On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues

TL;DR

This paper explores the relationship between Barnes' multiple gamma functions and their $p$-adic analogues, establishing algebraicity results and connections to Stark units and Gross-Stark units in the context of totally real fields.

Contribution

It proves the $p$-adic algebraicity of products of ratios of Barnes' gamma functions and their $p$-adic analogues, extending previous algebraicity results and relating them to Stark units.

Findings

Proved the $p$-adic algebraicity of certain products involving Barnes' gamma functions.

Established a relation between ratios of exponential functions and Stark units.

Extended previous algebraicity results to the $p$-adic setting.

Abstract

Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $\iota$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,\iota)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $\zeta(s,c)$ has a canonical decomposition $\zeta'(0,c)=\sum_{\iota}X(c,\iota)$, where $\iota$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,\iota))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,\iota)$: In particular, we discussed the relation between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and Gross-Stark units. In a previous paper, the author proved the algebraicity of some products of $\exp(X(c,\iota))$'s. In this paper, we prove its $p$-adic analogue. Then, by using these algebraicity properties, we discuss the relation between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and Stark units.

Equations

$$C_{\mathfrak{f}}:=I_{\mathfrak{f}}/\{(\alpha )\in I_{\mathfrak{f}}\mid \alpha \in F^{\times },\alpha \equiv 1{\mathrm{mod}}^{\ast }\mathfrak{f},\alpha >>0\}.$$

$${\zeta }^{\prime }(0,c)=\sum _{\iota \in \mathrm{Hom}(F,\mathbb{R})}\left(\sum _{j\in J}\sum _{\mathbf{x}\in R(c,{\mathbf{v}}_j)}\log (\Gamma (\iota (\mathbf{x}{}^t{\mathbf{v}}_j),\iota ({\mathbf{v}}_j)))\right)+\text{ correction terms}.$$

$$X(c,\iota ):=\sum _{j\in J}\sum _{\mathbf{x}\in R(c,{\mathbf{v}}_j)}\log (\Gamma (\iota (\mathbf{x}{}^t{\mathbf{v}}_j),\iota ({\mathbf{v}}_j)))+W(c,\iota )+V(c,\iota ).$$

$${\zeta }^{\prime }(0,c)=\sum _{\iota \in \mathrm{Hom}(F,\mathbb{R})}X(c,\iota ).$$

$$\log \parallel u^{\tau }{\parallel }_w=-W_K{\zeta }_S^{\prime }(0,\tau )(\tau \in \mathrm{Gal}(K/F)).$$

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

$$\exp (2{\zeta }_S^{\prime }(0,\tau ))\in {\iota }_w({\mathcal{O}}_K^{\times }).$$

$${\zeta }_S^{\prime }(0,\tau )={\zeta }_R(0,\tau )\log N{\mathfrak{p}}_F.$$

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

$${\log }_p{N}_{K_{{\mathfrak{p}}_K}/{\mathbb{Q}}_p}({u^{\prime }}^{\tau })=-W{\zeta }_{p,S}^{\prime }(0,\tau )(\tau \in \mathrm{Gal}(K/F)).$$

$$\sin (\frac{a}{m}\pi {)}^{\tau }=\pm \sin (\tau (\frac{a}{m})\pi )(\frac{a}{m}\in \mathbb{Q}\cap (0,1),\tau \in \mathrm{Aut}(\mathbb{C})).$$

$$[a:b][c:d]:=[ac:bd],$$

$$[a:b]=[c:d]\text{ means }\frac{a}{c}=\frac{b}{d},$$

$$[a:b]\equiv [c:d]\mathrm{mod}{\mu }_{\infty }\text{ means }\frac{a}{c}\frac{d}{b}\in {\mu }_{\infty },$$

$$[\exp (X([(a)],\mathrm{id})):{\exp }_p(X_p([(a)],\mathrm{id}))]\equiv [\Gamma (\frac{a}{m})/\sqrt{2\pi }:{\Gamma }_p(\frac{a}{m})]\mathrm{mod}{\mu }_{\infty }.$$

$$[\Gamma (\frac{a}{m})/\sqrt{2\pi }:{\Gamma }_p(\frac{a}{m})][\Gamma (\frac{m-a}{m})/\sqrt{2\pi }:{\Gamma }_p(\frac{m-a}{m})]\equiv [1:2\sin (\frac{a}{m}\pi )]\mathrm{mod}{\mu }_{\infty }.$$

$$[{\int }_{\gamma }{\eta }_{\frac{a}{m},\frac{b}{m}}:{\int }_{p,\gamma }{\eta }_{\frac{a}{m},\frac{b}{m}}][{\int }_{\gamma }{\eta }_{\frac{m-a}{m},\frac{m-b}{m}}:{\int }_{p,\gamma }{\eta }_{\frac{m-a}{m},\frac{m-b}{m}}]=[2\pi i:(2\pi i{)}_p],$$

$$\mathfrak{B}(\frac{a}{m},\frac{b}{m}):=\frac{\frac{\Gamma (\frac{a}{m})\Gamma (\frac{b}{m})}{\Gamma (\frac{a+b}{m})}}{{\int }_{\gamma }{\eta }_{\frac{a}{m},\frac{b}{m}}}\frac{{\int }_{p,\gamma }{\eta }_{\frac{a}{m},\frac{b}{m}}}{\frac{{\Gamma }_p(\frac{a}{m}){\Gamma }_p(\frac{b}{m})}{{\Gamma }_p(\frac{a+b}{m})}}\in B_{\mathrm{dR}}(p\nmid ab(a+b))$$

$${\Phi }_{\tau }(\mathfrak{B}(\frac{a}{m},\frac{b}{m}))\equiv p^{\frac{\mathrm{deg}\tau }{2}}\mathfrak{B}(\tau (\frac{a}{m}),\tau (\frac{b}{m}))\mathrm{mod}{\mu }_{\infty }(\tau \in W_p).$$

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

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

$${\zeta }_{\mathrm{fml}}(-m,\mathbf{v},\mathbf{x}{}^t\mathbf{v}):=(-1{)}^{r}m!\sum _{|\mathbf{l}|=m+r}\prod _{i=1}^r\frac{B_{l_i}(x_i)v_i^{l_i-1}}{l_i!}\in k.$$

$$\zeta (-m,\mathbf{v},\mathbf{x}{}^t\mathbf{v})={\zeta }_{\mathrm{fml}}(-m,\mathbf{v},\mathbf{x}{}^t\mathbf{v}).$$

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

$$P$$

$$1+M$$

$${\mu }_{(p)}$$

$$\displaystyle\mathbb{C}_{p}^{\times}\$$

$$z$$

$${\mathrm{ord}}_p:{\mathbb{C}}_p^{\times }\to \mathbb{Q},{\theta }_p:{\mathbb{C}}_p^{\times }\to {\mu }_{(p)}$$

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

$$⟨z⟩:=p^{-{\mathrm{ord}}_p(z)}{\theta }_p(z{)}^{-1}z.$$

$$z^s:=\sum _{k=1}^{\infty }\left(\genfrac{}{}{0ex}{}{s}{k}\right)(z-1{)}^k,$$

Full text

1 Introduction

Let $F$ be a totally real field, $\mathcal{O}_{F}$ the ring of integers of $F$, $\mathfrak{f}$ an integral ideal of $F$, $I_{\mathfrak{f}}$ the group of all fractional ideals of $F$ relatively prime to $\mathfrak{f}$. We consider the narrow ideal class group modulo $\mathfrak{f}$ defined as

[TABLE]

Here $\alpha>\!\!>0$ means that $\alpha$ is totally positive. We denote the class in $C_{\mathfrak{f}}$ of a fractional ideal $\mathfrak{a}$ by $[\mathfrak{a}]$. Shintani [Shin2] gave an explicit formula for the derivative value $\zeta^{\prime}(0,c)$ of the partial zeta function $\zeta(s,c):=\sum_{\mathcal{O}_{F}\supset\mathfrak{a}\in c}N\mathfrak{a}^{-s}$ with $c\in C_{\mathfrak{f}}$. The main term is a finite sum of $\log$ of Barnes’ multiple gamma functions $\Gamma(z,\bm{v})$ (Definition 2) of the following form:

[TABLE]

Here $\bm{v}_{j}$ are suitable vectors whose entries are totally positive integers of $F$, $R(c,\bm{v}_{j})$ are finite sets of vectors whose entries are positive rational numbers (for details, see Theorem 1, Definition 6). We denote by $\bm{x}\,{}^{t}\bm{v}_{j}$ the inner product, by $\mathrm{Hom}(F,\mathbb{R})$ the set of all real embeddings of $F$. Yoshida [Yo, Chap. III, (3.6)–(3.9)] discovered an appropriate decomposition $\sum_{\iota\in\mathrm{Hom}(F,\mathbb{R})}W(c,\iota)+V(c,\iota)$ of the “correction terms” and defined the class invariant

[TABLE]

It follows that we have

[TABLE]

Since Shintani’s expression (1) is not unique, Yoshida’s invariant has an ambiguity: $\exp(X(c,\iota))$ is well-defined up to multiplication by a rational power of a unit of $\iota(F)$. Yoshida conjectured that each value of Shimura’s period symbol $p_{K}$ ([Shim, Theorem 32.5]) can be expressed as a product of rational powers of $\exp(X(c,\iota))$’s [Yo, Chap. III, Conjecture 3.9]. Yoshida also studied the relation between Stark units and $\exp(X(c,\iota))$’s. On the other hand, Yoshida and the author defined and studied the $p$-adic analogue $X_{p}(c,\iota)$ in [KY1, KY2]. In particular we formulated a refinement of a $p$-adic analogue of the Stark conjecture by Gross. Let us explain more precisely: Let $K/F$ be an abelian extension of number fields, $S$ a finite set of places of $F$. We assume that

• •

$S$ contains all infinite places of $F$ and all ramified places in $K/F$.

• •

$S$ contains a distinguished place $v$ which splits completely in $K/F$.

• •

$\#S>2$. (This assumption is for simplicity. $\#S>1$ is essential.)

We fix a place $w$ of $K$ dividing $v$. Then the rank $1$ abelian Stark conjecture, which is a version of Stark’s conjectures in [St], implies the following statement.

Conjecture 1**.**

There exists a $v$-unit $u\in K$, which is called a Stark unit, satisfying

[TABLE]

Here $W_{K}$ is the number of roots of unity in $K$. We define the partial zeta function associated with $S,\tau$ by

[TABLE]

where $\mathfrak{a}$ runs over all integral ideals of $F$, relatively prime to each finite place in $S$, whose images under the Artin symbol $(\frac{K/F}{*})$ equal $\tau$. If $w$ is a finite place, we put $\parallel\!\!x\!\!\parallel_{w}:=Nw^{-\mathrm{ord}_{w}(x)}$. Otherwise we take an embedding $\iota_{w}\colon K\hookrightarrow\mathbb{C}$ associated with $w$ and put $\parallel\!\!x\!\!\parallel_{w}:=|\iota_{w}(x)|$ or $|\iota_{w}(x)|^{2}$ for a real or complex place $w$ respectively. Note that when $v$ is a real place, assuming the Stark conjecture, we can write a Stark unit explicitly as

[TABLE]

Here we see that $W_{K}=2$ since $K$ has a real place $w$.

Next we consider the case when $v$ is a finite place lying above a rational prime $p$. We additionally assume that

• •

$F$ is a totally real field, $K$ is a CM-field.

• •

$S$ contains all places of $F$ lying above $p$.

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 we denote by $\mathbb{C}_{p}$ the $p$-adic completion of the algebraic closure $\overline{\mathbb{Q}_{p}}$ of $\mathbb{Q}_{p}$. For a number field $L$, we denote by $\mathfrak{p}_{L}:=\mathfrak{p}_{L,\mathrm{id}}$ the prime ideal corresponding to the $p$-adic topology on $L$ induced by $\mathrm{id}\colon L\hookrightarrow\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$. We assume that

• •

$v=\mathfrak{p}_{F}$, $w=\mathfrak{p}_{K}$.

Put $R:=S-\{\mathfrak{p}_{F}\}$. We see that $\zeta_{S}(s,\tau)=(1-N\mathfrak{p}_{F}{}^{-s})\zeta_{R}(s,\tau)$ since $\mathfrak{p}_{F}$ splits completely in $K$. It follows that

[TABLE]

Hence, by $\zeta_{R}(0,\tau)\in\mathbb{Q}$, there exist a positive integer $W$ and a $\mathfrak{p}_{F}$-unit $u^{\prime}\in K$ which satisfy

[TABLE]

The Stark conjecture with $v=\mathfrak{p}_{F}$ states that we can take $W=W_{K}$. Gross conjectured the following property of $u^{\prime}$ as a $p$-adic number.

Conjecture 2** ([Gr2, Conjecture 3.13]).**

Let $u^{\prime}\in K$ be a $\mathfrak{p}_{F}$-unit satisfying (3). Then we have

[TABLE]

Here $\log_{p}$ denotes Iwasawa’s $p$-adic $\log$ function and $\zeta_{p,S}(s,\tau)$ denotes the $p$-adic interpolation function for $\zeta_{S}(s,\tau)$. Dasgupta-Darmon-Pollack [DDP] proved that Conjecture 2 holds true under certain conditions. In [KY1], we formulated a conjecture which expresses $\log_{p}{u^{\prime}}^{\tau}$ (without the norm $N_{K_{\mathfrak{p}_{K}}/\mathbb{Q}_{p}}$) by using not only $X_{p}(c,\iota)$ but also $X(c,\iota)$. More precisely, we see that the ratio $[\exp(X(c,\iota)):\exp_{p}(X_{p}(c,\iota))]$ is well-defined up to roots of unity (Corollary 1-(i)), and our conjecture (Conjecture 3) states that $u^{\prime}$ can be expressed as a product of $\frac{\exp(X(c,\iota))}{\exp_{p}(X_{p}(c,\iota))}$’s, up to $\ker\log_{p}$ (Proposition 6-(ii)). One of the main results in this paper states that we can express a Stark unit as another product of $\frac{\exp(X(c,\iota))}{\exp_{p}(X_{p}(c,\iota))}$’s (Corollary 1-(ii)). Roughly speaking, Stark units (2) and Gross-Stark units (3) can be written as a product of $\frac{\exp(X(c,\iota))}{\exp_{p}(X_{p}(c,\iota))}$’s uniformly (Remark 6).

To clarify the meaning of the main results in this paper, we briefly describe the results in [Ka2]. Let $F=\mathbb{Q}$, $v$ the unique real place of $\mathbb{Q}$. Then the Stark conjecture implies the following “reciprocity law” on the sine function (in other words, on cyclotomic units):

[TABLE]

Here we define $\tau(\tfrac{a}{m})\in\mathbb{Q}\cap(0,1)$ by $\tau(\zeta_{m}^{a})=\zeta_{m}^{\tau(\frac{a}{m})m}$ with $\zeta_{m}:=e^{\frac{2\pi i}{m}}$. In [Ka2] we proved a reciprocity law (5) on a period-ring valued beta function, which is a refinement of (4) in the following sense: For simplicity, we assume that $p\mid m$, $p\neq 2$ here. We use the following notation.

[TABLE]

where $\mu_{\infty}$ denotes the group of all roots of unity.

• •

Let $\Gamma_{p}\colon\mathbb{Q}_{p}\rightarrow\mathbb{C}_{p}^{\times}$ be a generalization [Ka2, Lemma 4.2] of Morita’s $p$-adic gamma function. When $F=\mathbb{Q}$, $\mathfrak{f}=(m)$, $c=[(a)]\in C_{(m)}$ $(a\in\mathbb{N},\ (a,m)=1)$, we see that

[TABLE]

Moreover we may regard this ratio as a refinement of a cyclotomic unit:

[TABLE]

• •

We consider the $m$th Fermat curve $F_{m}\colon x^{m}+y^{m}=1$ and its differential forms $\eta_{\frac{a}{m},\frac{b}{m}}:=x^{a-1}y^{b-m}\mathit{d}x$ ($0<a,b<m$, $a+b\neq m$). Although the period integral $\int_{\gamma}\eta_{\frac{a}{m},\frac{b}{m}}$ depends on the choice of a closed path $\gamma\subset F_{m}(\mathbb{C})$, the ratio $[\int_{\gamma}\eta_{\frac{a}{m},\frac{b}{m}}:\int_{p,\gamma}\eta_{\frac{a}{m},\frac{b}{m}}]$ is constant. Here $\int_{p,\gamma}\eta_{\frac{a}{m},\frac{b}{m}}\in B_{\mathrm{dR}}$ is the $p$-adic period defined by the $p$-adic Hodge theory, $B_{\mathrm{dR}}$ is Fontaine’s $p$-adic period ring. Moreover we have

[TABLE]

where $(2\pi i)_{p}\in B_{\mathrm{dR}}$ is the $p$-adic counterpart of $2\pi i$.

• •

Rohrlich’s formula in [Gr1] implies that $\frac{\int_{\gamma}\eta_{\frac{a}{m},\frac{b}{m}}}{\frac{\Gamma(\frac{a}{m})\Gamma(\frac{b}{m})}{\Gamma(\frac{a+b}{m})}}\in\overline{\mathbb{Q}}$. It follows that

[TABLE]

are well-defined up to $\mu_{\infty}$.

• •

Coleman’s formula in [Co] implies that [Ka2, Theorem 7.2-(ii)]

[TABLE]

Here we denote by $W_{p}\subset\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ the Weil group, by $\Phi_{\tau}$ the absolute Frobenius automorphism associated with $\tau\in W_{p}$, acting on a subring of $B_{\mathrm{dR}}$.

• •

We can express $2\sin(\tfrac{a}{m}\pi)$ as a product of rational powers of $\mathfrak{B}(\tfrac{a}{m},\tfrac{b}{m})$’s up to $\mu_{\infty},(2\pi i)_{p}$. Hence the reciprocity law (5) implies (4), up to $\mu_{\infty}$.

Summarizing the above, when $F=\mathbb{Q}$ we gave an alternative proof of a part of the Stark conjecture by studying the ratios $[$ $\Gamma$-function $:$ $p$-adic $\Gamma$-function $]$, $[$ periods of Fermat curves $:$ $p$-adic periods of Fermat curves $]$. The ratio $[\exp(X(c,\iota)):\exp_{p}(X_{p}(c,\iota))]$ is a generalization of the former one. The main result in this paper, which expresses Stark units and Gross-Stark units in terms of $\frac{\exp(X(c,\iota))}{\exp_{p}(X_{p}(c,\iota))}$’s, is a significant step toward the generalization of the above results in [Ka2].

Remark 1**.**

1. (i)

For a generalization of the ratios $[$ periods of Fermat curves $:$ $p$-adic periods of Fermat curves $]$, there are results by de Shalit **[dS]** on the ratios $[$ CM-periods $:$ $p$-adic CM periods $]$ under a certain condition. 2. (ii)

Yoshida’s conjecture **[Yo, Chap. III, Conjecture 3.9]** is a generalization of Rohrlich’s formula. There is a slight generalization **[Ka3, Conjecture 5.5]**. 3. (iii)

Yoshida and the author formulated conjectures in **[KY1, KY2]** which are generalizations of Coleman’ formula.

Remark 2**.**

Dasgupta formulated a conjecture [Da, Conjecture 3.21] which is also a refinement of Conjecture 2: A modified version of $u^{\prime}$ of (3) is expressed in terms of a $p$-adic measure associated with zeta values. This conjecture is a further refinement of Conjecture 3 by $\ker\log_{p}$. More precisely, we can show the following: Let $u_{T}(\mathfrak{b},\mathcal{D})$ be as in [Da, Conjecture 3.21], $Y_{p}(\tau,\mathrm{id})$ for $\tau\in\mathrm{Gal}(H/F)$ as in Proposition 6-(ii), and $(\frac{H/F}{*})$ the Artin symbol. For simplicity we take $T:=\{\eta\}$ with $\eta$ a prime ideal of $F$. Then we can show that ([Ka4])

[TABLE]

Here we note that $\mathfrak{f}$ in [Da] is $\mathfrak{f}\mathfrak{p}_{\mathrm{id}}$ in this paper.

The outline of this paper is structured as follows. In §2, we introduce some basic properties of Barnes’ multiple zeta functions, multiple gamma functions, and their $p$-adic analogues. In §3, we recall a Proposition by Shintani, which is needed in order to relate the derivative values of the classical (resp. $p$-adic) partial zeta functions to the classical (resp. $p$-adic) multiple gamma functions in Theorem 2 (resp. Theorem 3) in §4. We note that the classical multiple gamma functions and the $p$-adic analogues have not yet been mixed at this point. In §4, we recall the definition and some properties of Yoshida’s class invariant $X(c,\iota)$. We also provide their $p$-adic analogues by quite similar arguments. There are two applications of these properties: First, in §5, we prove the algebraicity of some products of $\exp_{p}(X_{p}(c,\iota))$’s, which is the $p$-adic analogue of the main results in [Ka3]. Then we can clarify the relation between Stark units and the ratios $[\exp(X(c,\iota)):\exp_{p}(X_{p}(c,\iota))]$. Next, in §6, we prove that a conjecture [KY1, Conjecture 5.10] on the ratios $[\exp(X(c,\iota)):\exp_{p}(X_{p}(c,\iota))]$ holds true, and also clarify the relation between Gross-Stark units and the same ratios.

2 Multiple zeta and gamma functions

In this section, we review some basic properties of Barnes’ multiple zeta and gamma functions, and their $p$-adic analogues. For omitted proofs and details, we refer to [Yo, Chap. I, §1], [Ka1]. We denote by $\mathbb{R}_{+}$ the set of all positive real numbers, by $\mathbb{Z}_{\geq 0}$ the set of all non-negative integers.

Definition 1**.**

1. (i)

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

[TABLE]

where $\bm{m}\,{}^{t}\bm{v}$ denotes the inner product. 2. (ii)

Let $k$ be a field of characteristic [math]. (We may assume that $k=\mathbb{C}$ or $\mathbb{C}_{p}$.) Let $\bm{v}=(v_{1},\dots,v_{r})\in(k^{\times})^{r}$, $\bm{x}=(x_{1},\dots,x_{r})\in k^{r}$. Then we define “formal multiple zeta values” for $m\in\mathbb{Z}_{\geq 0}$ as

[TABLE]

*Here we denote the *$l$th Bernoulli polynomial by $B_{l}(x)$ and $\bm{l}$ in the sum runs over all $\bm{l}=(l_{1},\dots,l_{r})\in\mathbb{Z}_{\geq 0}^{r}$ satisfying $|\bm{l}|:=l_{1}+\dots+l_{r}=m+r$.

Proposition 1**.**

The series in the definition of $\zeta(s,\bm{v},z)$ converges for $\mathrm{Re}(s)>r$, has a meromorphic continuation to $\mathbb{C}$, which is analytic at $s=0,-1,-2,\dots$. Moreover when $\bm{v}\in\mathbb{R}_{+}^{r}$, $\bm{x}\,{}^{t}\bm{v}\in\mathbb{R}_{+}$, $m\in\mathbb{Z}_{\geq 0}$ we have

[TABLE]

Definition 2**.**

Let $z\in\mathbb{R}_{+}$, $\bm{v}\in\mathbb{R}_{+}^{r}$. Then we define Barnes’ multiple gamma function by

[TABLE]

Note that this definition is modified from the original one $\frac{\Gamma(z,\bm{v})}{\rho(z)}=\exp\left(\frac{\partial}{\partial s}\zeta(s,\bm{v},z)|_{s=0}\right)$ with a correction term $\rho(z)$.

We fix embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$ throughout this paper. In particular $p^{r}\in\mathbb{C}_{p}$ is well-defined for each $r\in\mathbb{Q}$.

Definition 3**.**

1. (i)

We put

[TABLE]

and consider the following decomposition

[TABLE]

More precisely, let

[TABLE]

be unique group homomorphisms satisfying

[TABLE]

Then we put

[TABLE] 2. (ii)

For $z\in 1+M$, $s\in\mathbb{Z}_{p}$, we put

[TABLE]

where $\binom{s}{k}$ denotes the binomial coefficient. 3. (iii)

We denote the Iwasawa $p$-adic logarithmic function by $\log_{p}$, which is defined as

[TABLE] 4. (iv)

Let $r\in\mathbb{N}$. For a function $f\colon\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{p}$, we put

[TABLE]

whenever the limit exists. 5. (v)

Let $z\in\mathbb{C}_{p}^{\times}$, $\bm{v}=(v_{1},\dots,v_{r})\in\left(\mathbb{C}_{p}^{\times}\right)^{r}$. For simplicity we assume that

[TABLE]

Then we put

[TABLE]

and define the $p$-adic multiple zeta function by

[TABLE]

Proposition 2**.**

1. (i)

$\zeta_{p}(s,\bm{v},z)$* is analytic at $s=0$, and continuous on $s\in\mathbb{Z}_{p}-\{1,2,\dots,r\}$.* 2. (ii)

$\zeta_{p}(s,\bm{v},z)$* is continuous for $\bm{v},z$.* 3. (iii)

Let $z\in\overline{\mathbb{Q}}$, $\bm{v}=(v_{1},\dots,v_{r})\in\overline{\mathbb{Q}}^{r}$. We assume that

[TABLE]

Then $\zeta_{p}(s,\bm{v},z)$ satisfies the following $p$-adic interpolation property:

[TABLE]

Proof.

(i), (iii) follow from [Ka1, Lemma 5.3-1, Theorem 5.1] respectively. Let $z,z^{\prime}\in\mathbb{C}_{p}^{\times}$, $\bm{v},\bm{v}^{\prime}\in\left(\mathbb{C}_{p}^{\times}\right)^{r}$ satisfy (6). When $z^{\prime}\rightarrow z$, we may assume that $\mathrm{ord}_{p}(z)=\mathrm{ord}_{p}(z^{\prime})$, $\theta_{p}(z)=\theta_{p}(z^{\prime})$. Then we see that for $m\in\mathbb{Z}_{\geq 0}$

[TABLE]

Hence by [Ka1, (5.3)] we have

[TABLE]

Since non-positive integers are dense in $\mathbb{Z}_{p}$, the assertion (ii) is clear. ∎

Definition 4**.**

Let $z\in\mathbb{C}_{p}^{\times}$, $\bm{v}\in\left(\mathbb{C}_{p}^{\times}\right)^{r}$ satisfy (6). Then we define the $p$-adic $\log$ multiple gamma function by

[TABLE]

Proposition 3**.**

1. (i)

Let $z\in\mathbb{R}_{+}$, $\bm{v}\in\mathbb{R}_{+}^{r}$, $\alpha\in\mathbb{R}_{+}$. Then we have

[TABLE] 2. (ii)

Let $z\in\mathbb{C}_{p}^{\times}$, $\bm{v}\in\left(\mathbb{C}_{p}^{\times}\right)^{r}$ satisfy (6). Let $\alpha\in\mathbb{C}_{p}^{\times}$. Then we have

[TABLE]

Proof.

Follows immediately from Definitions 1-(i), 2, 3, 4. ∎

3 Shintani domains

Definition 5**.**

Let $F$ be a totally real field of degree $n$.

1. (i)

Let $\bm{v}_{1},\dots,\bm{v}_{r}\in\mathbb{R}^{n}$ be linearly independent vectors. We define an ($r$-dimensional open simplicial) cone with basis $\bm{v}_{1},\dots,\bm{v}_{r}$ as

[TABLE] 2. (ii)

We identify

[TABLE]

In particular, we regard $F$ as a subset of $F\otimes\mathbb{R}=\mathbb{R}^{n}$. We say that a cone $C(\bm{v}_{1},\dots,\bm{v}_{r})$ is a cone of $F$ if $\bm{v}_{1},\dots,\bm{v}_{r}\in\mathcal{O}_{F}$. 3. (iii)

Considering $F\subset F\otimes\mathbb{R}=\mathbb{R}^{n}$, we put

[TABLE]

Theorem 1** ([Shin1, Proposition 4]).**

We consider the natural action $E_{F,+}\curvearrowright F\otimes\mathbb{R}_{+}$, $u(\alpha\otimes\beta):=(u\alpha)\otimes\beta$. Then there exists a fundamental domain $D$ of $F\otimes\mathbb{R}_{+}/E_{F,+}$ which can be written as a finite disjoint union of cones of $F$. Namely there exist $v_{ij}\in\mathcal{O}_{F,+}$ $(j\in J,1\leq i\leq r(j),\ |J|<\infty,\ r(j)\in\mathbb{N})$ satisfying

[TABLE]

We call such a $D$ a Shintani domain.

4 The class invariants $X(c,\iota)$, $X_{p}(c,\iota)$

In this section, we summarize the definitions and some properties of $X(c,\iota)$ and $X_{p}(c,\iota)$. The class invariant $X(c)$ was introduced and studied by Yoshida [Yo] ($X(c,\iota)$ in this paper is almost equal to $X(\iota(c))$ in [Yo]). Yoshida and the author studied the $p$-adic analogue $X_{p}(c)$ in [KY1, KY2]. Throughout this section let $F$ be a totally real field of degree $n$, $\mathfrak{f}$ an integral ideal of $F$, $c\in C_{\mathfrak{f}}$, $\iota\in\mathrm{Hom}(F,\mathbb{R})$, and $D=\coprod_{j\in J}C(\bm{v}_{j})$ a disjoint union of cones with $\bm{v}_{j}\in\mathcal{O}_{F}^{r(j)}$. Unless otherwise specified, we do not assume that $D$ is a Shintani domain in the sense of Theorem 1. Let $\pi\colon C_{\mathfrak{f}}\rightarrow C_{(1)}$ be the natural projection. We take an integral ideal $\mathfrak{a}_{c}$ satisfying $\mathfrak{a}_{c}\mathfrak{f}\in\pi(c)$ for each $c\in C_{\mathfrak{f}}$.

4.1 The case of a totally positive $D$

In this subsection, we assume that $D=\coprod_{j\in J}C(\bm{v}_{j})$ ($\bm{v}_{j}\in\mathcal{O}_{F}^{r(j)}$) is totally positive, that is

[TABLE]

We introduce Yoshida’s invariants $G,W,V,X$ in [Yo, Chap. III, (3.6)–(3.9), (3.28)–(3.31)], which is slightly modified in [KY1, (4.3)], [Ka3, §2]. The equalities in Definitions 7, 9 follow from [Yo, Chap. II, Lemma 3.2].

Definition 6**.**

Let $\bm{v}=(v_{1},\dots,v_{r})\in\mathcal{O}_{F,+}^{r}$ be linearly independent. Then we put

[TABLE]

Definition 7**.**

We put

[TABLE]

Definition 8**.**

We define group homomorphisms $\log_{\iota}\colon I_{F}\rightarrow\mathbb{R}$ for $\iota\in\mathrm{Hom}(F,\mathbb{R})$ as follows: For each prime ideal $\mathfrak{p}$ of $F$, we choose a generator $\pi_{\mathfrak{p}}\in\mathcal{O}_{F,+}$ of the principal ideal $\mathfrak{p}^{h_{F,+}}$, where $h_{F,+}=|C_{(1)}|$ is the narrow class number. Then we put

[TABLE]

We extend this linearly to $\log_{\iota}\colon I_{F}\rightarrow\mathbb{R}$.

We consider a principal ideal $(\alpha)=\alpha\mathcal{O}_{F}$ with $\alpha\in F^{\times}$. The difference between $\log\iota(\alpha)$ and $\log_{\iota}(\alpha)$ is as follows: By definition there exists a generator $\alpha^{\prime}\in(\alpha)^{h_{F,+}}$ satisfying $\log_{\iota}(\alpha)=\frac{1}{h_{F,+}}\log\iota(\alpha^{\prime})$. Then we see that

[TABLE]

Definition 9**.**

We put

[TABLE]

Definition 10**.**

For $\iota,\iota^{\prime}\in\mathrm{Hom}(F,\mathbb{R})$ with $\iota\neq\iota^{\prime}$, we put

[TABLE]

Then we define

[TABLE]

Here $\iota^{\prime}$ runs over all $\iota^{\prime}\in\mathrm{Hom}(F,\mathbb{R})$ with $\iota^{\prime}\neq\iota$ in the first sum, $\iota^{\prime},\iota^{\prime\prime}$ run over all $\iota^{\prime},\iota^{\prime\prime}\in\mathrm{Hom}(F,\mathbb{R})$ with $\iota^{\prime}\neq\iota^{\prime\prime}$ in the second sum.

Definition 11**.**

We put

[TABLE]

If $D$ is a Shintani domain in the sense of Theorem 1, and if we fix $D,\mathfrak{a}_{c}$, then we put

[TABLE]

The following Theorem is essentially due to Yoshida: Yoshida transformed Shintani’s formula (1) in [Shin2] into a similar form. Yoshida and the author modified the $W$-term slightly in [KY1], [Ka3] into the present form.

Theorem 2** ([Yo, Chap. III, (3.11)], [Ka3, Theorem 2.5]).**

Let $D$ be a Shintani domain. Then we have

[TABLE]

The following Lemma also is essentially due to Yoshida, although $\exp(X(\iota(c)))$ in [Yo] was well-defined up to $\iota(F_{+})^{\mathbb{Q}}$.

Lemma 1** ([Yo, Chap. III, §3.6, 3.7], [Ka3, Lemma 3.11]).**

We consider the following operations on $D,\mathfrak{a}_{c}$.

1. (I)

Let $j_{0}\in J$. We decompose $C(\bm{v}_{j_{0}})=\coprod_{l\in L}C(\bm{v}_{l})$ and replace

[TABLE]

Note that a replacement of basis $(|L|=1)$ is included in this case. 2. (II)

Let $j_{0}\in J$, $\epsilon\in E_{F,+}$. We replace $C(\bm{v}_{j_{0}})$ by $C(\epsilon\bm{v}_{j_{0}})$, that is

[TABLE] 3. (III)

Let $\alpha\in\mathfrak{a}_{c}^{-1}\cap F_{+}$. We replace

[TABLE]

simultaneously. Here we take $\bm{v}_{j}^{\prime}$ satisfying $\alpha^{-1}C(\bm{v}_{j})=C(\bm{v}_{j}^{\prime})$, $\bm{v}_{j}^{\prime}\in\mathcal{O}_{F,+}^{r(j)}$ for each $j\in J$.

In order to simplify the following expressions, we put

[TABLE]

In particular

[TABLE]

have meanings.

1. (i)

$G(c,\iota,D,\mathfrak{a}_{c})$* changes as follows by the operations *(I), (II), (III).

1. (I)

Stays constant. 2. (II)

$\displaystyle G(c,\iota,D,\mathfrak{a}_{c})-\iota(Z_{j_{0}})\log\iota(\epsilon)$. 3. (III)

$\displaystyle G(c,\iota,D,\mathfrak{a}_{c})+\sum_{j\in J}\iota(Z_{j})\log\iota(\alpha)$. 2. (ii)

$W(c,\iota,D,\mathfrak{a}_{c})$ changes as follows by the operations (I), (II), (III).

1. (I)

Stays constant. 2. (II)

Stays constant. 3. (III)

$\displaystyle W(c,\iota,D,\mathfrak{a}_{c})-\sum_{j\in J}\iota(Z_{j})\log_{\iota}(\alpha)$. 3. (iii)

$V(c,\iota,D,\mathfrak{a}_{c})$ changes as follows by the operations (I), (II), (III).

1. (I)

Stays constant. 2. (II)

$\displaystyle V(c,\iota,D,\mathfrak{a}_{c})+\left(\iota(Z_{j_{0}})-\frac{\mathrm{Tr}_{F/\mathbb{Q}}(Z_{j_{0}})}{n}\right)\log\iota(\epsilon)$. 3. (III)

$\displaystyle V(c,\iota,D,\mathfrak{a}_{c})+\sum_{j\in J}\left(\iota(Z_{j})-\frac{\mathrm{Tr}_{F/\mathbb{Q}}(Z_{j})}{n}\right)\left(\frac{1}{n}\log(N_{F/\mathbb{Q}}(\alpha))-\log\iota(\alpha)\right)$. 4. (iv)

Additionally assume that $D$ is a Shintani domain. Then $X(c,\iota,D,\mathfrak{a}_{c})$ changes as follows by the operations (I), (II), (III).

1. (I)

Stays constant. 2. (II)

$\displaystyle X(c,\iota,D,\mathfrak{a}_{c})-\frac{\mathrm{Tr}_{F/\mathbb{Q}}(Z_{j})}{n}\log\iota(\epsilon)$. 3. (III)

$\displaystyle X(c,\iota,D,\mathfrak{a}_{c})-\frac{\zeta(0,c)}{h_{F,+}}\log\iota(u_{\alpha})$. Here we take $u_{\alpha}\in E_{F,+}$ is as in (7).

In particular, $\exp(X(c,\iota))\bmod\iota(E_{F,+})^{\mathbb{Q}}$ does not depend on $D$, $\mathfrak{a}_{c}$, where we put $\iota(E_{F,+})^{\mathbb{Q}}:=\{\iota(u)^{\frac{1}{N}}\mid u\in E_{F,+},\ N\in\mathbb{N}\}$. More precisely, for Shintani domains $D,D^{\prime}$, integral ideals $\mathfrak{a}_{c},\mathfrak{a}_{c}^{\prime}$ satisfying $\mathfrak{a}_{c}\mathfrak{f},\mathfrak{a}_{c}^{\prime}\mathfrak{f}\in\pi(c)$, there exist $N\in\mathbb{N}$, $u_{c,D,D^{\prime},\mathfrak{a}_{c},\mathfrak{a}_{c}^{\prime}}\in E_{F,+}$ satisfying

[TABLE]

Proof.

For comparison to the $p$-adic case (Lemma 4), we give a brief sketch of the poof of (i)-(II). Let $D^{\prime}:=(\coprod_{j\in J-\{j_{0}\}}C(\bm{v}_{j}))\coprod C(\epsilon\bm{v}_{j_{0}})$. By definition we have

[TABLE]

By noting that

[TABLE]

we can rewrite

[TABLE]

as desired. The other cases are similar. ∎

4.2 The case of a non-totally positive $D$

In this subsection, we fix an embedding $\iota_{0}\in\mathrm{Hom}(F,\mathbb{R})$. We put

[TABLE]

For each subset $A\subset F$, we put

[TABLE]

We introduce generalizations of Yoshida’s invariants in [Ka3].

Definition 12**.**

For each $\iota\in\mathrm{Hom}(F,\mathbb{R})$, we take $\nu_{\iota}\in\mathcal{O}_{F}$ satisfying

[TABLE]

and put

[TABLE]

Remark 3**.**

Let $H_{\mathfrak{f}}$ be the maximal ray class field modulo $\mathfrak{f}$ in the narrow sense, $\mathrm{Art}\colon C_{\mathfrak{f}}\rightarrow\mathrm{Gal}(H_{\mathfrak{f}}/F)$ the Artin map. Then $\mathrm{Art}(c_{\iota})$ is the complex conjugation at $\iota$ [Yo, Chap. III, the first paragraph of §5.1]. Hence the fixed subfield $H_{\mathfrak{f}}^{\langle\mathrm{Art}(c_{\iota})\rangle}$ is the maximal subfield where the real place $\iota$ splits completely.

Definition 13**.**

Let $\bm{v}=(v_{1},\dots,v_{r})\in\mathcal{O}_{F,(n-1)+}^{r}$ be linearly independent. Then we put

[TABLE]

Definition 14**.**

Let $\iota\in\mathrm{Hom}(F,\mathbb{R})$, $\neq\iota_{0}$. By replacing $c$ with $c\cup cc_{\iota_{0}}$ in Definition 7, we put

[TABLE]

Remark 4**.**

When $C(\bm{v})\subset F\otimes\mathbb{R}_{(n-1)+}$, we consider the pair $c,cc_{\iota_{0}}$ for the following reason: Let $z\in({\mathfrak{a}}_{c}\mathfrak{f})^{-1}\cap C(\bm{v})$ satisfy $(z){\mathfrak{a}}_{c}\mathfrak{f}\in c$. We write $\bm{v}=(v_{1},\dots,v_{r})$ and $z=\bm{x}\,{}^{t}\bm{v}$ with $\bm{x}\in\mathbb{Q}_{>0}^{r}$. If $C(\bm{v})\subset F\otimes\mathbb{R}_{+}$, then we see that

[TABLE]

by noting that $(z\pm v_{i}){\mathfrak{a}}_{c}\mathfrak{f}=(1\pm v_{i}/z)(z){\mathfrak{a}}_{c}\mathfrak{f}$. It follows that $\sum_{z\in({\mathfrak{a}}_{c}\mathfrak{f})^{-1}\cap C(\bm{v}),\,(z){\mathfrak{a}}_{c}\mathfrak{f}\in c}\iota(z)^{-s}=\sum_{\bm{x}\in R(c,\bm{v})}\zeta(s,\iota(\bm{v}),\iota(\bm{x}\,{}^{t}\bm{v}))$. If $C(\bm{v})\subset F\otimes\mathbb{R}_{(n-1)+}$, then it follows only that $(z\pm v_{i}){\mathfrak{a}}_{c}\mathfrak{f}\in c\cup cc_{\iota_{0}}$.

Definition 15**.**

For each $\iota\in\mathrm{Hom}(F,\mathbb{R})$ (including $\iota=\iota_{0}$), we put

[TABLE]

We note that $\zeta_{\mathrm{fml}}(0,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))=\zeta(0,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ if all entries of $\iota(\bm{v}_{j})$ are positive.

Definition 16**.**

1. (i)

Let $\bm{v}=(v_{1},\dots,v_{r})\in\mathcal{O}_{F}^{r}$ be linearly independent. Let $\bm{x}=(x_{1},\dots,x_{r})\in(\mathbb{Q}\cap(0,1])^{r}$, $\iota\in\mathrm{Hom}(F,\mathbb{R})$. We put

[TABLE]

Here for an $r$-dimensional vector $(a_{q})=(a_{1},\dots,a_{r})$, we denote by $(a_{q})_{q\neq p}$ the $(r-1)$-dimensional vector $(a_{1},\dots,a_{p-1},a_{p+1},\dots,a_{r})$. 2. (ii)

For each $\iota\in\mathrm{Hom}(F,\mathbb{R})$ (including $\iota=\iota_{0}$), we put

[TABLE]

Definition 17**.**

Let $\iota\in\mathrm{Hom}(F,\mathbb{R})$, $\neq\iota_{0}$. We put

[TABLE]

Lemma 2** ([Ka3, Lemmas 3.6, 3.7, 3.10]).**

Let $\iota\in\mathrm{Hom}(F,\mathbb{R})$, $\neq\iota_{0}$. Then the same statements as in Lemma 1 hold, by replacing $c$ with $c\cup cc_{\iota_{0}}$. In particular we replace $Z_{j}$ with

[TABLE]

For (iv)-(III), we replace $\zeta(0,c)$ with

[TABLE]

Lemma 3** ([Ka3, Lemma 3.12]).**

We assume that $D$ is a Shintani domain, and that $\iota\neq\iota_{0}$. Then the following assertions hold.

1. (i)

We have

[TABLE]

Here, in the symbol $X(cc_{\iota_{0}}\cup c,\iota,\nu_{\iota_{0}}^{-1}D,(\nu_{\iota_{0}})\mathfrak{a}_{c})$, the roles of $c,c_{\iota_{0}}$ are exchanged. We take $u_{\nu_{\iota_{0}}}\in E_{F,+}$ is as in (7). 2. (ii)

If $c_{\iota_{0}}=[(1)]$ or if $\mathcal{O}_{F,(n-1)+}^{\times}=\emptyset$, then we have

[TABLE]

Otherwise, we take an element $\epsilon\in\mathcal{O}_{F,(n-1)+}^{\times}$. Then we have

[TABLE]

Here we take $u_{\nu_{\iota_{0}}\epsilon^{-1}}\in E_{F,+}$ is as in (7).

4.3 $p$-adic analogues

We introduce the $p$-adic analogues of Yoshida’s invariants. The case when $D\subset F\otimes\mathbb{R}_{+}$ was studied in [KY1, KY2]. In the $p$-adic case, the Archimedean topology induced by $\iota\in\mathrm{Hom}(F,\mathbb{R})$ is not so important. Instead, we consider the prime ideal corresponding to the $p$-adic topology.

Definition 18**.**

Recall that we fixed embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$. We identify

[TABLE]

and define for $\iota\in\mathrm{Hom}(F,\mathbb{R})$ $(=\mathrm{Hom}(F,\mathbb{C}_{p}))$

[TABLE]

In other words, $\iota(\mathfrak{p}_{\iota})$ corresponds to the $p$-adic topology on $\iota(F)\subset\mathbb{C}_{p}$.

We fix $\iota_{0}\in\mathrm{Hom}(F,\mathbb{R})$ and define $F\otimes\mathbb{R}_{(n-1)+}\subset F\otimes\mathbb{R}$ as in the previous subsection. We note that $\iota$ may be equal to $\iota_{0}$. Throughout this subsection, we assume that

[TABLE]

Under this assumption $(\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ with $\bm{x}\in R(c,\bm{v}_{j})$ satisfy (6). Namely, the $p$-adic interpolation function $\zeta_{p}(s,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ of $\zeta(s,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ is well-defined. Hence we may consider the $p$-adic interpolation of $\sum_{z\in({\mathfrak{a}}_{c}\mathfrak{f})^{-1}\cap D,\,(z){\mathfrak{a}}_{c}\mathfrak{f}\in c}\iota(z)^{-s}$ in Definition 7. In the setting of Definition 14, $\zeta_{p}(s,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ for $\bm{x}\in R(c\cup cc_{\iota_{0}},\bm{v}_{j})$ are also well-defined whenever $\mathfrak{p}_{\iota}\mid\mathfrak{f}$, although $\zeta(s,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ with $\iota=\iota_{0}$ may not be well-defined.

Definition 19**.**

When $D\subset F\otimes\mathbb{R}_{(n-1)+}$, we put

[TABLE]

When $D\subset F\otimes\mathbb{R}_{+}$, we also define $G_{p}(c,\iota,D,\mathfrak{a}_{c})$ by replacing $R(c\cup cc_{\iota_{0}},\bm{v}_{j})$ with $R(c,\bm{v}_{j})$.

Definition 20**.**

Let $h_{F,+},\pi_{\mathfrak{p}}$ be as in Definition 8 and we put

[TABLE]

for prime ideals $\mathfrak{p}$ of $F$ and for $\iota\in\mathrm{Hom}(F,\mathbb{R})$. We extend this linearly to $\log_{\iota,p}\colon I_{F}\rightarrow\mathbb{C}_{p}$. When $D\subset F\otimes\mathbb{R}_{(n-1)+}$, we put

[TABLE]

When $D\subset F\otimes\mathbb{R}_{+}$, we also define $W_{p}(c,\iota,D,\mathfrak{a}_{c})$ similarly.

Definition 21**.**

We define $V_{p}(\bm{v},\bm{x},\iota)$ by replacing $\log|\cdots|$ in Definition 16 with $\log_{p}(\cdots)$. When $D\subset F\otimes\mathbb{R}_{(n-1)+}$, we put

[TABLE]

When $D\subset F\otimes\mathbb{R}_{+}$, we also define $V_{p}(c,\iota,D,\mathfrak{a}_{c})$ similarly.

Definition 22**.**

When $D\subset F\otimes\mathbb{R}_{(n-1)+}$, we put

[TABLE]

When $D\subset F\otimes\mathbb{R}_{+}$, we also define $X_{p}(c,\iota,D,\mathfrak{a}_{c})$ similarly. If $D$ is a Shintani domain, we put

[TABLE]

Proposition 4**.**

The definitions of $G_{p},W_{p},V_{p},X_{p}$ doses not depend on the choice of the cone decomposition $D=\coprod_{j\in J}C(\bm{v}_{j})$ when we fix $D$.

Proof.

We use a similar argument to the proof of [Ka3, Lemma 3.6]. We can reduce the problem to a refinement of a cone: Let $D=C(\bm{v})$ with $\bm{v}=(v_{1},\dots,v_{r})\in\mathcal{O}_{F}^{r}$. We abbreviate $\iota(\bm{v})$ as $\bm{v}$. It suffices to show that $G_{p},W_{p},V_{p}$ does not change under the following operations.

1. (I)

Change the order of the basis $\bm{v}$. 2. (II)

Replace $v_{1}$ by $nv_{1}$ with $n\in\mathbb{N}$. 3. (III)

Decompose $C(\bm{v})$ into $C(\bm{v}^{\sharp})\coprod C(\bm{v}^{\flat})\coprod C(\bm{v}^{\natural})$ with

[TABLE]

The case of $V_{p}$ is completely the same as [Ka3, Lemma 3.6]. The remaining cases follows from that

[TABLE]

for $\bm{x}=(x_{1},\dots,x_{r})\in R(c,\bm{v})$ or $\in R(c\cup cc_{\iota_{0}},\bm{v})$. Here we put

[TABLE]

When all $v_{i}$ are positive, these equations follow from the $p$-adic interpolation property, since the same equations obviously hold for $\zeta(s,\bm{v},z)$. Furthermore $\zeta_{p}(s,\bm{v},z)$ is continuous in the sense of Proposition 2-(ii), so we can generalize equations to all $\bm{v}$. ∎

We proved the following $p$-adic analogue of Theorem 2, although we will not use it in this paper.

Theorem 3** ([Ka1, Theorem 6.2], [KY1, Theorem 3.1]).**

Assume that $\mathfrak{p}$ divides $\mathfrak{f}$ for any prime ideal $\mathfrak{p}$ lying above $p$. If $p=2$, we further assume that $2\mathfrak{p}$ divides $\mathfrak{f}$ for any $\mathfrak{p}$ lying above $2$. When $D$ is a Shintani domain, we have

[TABLE]

Here $\zeta_{p}(s,c)$ is the $p$-adic interpolation function of $\zeta(s,c)$.

Lemma 4**.**

Let $\iota\in\mathrm{Hom}(F,\mathbb{R})$ satisfy $\mathfrak{p}_{\iota}\mid\mathfrak{f}$. Then the same statements as in Lemmas 1, 2, 3 hold, by replacing $X,G,W,V,\log,\log_{\iota}$ with $X_{p},G_{p},W_{p},V_{p},\log_{p},\log_{\iota,p}$, not excepting $\iota=\iota_{0}$. We also replace $\zeta(0,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ with $\zeta_{\mathrm{fml}}(0,\iota(\bm{v}_{j}),\iota(\bm{x}\,{}^{t}\bm{v}_{j}))$ when $\iota(\bm{v}_{j})$ has a negative entry.

Proof.

The same proof as those of [Yo, Chap. III, §3.6, 3.7], [Ka3, Lemmas 3.10, 3.11, 3.12] works, by using Propositions 3-(ii), 4. For example, consider the $p$-adic analogue of Lemma 1-(i)-(II): Let $D=\coprod_{j\in J}C(\bm{v}_{j})$, $D^{\prime}=(\coprod_{j\in J-\{j_{0}\}}C(\bm{v}_{j}))\coprod C(\epsilon\bm{v}_{j_{0}})$. Then we have

[TABLE]

We see that $R(c,\epsilon\bm{v}_{j_{0}})=R(c,\bm{v}_{j_{0}})$ by (8). Hence Proposition 3-(ii) states that

[TABLE]

as desired. The other cases can be proved similarly. ∎

5 The case when a real place splits completely

Theorem 4 is one of the main results in this paper. The Archimedean part (9) was proved in [Ka3]. The $p$-adic part (10) is a new result, although it is proved by quite similar arguments. Let $c_{\iota}=[(\nu_{\iota})]\in C_{\mathfrak{f}}$ be as in Lemma 12.

Theorem 4**.**

Let $c\in C_{\mathfrak{f}}$, $\iota_{0}\in\mathrm{Hom}(F,\mathbb{R})$. Then there exist $u\in E_{F,+}$, $m\in\mathbb{N}$ satisfying

[TABLE]

Proof.

(9) was proved in [Ka3, Proof of Theorem 3.1] by using Lemmas 1, 2, 3, and [Ka3, Lemma 3.4]. We can repeat the same argument for the $p$-adic analogue (10) by Lemma 4: Let $D$, $\nu$, $X_{t}$, $\epsilon_{t}$ ($t\in T$, $|T|<\infty$) be as in [Ka3, Lemma 3.4]. Then we have

• •

$D$ is a Shintani domain, $\nu\in F_{(n-1)+}$, $\epsilon_{t}\in E_{F,+}$.

• •

Each $X_{t}$ is a subset of $F\otimes\mathbb{R}_{(n-1)+}$ which can be expressed as a finite disjoint union of cones of $F$: $X_{t}=\coprod_{j\in J_{t}}C(\bm{v}_{j})$.

• •

We denotes by $\biguplus$ the multiset sum. Then we have

[TABLE]

Then, by Lemma 2-(iv)-(II) or its $p$-adic analogue in Lemma 4, we have for $*=\emptyset$ or $p$

[TABLE]

where we put

[TABLE]

First we assume that $c_{\iota_{0}}=[(1)]$ or $\mathcal{O}_{F,(n-1)+}^{\times}=\emptyset$. Then we have

[TABLE]

by Lemma 3-(ii), (i), (ii), Lemma 1-(iv), and their $p$-adic analogues respectively. Summarizing the above, we can write

[TABLE]

Next, we assume that $c_{\iota_{0}}\neq[(1)]$ and $\mathcal{O}_{F,(n-1)+}^{\times}\neq\emptyset$. We take an element $\epsilon\in\mathcal{O}_{F,(n-1)+}^{\times}$. Then we have

[TABLE]

by Lemma 3-(ii), (i), (ii), Lemma 1-(iv) (3 times), and their $p$-adic analogues respectively. Summarizing the above, we can write

[TABLE]

Thus the assertion holds in both cases. ∎

Corollary 1**.**

Let $p,\iota$ satisfy $\mathfrak{p}_{\iota}\mid\mathfrak{f}$. Let $\exp_{p}\colon\mathbb{C}_{p}\rightarrow\mathbb{C}_{p}^{\times}$ be any group homomorphism which coincides with the usual power series $\sum_{n=0}^{\infty}\frac{z^{n}}{n!}$ on a neighborhood of [math]. We denote by $\mu_{\infty}$ the group of all roots of unity.

1. (i)

The ratio $[\exp(X(c,\iota)):\exp_{p}(X_{p}(c,\iota))]\bmod\mu_{\infty}$ depends only on $c,\iota$. Strictly speaking, let $D,D^{\prime}$ be Shintani domains, and $\mathfrak{a}_{c},\mathfrak{a}_{c}^{\prime}$ integral ideals satisfying $\mathfrak{a}_{c}\mathfrak{f},\mathfrak{a}_{c}^{\prime}\mathfrak{f}\in\pi(c)$. Then we have

[TABLE] 2. (ii)

Whenever $\mathfrak{p}_{\iota}\mid\mathfrak{f}$ ($p$* may vary), we have*

[TABLE]

Proof.

We see that $\exp_{p}\circ\log_{p}=\mathrm{id}$ on a neighborhood of $1$. Hence we have for a large enough $N$

[TABLE]

In particular, when $\mathrm{ord}_{p}(z)=0$ we have

[TABLE]

Hence (i) follows from Lemmas 1, 4. The assertion (ii) follows from Theorems 2, 4 since $\prod_{\iota\in\mathrm{Hom}(F,\mathbb{R})}\iota(u)=N_{F/\mathbb{Q}}(u)=1$ for $u\in E_{F,+}$. ∎

6 The case when a finite place splits completely

In the previous section (especially in Corollary 1-(ii)), $\exp(X(c,\iota))$ are the main terms and $\exp_{p}(X_{p}(c,\iota))$ are the correction terms. Their roles are exchanged in this section.

6.1 A brief review of the results in [KY1]

Let $\mathfrak{p}_{\iota}$ be the prime ideal of $F$ corresponding to the $p$-adic topology of $F$ induced by $F\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}\mathbb{C}_{p}$ as in Definition 18.

Definition 23**.**

1. (i)

We denote by $H_{\mathfrak{f}}$ the maximal ray class field modulo $\mathfrak{f}$ in the narrow sense, by $\mathrm{Art}\colon C_{\mathfrak{f}}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathrm{Gal}(H_{\mathfrak{f}}/F)$ the Artin map. 2. (ii)

We denote the group of all characters of $C_{\mathfrak{f}}$ by $\hat{C}_{\mathfrak{f}}$. Let $\chi\in\hat{C}_{\mathfrak{f}}$. For an integral ideal $\mathfrak{g}$ we denote the associated character $\in\hat{C}_{\mathfrak{fg}}$ by $\chi_{\mathfrak{g}}$. Namely, $\chi_{\mathfrak{g}}$ is the composite map

[TABLE] 3. (iii)

Let $K$ be an extension field of $F$. For $\iota\in\mathrm{Hom}(F,\mathbb{R})$, we take a lift $\tilde{\iota}\colon K\rightarrow\mathbb{C}_{p}$ of $\iota\colon F\rightarrow\mathbb{C}_{p}$ and put

[TABLE]

Moreover we take a generator $\alpha_{K,\tilde{\iota}}$ of the principal ideal $\mathfrak{p}_{K,\tilde{\iota}}^{h_{K}}$, where $h_{K}$ is the class number of $K$. 4. (iv)

We denote by $\overline{\mathbb{Q}}\log\overline{\mathbb{Q}}^{\times}$ (resp. $\overline{\mathbb{Q}}\log_{p}\overline{\mathbb{Q}}^{\times}$) the $\overline{\mathbb{Q}}$-subspace of $\mathbb{C}$ (resp. $\mathbb{C}_{p}$) generated by $\log a$ (resp. $\log_{p}a$) with $a\in\overline{\mathbb{Q}}^{\times}$. For $\log a$, we take any branch of $\log$. 5. (v)

We define a $\overline{\mathbb{Q}}$-linear mapping $[\ ]_{p}$ by

[TABLE]

This map is well-defined **[KY1, Lemma 5.1]** by a well-known theorem of A. Baker.

Remark 5**.**

Let $H$ be an intermediate field of $H_{\mathfrak{f}}/F$, $\chi^{\prime}$ a character of $\mathrm{Gal}(H/F)$. Then there exists a character $\chi\in\hat{C}_{\mathfrak{f}}$ corresponding to $\chi^{\prime}$ via the Artin map. We note that $\chi$ may not be primitive. The relation between the Artin $L$-function $L(s,\chi^{\prime})$ and the Hecke $L$-function $L(s,\chi)$ can be written as

[TABLE]

Here $\mathfrak{q}$ runs over all prime ideal dividing $\mathfrak{f}$, not dividing the conductor $\mathfrak{f}_{\chi^{\prime}}$ of $\chi^{\prime}$. We denote by $\mathrm{Frob}_{\mathfrak{q}}\in\mathrm{Gal}(H/F)$ the Frobenius automorphism.

Proposition 5** ([KY1, Lemma 5.5, Proposition 5.6]).**

Let $\chi\in\hat{C}_{\mathfrak{f}}$. We assume that

[TABLE]

Then we have for $k\in\mathbb{N}$

[TABLE]

Moreover the quantity

[TABLE]

does not depend on the choices of $D$ and $\mathfrak{a}_{c}$’s.

Proof.

The case $k=1$ follows from [KY1, Lemma 5.5, Proposition 5.6]. We can use mathematical induction by [KY1, Lemmas 5.3, 5.4]. ∎

Definition 24**.**

Let $\chi\in\hat{C}_{\mathfrak{f}}$ satisfy $\mathfrak{p}_{\iota}\nmid\mathfrak{f}$, $\chi(\mathfrak{p}_{\iota})=1$. For $k\in\mathbb{N}$ we define

[TABLE]

It follows that

[TABLE]

since the $W,V$-terms are in $\overline{\mathbb{Q}}\log\overline{\mathbb{Q}}^{\times}$.

In [KY1], under the assumption $\chi(\mathfrak{p}_{\iota})=1$, we formulated the following two conjectures on the exact value of $Y_{p}(\chi_{\mathfrak{p}_{\iota}},\iota)$, which is a refinement of Conjecture 2. The latter one has now become a corollary of Theorem 4.

Conjecture 3** ([KY1, Conjecture A′]).

Let $K$ be a CM-field which is abelian over $F$ with $\mathfrak{f}_{K/F}$ the conductor. We assume that $\mathfrak{p}_{\iota}$ splits completely in $K/F$. (Hence $\mathfrak{p}_{\iota}\nmid\mathfrak{f}_{K/F}$.) Then for any odd character $\chi^{\prime}$ of $\mathrm{Gal}(K/F)$, we have

[TABLE]

Here $\rho$ denotes the unique complex conjugation on $K$. We take $\chi\in\hat{C}_{\mathfrak{f}_{K/F}}$ corresponding to $\chi^{\prime}$.

Theorem 5** ([KY1, Conjecture 5.10]).**

Let $\chi\in\hat{C}_{\mathfrak{f}}$. We assume that $\chi$ is not totally odd and satisfies $\chi(\mathfrak{p}_{\iota})=1$. (Hence $\mathfrak{p}_{\iota}\nmid\mathfrak{f}$.) Then we have

[TABLE]

Proof.

Since $\chi$ is not totally odd, there exists $\iota_{0}\in\mathrm{Hom}(F,\mathbb{R})$ satisfying $\chi(c_{\iota_{0}})=1$. Then we can write

[TABLE]

When $\iota\neq\iota_{0}$, we have $X_{p}(c,\iota)+X_{p}(cc_{\iota_{0}},\iota)-[X(c,\iota)+X(cc_{\iota_{0}},\iota)]_{p}=0$ by Theorem 4. Even when $\iota=\iota_{0}$, by Corollary 1 and $\chi(\mathfrak{p}_{\iota})=1$, we have

[TABLE]

Since $\chi$ is not totally odd, we see that $\mathrm{ord}_{s=0}L(s,\chi)\geq 1$, so $\frac{d}{ds}L(s,\chi)(1-N\mathfrak{p}_{\iota}^{-s})=0$. Hence the assertion is clear. ∎

We reformulate Conjecture 3 in the remaining of this subsection.

Lemma 5** ([KY1, Lemma 6.4]).**

Let $\chi\in\hat{C}_{\mathfrak{f}}$ satisfy $\chi(\mathfrak{p}_{\iota})=1$. Then for an integral ideal $\mathfrak{g}$ of $F$ we have

[TABLE]

Here $\mathfrak{q}$ runs over all prime ideals dividing $\mathfrak{g}$. In particular for $k\in\mathbb{N}$ we have

[TABLE]

Proposition 6**.**

Assume that $\mathfrak{p}_{\iota}\nmid\mathfrak{f}$. Let $H$ be the fixed subfield of $H_{\mathfrak{f}}$ under $\langle\mathrm{Art}([\mathfrak{p}_{\iota}])\rangle$. Conjecture 3 is equivalent to each statement below.

1. (i)

Let $\chi\in\hat{C}_{\mathfrak{f}}$. We assume that $\chi$ is totally odd and satisfies $\chi(\mathfrak{p}_{\iota})=1$. We denote the character of $\mathrm{Gal}(H/F)$ corresponding to $\chi$ by $\chi^{\prime}$. Then we have

[TABLE] 2. (ii)

Let $\tau\in\mathrm{Gal}(H/F)$. We put

[TABLE]

Here $c$ runs over all ideal classes whose images under the composite map $C_{\mathfrak{f}\mathfrak{p}_{\iota}}\rightarrow C_{\mathfrak{f}}\rightarrow\mathrm{Gal}(H_{\mathfrak{f}}/F)\rightarrow\mathrm{Gal}(H/F)$ equal $\tau$. Then we have

[TABLE]

Proof.

First we note that $\chi\in\hat{C}_{\mathfrak{f}}$ is totally odd if and only if the intermediate field of $H_{\mathfrak{f}}/F$ corresponding to $\mathrm{Art}(\ker\chi)$ is a CM-field. Hence the equivalence Conjecture 3 $\Leftrightarrow$ (i) follows from Lemma 5. If $\chi$ is not totally odd, then we have $L(0,\chi)=0$, so the equation in (i) follows from Theorem 5. Therefore the equivalence (i) $\Leftrightarrow$ (ii) follows from the orthogonality of characters. ∎

Remark 6**.**

Assume that $\mathfrak{p}_{\iota}\mid\mathfrak{f}$. Let $H$ be an intermediate field of $H_{\mathfrak{f}}/F$. First we assume that the real place $\iota$ splits completely in $H$. Then, by Remark 3 and Corollary 1-(ii), we see that the Stark conjecture (2) with $v=\iota$ implies

[TABLE]

where $\phi_{H}$ denotes the composite map

[TABLE]

Strictly speaking, we put $\prod\frac{\exp_{p}(X_{p}(c,\iota))}{\exp(X(c,\iota))}:=\frac{\prod\exp_{p}(X_{p}(c,\iota))}{\prod\exp(X(c,\iota))}$. Next we assume that $\mathfrak{p}_{\iota}$ splits completely in $H$. Then Proposition 6-(ii) states that Conjecture 3 implies

[TABLE]

Here $W$ is the least common multiple of the denominators of $\zeta(0,c^{-1})^{\prime}s$, $\alpha$ is a $\mathfrak{p}_{\iota}$-unit of $H$ defined as

[TABLE]

where we put $\mathfrak{f}_{0}$ to be the prime-to-$\mathfrak{p}_{\iota}$ part of $\mathfrak{f}$ and $\mathrm{Art}_{0}\colon C_{\mathfrak{f}_{0}}\rightarrow\mathrm{Gal}(H/F)$ denotes the Artin map. We note that $\ker\log_{p}$ is generated by rational powers of $p$ and the roots of unity.