$p$-adic measures associated with zeta values and $p$-adic $\log$ multiple gamma functions

TL;DR

This paper explores the relationship between two $p$-adic symbols related to Gross-Stark units, connecting $p$-adic gamma functions and multiplicative integration to deepen understanding of $p$-adic $L$-values and units.

Contribution

It establishes an explicit relation between Yoshida's $Y_p( au)$ symbol and Dasgupta's $u_T( au)$ symbol, linking two approaches to Gross-Stark units.

Findings

Derived an explicit formula relating $Y_p( au)$ and $u_T( au)$

Connected $p$-adic gamma functions with multiplicative integration methods

Enhanced understanding of $p$-adic units in abelian extensions

Abstract

We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $\tau \in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(\tau)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(\tau)$. Dasgupta constructed the symbol $u_T(\tau)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by $u_T(\tau)$. In this paper, we give an explicit relation between $Y_p(\tau)$ and $u_T(\tau)$.

Equations

$${\zeta }_S(s,\tau ):=\sum _{(\frac{K/F}{\mathfrak{a}})=\tau ,(\mathfrak{a},S)=1}N{\mathfrak{a}}^{-s}.$$

$$\log |u^{\tau }{|}_{\mathfrak{P}}=-W{\zeta }_S^{\prime }(0,\tau )(\tau \in \mathrm{Gal}(K/F)).$$

$${\log }_p{N}_{K_{\mathfrak{P}}/{\mathbb{Q}}_p}(u^{\tau })=-W{\zeta }_{p,S}(0,\tau ).$$

$${\log }_p(u_{\eta }(\mathfrak{b},{\mathcal{D}}_{\mathfrak{f}}))=-Y_p((\frac{H/F}{\mathfrak{b}}))+N\eta Y_p((\frac{H/F}{\mathfrak{b}{\eta }^{-1}})).$$

$$E_{\mathfrak{f},+}:=\{ϵ\in E_{+}\mid ϵ\equiv 1\mathrm{mod}\mathfrak{f}\}.$$

$$F\otimes \mathbb{R}={\mathbb{R}}^n,\sum _{i=1}^k{a}_i\otimes b_i\mapsto (\sum _{i=1}^k\iota (a_i)b_i{)}_{\iota \in \mathrm{Hom}(F,\mathbb{R})},$$

$$F\otimes {\mathbb{R}}_{+}:={\mathbb{R}}_{+}^n$$

$$C(\mathbf{v}):=\{\mathbf{t}{}^t\mathbf{v}\in F\otimes \mathbb{R}\mid \mathbf{t}\in {\mathbb{R}}_{+}^r\}.$$

$$D=\coprod _{i\in J}C({\mathbf{v}}_j)(|J|<\infty ,{\mathbf{v}}_j\in {\mathcal{O}}_{F,+}^{r(j)},r(j)\in \mathbb{N}).$$

$$F\otimes {\mathbb{R}}_{+}=\coprod _{ϵ\in E_{\mathfrak{f},+}}ϵD.$$

$$\zeta (s,\mathbf{v},z):=\sum _{\mathbf{m}\in {\mathbb{Z}}_{\ge 0}^r}(z+\mathbf{m}{}^t\mathbf{v}{)}^{-s}.$$

$$\Gamma (z,\mathbf{v}):=\exp \left(\frac{\partial }{\partial s}\zeta (s,\mathbf{v},z){|}_{s=0}\right).$$

$$|p^{-{\mathrm{ord}}_p(z)}{\theta }_p(z{)}^{-1}z{|}_p<1(z\in {\mathbb{C}}_p^{\times }).$$

$$z\in {\mathbb{R}}_{+},\mathbf{v}\in {\mathbb{R}}_{+}^r\text{ via the embedding }\overline{\mathbb{Q}}\hookrightarrow \mathbb{C},$$

$${\mathrm{ord}}_p(z)<{\mathrm{ord}}_p(v_1),\dots ,{\mathrm{ord}}_p(v_r)\text{ via the embedding }\overline{\mathbb{Q}}\hookrightarrow {\mathbb{C}}_p.$$

$${\zeta }_p(-m,\mathbf{v},z)=p^{-{\mathrm{ord}}_p(z)m}{\theta }_p(z{)}^{-m}\zeta (-m,\mathbf{v},z)(m\in {\mathbb{Z}}_{\ge 0}).$$

$$L{\Gamma }_p(z,\mathbf{v}):=\frac{\partial }{\partial s}{\zeta }_p(s,\mathbf{v},z){|}_{s=0}.$$

$$\mathrm{Hom}(F,\mathbb{R})=\mathrm{Hom}(F,{\mathbb{C}}_p).$$

$$C_{\mathfrak{f}}\cong \mathrm{Gal}(H_{\mathfrak{f}}/F).$$

$${\mathfrak{a}}_c\mathfrak{f}\in \pi (c).$$

$$R(c,\mathbf{v}):=R(c,\mathbf{v},{\mathfrak{a}}_c):=\{\mathbf{x}\in (\mathbb{Q}\cap (0,1]{)}^r\mid {\mathcal{O}}_F\supset (\mathbf{x}{}^t\mathbf{v}){\mathfrak{a}}_c\mathfrak{f}\in c\}.$$

$$G(c,\iota ):=G(c,\iota ,D,{\mathfrak{a}}_c):=\sum _{j\in J}\sum _{\mathbf{x}\in R(c,{\mathbf{v}}_j)}\log \Gamma (\iota (\mathbf{x}{}^t{\mathbf{v}}_j),\iota ({\mathbf{v}}_j)).$$

$${\mathfrak{p}}_{\iota }:=\{z\in {\mathcal{O}}_F\mid |\iota (z){|}_p<1\}.$$

$$G_p(c,\iota ):=G_p(c,\iota ,D,{\mathfrak{a}}_c):=\sum _{j\in J}\sum _{\mathbf{x}\in R(c,{\mathbf{v}}_j)}L{\Gamma }_p(\iota (\mathbf{x}{}^t{\mathbf{v}}_j),\iota ({\mathbf{v}}_j)).$$

$$[{]}_p:\overline{\mathbb{Q}}\log {\overline{\mathbb{Q}}}^{\times }\to \overline{\mathbb{Q}}{\log }_p{\overline{\mathbb{Q}}}^{\times },a\log b\mapsto a{\log }_{p}b,a\pi \mapsto 0(a,b\in \overline{\mathbb{Q}},b\ne 0).$$

$$\sum _{c\in C_{\mathfrak{fq}},\mathrm{Art}(\overline{c}){|}_H=\tau }G(c,\iota )\in \overline{\mathbb{Q}}\log {\overline{\mathbb{Q}}}^{\times }(\tau \in \mathrm{Gal}(H/F)).$$

$$\sum _{c\in C_{\mathfrak{fq}}}{\chi }_{\mathfrak{q}}(c)X(c,\iota )\in \overline{\mathbb{Q}}\log {\overline{\mathbb{Q}}}^{\times }(\chi \in {\hat{C}}_{\mathfrak{f}},\chi ([\mathfrak{q}])=1).$$

$$Y_p(\tau ,\iota ):=\sum _{c\in C_{\mathfrak{f}{\mathfrak{p}}_{\iota }},\mathrm{Art}(\overline{c}){|}_H=\tau }{G}_p(c,\iota )-[\sum _{c\in C_{\mathfrak{f}{\mathfrak{p}}_{\iota }},\mathrm{Art}(\overline{c}){|}_H=\tau }G(c,\iota ){]}_p(\tau \in \mathrm{Gal}(H/F)).$$

$$Y_p(\tau ,\iota )=\frac{-1}{h_H}\sum _{c\in C_{\mathfrak{f}}}\zeta (0,c^{-1}){\log }_p\tilde{\iota }\left({\alpha }_{H,\tilde{\iota }}^{\tau \mathrm{Art}(c)}\right).$$

$$R=\coprod _{i=1}^k\{({\mathbf{x}}_i+\mathbf{m}){}^t{\mathbf{v}}_i\mid \mathbf{m}\in {\mathbb{Z}}_{\ge 0}^{r_i}\}({\mathbf{x}}_i\in {\mathbb{Q}}_{+}^{r_i},{\mathbf{v}}_i\in F_{+}^{r_i}).$$

$${\zeta }_{\iota }(s,R)$$

Full text

1 Introduction

Let $F$ be a totally real field, $K$ a CM-field which is abelian over $F$, $S$ a finite set of places of $F$. We assume that

• •

$S$ contains all infinite places of $F$, all places of $F$ lying above a rational prime $p$, and all ramified places in $K/F$.

• •

Let $\mathfrak{p}$ be the prime ideal corresponding to the $p$-adic topology on $F$. (Hence $\mathfrak{p}\in S$.) Then $\mathfrak{p}$ splits completely in $K/F$.

For $\tau\in\mathrm{Gal}(K/F)$, we consider the partial zeta function

[TABLE]

Here $\mathfrak{a}$ runs over all integral ideals of $F$, relatively prime to any finite places in $S$, whose image under the Artin symbol $(\frac{K/F}{*})$ is equal to $\tau$. The series converges for $\mathrm{Re}(s)>1$, has a meromorphic continuation to the whole $s$-plane, and is analytic at $s=0$. Moreover, under our assumption, we see that

• •

There exists the $p$-adic interpolation function $\zeta_{p,S}(s,\tau)$ of $\zeta_{S}(s,\tau)$.

• •

$\mathrm{ord}_{s=0}\zeta_{S}(s,\tau),\mathrm{ord}_{s=0}\zeta_{p,S}(s,\tau)\geq 1$.

• •

There exist a natural number $W$, a $\mathfrak{p}$-unit $u$ of $K$, which satisfy

[TABLE]

Here $\mathfrak{P}$ denotes the prime ideal corresponding to the $p$-adic topology on $K$, $|x|_{\mathfrak{P}}:=N\mathfrak{P}^{-\mathrm{ord}_{\mathfrak{P}}x}$.

Gross conjectured the following $p$-adic analogue of the rank $1$ abelian Stark conjecture:

Conjecture 1** ([Gr, Conjecture 3.13]).**

Let $u$ be a $\mathfrak{p}$-unit characterized by (1) up to roots of unity. Then we have

[TABLE]

Dasgupta-Darmon-Pollack [DDP] proved a large part of Conjecture 1. Yoshida and the author, and independently Dasgupta formulated refinements of Conjecture 1: Let $\mathfrak{f}$ be an integral ideal of a totally real field $F$ satisfying $\mathfrak{p}\nmid\mathfrak{f}$, $H_{\mathfrak{f}}$ the narrow ray class field modulo $\mathfrak{f}$, $H$ the maximal subfield of $H_{\mathfrak{f}}$ where $\mathfrak{p}$ splits completely. Yoshida and the author [KY1] essentially constructed the invariant $Y_{p}(\tau)$ (Definition 6) for $\tau\in\mathrm{Gal}(H/F)$ by using $p$-adic $\log$ multiple gamma functions. Then [KY1, Conjecture A*′*] states that $\log_{p}u^{\tau}$ (without $N_{K_{\mathfrak{P}}/\mathbb{Q}_{p}}$) can be expressed by $Y_{p}(\tau)$. On the other hand, Dasgupta constructed the invariant $u_{T}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ (Definition 12-(iv)) by using the multiplicative integration for $p$-adic measures associated with Shintani’s multiple zeta functions. Then [Da, Conjecture 3.21] states that a modified version of $u^{\tau}$ can be expressed by $u_{T}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$. In [Ka3, Remark 2], the author announced the following relation between these refinements.

Theorem** (Theorem 1).**

Let $\eta$ be a “good” prime ideal in the sense of Definition 11. We put $T:=\{\eta\}$. Then we have

[TABLE]

In particular, we see that two refinements are consistent (roughly speaking, [Da, Conjecture 3.21] is a further refinement of [KY1, Conjecture A*′*] by $\ker\log_{p}$). The aim of this paper is to prove this Theorem.

Let us explain the outline of this paper. In §2, we introduce Shintani’s technique of cone decompositions. We obtain a suitable fundamental domain of $F\otimes\mathbb{R}_{+}/E_{\mathfrak{f},+}$, where $F\otimes\mathbb{R}_{+}$ denotes the totally positive part of $F\otimes\mathbb{R}$, $E_{\mathfrak{f},+}$ is a subgroup of the group of all totally positive units. We need such fundamental domains in order to construct both of the invariants $Y_{p}$, $u_{T}$. In §3, we recall the definition and some properties of $Y_{p}$, which is essentially defined in [KY1] and slightly modified in [Ka3]. The classical or $p$-adic $\log$ multiple gamma function is defined as the derivative values at $a=0$ of the classical or $p$-adic Barnes’ multiple zeta function, respectively. Then the invariant $Y_{p}(\tau,\iota)$ is defined in Definition 6, as a finite sum of the “difference” of $p$-adic $\log$ multiple gamma functions and classical $\log$ multiple gamma functions. Conjecture 2 predicts exact values of $Y_{p}(\tau,\iota)$. In §4, we also recall some results in [Da]. Dasgupta introduced $p$-adic measures $\nu_{T}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ associated with special values of Shintani’s multiple zeta functions, and defined $u_{T}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ as the multiplicative integration $\times\int_{\mathbf{O}}x\ d\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}},x)$ with certain correction terms. Dasgupta formulated a conjecture (Conjecture 3) on properties of $u_{T}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$. In §5, we state and prove the main result (Theorem 1) which gives an explicit relation between $Y_{p}(\tau,\mathrm{id})$ and $\log_{p}(u_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}}))$. Then we will see that Conjectures 2, 3 are consistent in the sense of Corollary 1. The key observation is Lemma 3: Dasgupta’s $p$-adic measure $\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ is originally associated with Shintani’s multiple zeta functions. By this Lemma, we can relate $\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ to Barnes’ multiple zeta functions and $p$-adic analogues as in Lemma 4.

2 Shintani domains

Let $F$ be a totally real field of degree $n$, $\mathcal{O}_{F}$ the ring of integers of $F$, $\mathfrak{f}$ an integral ideal of $F$. We denote by $F_{+}$ the set of all totally positive elements in $F$ and put $\mathcal{O}_{F,+}:=\mathcal{O}_{F}\cap F_{+}$, $E_{+}:=\mathcal{O}_{F}^{\times}\cap F_{+}$. We consider subgroups of $E_{+}$ of the following form:

[TABLE]

We identify

[TABLE]

where $\mathrm{Hom}(F,\mathbb{R})$ denotes the set of all real embeddings of $F$. In particular, the totally positive part

[TABLE]

has a meaning. On the right-hand side, $\mathbb{R}_{+}$ denotes the set of all positive real numbers. Let $v_{1},\dots,v_{r}\in\mathcal{O}_{F}$ be linearly independent. Then we define the cone with basis $\bm{v}:=(v_{1},\dots,v_{r})$ as

[TABLE]

Here we $\bm{t}\,{}^{t}\bm{v}$ denotes the inner product.

Definition 1**.**

1. (i)

We call a subset $D\subset F\otimes\mathbb{R}_{+}$ is a Shintani set if it can be expressed as a finite disjoint union of cones:

[TABLE] 2. (ii)

We consider the natural action $E_{\mathfrak{f},+}\curvearrowright F\otimes\mathbb{R}_{+}$, $u(a\otimes b):=(ua)\otimes b$. We call a Shintani set $D$ a Shintani domain $\bmod E_{\mathfrak{f},+}$ if it is a fundamental domain of $F\otimes\mathbb{R}_{+}/E_{\mathfrak{f},+}$:

[TABLE]

When $\mathfrak{f}=(1)$, we write $\bmod E_{+}$ instead of $\bmod E_{(1),+}$.

Shintani [Sh, Proposition 4] showed that there always exists a Shintani domain.

3 $p$-adic $\log$ multiple gamma functions

We recall the definition and some properties of the symbol $Y_{p}$ defined in [KY1], [Ka3]. We denote by $\mathbb{R}_{+}$ the set of all positive real numbers.

Definition 2**.**

Let $z\in\mathbb{R}_{+}$, $\bm{v}\in\mathbb{R}_{+}^{r}$. Barnes’ multiple zeta function is defined as

[TABLE]

This series converges for $\mathrm{Re}(s)>r$, has a meromorphic continuation to the whole $s$-plane, and is analytic at $s=0$. Then Barnes’ multiple gamma function is defined as

[TABLE]

Note that this definition is modified from that given by Barnes. For the proof and details, see [Yo, Chap I, §1]. Throughout this paper, we regard each number field as a subfield of $\overline{\mathbb{Q}}$, and fix two embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$. Here $\mathbb{C}_{p}$ denotes the $p$-adic completion of the algebraic closure of $\mathbb{Q}_{p}$. We denote by $\mu_{(p)}$ the group of all roots of unity of prime-to-$p$ order. Let $\mathrm{ord}_{p}\colon\mathbb{C}_{p}^{\times}\rightarrow\mathbb{Q}$, $\theta_{p}\colon\mathbb{C}_{p}^{\times}\rightarrow\mu_{(p)}$ be unique group homomorphisms satisfying

[TABLE]

Definition 3**.**

Let $z\in\overline{\mathbb{Q}}$, $\bm{v}\in(\overline{\mathbb{Q}}^{\times})^{r}$. We assume that

[TABLE]

Then we denote by $\zeta_{p}(s,\bm{v},z)$ $(s\in\mathbb{Z}_{p}-\{1,2,\dots,r\})$ the $p$-adic multiple zeta function characterized by

[TABLE]

We define the $p$-adic $\log$ multiple gamma function as

[TABLE]

The construction of $\zeta_{p}(s,\bm{v},z)$ is due to Cassou-Noguès [CN1]. The author defined and studied $L\Gamma_{p}(z,\bm{v})$ in [Ka1]. See [Ka3, §2] for a short survey.

Definition 4**.**

Let $F$ be a totally real field, $\mathfrak{f}$ an integral ideal, $D=\coprod_{j\in J}C(\bm{v}_{j})$ $(\bm{v}_{j}\in\mathcal{O}_{F,+}^{r(j)})$ a Shintani domain $\bmod E_{+}$. We denote by $\mathrm{Hom}(F,\mathbb{R})$ (resp. $\mathrm{Hom}(F,\mathbb{C}_{p})$) the set of all embeddings of $F$ into $\mathbb{R}$ (resp. $\mathbb{C}_{p}$). Since we fixed embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$, we may identify

[TABLE]

1. (i)

We denote by $C_{\mathfrak{f}}$ the narrow ideal class group modulo $\mathfrak{f}$, by $H_{\mathfrak{f}}$ the narrow ray class field modulo $\mathfrak{f}$. In particular, the Artin map induces

[TABLE] 2. (ii)

Let $\pi\colon C_{\mathfrak{f}}\rightarrow C_{(1)}$ be the natural projection. For each $c\in C_{\mathfrak{f}}$, we take an integral ideal $\mathfrak{a}_{c}$ satisfying

[TABLE] 3. (iii)

For $c\in C_{\mathfrak{f}}$, $\bm{v}\in\mathcal{O}_{F}^{r}$, we put

[TABLE] 4. (iv)

For $c\in C_{\mathfrak{f}}$, $\iota\in\mathrm{Hom}(F,\mathbb{R})$, we define

[TABLE] 5. (v)

For $\iota\in\mathrm{Hom}(F,\mathbb{R})$ $(=\mathrm{Hom}(F,\mathbb{C}_{p}))$, we put

[TABLE]

Note that the prime ideal $\iota(\mathfrak{p}_{\iota})$ corresponds to the $p$-adic topology on $\iota(F)\subset\mathbb{C}_{p}$. 6. (vi)

Assume that $\mathfrak{p}_{\iota}\mid\mathfrak{f}$. For $c\in C_{\mathfrak{f}}$, $\iota\in\mathrm{Hom}(F,\mathbb{R})$, we define

[TABLE]

Note that $(\iota(\bm{x}\,{}^{t}\bm{v}_{j}),\iota(\bm{v}_{j}))$ satisfies the assumption (3) whenever $\mathfrak{p}_{\iota}\mid\mathfrak{f}$, $\bm{x}\in R(c,\bm{v}_{j})$.

The following map $[\ ]_{p}$ is well-defined by [KY1, Lemma 5.1].

Definition 5**.**

We denote by $\overline{\mathbb{Q}}\log_{p}\overline{\mathbb{Q}}^{\times}$ (resp. $\overline{\mathbb{Q}}\log\overline{\mathbb{Q}}^{\times}$) the $\overline{\mathbb{Q}}$-subspace of $\mathbb{C}_{p}$ (resp. $\mathbb{C}$) generated by $\log_{p}b$ (resp. $\pi$, $\log b$) with $b\in\overline{\mathbb{Q}}^{\times}$. We define a $\overline{\mathbb{Q}}$-linear map $[\ ]_{p}$ by

[TABLE]

Lemma 1**.**

Let $H$ be an intermediate field of $H_{\mathfrak{f}}/F$, $\mathfrak{q}$ a prime ideal of $F$, relatively prime to $\mathfrak{f}$, splitting completely in $H/F$. Then we have

[TABLE]

Here $c$ runs over all ideal classes whose images under the composite map $C_{\mathfrak{f}\mathfrak{q}}\rightarrow C_{\mathfrak{f}}\rightarrow\mathrm{Gal}(H_{\mathfrak{f}}/F)\rightarrow\mathrm{Gal}(H/F)$ is equal to $\tau$.

Proof.

We put $W(c,\iota):=W(\iota(c))$ in [KY1, (4.3)], $V(c,\iota):=V(\iota(c))$ in [KY1, (1.6)], and $X(c,\iota):=G(c,\iota)+W(c,\iota)+V(c,\iota)$. Here we consider the ideal class group $C_{\iota(\mathfrak{f})}$ of $\iota(F)$ modulo $\iota(\mathfrak{f})$. Then $\iota(c)$ denotes the image of $c\in C_{\mathfrak{f}}$ in $C_{\iota(\mathfrak{f})}$ under the natural map. By the definition [KY1, (4.3)] and [KY2, Appendix I, Theorem], we have $W(c,\iota),V(c,\iota)\in\overline{\mathbb{Q}}\log\overline{\mathbb{Q}}^{\times}$. Moreover [KY1, Lemma 5.5] states that

[TABLE]

Here $\chi_{\mathfrak{q}}$, $[\mathfrak{q}]$ denote the composite map $C_{\mathfrak{f}\mathfrak{q}}\rightarrow C_{\mathfrak{f}}\stackrel{{\scriptstyle\chi}}{{\rightarrow}}\mathbb{C}^{\times}$, the ideal class $\in C_{\mathfrak{f}}$ of $\mathfrak{q}$, respectively. Therefore, when $H$ is the fixed subfield under $(\frac{H_{\mathfrak{f}}/F}{\mathfrak{q}})$, it follows from the orthogonality of characters. The general case follows from this case immediately. ∎

Definition 6**.**

Let $H$ be an intermediate field of $H_{\mathfrak{f}}/F$. Assume that $\mathfrak{p}_{\iota}\nmid\mathfrak{f}$ and that $\mathfrak{p}_{\iota}$ splits completely in $H/F$. Then we define

[TABLE]

When $\iota=\mathrm{id}$, we drop the symbol $\iota$: $Y_{p}(\tau):=Y_{p}(\tau,\mathrm{id})$.

By [KY1, Proposition 5.6] (and the orthogonality of characters), we see that $Y_{p}(\tau,\iota)$ depends only on $H,\mathfrak{f},\tau,\iota$, not on $D$, $\mathfrak{a}_{c}$’s. We formulated a conjecture [KY1, Conjecture A*′*], which is equivalent to the following Conjecture 2 by [Ka3, Proposition 6-(ii)].

Conjecture 2**.**

Let $H_{\mathfrak{f}}/H/F$ be as above: we assume that

$\mathfrak{p}_{\iota}$* does not divide $\mathfrak{f}$, splits completely in $H/F$.*

We take a lift $\tilde{\iota}\colon H\rightarrow\mathbb{C}_{p}$ of $\iota\colon F\rightarrow\mathbb{C}_{p}$ and put $\mathfrak{p}_{H,\tilde{\iota}}:=\{z\in\mathcal{O}_{H}\mid|\tilde{\iota}(z)|_{p}<1\}$. Let $\alpha_{H,\tilde{\iota}}$ be a generator of the principal ideal $\mathfrak{p}_{H,\tilde{\iota}}^{h_{H}}$, where $h_{H}$ denotes the class number. Then we have

[TABLE]

Remark 1**.**

Roughly speaking, the above conjecture states a relation between the ratios $[$ $p$-adic multiple gamma functions $:$ multiple gamma functions $]$ and Stark units associated with the finite place $\mathfrak{p}_{\iota}$. We also studied a relation between the same ratios and Stark units associated with real places in [Ka3]. We found a more significant relation between the ratios $[$ $p$-adic gamma function $:$ gamma function $]$ and cyclotomic units in [Ka2].

We rewrite the definition of $Y_{p}$ for later use.

Definition 7**.**

Let $R$ be a subset of $F_{+}$. We assume that $R$ can be expressed in the following form:

[TABLE]

1. (i)

We define

[TABLE] 2. (ii)

Additionally we assume that each $(\iota(\bm{x}_{i}\,{}^{t}\bm{v}_{i}),\iota(\bm{v}_{i}))$ satisfies (3). Then there exists the $p$-adic interpolation function

[TABLE]

of $\zeta_{\iota}(s,R)$. We define

[TABLE]

When $\iota=\mathrm{id}$, we drop the symbol $\iota$.

It follows that, for any Shintani domain $D$ $\bmod E_{+}$ and for any integral ideals $\mathfrak{a}_{c}$ satisfying $\mathfrak{a}_{c}\mathfrak{f}\in\pi(c)$, we have

[TABLE]

where we put $R_{c}:=\{z\in D\mid\mathcal{O}_{F}\supset z\mathfrak{a}_{c}\mathfrak{f}\mathfrak{p}_{\iota}\in c\}$. We will use the following properties of the classical or $p$-adic multiple gamma functions in the proof of Theorem 1.

Proposition 1**.**

1. (i)

Let $R$ be as in Definition 7-(i), $\alpha\in F_{+}$. Then we have

[TABLE] 2. (ii)

Let $R$ be as in Definition 7-(ii), $\alpha\in F_{+}$. Then we have

[TABLE]

Proof.

The assertions follow from $\zeta_{\iota}(s,\alpha R)=\iota(\alpha)^{-s}\zeta_{\iota}(s,R)$ immediately. ∎

We also recall Shintani’s multiple zeta functions in [Sh, (1.1)] which we need in subsequent sections.

Definition 8**.**

1. (i)

Let $A=(a_{ij})$ be an $(l\times r)$-matrix with $a_{ij}\in\mathbb{R}_{+}$, $\bm{x}\in\mathbb{R}_{+}^{r}$, $\bm{\chi}=(\chi_{1},\dots,\chi_{r})\in(\mathbb{C}^{\times})^{r}$ with $|\chi_{i}|\leq 1$. Then Shintani’s multiple zeta function is defined as

[TABLE]

This series converges for $\mathrm{Re}(s)>\frac{r}{l}$, has a meromorphic continuation to the whole $s$-plane, is analytic at $s=0$. 2. (ii)

Let $\bm{x}$, $\bm{\chi}$ be as in (i). For $\bm{v}=(v_{1},\dots,v_{r})\in F_{+}^{r}$, we consider two kinds of Shintani’s multiple zeta functions:

1. (a)

Shintani’s multiple zeta function with $l=1$:

[TABLE]

Here we consider $v_{i}\in F_{+}\stackrel{{\scriptstyle\mathrm{id}}}{{\hookrightarrow}}\mathbb{R}_{+}$. 2. (b)

Let $A$ be the $(n\times r)$-matrix whose raw vectors are $\iota(\bm{v}_{i})$ $(\iota\in\mathrm{Hom}(F,\mathbb{R}),\ n:=[F:\mathbb{Q}])$. Then we put

[TABLE] 3. (iii)

Let $R$ be as in Definition 7-(i). We define

[TABLE]

4 $p$-adic measures associated with zeta values

We consider the following two kinds of integration.

Definition 9**.**

Let $K$ be a finite extension of $\mathbb{Q}_{p}$, $O$ the ring of integers of $K$, $P$ the maximal ideal of $O$.

1. (i)

We say $\nu$ is a $p$-adic measure on $O$ if for each open compact subset $U\subset O$, it takes the value $\nu(U)\in K$ satisfying

1. (a)

$\nu(U\coprod U^{\prime})=\nu(U)+\nu(U^{\prime})$* for disjoint open compact subsets $U,U^{\prime}$.* 2. (b)

$|\nu(U)|_{p}$’s are bounded.

We say a $p$-adic measure $\nu$ is a $\mathbb{Z}$-valued measure if $\nu(U)\in\mathbb{Z}$. 2. (ii)

Let $\nu$ be a $p$-adic measure, $f\colon O\rightarrow O$ a continuous map. We define

[TABLE] 3. (iii)

Let $\nu$ be a $\mathbb{Z}$-valued measure, $f\colon(O-P^{e})\rightarrow(O-P^{e})$ a continuous map $(e\in\mathbb{N})$. We define

[TABLE]

We recall the setting in [Da]. Let $F$ be a totally real field of degree $n$, $\mathfrak{f}$ an integral ideal of $F$, $\mathfrak{p}:=\mathfrak{p}_{\mathrm{id}}$ the prime ideal corresponding to the $p$-adic topology on $F$ induced by $\mathrm{id}\colon F\hookrightarrow\mathbb{C}_{p}$. We assume that $\mathfrak{p}\nmid\mathfrak{f}$.

Definition 10** ([Da, Definitions 3.8, 3.9]).**

Let $\eta$ be a prime ideal of $F$.

1. (i)

We say $\eta$ is good for a cone $C(v_{1},\dots,v_{r})$ if $v_{i}\in\mathcal{O}_{F}-\eta$ and if $N\eta$ is a rational prime (i.e., the residue degree $=1$). 2. (ii)

We say $\eta$ is good for a Shintani set $D$ if it can be expressed as a finite disjoint union of cones for which $\eta$ is good.

Definition 11** ([Da, Definitions 3.13, 3.16, Conjecture 3.21]).**

We take an element $\pi\in\mathcal{O}_{F,+}$, a prime ideal $\eta$, a Shintani domain $\mathcal{D}_{\mathfrak{f}}$ $\bmod E_{\mathfrak{f},+}$ satisfying the following conditions.

1. (i)

Let $e$ be the order of $\mathfrak{p}$ in $C_{\mathfrak{f}}$. We fix a generator $\pi\in\mathfrak{p}^{e}$ satisfying $\pi\in\mathcal{O}_{F,+}$, $\pi\equiv 1\bmod\mathfrak{f}$. 2. (ii)

$N\eta\geq n+2$* and $(N\eta,\mathfrak{f}\mathfrak{p})=1$.* 3. (iii)

The residue degree of $\eta$ $=1$ and the ramification degree of $\eta$ $\leq N\eta-2$. 4. (iv)

$\eta$* is “simultaneously” good for $\mathcal{D}_{\mathfrak{f}},\pi^{-1}\mathcal{D}_{\mathfrak{f}}$ in the following sense: There exist vectors $\bm{v}_{j}\in(\mathcal{O}_{F,+}-\eta)^{r(j)}$, units $\epsilon_{j}\in E_{\mathfrak{f},+}$ $(j\in J^{\prime},\ |J^{\prime}|<\infty)$ satisfying*

[TABLE]

Remark 2**.**

Dasgupta [Da] took a suitable set $T$ of prime ideals instead of one prime ideal $\eta$. In this article, we assume that $|T|=1$ for simplicity.

We denote by $F_{\mathfrak{p}},\mathcal{O}_{F_{\mathfrak{p}}}$ the completion of $F$ at $\mathfrak{p}$, the ring of integers of $F_{\mathfrak{p}}$ respectively.

Definition 12** ([Da, Definitions 3.13, 3.17]).**

Let $\pi,\eta,\mathcal{D}_{\mathfrak{f}}$ be as in Definition 11, $\mathfrak{b}$ a fractional ideal of $F$ relatively prime to $\mathfrak{f}\mathfrak{p}N\eta$. We put

[TABLE]

1. (i)

For an open compact subset $\mathcal{U}\subset\mathcal{O}_{F_{\mathfrak{p}}}$, a Shintani set $D$, we put

[TABLE]

Here $\zeta_{N}(s,R)$ is defined in Definition 8. By [Da, Proposition 3.12] we see that

• •

When $\eta$ is good for $D$, we have $\nu_{\eta}(\mathfrak{b},D,\mathcal{U})\in\mathbb{Z}[N\eta^{-1}]$.

• •

When $\eta$ is good for $D$ and $N\eta\geq n+2$, we have $\nu_{\eta}(\mathfrak{b},D,\mathcal{U})\in\mathbb{Z}$. 2. (ii)

Assume that $\eta$ is good for $D$ and that $\eta\nmid p$. We define a $p$-adic measure $\nu_{\eta}(\mathfrak{b},D)$ on $\mathcal{O}_{F_{\mathfrak{p}}}$ by

[TABLE]

Under the assumption of Definition 11, $\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ is a $\mathbb{Z}$-valued measure. 3. (iii)

For $\tau\in\mathrm{Gal}(H_{\mathfrak{f}}/F)$, we put

[TABLE]

Here $H_{\mathfrak{f}}$ denotes the narrow ray class field modulo $\mathfrak{f}$. 4. (iv)

We define

[TABLE]

where $\mathbf{O}:=\mathcal{O}_{F_{\mathfrak{p}}}-\pi\mathcal{O}_{F_{\mathfrak{p}}}$. The product in the first line is actually a finite product since $\nu_{\eta}(\mathfrak{b},\epsilon\mathcal{D}_{\mathfrak{f}}\cap\pi^{-1}\mathcal{D}_{\mathfrak{f}},\mathcal{O}_{F_{\mathfrak{p}}})=0$ for all but finite $\epsilon\in E_{\mathfrak{f},+}$.

Conjecture 3** ([Da, Conjecture 3.21]).**

Let $\pi,\eta,\mathcal{D}_{\mathfrak{f}}$ be as in Definition 11, $H$ the fixed subfield of $H_{\mathfrak{f}}$ under $(\frac{H_{\mathfrak{f}}/F}{\mathfrak{p}})$.

1. (i)

Let $\tau\in\mathrm{Gal}(H/F)$. For a fractional ideal $\mathfrak{b}$ relatively prime to $\mathfrak{f}\mathfrak{p}N\eta$ satisfying $(\frac{H/F}{\mathfrak{b}})=\tau$, we put

[TABLE]

Then $u_{\eta}(\tau)$ depends only on $\mathfrak{f},\tau,\eta$, not on the choices of $\mathcal{D}_{\mathfrak{f}},\mathfrak{b}$. 2. (ii)

For any $\tau\in\mathrm{Gal}(H/F)$, $u_{\eta}(\tau)$ is a $\mathfrak{p}$-unit of $H$ satisfying $u_{\eta}(\tau)\equiv 1\bmod\eta$. 3. (iii)

For any $\tau,\tau^{\prime}\in\mathrm{Gal}(H/F)$, we have $u_{\eta}(\tau\tau^{\prime})=u_{\eta}(\tau)^{\tau^{\prime}}$.

5 The main results

We keep the notation in the previous sections: Let $F$ be a totally real field of degree $n$, $H_{\mathfrak{f}}$ the narrow ray class field modulo $\mathfrak{f}$. We assume that the prime ideal $\mathfrak{p}$ corresponding to the $p$-adic topology on $F$ does not divide $\mathfrak{f}$. Let $H$ be the fixed subfield of $H_{\mathfrak{f}}$ under $(\frac{H_{\mathfrak{f}}/F}{\mathfrak{p}})$. For $\tau\in\mathrm{Gal}(H/F)$, let $Y_{p}(\tau):=Y_{p}(\tau,\mathrm{id})$ be as in Definition 6. For a fractional ideal $\mathfrak{b}$ relatively prime to $\mathfrak{f}\mathfrak{p}N\eta$, let $u_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}})$ be as in Definition 12-(iv).

Theorem 1**.**

We have

[TABLE]

Corollary 1**.**

Conjecture 2* implies*

[TABLE]

where $\mathfrak{p}_{H}$, $h_{H}$, $\alpha_{H}$ are the prime ideal of $H$ corresponding to the $p$-adic topology on $H$, the class number of $H$, a generator of $\mathfrak{p}_{H}^{h_{H}}$.

We prepare some Lemmas in order prove this Theorem.

Lemma 2** ([CN2, Théorème 13]).**

Let $\bm{v}\in F_{+}^{r}$, $z\in F_{+}$, $\bm{\xi}=(\xi_{1},\dots,\xi_{r})$ with $\xi_{i}$ roots of unity, $\neq 1$. For $k\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Here we put $(\bm{1}-\bm{\xi})^{\bm{m}}:=\prod_{i=1}^{r}(1-\xi_{i})^{m_{i}}$, $\begin{Bmatrix}\bm{m}\\ \bm{l}\end{Bmatrix}:=\prod_{i=1}^{r}\left((-1)^{l_{i}-1}\binom{m_{i}-1}{l_{i}-1}\right)$ with the binomial coefficient $\binom{m_{i}-1}{l_{i}-1}$. The sum over $\bm{m}$ is actually a finite sum since we have $\sum_{\bm{l}}\begin{Bmatrix}\bm{m}\\ \bm{l}\end{Bmatrix}N(z-\bm{l}\,{}^{t}\bm{v})^{k}=\sum_{\bm{l}}\begin{Bmatrix}\bm{m}\\ \bm{l}\end{Bmatrix}(z-\bm{l}\,{}^{t}\bm{v})^{k}=0$ if $m_{i}$ is large enough.

Lemma 3**.**

Let $\nu_{\eta}(\mathfrak{b},D,\mathcal{U})$ be as in Definition 12-(i). Assume that $\eta$ is good for $D$. Then we have

[TABLE]

Proof.

It is enough to show the statement when

• •

$D$ is a cone $C(\bm{v})$ with $\bm{v}=(v_{1},\dots,v_{r})$, $v_{i}\in\mathcal{O}_{F}-\eta$.

• •

$\mathcal{U}$ is of the form $a+\mathfrak{p}^{m}\mathcal{O}_{F_{\mathfrak{p}}}$ ($m\in\mathbb{N}$, $a\in\mathcal{O}_{F_{\mathfrak{p}}}$).

Put $R:=F^{\times}_{\mathfrak{f}}\cap\mathfrak{b}^{-1}\cap C(\bm{v})\cap(a+\mathfrak{p}^{m}\mathcal{O}_{F_{\mathfrak{p}}})$. By definition we have

[TABLE]

Let $L$ be a positive integer satisfying $L\in\mathfrak{f}\mathfrak{p}^{m}\mathfrak{b}^{-1}$, $(\eta,L)=1$. Then we have

[TABLE]

Since $N\eta$ is a rational prime, the following homomorphism is a surjection.

[TABLE]

Here we denote the localization of $\mathcal{O}_{F}$ at $\eta$ by ${\mathcal{O}_{F}}_{(\eta)}$. Hence for each $\bm{x}\in R_{a}$, there exists an integer $n_{\bm{x}}$ satisfying $\bm{x}\,{}^{t}\bm{v}\equiv n_{\bm{x}}\bmod\eta{\mathcal{O}_{F}}_{(\eta)}$. Similarly we take $n_{i}$ satisfying $Lv_{i}\equiv n_{i}\bmod\eta{\mathcal{O}_{F}}_{(\eta)}$ and put $\bm{n}_{L\bm{v}}:=(n_{1},\dots,n_{r})$. Then the following are equivalent:

[TABLE]

Let $\zeta$ be a primitive $N\eta$th root of unity. We put $\xi_{\bm{x}}:=\zeta^{n_{\bm{x}}}$, $\xi_{i}:=\zeta^{n_{i}}$, $\bm{\xi}_{L\bm{v}}:=(\xi_{1},\dots,\xi_{r})$. Note that $\xi_{i}\neq 1$ for any $i$. Then we have

[TABLE]

Here we put $\bm{\xi}_{L\bm{v}}^{\bm{m}}:=\prod_{i=1}^{r}\xi_{i}^{m_{i}}$. It follows that

[TABLE]

Similarly we obtain

[TABLE]

By Lemma 2, we have for $\bm{x}\in R_{a}$

[TABLE]

Then the assertion follows from (6), (7), (8), (9). ∎

Dasgupta’s $p$-adic integration $\int d\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}},x)$ is originally associated with special values of multiple zeta functions “with the norm” $\zeta_{N}(\cdots)$. By the above Lemma, we can rewrite it in terms of special values of multiple zeta functions “without the norm” $\zeta(\cdots)$. This observation is one of the main discoveries in this paper.

Lemma 4**.**

Let $\nu_{\eta}(\mathfrak{b},D,\mathcal{U})$, $\mathbf{O}=\mathcal{O}_{F_{\mathfrak{p}}}-\pi\mathcal{O}_{F_{\mathfrak{p}}}$ be as in Definition 12. Assume that $\eta$ is good for $D$. Then we have for $k\in\mathbb{Z}_{\geq 0}$, $s\in\mathbb{Z}_{p}$

[TABLE]

Here we put $\langle x\rangle:=p^{-\mathrm{ord}_{p}x}\theta_{p}(x)^{-1}x$ by using $\mathrm{ord}_{p},\theta_{p}$ in (2).

Proof.

It is enough to show the statement when $D=C(\bm{v})$ with $\bm{v}=(v_{1},\dots,v_{r})$, $v_{i}\in\mathcal{O}_{F}-\eta$. By definition we can write

[TABLE]

By Lemmas 2, 3, we have

[TABLE]

where $L,R_{a},\xi_{\bm{x}},\bm{\xi}_{L\bm{v}}$ are as in the proof of Lemma 3. On the other hand, by Lemma 2 again, we obtain

[TABLE]

By definition, we see that $L\in\mathfrak{p}^{m}$, $\bm{x}\,{}^{t}(L\bm{v})\equiv a\bmod\mathfrak{p}^{m}\mathcal{O}_{F_{\mathfrak{p}}}$ for $\bm{x}\in R_{a}$. It follows that

[TABLE]

Hence the first assertion is clear. The second assertion follows from the $p$-adic interpolation property (4). ∎

Proof of Theorem 1.

For a fractional ideal $\mathfrak{b}$, a Shintani set $D$, an open compact subset $\mathcal{U}\subset\mathcal{O}_{F_{\mathfrak{p}}}$, and for $*=\emptyset,p$, we put

[TABLE]

whenever each function is well-defined. It suffices to show the following three equalities:

[TABLE]

Let $\bm{v}_{j}$, $\epsilon_{j}$ ($j\in J^{\prime}$) be as in Definition 11-(iv). Since $\mathcal{D}_{\mathfrak{f}},\pi^{-1}\mathcal{D}_{\mathfrak{f}}$ are fundamental domains of $F\otimes\mathbb{R}_{+}/E_{\mathfrak{f},+}$, we see that

[TABLE]

Namely we have

[TABLE]

By Lemma 3, we can write

[TABLE]

Therefore by Proposition 1-(i) we obtain

[TABLE]

We easily see that

[TABLE]

Hence, by Proposition 1-(i) again, we get

[TABLE]

Since $(F^{\times}_{\mathfrak{f}}\cap\mathfrak{b}^{-1}\cap\mathcal{D}_{\mathfrak{f}}\cap\mathbf{O})\coprod(F^{\times}_{\mathfrak{f}}\cap\pi\mathfrak{b}^{-1}\cap\mathcal{D}_{\mathfrak{f}}\cap\pi\mathcal{O}_{F_{\mathfrak{p}}})=F^{\times}_{\mathfrak{f}}\cap\mathfrak{b}^{-1}\cap\mathcal{D}_{\mathfrak{f}}\cap\mathcal{O}_{F_{\mathfrak{p}}}$, we have

[TABLE]

Then the assertion (11) follows from (5), (15), (16).

Next, differentiating (10) at $s=0$, we obtain

[TABLE]

By definition, we have $\log_{p}(\times\int_{\mathbf{O}}x\ d\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}},x))=\int_{\mathbf{O}}\log_{p}x\,d\nu_{\eta}(\mathfrak{b},\mathcal{D}_{\mathfrak{f}},x)$. Hence the assertion (12) is clear.

Finally we prove (13). Let $D$ be a Shintani domain $\bmod E_{+}$. For each $c\in C_{\mathfrak{f}\mathfrak{p}}$, we take an integral ideal $\mathfrak{a}_{c}$ satisfying $\mathfrak{a}_{c}\mathfrak{f}\in\pi(c)$, and put $R_{c}:=\{z\in D\mid\mathcal{O}_{F}\supset z\mathfrak{a}_{c}\mathfrak{f}\mathfrak{p}\in c\}$. By (5) we can write

[TABLE]

Since $H$ is the fixed subfield under $(\frac{H_{\mathfrak{f}}/F}{\mathfrak{p}})$, we may replace

[TABLE]

where $\overline{c}$ denotes the image under $C_{\mathfrak{f}\mathfrak{p}}\rightarrow C_{\mathfrak{f}}$, $[\mathfrak{a}]$ denotes the ideal class in $C_{\mathfrak{f}}$ of a fractional ideal $\mathfrak{a}$. On the other hand, we can write for $*=\emptyset,p$

[TABLE]

Therefore it suffices to show that we have for each $k$

[TABLE]

We fix $k$. Whenever $\overline{c}=[\mathfrak{b}\mathfrak{p}^{-k}]$, $\pi(c)\in C_{(1)}$ is constant, so we may put $\mathfrak{a}_{c}$ to be a fixed integral ideal $\mathfrak{a}_{0}$. Then we have

[TABLE]

Let $\alpha_{0}\in F_{+}$ be a generator of the principal ideal $(\mathfrak{a}_{0}\mathfrak{f}\mathfrak{p})(\mathfrak{b}\mathfrak{p}^{-k})^{-1}$. Then the following are equivalent:

$[z\mathfrak{a}_{0}\mathfrak{f}\mathfrak{p}]=[\mathfrak{b}\mathfrak{p}^{-k}]\Leftrightarrow[(z\alpha_{0})]=[(1)]\Leftrightarrow\exists\epsilon\in E_{+}$ s.t. $z\epsilon\alpha_{0}\equiv 1\bmod\mathfrak{f}$.

Hence, taking a representative set $E_{0}$ of $E_{+}/E_{\mathfrak{f},+}$, we can write

[TABLE]

Namely we have for $*=\emptyset,p$

[TABLE]

On the other hand, $\mathcal{D}_{\mathfrak{f}}^{\prime}:=\coprod_{\epsilon\in E_{0}}\epsilon\alpha_{0}D$ becomes another Shintani domain $\bmod E_{\mathfrak{f},+}$, and we can write for $*=\emptyset,p$

[TABLE]

Then the assertion (17), replacing $\mathcal{D}_{\mathfrak{f}}$ with $\mathcal{D}_{\mathfrak{f}}^{\prime}$, follows from (18), (19) and Proposition 1. We conclude the proof of (13) by showing that

[TABLE]

Note that the independence on the choice of $\mathcal{D}_{\mathfrak{f}}$ is also discussed in [Da, §5.2] under certain conditions. Similarly to [Yo, Chap. III, Lemma 3.13], we see that there exist cones $C(\bm{v}_{j})$ and units $u_{j}\in E_{\mathfrak{f},+}$ ($j\in J^{\prime\prime}$) which satisfy

[TABLE]

Therefore it suffices to show that

[TABLE]

It follows from Proposition 1 since $F_{\mathfrak{f}}^{\times}\cap\mathfrak{b}^{-1}\cap u_{j}C(\bm{v}_{j})\cap\mathfrak{p}^{k}\mathcal{O}_{F_{\mathfrak{p}}}^{\times}=u_{j}(F_{\mathfrak{f}}^{\times}\cap\mathfrak{b}^{-1}\cap C(\bm{v}_{j})\cap\mathfrak{p}^{k}\mathcal{O}_{F_{\mathfrak{p}}}^{\times})$. ∎