathrm{max})$
and $\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i}}(F_{i,\mathrm{max}}) \hookrightarrow \mathrm{Sel}_{A_{\psi_i}}(F_{i,\mathrm{max}})$
are pseudo-isomorphisms by Lemma <ref>.
Hence it is sufficient to prove the descent argument

\begin{align}\label{equation:selmer_str_descent}
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} (F_i F_\mathrm{max} /F_i)]]}\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i}}(F_i F_\mathrm{max})^\vee
\mathrm{pr}_i \left(
\mathrm{char}_{\mathcal{O}[[\mathrm{Gal} (F_{i,\mathrm{max}} /F_i)]]}
\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i}}(F_{i ,\mathrm{max}})^\vee \right)
\end{align}

for each $i$.

By construction, the Galois group of $F_{i ,\mathrm{max}} /F_i F_{\mathrm{max}}$ is a free $\mathbb{Z}_p$-module, whose $\mathbb{Z}_p$-rank $r_i$ is calculated as $d( d_i -1) + \delta_{F_i} -\delta_F$; here $\delta_F $ and $\delta_{F_i}$ respectively denote the $p$-Leopoldt defects of $F$ and $F_i$. Note that $\delta_{F_i}\geq \delta_F$ holds because $F_i$ is an extension of $F$.
Let us take a $\mathbb{Z}_p$-basis $\gamma_1 , \dotsc , \gamma_{r_i}$ of $\mathrm{Gal}(F_{i ,\mathrm{max}} /F_i F_{\mathrm{max}})$, and set $x_j=\gamma_j-1$ for $j=1,\dotsc,r_i$. Then $x_1,x_2,\dotsc,x_{r_i}$ forms a regular sequence of $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i )]]$ satisfying $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/(x_1 , \ldots , x_{r_i}) \cong \mathcal{O}[[\mathrm{Gal} (F_i F_{\mathrm{max}} /F_i )]] $. For each $j=1,\dotsc,r_i$, let $\mathfrak{A}_j$ denote the ideal of $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i )]]$ generated by $x_1 ,\dotsc, x_j$.

Let $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp$ be
a free $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]$-module of rank one on which $\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i)$ acts
by the tautological character $\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) \hookrightarrow \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\times$.
Let $\mathcal{T}_i $ denote the tensor product $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}$ on which $G_{F_i} $ acts diagonally.
We define $\mathcal{A}_i $ to be the discrete Galois module $\mathcal{A}_i = \mathcal{T}_i \otimes_{\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]} \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\vee$.
By Shapiro's lemma, $\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i}} (F_{i ,\mathrm{max}})$ is isomorphic to
the Selmer group $\mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i} (F_i )$
which is defined by

\begin{align*}
\mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i} (F_i )= \mathrm{Ker}
\left[
H^1 ((F_i )_{S_{F_i}}/F_i ,\mathcal{A}_i ) \longrightarrow
\underset{\substack{\lambda \in S_{F_i} \\  \lambda \nmid p\infty}}{\prod}
H^1 (F_{i, \lambda}^{\mathrm{ur}} ,\mathcal{A}_i)
\times \underset{\substack{\mathfrak{p}\in \Sigma^c_{p,F_i}}}{\prod}
H^1 (D_{\mathfrak{p}} ,\mathcal{A}_i)
\right].
\end{align*}

For each $j=1,\dotsc,r_i$, let us define $\mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_j ]} (F_i )$ in a similar manner. Then $\mathrm{Sel}^{\mathrm{str}}_{A_{\psi_i} }(F_i F_\mathrm{max})$ is isomorphic to
$\mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_{r_i} ]} (F_i )$ by Shapiro's lemma again.
To prove (<ref>), it suffices to show that we can choose the regular sequence $x_1,\dotsc,x_{r_i}$
of $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]$ so that the following two statements are fulfilled for each $j$;

* (control theorem)
the kernel and the cokernel of the map

\begin{align*}
s_j\colon \mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_j ]} (F_i ) \longrightarrow \mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_{j-1} ]} (F_i )
\end{align*}

induced by the natural inclusion $\mathcal{A}_i[\mathfrak{A}_{j-1}]\hookrightarrow \mathcal{A}_i[\mathfrak{A}_j]$ are both copseudonull (as $\mathcal{O}[[\mathrm{Gal}
(F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_j$-modules) where $\mathfrak{A}_j = (x_1,\dotsc,x_j ) $;

* the $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_j$-module
$\mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_j ]} (F_i )^\vee$ has no nontrivial pseudonull submodules.

First, we will show that the assertion (1) holds true for any regular sequence $x_1,\dotsc,x_{r_i}$
of $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]$ (not necessarily of the form $x_j =\gamma_j -1$ as above). We consider the commutative diagram

\begin{align*}
\begin{CD}
0 @>>> \mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_j ]}(F_i ) @>>> H^1 ((F_i )_{S_{F_i}}/F_i , \mathcal{A}_i [\mathfrak{A}_j ] )  @>>>
\mathrm{Loc}^{\mathrm{str},1}_{\mathcal{A}_i [\mathfrak{A}_j ]} (F_i )
\\
@. @V{s_j}VV @VV{t_j}V @VV{u_j}V \\
0 @>>> \mathrm{Sel}^{\mathrm{str}}_{\mathcal{A}_i [\mathfrak{A}_{j-1} ]}(F_i )[x_j ] @>>> H^1 ((F_i )_{S_{F_i}}/F_i , \mathcal{A}_i [\mathfrak{A}_{j-1} ] )[x_j]
@>>> \mathrm{Loc}^{\mathrm{str},1}_{\mathcal{A}_i [\mathfrak{A}_{j-1} ]} (F_i )[x_j ]
\\
\end{CD}
\end{align*}

where $\mathrm{Loc}^{\mathrm{str},1}_{\mathcal{A}_i [\mathfrak{A}_j ]} (F_i )$ denotes $\underset{\substack{\lambda \in S_{F_i} \\  \lambda \nmid p\infty}}{\prod}
H^1 (F_{i, \lambda}^{\mathrm{ur}} ,\mathcal{A}_i [\mathfrak{A}_j ])
\times \underset{\substack{\mathfrak{p}\in \Sigma^c_{p,F_i}}}{\prod}
H^1 (D_{\mathfrak{p}} ,\mathcal{A}_i [\mathfrak{A}_j ]) $. All of the vertical maps are induced by the natural inclusion $\mathcal{A}_i[\mathfrak{A}_{j}]\hookrightarrow \mathcal{A}_i[\mathfrak{A}_{j-1}]$.
By the above commutative diagram, the kernel of $s_j$ is a subgroup of the kernel of $t_j$.
Also, since $t_j$ is surjective due to the cohomological long exact sequence associated to the short exact sequence

\begin{align*}
\xymatrix{0\ar[r] & \mathcal{A}_i[\mathfrak{A}_j] \ar[r] & \mathcal{A}_i[\mathfrak{A}_{j-1}]\ar[r]^-{x_j} & \mathcal{A}_i[\mathfrak{A}_{j-1}] \ar[r] & 0},
\end{align*}

the cokernel of $s_j$ is a subquotient of the kernel of $u_j$ by the snake lemma.
First, we show that the kernel of $t_j$ is a pseudo-null $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_{j}$-module.
The kernel of $t_j$ is isomorphic to
$H^0  ((F_i )_{S_{F_i}}/F_i , \mathcal{A}_i [\mathfrak{A}_{j-1} ] )/x_j H^0  ((F_i )_{S_{F_i}}/F_i , \mathcal{A}_i [\mathfrak{A}_{j-1} ] )$.
This module is $x_j$-torsion subgroup of the Pontrjagin dual of the following module

\begin{equation}\label{equation:kernel:controltheorem}
\left(
\left. \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}\right/
\mathfrak{A}_{j-1} \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}
\right)_{{\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) }}
\end{equation}

where $(\ )_{{\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) }}$ denotes the largest ${{\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i)}}$-coinvariant quotient.
Let $U$ be a sufficiently small open subgroup of ${\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) }$ which acts trivially on
$T_{\psi_i}$ and $F_U$ the finite extension $(F_{i ,\mathrm{max}})^U$ of $F_i$. Then the module given in (<ref>)
is a quotient of $ \mathcal{O}[[\mathrm{Gal} (F_U /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}$, which is a finitely generated
$\mathcal{O}$-module. Hence the $x_j$-torsion subgroup of the module given in (<ref>) is also
a finitely generated $\mathcal{O}$-module.
Since the Krull dimension of $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_{j}$
is greater than $2$, every $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_{j}$-module which is finitely generated $\mathcal{O}$-module
is pseudo-null.
Next, we show that the kernel of $u_j$ is a pseudo-null $\mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]/\mathfrak{A}_{j}$-module.
The kernel of $u_j$ is isomorphic to $x_j$-torsion part of the following modules

\begin{multline}\label{equation:cokernel:controltheorem}
\underset{\substack{\lambda \in S_{F_i} \\  \lambda \nmid p\infty}}{\prod}
\left(
\left. \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}\right/
\mathfrak{A}_{j-1} \mathcal{O}[[\mathrm{Gal} (F_{i ,\mathrm{max}} /F_i) ]]^\sharp \otimes_{\mathcal{O}} T_{\psi_i}
\right)_{I_\lambda}
\\
\oplus