数理解析研究所講究録1052 巻 1998 年 166-200

166

ON ABSOLUTE CM-PERIODS II
BY HIROYUKI YOSHIDA
Abstruct. For a CM-field $K$ , Shimura defined the period symbol
by factorizing periods of abelianvarieties with complex multiplication. We define the absolute period symbolusing division valuesof the multiple gamma function and conjecture thatup to the multiplication bycoincides withalgebraic numbers. Taking the action of
into account, we present a refined version of thisconjecture. We show that these conjectures are consistently formulated and discuss various numericalexamples which support our conjectures strongly.
$p_{K}$$g_{K}$$p_{K}$$g_{K}$$\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$In our previous paper [Y2], we formulated a conjecture which gives an expression of
the derivatives of Artin -functions at $s=0$ by CM-periods. However we could not
express CM-periods themselves by such a conjecture.1 In the present paper, we shall givea conjecture which expresses CM-periods by the values of the multiple gamma function atdivision points, and present various numerical examples which support it.
Let us explain our ideas and the contents of this paper more precisely. Let $K$ be a
CM-field,be the set of all isomorphisms of $K$ intoandbe the free abeliangroup generated by. For every, Shimura defined ([S2], [S3]) the CM-period, which is uniquely determined. The fundamental properties ofthe period symbol
will be reviewed in \S 1. For, let us write $a\sim b$ if $b\neq 0$ and. Using , we can write the Chowla-Selberg formula as$\mathrm{L}$$J_{K}$$J_{K}$$\sigma,$$p_{K}(\sigma, \tau)\in \mathrm{C}^{\cross}$$\tau\in I_{K}$$\mathrm{m}\mathrm{o}\mathrm{d} \overline{\mathrm{Q}}^{\mathrm{x}}$$p_{K}$$a/b\in\overline{\mathrm{Q}}$$I_{K}$$\mathrm{C}$$a,$$b\in \mathrm{C}$$p_{K}$(1)$\pi p_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})^{2}\sim\prod^{d-1}\Gamma a=1(\frac{a}{d})^{w}x(a)/2h$.

Here $K$ is an imaginary quadratic field of discriminant $-d,$is the number of roots ofunity contained in $K,$ is the class number of $K$ and is the Dirichlet character whichcorresponds to $K$ .
Now assume that $K$ is abelian over a totally real field $F$ . Put $n=[F : \mathrm{Q}],$ $G=$
and let $\rho\in G$ be (the induced map from) the complex conjugation. Letbethe set of all characters of such that $\chi(\rho)=-1$ . Let
and extend
to an element of, for which we use the same symbol . Our previous conjecture,that is essentially equivalent to a conjecture of Colmez [C], can be brought to the form
$w$

$h$

$\chi$$\hat{G}_{-}$$\mathrm{G}\mathrm{a}1(K/F)$$\chi$$J_{K}$$\sigma_{i}$

(2)

$G$

$J_{F}=\{\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}\}$$\sigma_{i}$,$\prod_{i=1}^{n}p_{K(\tau}\sigma_{i,i}\sigma)\sim\pi^{-n\mu(\mathcal{T})}/2x\in\prod\exp(\frac{\chi(\tau)}{|G|}\hat{c}_{-}\frac{L’(0,x)}{L(0,x)})$1This point will be shown explicitly by an example in \S 7. See also the[Y2], \S 2.

$\tau\in G$$\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{C}}\mathrm{u}\mathrm{S}.\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}$,in the beginnin. of$\mathrm{g}$167where $\mu(\tau)=1$ (resp. $-1$ ) if $\tau--1$ (resp. $\tau=\rho$ ) and $\mu(\tau)=0$ otherwise (cf. \S 3).
A property of the period symbol implies that.
, [Sh2], [Sh3]) expressin terms of the multiple gamma
Shintani’s results (function introduced by Barnes ([Bal], [Ba2]), which will be reviewed in \S 2. Inserting, we shall realize that the right hand side of (2) can be
Shintani’s formula forfactorized naturally in accord with the factorization of the left hand side. Thus, in \S 4, weusing the multiple gamma function so thatshall defineforis equal to the right hand side of (2) and shall predict$\prod_{i=1}^{n}pK(\sigma_{i}, \prime \mathrm{r}\sigma i)\sim\prod_{i=1}^{n}p_{K}\sigma_{i}(\mathrm{i}\mathrm{d}, \sigma_{i^{-1}}\tau\sigma_{i})$$L’(\mathrm{O}, x)$$[\mathrm{s}_{}\mathrm{h}1]$$L’(\mathrm{O}, x)$$g_{K}(\mathrm{i}\mathrm{d}, \tau)\in \mathrm{C}^{\cross}$$\tau\in \mathrm{G}\mathrm{a}1(K/F)$$\prod_{i=1}^{n}gK^{\sigma}i(\mathrm{i}\mathrm{d}, \sigma_{i}^{-1}\tau\sigma i)$

Conjecture A. $p_{K}(id, \tau)\sim g_{K}(id, \tau)$

for $\tau\in G\mathrm{a}l(K/F)$ .

, Conjecture A follows from a result of Anderson [A].
We note that if
We can sharpen Conjecture A in a similar way as in [Y2], \S 5. By virtue of a theorem,of typeof $K$ andof Shimura (cf. \S 1), we can choose Gr\"ossencharacters$m$ so that,
integers$F=\mathrm{Q}$$\lambda_{1},$$\epsilon_{1},$$\epsilon_{2},$$\cdots$$\lambda_{2},$$\cdots$$A_{0}$$\lambda_{q}$$\epsilon_{q},$$\prod_{i=1}L(m/2, \lambda_{i}q)\epsilon_{i}\sim\pi^{Ae}p_{K}(\mathrm{i}\mathrm{d}, \tau)$holds with

$A\in 2^{-1}\mathrm{Z},$$e\in$

Z. Then we expect

Conjecture B. For every$\sigma\in A\mathrm{u}t(\mathrm{C})$, there exis$ts$

a root of unity$( \frac{\prod_{i=1}^{q}L(m/2,\lambda_{i})^{\epsilon_{i}}}{\pi^{A}g_{K}(id,\tau)^{e}})^{\sigma}=\zeta\cdot\frac{\prod_{i=1}^{q}L(m/2,\lambda_{i}\sigma)^{\epsilon}i}{\pi^{A}g_{K^{\sigma}}(id,\sigma-1_{\mathcal{T}}\sigma)e}$$\zeta$such th at.

ex.press

in
mod
In \S 4, we shall also show that Conjecture A is sufficient to..
terms of the multiple gamma function for all.
Let us describe our factorization of (2) explicitly in the simplest case $n=2,$ $[K : F]=2$ ,$\tau=1$ . Let,
be the generator of the group of totally positive units of $F$ . Let$F$...,
be thebe integral ideals which represent narrow ideal classes of . Letandare the archimedean primes ofconductor of $K$ as a class field over $F$ , where$F$ . Let, takeso thatof $F$ . For
be the ideal class group of conductor
belong to the same narrow ideal class and put
and$\sigma,$$\overline{\mathrm{Q}}^{\cross}$$p_{K(\sigma,\mathcal{T})}$$\tau\in I_{K}$$\epsilon>1$$a_{1},$$\alpha_{h_{0}}$$\infty_{1}$$\infty_{2}$$c\in C_{\mathrm{f}}$$\mathrm{f}\infty_{1}\infty_{2}$$C_{\mathrm{f}}$$c$$\alpha_{2}$$\mathrm{f}\infty_{1}\infty_{2}$$a_{\mu}$$a_{\mu}\mathrm{f}$$R(\epsilon, c)=${$z=x+y\epsilon\in(\alpha_{\mu}\mathrm{f})^{-1}|x,$ $y\in \mathrm{Q},$and letRegardas a character of
By Shintani’s formula, (2) can be written as$\chi\in\hat{G}_{-}$$C_{\mathrm{f}}$$0<x\leq 1,0\leq y<1,$ $(z)a_{\mu}\mathrm{f}=c$$\Gamma_{2}(z, (1, \epsilon))$

.

in

.

}.

be the double gamma function.

$i=1 \square p_{K}(\sigma_{i}, \sigma i)\sim\pi-1\frac{1}{2L(0,\chi)}2\exp(\sum_{c\in C_{\mathrm{f}}}\chi(c)z=x+y\epsilon\sum_{\epsilon\in R(,c)}$(3)$C_{\mathrm{f}}$$[ \log\frac{\Gamma_{2}(\mathcal{Z},(1,\epsilon))\Gamma_{2}(Z’,(1,\epsilon’))}{\rho_{2}((1,\epsilon))\rho_{2}((1,\epsilon))},+\frac{\epsilon’-\epsilon}{2}\log\epsilon B_{2(x})$$- \log N(a_{\mu}\mathrm{f})\{\frac{\epsilon+\epsilon’}{4}(B_{2}(x)+B_{2}(y))+B_{1}(x)B_{1}(y))\}])$.

168
Here, formial. (For$\alpha\in F,$$\rho_{2}((1,$$\alpha’$$\epsilon))$denotes its conjugate and $B_{m}(x)$ denotes the m-th Bernoulli polyno, see \S 2.) We define$g_{K}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi-1/2\exp(\frac{1}{2L(0,x)}\sum_{\int C\in C}\chi(c)z=x+y\epsilon\in R\sum_{\epsilon(,c)}[\log\frac{\Gamma_{2}(_{Z},(1,\epsilon))}{\rho_{2}((1,\epsilon))}$,$+ \frac{\epsilon’-\epsilon}{4}\log\epsilon B2(_{X})-\frac{1}{4}\log N(\alpha_{\mu}\mathrm{f})\{\epsilon’B2(X)+\epsilon B2(y)+2B1(X)B_{1}(y))\}])$$g_{K^{\sigma_{2}}}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi-1/2\exp(\frac{1}{2L(0,\chi)}\sum_{c\in c_{\mathrm{f}}}x(C)\sum_{(\mathcal{Z}=x+y\epsilon\in R\epsilon,C)}[\log\frac{\Gamma_{2}(z’,(1,\epsilon’))}{\rho_{2}((1,\epsilon’))}$,$+ \frac{\epsilon’-\epsilon}{4}\log\epsilon B_{2}(x)-\frac{1}{4}\log N(a_{\mu}\mathrm{f})\{\epsilon B_{2}(X)+\epsilon^{J}B2(y)+2B1(_{X)}B_{1}(y))\}])$

where$\sigma_{2}|F\neq \mathrm{i}\mathrm{d}$. Put$a= \frac{1}{L(0,\chi)}\sum_{C\in C;}x(_{C)}\sum\frac{\epsilon’-\epsilon}{4}B_{2}(X)z=x+y\epsilon\in R(\epsilon,c)$.

Then Conjecture A predicts (cf. Lemma 4)(4)$\pi p_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})2\sim\epsilon\prod ac\in c\int\{_{z\in R(,C)}\prod_{\epsilon}\frac{\Gamma_{2}(z,(1,\epsilon))}{\rho_{2}((1,\epsilon))}\}^{\chi(_{C})/L(}0,x)$$0<d\in \mathrm{Q},$Ifis of the formwith
. In our examples discussed in
the text, this quantity will play an important role. Note that
, whereis the relative class number of $K/F$ and is the index of the unit group of $F$ in the unitgroup of $K$ . The analogy of (4) to (1) is now manifest.
In the general case, Shintani’s formula depends not only on the choice of representativesof narrow ideal classes but also on a cone decomposition of
so does our definition of“absolute period”. In \S 5, we shall study the dependence ofand showthat the validity of Conjectures A and does not depend on these auxiliary data. A careful
reader will notice that our definition ofis almost uniquely forced to satisfy thisdemand of canonicality and the factorization of (2).
In $\S 6\sim\S 8,$ we shall discuss numerical examples to convince ourselves of the truth ofConjectures A and B. In \S 6, we examine the case when $F$ is real quadratic, $[K:F]=2,$ $K$
is not normal over Q. In \S 7, we examine the normal closures over of CM-fields discussed
in \S 6, which are cyclic extensions of degree 4 of real quadratic fields. These sections
correpond to the classical case treated by Hecke and can be regarded as a sharpening of
the experiments made in [Y2]. Basic procedures of present experiments are the same as
before and though we shall describe our results fairly explicitly, the reader is advised tosee [Y2], \S 3, \S 4 for more technical details. In \S 8, we shall present an example when $F$ isa cubic field and $[K : F]=2$ . This example is important for us to believe the validity of$F=\mathrm{Q}(\sqrt{d}),$$b\sqrt{d}$$a$$b\in \mathrm{Q}$$a$$L(\mathrm{O}, x)=2h/\dot{i}$$\dot{i}$$\mathrm{R}_{+}^{n};$$g_{K}(\mathrm{i}\mathrm{d}, \mathcal{T})$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\mathrm{B}$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\mathrm{Q}$

$h$

169
Conjectures A and in the general case, since the term $V(c)$ in the definition of(cf. (4.4)) is too simple when $n=2$ .
Our conjectures will produce a number of new problems. The author would like todiscuss some of them on a future occasion.
$\mathrm{B}$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$

Notation and Terminology. Throughout the paper, we fix an algebraicofin C. By an algebraic number field, we understand an algebraic extension ofof finite
degree contained in . We denote by the complex conjugation. For an algebraic number
field $K,$
denotes the set of all isomorphisms of $K$ into anddenotes the free abeliangroup generated by. The ring of integers, the unit group and the class number of $K$
are denoted byand byrespectively. We denote by $I(K)$ the ideal group of$K$ .
For an integral ideal of $K,$
denotes the ideal group of $K$ modulo , i.e. thegroup of fractional ideals which are relatively prime to . We abbreviate
to if noconfusion is likely. For an extension of $K$ of finite degree,denotes the restriction
homomorphism fromtodenotes the inflation homomorphism fromtosuch that, foris the sum of all elements ofwhose restrictions to$K$ coincide with. The norm (resp. trace) map from to $K$ is denoted by $N_{L/K}$ (resp.
. We abbreviate(resp.
) to $N$ (resp. Tr). The relative discriminant of$K$over is denoted by $D(L/K)$ . For an abelian extension of $K$ of finite degree and fora fractional ideal of $K$ , which is relatively prime to the conductor of $L,$
denotes$\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{S}\mathrm{u}\mathrm{r}\mathrm{e}\overline{\mathrm{Q}}$$\mathrm{Q}$$\mathrm{Q}$$\overline{\mathrm{Q}}$$\rho$$J_{K}$$\mathrm{C}$$I_{K}$$J_{K}$$E_{K}$$\mathrm{J}\supset_{K},$$h_{K}$$I_{\mathrm{f}}(K)$$\mathrm{f}$$\mathrm{f}$$\rho|K$$\mathrm{f}$$L$$I_{L}$$I_{L}$$\sigma\in J_{K},$$\rho$${\rm Res}_{L/K}$$I_{K;}\mathrm{I}\mathrm{n}\mathrm{f}_{L/K}$$I_{K}$$\mathrm{I}\mathrm{n}\mathrm{f}_{L/}K(\sigma)$$J_{L}$$L$$\sigma$$\mathrm{T}\mathrm{r}_{L/K})$$N_{K/\mathrm{Q}}$$\mathrm{R}_{K/\mathrm{Q}}$

$L$

$L$

$X$

$( \frac{L/K}{X})$the Artin symbol. We denote by
the idele group of $K$ .
For a totally real algebraic number field $F$ of degree , we denote by
the archimedean primes of $F$ . We identify every
with an element ofand chooseso that it corresponds to the identity embedding of $F$ into C. For $a\in F,$
meansthat is totally positive. We put. By a CM-field, we understanda totally imaginary quadratic extension of a totally real algebraic number field. For a
CM-field $K,$
is called a CM-type if
is the sum of all elements in. A
representationof, where $K$ is a CM-field which is a finite Galois extensionof a totally real algebraic number field $F$ , is called odd if. For a finite group$H$, a subgroup $H$ ofand a representation of , the induced representation fromis denoted by. For elementsof a group,
denotes the
subgroup generated by. For a set $S,$
denotes the cardinality of . For ,, we write $a\sim b$ if $b\neq 0$ and. We put $\mathrm{R}_{+}=\{x\in \mathrm{R}|x>0\}$ .
$K_{A}^{\cross}$$n$$\infty_{1},$$\infty_{2},$$\cdots,$$J_{F}$$\infty_{i}$$\infty_{n}$$\infty_{1}$$a\gg \mathrm{O}$$E_{F}^{+}=\{\epsilon\in E_{F}|\epsilon\gg 0\}$$a$$\Phi\in I_{K}$$\psi$$\Phi+\Phi\rho$$J_{K}$$\mathrm{G}\mathrm{a}1(K/F)$$\psi(\rho)=-\mathrm{i}\mathrm{d}$

$G$

$G$

$\psi$$\psi$$\mathrm{I}\mathrm{n}\mathrm{d}_{H}^{G}\psi$$g_{1},$$g_{1},$$g_{2},$$\cdots,$$b\in \mathrm{C}$$g_{2},$$\cdots,$$\langle g_{1}, g_{2}, \cdots, g_{n}\rangle$$g_{n}$

$S$

$|S|$

$g_{n}$$a$$a/b\in\overline{\mathrm{Q}}$\S 1. Two theorems of Shimura
In this section, we quote two theorems of Shimura on CM-periods which are crucial for
succeeding sections.
Theorem Sl ([S3], Theorem 1.1, [S5], Theorem 32.5). For every CM-field $K$ , there
exists a map:
with the following properti es.
$K$(1) If is a CM-type of andfor every, where is$(K,\Phi)$
any abelian variety definedof typeandistional holomorphic
differential 1-form on A which is multiplied byunder the action of $a\in K\cap End(A)$ .
$p_{K}$$\Phi$$I_{K}\cross I_{K}arrow \mathrm{C}^{\cross}$$\sigma\in\Phi,$$\int_{c}\omega_{\sigma}\sim\pi p_{K}(\sigma, \Phi)$$c\in H_{1}(A, \mathrm{Z})$$over\overline{\mathrm{Q}}$$\mathrm{a}n\mathrm{y}\overline{\mathrm{Q}}- r\mathrm{a}$$\omega_{\sigma}$$a^{\sigma}$

$A$

170
(2)

$p_{K}(\xi_{1}+\xi_{2}, \eta)\sim p_{K}(\xi_{1}, \eta)p_{K}(\xi 2, \eta),$ $p_{K}(\xi, \eta 1+\eta_{2})\sim pK(\xi, \eta 1)pK(\xi, \eta_{2})$$\xi_{2},$$\eta,$$\eta_{1},$$\eta_{2}\in I_{K}$.
for every(3)
and $K\subset L,$(4) $pK(\xi, ResL/K(\zeta))\sim pL(InfL/K(\xi), \zeta)$ if$\xi)\sim p_{L}(\zeta, Inf_{L}/K(\xi))$ if
and $K\subset L,$(5)
if is an isomorphism of $K’$ onto $K$ .
(6)$p_{K}({\rm Res}_{L/K(\zeta}),$$p_{K’}(\gamma\xi, \gamma\eta)\sim pK(\xi, \eta)$$I_{\mathrm{f}}(K)$$\xi_{1}$,$\xi,$$\eta\in I_{K}$$\xi\in I_{K},$$\zeta\in I_{L}$$L$$\xi\in I_{K},$$\zeta\in I_{L}$$L$

is a CM-field.
is a CM-field.

$\gamma$be a CM-type andsuch that$\Phi$$\xi,$.

$p_{K}(\xi\rho, \eta)\sim pK(\xi, \eta\rho)\sim p_{K}(\xi, \eta)^{-1}$Letfor every$\mathrm{f}$be an integral ideal of $K$ . Let$\lambda((\alpha))=\prod_{\sigma\in\Phi}(\alpha^{\sigma}\rho/|\alpha|\sigma)^{t_{\sigma}}$if$\lambda$$\alpha\equiv 1$

be a Gr\"ossencharacter of$\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{f}$,denote the -function attached to
are non-negative integers. Let
when we emphasize the dependence on $K$ .
as
Theorem S2 ([S1], Theorem 2 combined with [S3], Theorem 1.1; or [S5], Thefor every
orem 32.12). For every integer $m$ such th- at
.
, we

where
. We write$t_{\sigma},$$\lambda$$L$$L(s, \lambda)$$\sigma\in\Phi$$L(s, \lambda)$$L_{K}(s, \lambda)$$m-t_{\sigma}.\in 2\mathrm{Z}$$\sigma\in\Phi$$a\mathrm{n}d-t_{\sigma}<m\leq t_{\sigma}$$h\mathrm{a}ve$.

$L(m/2, \lambda)\sim\pi.pK(e/.2\sum_{\Phi\sigma\in}t\cdot\sigma, \Phi)\sigma$ ’

where$e=m[F: \mathrm{Q}]+\sum_{\sigma\in\Phi}t_{\sigma}$.

\S 2. Review of Shintani’s results.

In this section, we recall Shintani’s formula which expresses the derivative of a partial
zeta function of a totally real algebraic number field at $s=0$ in terms of the multiple
gamma function. We follow Shintani [Sh3], [Sh5] faithfully.
, we$x>0$ . For
Let be a natural number and let
define the multiple Riemann zeta function by$s\in \mathrm{C}$$\omega--(\omega_{1}, \omega_{2}, \cdots, \omega_{r})\in \mathrm{R}_{+}^{r},$$r$$\zeta_{r}(_{S,\omega,X})=.\sum_{m_{rr}\Omega=m1\omega 1+m_{2}\omega_{2}+\cdots+\omega}(_{X}+\Omega)-S$.

Here $(m_{1}, m_{2}, \cdots, m_{r})$ extends over all -tuples of non-negative integers. This series
converges when $\Re(s)>r$ and can be continued meromorphically to the whole s-plane;, . We put
is holomorphic except for simple poles at $s=1,2,$$r$$\cdots$$\zeta_{r}(s, \omega, x)$$r$$- \log\rho_{r}(\omega)=\lim_{arrow x+0}\{\frac{\partial}{\partial s}\zeta r(s,\omega, x)|_{s}=0+\log x\}$$\frac{\partial}{\partial s}\zeta_{r}(s,\omega, X)|_{s}=0=\log\{\frac{\Gamma_{r}(x,\omega)}{\rho_{r}(\omega)}\}$,.

can be.
is the -ple gamma function introduced by Barnes
holomorphic
is-plane;wholetheanalytically continued as an entire function toat if is not of the form $x=-(m_{1}\omega_{1}+m_{2}\omega_{2}+\cdots+m_{r}\omega_{r})$ with non-negative integers$\Gamma_{r}(x, \omega)$$([\mathrm{B}\mathrm{a}2])$

$r$

$x$

$x$

$x$

$\Gamma_{r}(x, \omega)-1$$\Gamma_{r}(x, \omega)$171$m_{1},$$m_{2},$$\cdots,$$m_{r}$. For$\dot{i},$$1\leq\dot{i}\leq r$, put$\check{\omega}(\dot{i})=(\omega_{1}, \cdots, \omega_{i-1},\omega_{i+}1, \cdots,\omega_{r})$. Then we havethe difference equation(2.1)$\log\frac{\Gamma_{r}(x+\omega i,\omega)}{\rho_{r}(\omega)}-\log\frac{\Gamma_{r}(x,\omega)}{\rho_{r}(\omega)}=-\log\frac{\Gamma_{r-1}(X,\check{\omega}(\dot{i}))}{\rho_{r-1}(\check{\omega}(\dot{i}))}$.

$1\leq j\leq r$ be an-matrix. We assume $a_{ij}>0$ for all
Let $A=(a_{ij}),$$t_{X}=(x_{1},x_{2},\cdots, x_{n}),$for all and thatand . Let be a column vector such that$x\neq 0.$
put$1\leq\dot{i}\leq n,$$j$$n\cross r$$\dot{i}$$x_{i}\geq 0$$x$$\dot{i}$We.

$A=$ ,
For$s\in \mathrm{C}$$Ax=$ .

, we define(2.2)$\zeta(s, A, x)=\sum_{z_{1},\cdots,z_{r^{=0}}i}^{\infty}\square =n1\{\sum_{=j1}aij(rz_{j}+x_{j})\}^{-S}’,\cdot$

This series converges if $\Re(s)>r/n$ and can be continued meromorphically to the whole$1\leq j,$ $k\leq n,$ $j\neq k$ ,-plane; it is holomorphic at $s=0$ . For $l=(l_{1}, l_{2}, \cdots, l_{r}),$we put$0\leq l_{i}\in \mathrm{Z},$

$s$

(2.3)

$C_{l,j,k}(A)= \int_{0}^{1}\{\prod_{m=1}^{r}(a_{j}m+a_{km}u)^{l_{m}-1} - \prod_{m=1}^{r}a_{j}^{l_{m}}-1\}m\frac{du}{u}$$C_{l}(A)-- \sum_{(j,k)}C_{l,j},k(A..)$.

.

Here $(j, k)$ extends over all ordered pairs of integers such that $1\leq j,$ $k\leq n,$ $j\neq k$ . The
Bernoulli polynomials are defined by$\frac{e^{xt}}{e^{t}-1}=\sum_{m=0}^{\infty}\frac{B_{m}(x)}{m!}tm-1$. Now Shintani’s formulas

are

.

(2.4)

$((0, A, X)= \sum_{=i1}\zeta r(0, A_{i}, (AX)_{i})/n=\frac{(-1)^{r}}{n}\sum\sum_{jli=1}\prod\{\frac{a_{ij}^{l_{j}-1}B_{l}(_{X)}jj}{l_{j}!}nn=1r\}$(2.5)$\frac{\partial}{\partial s}\zeta(_{S,A,x})|_{S}=0=\log\{\prod_{i=1}\frac{\Gamma_{r}((Ax)iAi)}{\rho_{r}(A_{i})}n,\}+\frac{(-1)^{r}}{n}\sum_{\iota}cl(A)\prod_{j=1}^{r}\{\frac{B_{l_{\mathrm{j}}}(x_{j})}{l_{j}!}\}$.

Here $l=(l_{1}, l_{2}, \cdots, l_{r})$ extends over all -tuples of non-negative integers such that $l_{1}+l_{2}+$

.. .

$r$

$+l_{r}=r$.

172
Now let $F$ be a totally real algebraic number field of degree . Let
We putfor $x\in F$ and embed $F$ intoby

$J_{F}=\{\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}\}$$n$$x^{(i)}=x^{\sigma_{i}}$$F\ni xarrow(x^{(1)()}, x2, \cdots, x^{(n)})\in \mathrm{R}^{n}$

For

$r$

.

$\mathrm{R}^{n}$linearly independent vectors$v_{1},$$v_{2},$$v_{r}\in \mathrm{R}^{n}$$\cdots,$

.

, put

$C(v_{1}, v_{2}, \cdots, v_{r})=\{\sum_{i=1}^{r}t_{i}v_{i}|t_{1}, t_{2}, \cdots, t_{r}>0\}$and callan -dimensional open simplicial cone with basis.
There exists a finite set and an $r(j)$ -dimensional open simplicial conewith basis fromfor every such that (cf. [Shl], Proposition 4)$C(v_{1}, v_{2}, \cdots, v_{r})$$r$$v_{1},$$J$$v_{2},$$\cdots,$$v_{r}$$C_{j}$$j$$\mathrm{J}\supset_{F}$(2.6)(disjoint union).

$\mathrm{R}_{+}^{n}=\bigcup_{\epsilon\in E_{F}^{+}}\epsilon(\bigcup_{j\in}{}_{J}C_{j})$We put$C_{j}=C(v_{j1}, v_{j2}, \cdots, v_{jr(j)})$and define an $n\cross r(j)$ -matrix$z\in C_{j}$ , we put$A_{j}$

by,$v_{j1},$ $v_{j2},$ $\cdots,$ $v_{jr(j)}\in \mathrm{J}\supset_{F}$$A_{j}=(v_{jm}^{(l)})$

(2.7)

$a$

$A_{j}$

is$v_{jm}^{(l)}$). For$t_{X(z)=}(X_{1}(z), X2(Z),$ $\cdots,$ $x_{r(j)(_{\mathcal{Z}}))}\in \mathrm{Q}^{r(j)}$$z=x_{1}(z)vj1+X2(_{Z})vj2+\cdots+x_{r}(j)(z)v_{j(j)}r$ ’

For a hactional ideal(the $(l,$ $m)$ component of.

of $F$ , we put$R(C_{j}, a)=\{z\in\alpha\cap C_{j}|0<x_{1}(z), X_{2}(z), \cdots, x_{r(j)}(z)\leq 1\}$.

Letbe the class number of $F$ in the narrow sense and let
be integral$F$
ideals which represent narrow ideal classes. Let be an integral ideal ofand letdenote the ideal class group modulo, take. For
so that andbelong to the same narrow ideal class and put$h_{0}$$a_{1},$$\cdots,$$\alpha_{2},$$a_{h_{\mathrm{O}}}$$\mathrm{f}$$\mathrm{f}\infty_{1}\infty 2\ldots\infty_{n}$(2.8)$R(C_{j}, c)=${ $z\in R(C_{j},$$C_{\mathrm{f}}$$c\in C_{\mathrm{f}}$$(a_{\mu}\mathrm{f})^{-1})|(z)\mathrm{o}_{\mu}\mathrm{f}=c$$\alpha_{\mu}$in$C_{\mathrm{f}}$$c$$a_{\mu}\mathrm{f}$}.

Then we have
(2.9)

{

$z\in(a_{\mu}\mathrm{f})^{-1}\cap C_{j},$$(z)a_{\mu}\mathrm{f}=c$in$C_{\mathrm{f}}$}$= \bigcup_{z\in}R(C_{j^{C)}},\{z+x_{1}v_{j1}+x_{2}v_{j2}+\cdots+xvr(j)jr(j)|0\leq x_{1}, x_{2,(j)}\ldots X_{\Gamma}\in \mathrm{Z}\}$Let$\zeta_{F}(s, c)=\sum_{\mathrm{n}\mathrm{o},,a=c\mathrm{i}C_{\mathrm{f}}}N(a)^{-s}$be the partial zeta function of the class . By (2.9), we have$c$

(2.10)$\zeta_{F}(s, c)=\sum_{j\in Jz\in R}\sum_{(Cj,c)}N(a_{\mu}\mathrm{f})^{-}s\zeta(S, A_{j}, x(z))$

.

.

173
Rom (2.4), (2.5) and (2.10), we obtain the following expressions of
For $j\in J$ and $z\in R(C_{j}, c)$ , we put(2.11)$\zeta_{F}(0, C)$$H_{j}(z)= \sum_{l}[_{m}^{r}(\prod_{=1}j)\{\frac{B_{l_{m}}(x(\mathcal{Z})_{m})}{l_{m}!}\}\mathrm{R}F/\mathrm{Q}\{\prod_{m=1}^{)}(v^{l_{m}1}-)\}jm]r(j$(2.12)$T_{j}(z)= \prod_{k=1}^{n}\{\frac{\Gamma_{r(j)}(_{Z}(k)A_{j}^{()})k}{\rho_{r(j})(A(k))j},\}$(2.13)

and$\zeta_{F}’(\mathrm{o}, C)$

.

,

,

.

$S_{j}(z)= \sum_{l}C_{l}(A_{j})r(\prod_{m=1}j)\{\frac{B_{l_{m}}(x(z)_{m})}{l_{m}!}\}$$l=(l_{1}, l_{2}, \cdots, l_{r(j)})$ extends over all $r(j)$ -tuples of non-negative integers
Here insuch$l_{1}+l_{2}+\cdots+l_{r(j)}=r(j);A_{j}^{(k)}$thatdenotes the k-th row of . For, we put$\sum_{l},$$A_{j}$$c\in C_{\mathrm{f}}$(2.14)$H(c)= \sum_{j\in j}\frac{(-1)^{r(}j)}{n}z\in R(cj_{)})\sum_{C}Hj(Z)$(2.15)$T(c)= \prod_{j\in Jz\in R(}\prod_{cc_{j})},\tau j(Z)\exp\{\sum\frac{(-1)^{r(}j)}{n}j\in J\in R(\sum_{zC_{j},C)}s_{j(z})\}$,.

Then we have,

(2.16)$\zeta_{F}(0, C)=H(c)$(2.17)$\zeta_{F}’(\mathrm{o}, C)=\log T(c)-\log N(\alpha_{\mu}\mathrm{f})\zeta_{F}(\mathrm{o}, C)$.

\S 3. Reformulation of our previous conjecture
Let us begin by recalling a conjecture in [Y2]. Let be a CM-field which is normal overQ. Putand let $\rho\in G$ be the complex conjugation. Let
be the set of
equivalence classes of all irreducible representations ofand letbe the subset ofwhich consists of the equivalence classes of all irreducible odd representations. For,
let $L(s, \eta)$ be the Artin -function attached to . Conjecture 2.2 of [Y2] states$\mathrm{L}$$\hat{G}$$G=\mathrm{G}\mathrm{a}1(L/\mathrm{Q})$$G$$\hat{G}_{-}$$\hat{G}$$\eta\in\hat{G}$$\mathrm{L}$$\eta$

174
Conjecture 1. Let$c$be a conjugacy class in G. Then$\prod_{\sigma\in c}pL(id, \sigma)\sim\pi-\mu(c)/2\prod_{-}\exp\eta\in\hat{G}(\frac{|c|\chi_{\eta}(C)}{[L.\mathrm{Q}]}.\frac{L’(0,\eta)}{L(0,\eta)})$where$\chi_{\eta}$

is the character of$\eta$

,

and
1

if

if

$\mu(c)=\{$ $-1$$0$$c=\{1\}$if,$c=\{\rho\}$,$c\neq\{1\},$ $\{\rho\}$.

Now let $F$ be a totally real algebraic number field such that $[F:\mathrm{Q}]=n$ . Let $K$ be a
Galois extension of $F$ of finite degree. We assume that $K$ is a CM-field and let be the.
normal closure of $K$ over Q. We put
For $\tau\in G$ , we put.
$L$$G=\mathrm{G}\mathrm{a}1(L/\mathrm{Q}),$$H=\mathrm{G}\mathrm{a}1(L/F),$$H_{0}=\mathrm{G}\mathrm{a}1(L/K)$$P_{\tau}= \prod_{Hx\in 0\backslash c}p_{K}(x|K, (\mathcal{T}X|K))$

if $\tau\in H$ . We are going to compute
We note that
Conjecture 1. By Theorem Sl, we have
$P_{\tau}= \prod_{\sigma\in J\kappa}pK(\sigma, \mathcal{T}\sigma)$$P_{\tau}$assuming$h\mathcal{T}x|H0|$$P_{\tau} \sim\square p_{K}(X|K, (\mathcal{T}X|K)x\in G)^{1/||}H0\sim\prod_{x\in Gh\in H_{0}}\square p_{L()^{1/}}X,$$\sim\prod_{x\in G}\prod_{h\in H0}pL(\mathrm{i}\mathrm{d}, x-1h\tau x)^{1/}|H0|\sim\prod_{h\in H0\sigma\in}\prod_{c(h\mathcal{T})}pL(\mathrm{i}\mathrm{d}, \sigma)^{|G|/|0||}HC(h\mathcal{T})|$

Here for $g\in G,$ $C(g)$ denotes the conjugacy class of
formula, we get

$g$

in

$G$

. Applying Conjecture 1 to this$P_{\tau} \sim\prod_{H_{\mathrm{O}}h\in}\{\pi^{-\mu(h)|}\prod_{\hat{G}}\mathcal{T}c|/2|C(h_{\mathcal{T}})||H\mathrm{o}|\mathrm{e}\eta\in-\mathrm{x}\mathrm{p}(\frac{|C(h\mathcal{T}).|x\eta(h\mathcal{T})}{[L.\mathrm{Q}]}\frac{L’(0,\eta)}{L(0,\eta)}\frac{|G|}{|H_{0}||C(h\mathcal{T})|})\}$.

We put $\mu(\tau)=1$ (resp. $\mu(\tau)=-1$ ) if there exists an $h\in H_{0}$ such that $h\tau=1$ (resp.
$h\tau=\rho)$ and $\mu(\tau)=0$ otherwise, using the same letter. Then we have
$\mu$

(3.1)

We have$P_{\tau} \sim\pi^{-\mu(}\prod_{\eta\in\hat{c}}\mathcal{T})|G|/2|H\mathrm{o}|\exp-(\frac{1}{|H_{0}|}\sum_{h\in H0}\chi_{\eta}(h\tau)\cdot\frac{L’(0,\eta)}{L(0,\eta)})$$\sum_{h\in H_{\mathrm{O}}}x_{\eta}(h\mathcal{T})=0$$\{\eta_{1}, \eta_{2}, \cdots, \eta_{l}\}$be the set of allif$\eta|H_{0}$$\eta\in\hat{G}_{-}$

does not contain the trivial representation. Let
contains the trivial representation.
such that$\eta|H_{0}$Then we get
(3.2)

.

.

$P_{\tau} \sim\pi^{-1^{K}:}\prod^{\iota}\mathrm{Q}]\mu(\tau)/2i=1\exp(\frac{1}{|H_{0}|}h\in\sum_{\mathrm{o}}H\chi_{\eta_{i}}(h\tau)\cdot\frac{L’(0,\eta i)}{L(0,\eta_{i})})$

175
as a representation of $H$ . By the Robenius reciprocity,
we can write
and, whereis a representation of $H$ such that
does not contain the trivial representation. Now$\tau\inH$
let us assume that
. Then we haveTake$\omega\in(H\overline{/H_{0})}$-and regard$\omega$$\mathrm{I}\mathrm{n}\mathrm{d}_{H}^{c}\omega\cong\sum_{i}=1ci,\omega\eta_{i}l,$$c_{i,\omega}\in \mathrm{Z}$$\xi_{i}$$\eta_{i}|H\cong\sum H\overline{/H\mathrm{o})}-C\omega\in(i,\omega\omega\oplus\xi_{i}$$\xi_{i}|H_{0}$$\sum_{h\in H_{0}}\chi_{\eta_{i}}(h\tau)=|H_{01\chi_{\omega}()}\omega\in(H\frac{\sum}{/H_{0})}-Ci,\omega\tau$

Since$L(s, \omega)=\prod_{i=1}^{l}L(s, \eta i)^{C_{i}},\omega$.

, we have$\sum_{i=1}^{l}C_{i},\omega^{\frac{L’(0,\eta_{i})}{L(0,\eta_{i})}}=\frac{L’(0,\omega)}{L(0,\omega)}$.

Hence we get$\sum_{i=1}^{l}\frac{1}{|H_{0}|}h\in H_{0}\sum x\eta_{i}(h\mathcal{T})\frac{L’(0,\eta i)}{L(0,\eta_{i})}=\chi_{\omega}\omega\in(H^{\frac{\sum}{/H_{0})}}-(\tau)\frac{L’(0,\omega)}{L(0,\omega)}$.

Inserting this formula in (3.2), we get(3.3)$P_{\tau} \sim\pi^{-[K:\mathrm{Q}}\exp]\mu(\mathcal{T})/2\omega\in(H\frac{\prod}{/H_{0})}-(\chi\omega(\tau)\frac{L’(0,\omega)}{L(0,\omega)})$.

Summing up the calculations above, we obtain:
Proposition 1. Let $K$ be a CM-field which is a Galois extension of $F$ of finite degree.
Put $G=G\mathrm{a}l(K/F)$ and let
be the set ofclasses of all irred cible odd$\tau\inG$, let $\mu(\tau)=1$ (resp. $-1$ ) if $\tau=1$ (resp. $\sim’=\rho$ ) andrepresenta tions of G. For$\mu(\tau)=0$ otherwise. If Conjectholds, then we have$\hat{G}_{-}$$eq\mathrm{u}iv\overline{\mathrm{a}}l\mathrm{e}nce$$\mathrm{u}$$\mathrm{u}re\mathit{1}$

(3.4)

for

$\prod_{\sigma\in J_{K}}pK(\sigma, \tau\sigma)\sim\pi^{-[K:}\prod \mathrm{Q}]\mu(\mathcal{T})/2(\chi\omega(\exp\tau)\frac{L’(0,\omega)}{L(0,\omega)})\omega\in\hat{c}_{-}$

Corollary. Let
and se the same letter$K$. We assume that is abelian over F. Then we haveto$J_{F}=\{\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}\}$$\mathrm{u}$$\sigma_{i}$$\tau\in G$.

for any extension of$\sigma_{i}$$J_{K}$(3.5)$\prod_{i=1}^{n}pK(\sigma i, \mathcal{T}\sigma i)\sim\pi-n\mu(\mathcal{T})/2\omega\in\prod_{-}\hat{G}\exp(\frac{\omega(\tau)}{|G|}\frac{L’(0,\omega)}{L(0,\omega)})$in (3.5) does not depend on the choice of the extenmod
in view of Theorem Sl, (6). Clearly (3.4) includes Conjecture 1 as a special caseWe note thatsion$\sigma_{i}$.

$p_{K}(\sigma_{i,i}\tau\sigma)$$\overline{\mathrm{Q}}^{\cross}$

176
where
. Actually we can prove that Corollary to Proposition 1 implies Conjecture
1 by a similar consideration to the above.
$F=\mathrm{Q}$\S 4. Main conjectures
Throughout this section, let $F$ be a totally real algebraic number field of degree andbe an abelian extension of $F$ ; we assume that $K$ is a CM-field. Let
be
$K$
$F$
the conductor ofas a class field over . Set. We begin by writing theright hand side of (3.5) using Shintani’s formula (2.17). For an integral ideal of $F,$
denotes the ideal class group modulo. For, regard as a characterby the Artin map and letofbe the conductor of . For an integral, we set
ideal
and regard
as a character of .
.
We have$n$$K$$\sim \mathrm{f}\infty_{1}\infty 2\ldots\infty_{n}$$G=\mathrm{G}\mathrm{a}1(K/F)$$\mathrm{f}$$\mathrm{f}\infty_{1}\infty 2\ldots\infty_{n}$$\omega$$\mathrm{f}_{\omega}\infty_{1}\infty_{2}\cdots\infty n$$C_{\overline{\mathrm{f}}}$$\mathrm{f}|\mathrm{f}\sim$$(\hat{G}_{-})_{\mathrm{f}}=\{\omega\in\hat{G}_{-}|.\mathrm{f}\omega$$C_{\mathrm{f}}$$\omega\in\hat{G}_{-}$$\omega$$\omega\in(\hat{G}_{-})_{\mathrm{f}}$$=\mathrm{f}\}$$C_{\mathrm{f}}$$\sum_{\omega\in\hat{G}_{-}}\frac{\omega(\tau)}{L(0,\omega)}L’(\mathrm{o}, \omega)=\sum_{\mathrm{f}1^{\sim}\mathrm{f}\omega(}\sum_{\in\hat{G}_{-)}\mathrm{f}}\frac{\omega(\tau)}{L(0,\omega)}L’(0, \omega)$$= \sum_{\mathrm{f}1\overline{\mathrm{f}}\omega\in}\sum\frac{\omega(\tau)}{L(0,\omega)}(\hat{c}_{-)}\mathrm{f}c\in\sum_{\int}\omega c(c)\zeta_{F}’(0, c)$(It can happen that
(4.1)

Writing$(\hat{G}_{-})_{\mathrm{f}}=\emptyset$

for some$\sim$$\mathrm{f}|\mathrm{f}$.

. ) Thus (3.5) can be written as$\prod_{i=1}^{n}pK(\sigma i, \tau\sigma i)\sim\pi-n\mu(\tau)/2(\exp\frac{1}{|G|}\sum\sum\frac{\omega(\tau)}{L(0,\omega)}\sum_{c\mathrm{f}|^{\sim}\mathrm{f}\omega\in(\hat{c}-)\mathrm{f}\in c_{\mathrm{f}}}\omega(C)\zeta’F(0, C))$$\zeta_{F}’(\mathrm{o}, C)$.

in full using (2.16) and (2.17), we have,$\zeta_{F}’(0, C)=\sum_{=k1}nj\in\sum_{jz\in R(}\sum_{j,C)}\log c\frac{\Gamma_{r(j)}(_{Z}(k)A_{j}^{()})k}{\rho_{r(j})(A(k))j}$(4.2)$+ \sum_{j\in J}\frac{(-1)^{r(}j)}{n}[\sum_{l}c\iota(Aj)\sum_{(z\in Rc_{\mathrm{j}},C)}r\prod_{m=1}^{(j})\{\frac{B_{l_{m}}(x(z)_{m})}{l_{m}!}\}$.

$- \log N(a\mathrm{f}\mu)\sum lz(\sum_{\in Rc_{j},C)}\prod^{r}m=(j)1\{\frac{B_{l_{m}}(X(_{Z)_{m}})}{l_{m}!}\}\mathrm{n}F/\mathrm{Q}\dagger_{m=1}^{r(}\prod^{j)}(v^{l_{m}1}-)jm\}]$We can factorize the right hand side of (4.1) naturally according as the factorization of
the left hand side. Note that Shintani’s formula (4.2) does not depend on the numberingof.
Now we choose
. For
, put
$J_{F}=\{\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}\}$$\sigma_{1}=\mathrm{i}\mathrm{d}$(4.3)$c\in C_{\mathrm{f}}$$G(c)=j \in\sum jz\in R(\sum_{\mathrm{j}c,C)}\log\frac{\Gamma_{r(j)}(_{Z}(1)A_{j}^{()})1}{\rho_{r(j)}(A(1))j},$

,

177

$V(c)= \sum(-j\in J1)r(j)\sum l[\frac{1}{n}\sum(Cl,1,k(A_{j})+Cl,k,1(Aj))-\frac{1}{n^{2}}\sum_{ik=21\leq,k\leq n,i\neq k}cl,i,k(Aj)]n$(4.4)$\cross\sum_{jz\in R(c)}\prod_{mc,,-arrow 1}^{)}r(j\frac{B_{l_{m}}(x(z)_{m})}{l_{m}!}$

(4.5)

,

,

$W(c)=- \frac{1}{n}\log N(a\mathrm{f}\mu)\sum_{Jj\in}(-1)r(j)\sum\sum_{c_{j}lz\in R(,c)}\prod_{m}^{r}(=1j)\frac{B_{l_{m}}(x(z)_{m})}{l_{m}!}(v_{j}^{()})^{l_{m}}m1-1$(4.6)$X(c)=G(c)+V(c)+W(C)$ .

For $\tau\in G$ , we put
(4.7)

.

$g_{K}( \mathrm{i}\mathrm{d}, \mathcal{T})=\pi^{-}\mu(\mathcal{T})/2\exp(\frac{1}{|G|}\sum_{\sim,\mathrm{f}1\mathrm{f}}\omega\in(\sum\hat{G}_{-})\mathrm{f}\frac{\omega(\tau)}{L(0,\omega)}\sum_{C\in c_{\mathrm{f}}}\omega(_{C})x(C))$

In (4.2), (4.4) and (4.5),
is taken over all $r(j)$ -tuples of non-negative integers $l=$such that $l_{1}+l_{2}+\cdots+l_{r(j)}--r(j)$ . Hereafter
will always have this.
meaning.
$\sum_{l}$$(l_{1}, l_{2}, \cdots, l_{r(j)})$$\sum_{l}$Conjecture A. For $\tau\in G$ , we$h\mathrm{a}Vep_{K(d,)g_{K}}\mathrm{j}\mathcal{T}\sim(id, \tau)$.

Roughly speaking, Conjecture A expresses
in terms of the multiple gammafunction. Admitting Conjecture , let us show that we can always expressby the multiple gamma function. Let
be the normal closure of$K$ over Q. We may
assume that
. Takeso that. By Theorem
Sl, we have
. Hence it suffices to considerfor. We may regard as an element of. Let
be thefixed field of. Thenis totally real and is abelian over
. Now Conjecture A
expresses
modby the multiple gamma function.
We are going to formulate Conjecture A in covariant forms under the action of
in a similar way as Conjecture 5.1 of [Y2]. Let $K$ be as above and let be an integral ideal
of $K$ . For, let
be a CM-type of $K$ andbe a character ofsuch that$p_{K}(\mathrm{i}\mathrm{d}, \mathcal{T})$$\mathrm{A}$$p_{K}(\sigma, \mathcal{T})$$\mathrm{m}\mathrm{o}\mathrm{d} \overline{\mathrm{Q}}^{\mathrm{x}},$$\sigma,$$\tau\in I_{K}$$L$$\sigma,$$\tau\in J_{K}$$\tilde{\sigma}\in J_{L}$$\tilde{\sigma}|K=\sigma$$p_{K}(\sigma, \tau)\sim p_{L}(\tilde{\sigma}, \mathrm{I}\mathrm{n}\mathrm{f}L/K(\tau))\sim p_{L}(\mathrm{i}\mathrm{d},\tilde{\sigma}^{-1}\mathrm{I}\mathrm{n}\mathrm{f}_{L/K}(\tau))$$p_{L}(\mathrm{i}\mathrm{d}, \alpha)$$\alpha\in J_{L}$$\langle\alpha, \rho\rangle$$\alpha$$\mathrm{G}\mathrm{a}1(L/\mathrm{Q})$$L$$F_{\alpha}$$F_{\alpha}$$F_{\alpha}$$\overline{\mathrm{Q}}^{\mathrm{x}}$$p_{L}(\mathrm{i}\mathrm{d}, \alpha)$$\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$$\mathrm{q}$$1\leq\dot{i}\leq q$$\Phi_{i}$$\lambda_{i}$$\lambda_{i}((\alpha))=\square (\alpha^{\sigma\rho}/\sigma\in\Phi_{i}|\alpha\sigma|)t^{(}\sigma i)$Here$t_{\sigma}^{(i)}$if$I_{\mathrm{q}}(K)$$\alpha\equiv 1$$\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{q}$.

are positive integers. We assume that there exists an integer $m$ which satisfies$m-t_{\sigma}^{(i)}\in 2\mathrm{Z}$and$-t_{\sigma}^{(i)}<m\leq t_{\sigma}^{(i)}$178
for every and every a$\in\Phi_{i}$$\dot{i}$

(4.8)

Let
that

. By Theorem S2, we have$\Phi_{i})$$L(m/2, \lambda_{i})\sim\pi^{e_{i}/2}pK(\sigma\sum_{\in\Phi_{i}}t^{(}\sigma\sigma i).,$$L$be the normal closure of $K$ overand put$\mathrm{Q}$and,$e_{i}=m[F: \mathrm{Q}]+\sum_{\sigma\in\Phi i}t(i)\sigma$

be a CM-type of$S$$\eta_{i}=\mathrm{I}\mathrm{n}\mathrm{f}_{L}/K(\sum_{\sigma}\in\Phi_{i}t_{\sigma}\cdot\sigma)(i),$${\rm Res}_{L/K}(\Phi_{i}0)=\Phi_{i}$. Then for integers$\sum_{\sigma\in s^{l}}(\sigma i)_{\sigma}+m_{\sigma}^{(i)}\sigma\rho$$\epsilon_{1},$$\epsilon_{2},$$\cdots,$$\epsilon_{q}$$L$. Take.

$\Phi_{i}^{0}\in I_{L}$so$\eta_{i}^{-1}\Phi_{i}^{0}=\sum_{\gamma\eta_{i},\Phi_{i}^{0}}\in\delta\in\gamma-1\delta=$, we have$\prod_{i=1}^{q}pK(\sum t_{\sigma}^{(}\cdot\sigma\sigma\in\Phi ii),$$\Phi_{i})^{\epsilon_{i}}\sim p_{L}(\mathrm{i}\mathrm{d},\sum\sigma\in Sn_{\sigma}\cdot\sigma)$with$n_{\sigma}= \sum_{i=1}^{q}\epsilon_{i}(l_{\sigma}(i)-m_{\sigma}^{(i)})$. Then we haveinteger such that$l_{\sigma}’-m_{\sigma}’$. For $\tau\in G$ , put$\mathrm{I}\mathrm{n}\mathrm{f}_{L/K}(\mathcal{T})=\sum_{\sigma\in S}l_{\sigma}’\sigma+m_{\sigma}’\sigma\rho,$$p_{K}( \mathrm{i}\mathrm{d}, \tau)\sim p_{L}(\mathrm{i}\mathrm{d}, \sum\sigma\in Sn_{\sigma}’\sigma)$$n_{\sigma}’=$. We assume that there exists an

$e$

(4.9)

$\sum_{\sigma\in S}n_{\sigma}\sigma=e\sum_{\in\sigma S}n’\sigma\sigma$.

Then we have$\prod_{i=1}^{q}L(m/2, \lambda i)^{\epsilon}i\sim\pi^{Ae}p_{K}(\mathrm{i}\mathrm{d}, \tau)$,$A= \sum_{i=1}^{q}\frac{\epsilon_{i}e_{i}}{2}$if the left hand side is meaningful.
.
and $C_{j}=C(v_{j1}, v_{j2}, \cdots, v_{jr(j)})$ , put
Forincluding the ordering of the basis though the ordering has(To fix our idea, we define
no effects on our definition as will be shown in \S 5.) Then we have the decomposition$C_{j}^{\sigma}=C(v_{j1}^{\sigma}, v_{j2}\sigma, \cdots, v_{jr(j)}^{\sigma})$$\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$$C_{j}^{\sigma}$$\mathrm{R}_{+}^{n}=\bigcup_{\epsilon\in E_{F^{\sigma}}^{+}}\epsilon(\cup j\in JC^{\sigma})j$

A natural way to define$1\leq\mu\leq h_{0}$ . We then say thatandare given,
and
When

$g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \sigma^{-1}\mathcal{T}\sigma)$$a_{\mu}^{\sigma},$$\tau\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$is to use the data$\alpha_{\mu}$depends only on$\sigma|F$and th at
if
th at
is covariantly defined. Then for every
such th at$Ass\mathrm{u}me$$L(m/2, \lambda_{i})^{\epsilon_{i}}\neq 0$

exists a root of unity

defined.

.

$\epsilon_{i}<0$$\sigma\in G\mathrm{a}l(\overline{\mathrm{Q}}/\mathrm{Q})$$\{g_{K^{\sigma}}(id, \sigma^{-1}\mathcal{T}\sigma)\}\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$$\zeta$$\{C_{j}^{\sigma}\}_{j\in J}$$\{g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \sigma^{-1}\mathcal{T}\sigma)\}\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})\dot{i}Scovar\dot{i}antly$$g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \sigma^{-1}\mathcal{T}\sigma)$$\{C_{j}\}_{j\in J}$

Conjecture B.

for(disjoint union).

$( \frac{\prod_{i=1}^{q}L(m/2,\lambda_{i})^{\epsilon_{i}}}{\pi^{A}g_{K}(id,\mathcal{T})^{e}})^{\sigma}=\zeta\cdot\frac{\prod_{i=1}^{q}L(m/2,\lambda^{\sigma})^{\epsilon}ii}{\pi^{A}g_{K^{\sigma}}(id,\sigma-1\tau\sigma)^{e}}$, there.

are even. For the general case, see [Y2], \S 3. In Conjecture
if allHeredepends on and the determination of can well be called a reciprocity law. A special
case of this problem is discussed in [Y2], \S 6, \S 7.
$t_{\sigma}^{(i)}$$\lambda_{i}^{\sigma}(a)=(\lambda_{i}(a))^{\sigma}$$\mathrm{B},$$\zeta$$\sigma$$\zeta$179
By Theorems Sl and S2, we can easily derive the relation2$\prod_{i=1}^{q}L(m/2, \lambda_{i}^{\sigma})^{\epsilon}i\sim\pi^{A}p_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \sigma-1_{\mathcal{T}\sigma})^{e}$from the assumption (4.9). Hence Conjecture
but also refines it.

$\mathrm{B}$is not only consistent with Conjecture A\S 5. On consistency of the main conjectures
In the formulation of Conjectures A anddepends on the choice of a cone
decomposition
and the representativesof the narrow ideal classes. In thissection, we shall show that if Conjecture A (resp. B) holds for one choice of these data,then it holds for any other choices.
Let us first show that if
is covariantly defined, then$\mathrm{B},$$\{C_{j}\}_{j\in}J$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\{\alpha_{\mu}\}$$\{g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \sigma^{-1}\mathcal{T}\sigma)\}\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$(5.1)

We putby$i=1 \square gK^{\sigma}i(\mathrm{i}\mathrm{d}, \sigma_{i}-1\tau\sigma i)=\pi^{-}\exp(nn\mu(\mathcal{T})/2\frac{1}{|G|}\sum_{\mathrm{f}\mathrm{f}1\omega}\sum_{\in(\hat{c}}\sim-)_{\mathrm{f}}\frac{\omega(\tau)}{L(0,\omega)}\sum_{CC\in \mathrm{f}}\omega(c)\zeta_{F}’(\mathrm{o}, C))$

$G_{i}=\mathrm{G}\mathrm{a}1(K^{\sigma_{i}}/F^{\sigma_{i}})$. Then we have$\omega^{(i)}(X)=\omega(\sigma iX\sigma^{-1})i’ x\in G_{i}$$G_{i}=\sigma^{-1}G\sigma ii$. For$\omega\in\hat{G}_{-}$. We have$\omega^{(i)}(c^{\sigma_{i}})=\omega(C)$,$c\in C_{\mathrm{f}}$,$L(_{S,\omega^{(i}}))=L(S,\omega)$, define.

$\omega^{(i)}\in(\hat{G}_{i})_{-}$.

By definition, we have$g_{K^{\sigma_{i}}}( \mathrm{i}\mathrm{d}, \sigma^{-}\sigma)ii1_{\mathcal{T}=\pi^{-}\mathrm{e}\mathrm{x}}\mu(\sigma\tau i^{-1}\sigma_{i})/2\mathrm{p}(\frac{1}{|G_{i}|}\mathrm{f}\sigma\sum_{i1}\sim_{\sigma_{i}}\mathrm{f}\omega\in(\hat{G})_{\mathrm{f}^{\sigma_{i}}}\sum_{i,-}\frac{\omega(\sigma_{i}^{-1}\tau\sigma_{i})}{L(0,\omega)}\sum_{\mathrm{f}^{\sigma_{i}}}\omega(c)xc\in C(C))$$= \pi^{-\mu(_{\mathcal{T}})}/2\exp(\frac{1}{|G|}\sum_{1\mathrm{f}\mathrm{f}}\sim\omega\sum_{\in(\hat{G}_{-})_{\mathrm{f}}}\frac{\omega(\tau)}{L(0,\omega)}c\in\sum_{C\mathrm{f}\sigma_{i}}\omega(c^{\sigma_{i}}-1)X(C))$

.

Hence
(5.2)

$gK^{\sigma_{i}}( \mathrm{i}\mathrm{d}, \sigma_{i^{-1}}\tau\sigma i)=\pi^{-\mu(\mathcal{T}})/2\exp(\frac{1}{|G|}\sum\sum\frac{\omega(\tau)}{L(0,\omega)}\mathrm{f}|\sim \mathrm{f}\omega\in(\hat{c}-)\mathrm{f}\sum_{c\in c_{\mathrm{f}}}\omega(C)x(C^{\sigma_{i}}))$and it suffices to show
(5.3)
$2L(m/2, \lambda_{i})\neq 0$$\sum_{i=1}^{n}x(C\sigma_{i})=\zeta\prime F(\mathrm{o}, C)$

implies$L(m/2, \lambda_{i}^{\sigma})\neq 0$.

. (cf. [S4], Corollary 6.2.)

180
Using (2.16), we have$\sum_{i=1}^{n}W(C)\sigma i-=\log N(a_{\mu}\mathrm{f})\zeta F(0, C)$.

Hence (5.3) is reduced to
(5.4)

Let

$\sum_{i=1}^{n}V(c^{\sigma_{i}})=\sum_{Jj\in}\frac{(-1)^{r(}j)}{n}\sum C_{l()\sum_{j}}Aljz\in R(c,C)mr(\prod^{j)}\{1\frac{B_{l_{m}}(x(Z)_{m})}{l_{m}!}\}=$

be the $n\cross r(j)$ -matrix determined by the cone
. Then (5.4) follows$A_{j,i}$$(l_{1}, l_{2}, \cdots, l_{r(j)})$Fix $j\in J$ and $l=$.

$\mathrm{h}\mathrm{o}\mathrm{m}$

.

(5.5)

We

$C_{j}^{\sigma_{i}}$.

$\sum_{i=1}^{n}(\sum_{k=2}^{n}(C_{l},1,k(A_{j,i})+cl,k,1(Aj,i))-\frac{1}{n}\sum_{\neq 1\leq p)\leq kn,pk}c_{l},k(\mathrm{P},j)))-Ai-c_{l}(A_{j})$$\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{y}-$have,$\frac{1}{n}\sum n$$i=11 \leq_{\mathrm{P}^{k}},\leq\sum_{knp\neq},\mathit{0}_{l},k(p,j,i)A=c_{l}(A_{j})$,$\sum_{i=1}^{n}\sum_{k=2}nCl,1,k(A_{j,i})+C_{l,k,1}(A_{j,i})=\sum_{=i1}n(\sum_{k1\leq\leq n,k\neq i}(C_{l,k},i(A_{j})+C_{l,i,k}(A_{j})))=2C_{l}(A_{j})$Hence (5.1) is established.
Now let us consider the dependence of
on
and
open
simplicial.
familya finiteofcones with basis in
We call
if everyis contained in someand if$\{C_{j}\}_{j}\in J$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\mathrm{R}_{+}^{n}=\bigcup_{\epsilon\in E}+^{\epsilon(\cup}Fl\in Lc_{l}’)$$\bigcup_{j\in}{}_{J}C_{j}=\bigcup_{l\in L}c_{l}’$Lemma 2. Let
asis insuch th at$\{C_{j}\}_{j\in J}$$b$$\{C_{l}’\}l\in L$$C_{j}$$C_{l}’$In this case, we have

. Let
be
a refinement of$\{C_{l}^{\gamma}\}l\in L$$\mathrm{J}\supset_{F}$$\{C_{j}\}_{j}\in J$$\{a_{\mu}\}$.

and$\{D_{i}\}_{i\in I}$(disjoint union).

.
\’oe two finite families of open simplicial cones with$\mathfrak{O}_{F}$$\mathrm{R}_{+}^{n}=\bigcup_{\epsilon\in E_{F}^{+}}\epsilon(\bigcup_{j\in}{}_{j}Cj)$(disjoint union),(disjoint union).

$\mathrm{R}_{+}^{n}=\bigcup_{\epsilon\in E_{F}^{+}}\epsilon(\cup i\in IDi)$

Then there exists a finite familyof open simplicial cones with asis inand afamily of units
such th at
is a finement ofand th at.
is a refinement of$b$$\{C_{l}’\}_{l\in L}$$\{\epsilon_{l}\}_{l\in L},$$\{\epsilon\iota C_{l}’\}l\in L$$\epsilon_{l}\in E_{F}^{+}$$\{C_{l}’\}_{l}\in L$$r\mathrm{e}$$\{C_{j}\}_{j\in J}$$\{D_{i}\}_{i\in I}$

Proof. We have
(5.6)

$C_{j}= \bigcup_{i\in I}\bigcup_{\epsilon\in E_{F}^{+_{c_{j}}}},\cap\epsilon Di\neq\emptyset(C_{j}\cap\epsilon D_{i})$(disjoint union),$\mathfrak{O}_{F}$

181
(5.7)

$D_{i}= \cup j\in J\bigcup_{\epsilon i\epsilon F}-\in E^{+},D\cap 1cj\neq\emptyset(Di^{\cap\epsilon C_{j}})-1$(disjoint union).

Fix $j\in J$ and. Let us show that there are only finitely manywhichsatisfy. Set $H=\{(x_{1}, x_{2}, \cdots, x_{n})\in \mathrm{R}_{+}^{n}|x_{1}x_{2n}\ldots X=1\}$ . Then
is a discrete subgroup of $H$ . For an open simplicial cone $C$ , put $C^{1}=C\cap H$ . Clearly
if and only if. It suffices to show that the closures
and$C=C(v_{1},v_{2},\cdots,v_{r})$are compact. Let
be an -dimensional open simplicial cone with
$1\leq k\leq r$ . If
, then we see immediately that is of
the form $x= \sum_{k=1}^{r}t_{k}v_{k},$ $0\leq t_{k}\leq 1/\sqrt[n]{v_{k1}v_{k2}v_{kn}}.$ Henceis compact.
By Shintani [Shl], Corollary to Lemma 2,
can be written as a disjoint unionof finitely many open simplicial
es withis in. Now the assertion follows from
(5.6) and (5.7).
$\dot{i}\in I$$\epsilon\in E_{F}^{+}$$C_{j}\cap\epsilon D_{i}\neq\emptyset$$E_{F}^{+}$$C_{j}\cap\epsilon D_{i}\neq\emptyset$$C_{j}^{1}\cap\epsilon D_{i}^{1}\neq\emptyset$$\overline{c_{j}^{1}}$$\overline{D_{i}^{1}}$$r$$x\in\overline{C^{1}}$$v_{k}=(v_{k1}, v_{k2}, \cdots, v_{kn})\in \mathrm{R}_{+}^{n},$$x$$\overline{C1}$$C_{j}\cap\epsilon D_{i}$ba..s$\mathrm{c}\mathrm{o}\mathrm{n}\backslash$$\iota\supset_{F}$By Lemma 2, to see the dependence ofon, it suffices to consider the.
following two operations.
(1) Replace
by a refinement.
(2) Replace one member
of
by
with
.
Now fix
and takeso that the narrow ideal class of is the same as that of. For $j\in J$ and for, put$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\{C_{j}\}_{j}\in J$$\{C_{\iota}’\}\iota\in L$$\{C_{j}\}_{j\in}J$$C_{j}$$c\in C_{\mathrm{f}}$$\{C_{j}\}_{j\in J}$$\epsilon_{j}\in E_{F}^{+}$$\epsilon_{j}C_{j}$$c$$\alpha_{\mu}$$1\leq\dot{i}\leq n$$a_{\mu}\mathrm{f}$$\zeta^{(i)}(_{S}, c_{j}, \alpha)\mu\sum_{z}=z\in(a_{\mu}\mathrm{f})-1\cap C_{j},()a\mathrm{f}\mu C\equiv(_{Z}(i))-S$.

Here, since $z^{(i)}>0,$ $(z^{(i)})-S$ is naturally defined by $\exp(-s\log z(i))$ taking the principal
branch ofandthe equality in . By (2.9), we have$\log z^{(i)}$$\equiv \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$$C_{\mathrm{f}}$$\zeta^{(i)}(_{S}, Cj, a)\mu=z\in R\sum_{(c_{j},c)}\zeta(s, A_{j}(i),$$X(Z(i)))$.

By (2.4) and (2.5), we obtain$r(j)$

(5.8)

(5.9)

,

$\zeta^{(i)}(\mathrm{o}, C_{j}, \alpha)\mu=(-1)^{r(j})\sum_{(z\in RCjc)},\sum_{l}\prod_{m=1}\frac{B_{l_{m}}(x(Z)_{m})}{l_{m}!}(v)^{\iota_{m}-1}jm(i)$$\frac{d}{ds}\zeta^{(i)}(_{S,ca)}j,\mu|_{s=0=\sum_{R(},\mathrm{o}\mathrm{g}}z\in c_{j}C)\mathrm{l}\frac{\Gamma_{r(j)}(Z^{(i)},A_{j}^{()})i}{\rho_{r(j})(A(i))j}$

For $!\leq i,$ $k\leq n,\dot{i}\neq k$ , we put.

$\zeta^{(i,k)}(_{S,c_{j}}, a)\mu\sum_{C}=(zZ)^{-}z\in(a_{\mu}\mathrm{f})-1_{\cap}c_{j},(z)\alpha \mathrm{f}\mu\equiv(i)(k)s$.

182
Using (2.9), (2.4) and (2.5) similarly to the above, we obtain(5.10),$\zeta^{(i,k)}(\mathrm{o}, \mathit{0}_{j’\mu}\alpha)=\frac{(-1)^{r(}j)}{2}\sum_{jz\in R(cC)},\sum\prod^{r(}lm=1j)\frac{B_{l_{m}}(X(Z)_{m})}{l_{m}!}\{(v^{(i)})^{\iota 1}jmm^{\neg}+(v_{jm}^{(k)-})l_{m}1\}$$\frac{d}{ds}\zeta^{(i,k)}(_{S,c_{j}}, a)\mu|_{S}=0=\sum_{Cz\in R(C_{j},)}$(5.11)$+ \frac{(-1)^{r(}j)}{2}\sum_{l}(c_{\iota,i,k}(A_{j})+C_{l,k},i(A_{j}))\prod_{=m1}^{r}\frac{B_{l_{m}}(x(_{Z)_{m}})}{l_{m}!}](j)$.

Therefore we have
(5.12)$G(c)=j \sum_{\in J}\frac{d}{ds}\zeta(1)(s, c_{j,\mu}a)|s=0$,(5.13)$V(c)= \sum_{j\in j}(\frac{2}{n}\sum_{k=2}(\frac{d}{ds}\zeta(1,k)(s, Cj, \alpha)\mu|_{s=}0-\frac{d}{ds}\zeta(1)(_{S}, Cj, a)\mu|S=0-\frac{d}{ds}\zeta^{(k})(_{S}, C_{j’\mu}a)|_{\mathit{8}0}=)n$,$- \frac{1}{n^{2}}\sum_{i1\leq i,k\leq n,\neq k}(\frac{d}{ds}\zeta^{(i}’ k)(s, \mathit{0}_{j}, a)\mu|S=0^{-}\frac{d}{ds}\zeta(i)(_{S}, cj, \alpha)\mu|_{S}=0^{-}\frac{d}{ds}\zeta(k)(_{S,C_{j,\mu}}a)|_{s}=0))$(5.14)$W(c)=- \frac{1}{n}\log N(\alpha_{\mu}\mathrm{f})\sum_{\in jJ}\zeta(1)(\mathrm{o}, Cj, a)\mu$.

does not change when we replace
By (5.12) $\sim(5.14)$ , it is evident thatis ofby its refinement. Also when we change the ordered choice of thedoes not change.
. We havewithis replaced by
Now assume that$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\mathrm{b}\mathrm{a}s$$C_{j}$$\epsilon C_{j}$$C_{j},$$\{C_{j}\}_{j\in J}$$g_{K}(\mathrm{i}\mathrm{d}, \tau)$$\epsilon\in E_{F}^{+}$$\zeta^{(i)}(S, \epsilon cj, a)\mu=(\epsilon^{()})^{-s}i\zeta(i)(s, C_{j}, \alpha_{\mu})$,$\zeta^{(i,k)}(s, \epsilon \mathit{0}_{j}, \alpha)\mu=(\epsilon^{(}\epsilon)^{-s}i)(k)\zeta(i,k)(_{S}, C_{j}, \alpha_{\mu})$.

by (5.8) and (5.10).
Note that we have
By a direct computation using $(5.12),(5.13)$ and (5.14), we see that $G(c)+V(c)$ is changed
to$\zeta^{(i,k)}(\mathrm{o}, Cj’ a_{\mu})=\frac{1}{2}(\zeta^{(i)}(\mathrm{o}, C_{j}, \alpha_{\mu})+\zeta^{(k)}(0, C_{j}, a_{\mu}))$$G(_{C})+V(_{C})- \frac{1}{n}\log\epsilon^{(}\sum_{k1}^{n}1)\zeta(k)(0, cj’ a_{\mu})=$183
and that $W(c)$ does not change. Therefore$g_{K}(\mathrm{i}\mathrm{d}, \tau)$

is multiplied by$(\epsilon^{(1)})^{-a}$$a= \frac{1}{|G|}\sum_{\mathrm{f}\int 1\omega}\sum_{\in(\hat{G}_{-)}\int}\frac{\omega(\tau)}{L(0,\omega)}\sim c\in\sum_{c\mathrm{f}}\omega(c)\zeta(0,\mathit{0}_{j,\mu}\alpha)$

where

,

(5.15)$\zeta(s, c_{j,\mu}a)=z\in(a_{\mu}\mathrm{f})-1\cap\sum N(_{Z})-s=\sum_{(Cj,(z)a\mu \mathrm{f}\equiv cz\in RC_{j},C)}\zeta(S, A_{j},x(Z))i$

Since$\zeta(0, C_{j}, a_{\mu})\in \mathrm{Q}$, we see easily that.

; we find similarly that
holds for, it also holds when we$a\in \mathrm{Q}$$g_{K^{\sigma_{i}}}(\mathrm{i}\mathrm{d}, \sigma_{i}^{-1}\tau\sigma_{i})$is multiplied by. Hence if Conjecture
replace a member
by.
Next suppose thatis replaced bywhereis a totally positive element of $F$ .
Since the consistency of the main conjectures with the change of a cone decomposition isalready established, it suffices tosimultaneous change3$(\epsilon^{(i)})^{-a}$$C_{j}$$\mathrm{B}$$\{C_{j}\}_{j\in J}$$\epsilon C_{j}$$(\alpha_{\mu})\alpha_{\mu}$$\alpha_{\mu}$$\alpha_{\mu}$consid.e$\alpha_{\mu}arrow(\alpha_{\mu})\alpha_{\mu}$,$\mathrm{r}\mathrm{t}.\mathrm{h}\mathrm{e}$:$\{C_{j}\}_{j}\in Jarrow\{(\alpha)\mu C_{j}-1\}j\in J$.

We have.
$\zeta^{(i)}(S, \alpha_{\mu}^{-}C_{j}1, (\alpha\mu)a_{\mu})=$

.

$.|$

$(z^{(i)})^{-s}$$\sum$$z\in((\alpha_{\mu})a\mathrm{f}\mu)-1\cap\alpha^{-}c_{j}(\mu)z)1(\alpha)\mu a_{\mu}\mathrm{f}\equiv c$$=)-1_{\cap C_{j}}(z \sum_{z\in(a_{\mu}\mathrm{f})\alpha_{\mu}\mathrm{f}\equiv C},((\alpha_{\mu})^{-}(i)1(i)s(i)s(i)c)z)^{-}=(\alpha_{\mu j,\mu})\zeta(s,a.\cdot$

Similarly we obtain$\zeta^{(i,k})(s, \alpha_{\mu}^{-}c_{j}1, (\alpha_{\mu})a\mu)=(\alpha_{\mu\mu}\alpha(i)(k))^{S}\zeta^{(k)}i,(_{S}, C_{j}, a_{\mu})$.

By (5.12), we see that $G(c)$ changes to$G(_{C})+jJ \sum_{\in}10..\mathrm{g}\alpha_{\mu j,\mu}^{()}\zeta^{(1)}1(0, C\alpha)$.

By (5.13), we see that $V(c)$ changes to$V(c)- \sum_{\in jJ}[\log\alpha_{\mu}^{(1)(1)}\zeta(0, C_{j,\mu}\alpha)-\frac{1}{n}\log\alpha_{\mu}(1)k=1\sum^{n}\zeta^{(}k)(\mathrm{o}, Cj, \alpha)\mu$$- \frac{1}{n}\log N(\alpha)\mu\zeta(1)(0, c_{j}, \alpha)\mu+\frac{1}{n^{2}}\log N(\alpha)\mu k\sum_{1=}^{n}\zeta^{(k})(0, o_{j,\mu}a)]$By (5.14), we see that $W(c)$ changes to$W(c)- \frac{1}{n}\sum_{Jj\in}\log N(\alpha_{\mu})\zeta^{()}1(\mathrm{o}, c_{j,\mu}\alpha)$

3We may assume that$(\alpha_{\mu})^{-1}cj$

has a basis in$\mathrm{J}\supset_{F}$

.

.

.

184
Hence $X(c)$ changes to$X(c)+ \log\alpha_{\mu}^{(}\zeta 1)F(0, C)-\frac{1}{n}\log N(\alpha\mu)\zeta F(0, C)$since$\zeta_{F}(0, C)=\frac{1}{n}\sum_{j\in J}\sum_{k=}^{n}1\zeta^{(}k)(0, C_{j}, \alpha_{\mu})$$(\alpha_{\mu}^{(1)}N(\alpha\mu)^{-1/}n)^{b}$. Therefore,

is multiplied by$g_{K}(\mathrm{i}\mathrm{d}, \tau)$where$b= \frac{1}{|G|}\sum_{\mathrm{f}\mathrm{f}1^{\sim}\omega\in(}\sum\frac{\omega(\tau)}{L(0,\omega)}\hat{c}_{-)_{\mathrm{f}}}\sum_{\mathrm{C}\in c_{\mathrm{f}}}\omega(_{C)}\zeta_{F}(\mathrm{o}, c)$$(5.\cdot 1.\cdot 6.)$$= \frac{1}{|G|}\sum_{\omega\in\hat{c}_{-}}\omega(\mathcal{T})=$

which is the sum of given by (5.15) over all $j\in J$ . In other words, $b=\mu(\tau)/2$ . We findsimilarly thatis multiplied by
Therefore Conjecture.
holds also for the new dataandof Conjecture a reciprocity map. We deduce
Let us call the map
the above considerations the following:$a$$(\alpha_{\mu}^{(i)/n}N(\alpha)^{-}\mu 1)^{b}$$g_{K^{\sigma_{i}}}(\mathrm{i}\mathrm{d}, \sigma_{i}^{-}\tau\sigma i)1$$\mathrm{B}$$\{\alpha_{\mu}^{-1}C_{j}\}_{jJ}\in$$(\alpha_{\mu})a_{\mu}$$\sigmaarrow\zeta=\zeta(\sigma)$$\mathrm{h}\mathrm{o}\mathrm{m}$$\mathrm{B}$

Proposition 3.
and(1) The reciprocity
does not depend on the ordered choice of the basis ofby its reffiement.
does not change when we replace(2) When we change a member
changes to
by
of
with
$C_{j}$$m\mathrm{a}p$$\{C_{j}\}_{j\in J}$$\{C_{j}\}_{j\in J}$$C_{j}$$\{\frac{(\epsilon^{a})^{\sigma}}{(\epsilon^{\sigma})^{a}}\}^{e}\zeta(\sigma)$, where$a\in \mathrm{Q}$(3) When we change an ideal
totally positive element$b\in \mathrm{Q}$$\alpha_{\mu}$is given by$a_{\mu}$

toof $F,$$(\mathit{5}.\mathit{1}\mathit{5}f$$(\alpha_{\mu})a_{\mu}$$\zeta(\sigma)$

and$\epsilon C_{j}$$\epsilon\in E_{F}^{+},$$\zeta(\sigma)$.
$\{C_{j}\}_{j\in}J$changes totowith a$\{(\alpha_{\mu})^{-}1C_{j}\}j\in J$$\{\frac{(\alpha_{\mu}^{\sigma})^{b}(N(\alpha_{\mu})^{b/n})^{\sigma}}{(\alpha_{\mu}^{b})^{\sigma}N(\alpha)^{b/}\mu n}\}^{e}\zeta(\sigma)$, whereis given by (5.16).

We also have to consider the dependence on the choice of $m$ andin Conjecture B.
$m$
The consistency with changes ofandfollows easily from considerations in [Y2], \S 7,
using Deligne’s conjecture proved by Blasius [B1].
$\lambda_{i}$$\lambda_{i}$\S 6. Numerical examples I: Quadratic extensions of real quadratic fields
Let $F$ be a real quadratic field. We are going to write Conjecture more explicitly.
be the generator of
denotes its conjugate. Let
For an element or an ideal of $F,$
$C_{2}=C(1,\epsilon)$.
. Let be an integral ideal of $F$ and. We may take $C_{1}=C(1),$Following Shintani [Sh2], we put$\mathrm{B}$

$z$

$z’$

$E_{F}^{+}$$\epsilon>1$$c\in C_{\mathrm{f}}$$\mathrm{f}$$R(\epsilon, c)=${$z=x+y\epsilon\in(\alpha_{\mu}\mathrm{f})^{-1}|x,$ $y\in \mathrm{Q},$$0<x\leq 1,0\leq y<1,$ $(z)a_{\mu}\mathrm{f}=c$in$C_{\mathrm{f}}$

},

185
where

$a_{\mu}$

is taken as before. We have,$\sum_{z\in R(c1,c)}\log\frac{\Gamma_{1}(Z,1)}{\rho_{1}(1)}=\sum_{-r^{-}}\mathrm{l}z=x+y\epsilon\in R(\epsilon,C),0\mathrm{o}\mathrm{g}\frac{\Gamma_{1}(z,1)}{\rho_{1}(1)}$$\sum_{z\in R(c2,c)}\log\frac{\Gamma_{2}(_{\mathcal{Z}},(1,\epsilon))}{\rho_{2}((1,\epsilon))}=z=x+y\epsilon\in R(\epsilon,C)\sum_{10<x\leq,0<y<1},\log\frac{\Gamma_{2}(_{Z},(1,\epsilon))}{\rho_{2}((1,\epsilon))}$.

$+ \sum_{=z=x+y\epsilon\in R(\epsilon,c),y0}\log\frac{\Gamma_{2}(_{Z+}\epsilon,(1,\epsilon))}{\rho_{2}((1,\epsilon))}$’By the difference equation (2.1), we obtain(6.1)$G(c)= \sum_{z\in R(\epsilon,c)}\log\frac{\Gamma_{2}(_{Z},(1,\epsilon))}{\rho_{2}((1,\epsilon))}$Here, for simplicity, we write

$z$

for

$z^{(1)}$.

. We have $C_{1}(A_{1})=0$ . Hence$V(c)= \frac{1}{4}\sum_{\iota}c_{l()}A_{2}z\in R(\sum c_{2^{C)}})m\prod_{1=}\frac{B_{l_{m}}(x(\mathcal{Z})_{m})}{l_{m}!}2$

If

$A=,$

$\alpha,$$\beta\in F,$$\alpha,$$\beta\gg 0$,$c_{(2,0)()=}A \frac{\alpha’\beta-\alpha\beta\prime}{\beta\beta’}\log\frac{\beta’}{\beta}$.

, then$c_{(1,1)}(A)=0$ ,$C_{(0,2)}(A)= \frac{\beta’\alpha-\beta\alpha\prime}{\alpha\alpha’}\log\frac{\alpha’}{\alpha}$Hence we obtain.

$V(c)= \frac{1}{2}(\epsilon’-\epsilon)\log\epsilon\sum,\frac{B_{2}(x(_{Z)_{1}})}{2}z\in R(C2^{C})$

Then we easily get
(6.2)

.

$V(c)= \frac{1}{4}(\epsilon’-\epsilon)\log\epsilon\sum_{)\mathcal{Z}=x+y\epsilon\in R(\epsilon,c}B_{2}(X)$

We also see easily that
(6.3)

$W(c)=- \frac{1}{4}\log N(a\mathrm{f}\mu)z=x+y\epsilon\sum_{)\in R(\epsilon,C}(\epsilon’B_{2}(x)+\epsilon B_{2}(y)+2B_{1}(x)B1(y))$

.

.

186
Summing up, we have$X(_{C})= \sum_{\epsilon z=x+y\epsilon\in R(,c)}\{$$\log\frac{\Gamma_{2}(_{\mathcal{Z}},(1,\epsilon))}{\rho_{2}((1,\epsilon))}+\frac{1}{4}(\epsilon’-\epsilon)\log\epsilon B_{2}(x)$(6.4)$- \frac{1}{4}\log N(\alpha \mathrm{f}\mu)(\epsilon JB2(X)+\epsilon B_{2}(y)+2B_{1}(x)B_{1}(y))\}$

Shintani’s formulas (2.16) and (2.17) simplify in this case to(6.5)$([\mathrm{S}\mathrm{h}4],$(16) $,$(17)$\zeta_{F}(0, C)=\sum_{\epsilon z=x+y\epsilon\in R(,c)}\{\frac{\epsilon+\epsilon’}{4}(B_{2}(_{X})+B2(y))+B1(x)B1(y))\}$

.

$)$

,

$\zeta_{F}’(\mathrm{o}, c)=\sum_{\epsilon z=x+y\epsilon\in R(,c)}\{\log\frac{\Gamma_{2}(\mathcal{Z},(1,\epsilon))\Gamma_{2}(z;,(1,\epsilon’))}{\rho_{2}((1,\epsilon))\rho_{2}((1,\epsilon))},+\frac{\epsilon’-\epsilon}{2}\log\epsilon B2(X)\}$(6.6)$-\log N(\alpha \mathrm{f}\mu)\zeta F(\mathrm{o}, C)$.

By (5.3), we get$X(_{C^{J}})= \sum_{\epsilon z=x+y\epsilon\in R(,c)}\{$$\log\frac{\Gamma_{2}(Z’,(1,\epsilon))}{\rho_{2}((1,\epsilon’))}+\frac{1}{4}(\epsilon;-\epsilon)\log\epsilon B_{2}(x)$(6.7)$- \frac{1}{4}\log N(a\mathrm{f}\mu)(\epsilon B_{2}(x)+\epsilon’B_{2}(y)+2B_{1}(X)B_{1}(y))\}$.

The following Lemma would be of some interest.
Lemma 4. We

and$h \mathrm{a}ve\sum z=x+y\epsilon\in R$ $( \epsilon,c)2(BX)=\sum_{z=x+,\in}y\epsilon R(\epsilon,C)B_{2}(y)$$W(c)=- \frac{1}{2}\log N(\alpha_{\mu}\mathrm{f})\zeta_{F}(0, C)$.

.

. Then$y+x(\epsilon+\epsilon’)+(1-x)\epsilon$ satisfies
so that $0<x_{1}\leq 1$ ,in . We take.
. Then we havemod and put $y_{1}=1-x,$
Put $\varphi(z)=z_{1}$ . Clearly the mapping is injective, hence bijective. Now thefollows from $B_{2}(1-X)=B_{2}(x)$ .

Proof. It suffices to show the first assertion. Let$(\epsilon^{-1}z+\epsilon)a_{\mu}\mathrm{f}=c$$x_{1}\equiv y+x(\epsilon+\epsilon’)$$z=x+y\epsilon\in R(\epsilon, c)$$C_{\mathrm{f}}$$x_{1}\in \mathrm{Q}$$z_{1}\in R(\epsilon, c)$$z_{1}=x_{1}+y_{1}\epsilon$$\mathrm{Z}$$\epsilon^{-1}z+\epsilon=$$\varphi$$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}_{0}.\mathrm{n}$in general, we haveThough it can well be the case that
not examined this problem yet.
be the conductor
Let $K$ be a totally imaginary quadratic extension of $F$ and let$F$ . Let$K$
corresponds
to $K$ .
which
be
the
Hecke
character
of
over
a
class
field
of as
Then (4.7) can be written as$W(c)=- \frac{1}{n}\log N(\alpha \mathrm{f}\mu)\zeta F(0, C)$$\mathrm{f}\infty_{1}\infty_{2}$$\chi$(6.8)$F_{A}^{\cross}$$\cdot$$g_{K}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi-1/2\exp(\frac{1}{2L(0,x)}\sum_{c\in c_{\mathrm{f}}}x(C)X(c))$

,

187
(6.9)

if

$g_{K^{\sigma}}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi^{-1/2}\exp(\frac{1}{2L(0,x)}\sum\chi(c)x(_{C’))}c\in c_{\mathrm{f}}$induces the’ nontrivial automorphism on $F$ . In this case, we have$\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$.

$g_{K}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})g_{K}\sigma(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi^{-1}\exp(\frac{L’(\mathrm{o},\chi)}{2L(0,x)})$Let$F=\mathrm{Q}(\sqrt{d}),$$\xi=\sqrt{x+y\sqrt{d}}\dot{i},$Defineletand$\sigma\in J_{K}$$\lambda_{a,b}^{(1)}$$0<d\in \mathrm{Q}$and $x+y\sqrt{d},$ $x,$$\xi’=\sqrt{x-y\sqrt{d}}\dot{i},$$K=\mathrm{Q}(\xi)$$y\in \mathrm{Q}$be a totally positive element. Set. We assume that $K$ is not normal over Q.

. Let be an integral ideal of $K$ . For positive integers
bybe characters ofsuch that$\xi^{\sigma}=\xi’$$\mathrm{q}$$\lambda_{a,b}^{(2)}$$a$$I_{\mathrm{q}}(K)$$\lambda_{a,b}^{(1)}((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})^{a}(\frac{\alpha^{\sigma\rho}}{|\alpha^{\sigma}|})^{b}$,$\lambda_{a,b}^{(2)}((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})^{a}(\frac{\alpha^{\sigma}}{|\alpha^{\sigma\rho}|})^{b}$$\alpha\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{q}$,,$\alpha\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{q}$.

By Theorem S2, we have
$L_{K}(1, \lambda^{(1})a,bp_{K})(4+a+b)/\sim\pi 2(a\cdot \mathrm{i}\mathrm{d}+b\cdot\sigma, \mathrm{i}\mathrm{d}+\sigma)$,.

$L_{K}(1, \lambda^{(2}))a,bpK(\sim\pi/(4+a+b)2\mathrm{d}a\cdot \mathrm{i}+b\cdot\sigma\rho, \mathrm{i}\mathrm{d}+\sigma\rho)$

We note that

$L_{K}(1, \lambda^{(i)})a,b\neq 0$

for every$\dot{i},$

$a,$

$b$

. By Theorem Sl, we have

(6.10)$\frac{L_{K()}1,\lambda_{4,2}(1)(1,\lambda(L_{K})2)4,2}{L_{K()}1,\lambda_{2,2}(1)(1,\lambda(2)L_{K})2}\sim\pi^{2}p_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})^{4}$(6.11)$\frac{L_{K()}1,\lambda_{2,4}(1)(1,\lambda(L_{K})2)2,4}{L_{K()}1,\lambda_{2,2}(1)(1,\lambda(2)L_{K})2}\sim\pi^{2}p_{K}(\sigma, \sigma)^{4}$

In this case, Conjecture$\mathrm{B}$,.

states that$(, \frac{L_{K}(1,\lambda_{4}^{(1)}2)LK(1,\lambda_{4,2}(2))LK(1,\lambda_{2,2}(1))^{-}1LK(1,\lambda_{2}(2))2-1}{\pi^{2}g_{K}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{4}},)^{\alpha}$(6.12)$= \zeta\cdot\frac{L_{K}(1,(\lambda_{4,2}(1))\alpha)LK(1,(\lambda(2))4,2)\alpha LK(1,(\lambda(1))2,2)^{-}\alpha 1LK(1,(\lambda_{2,2}(2))^{\alpha})^{-}1}{\pi^{2}g_{K^{\alpha}}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{4}}$for every$\alpha\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$with a root of unity . Put$\zeta$,$Q_{1}=’ \frac{L_{K}(1,\lambda_{4}^{(1)}2)LK(1,\lambda_{4,2}(2))LK(1,\lambda_{2,2}(1))^{-}1LK(1,\lambda_{2}(2))2-1}{\pi^{2}g_{K}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{4}},$(6.13)$Q_{2}=’ \frac{L_{K}(1,\lambda_{2}^{(1)}4)LK(1,\lambda_{2,4}(2))LK(1,\lambda_{2,2}(1))^{-}1LK(1,\lambda_{2}(2))2-1}{\pi^{2}g_{K^{\sigma}}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{4}},$

.

and ,
$b$

188
If $h_{K}=1$ and the conductors of
be written as(6.14)$\lambda_{a,b}^{(i)}$$(Q_{1})^{\alpha}=\zeta 1^{\cdot}Q_{1}$,$(Q_{2})^{\alpha}=\zeta 2^{\cdot}Q_{2}$,$(Q_{1})^{\alpha}=\zeta 1^{\cdot}Q_{2}$,$(Q_{2})^{\alpha}=\zeta 2^{\cdot}Q_{1}$

,

$\{$

with roots of unity
Example 1.
$\mathrm{f}=(\frac{13+\sqrt{5}}{2}),$are (1) in (6.13), then we see easily that (6.12) canand$\zeta_{1}$Let$\zeta_{2}$which depend on$F=\mathrm{Q}(\sqrt{5}),$$N(\mathrm{f})=41,$$\alpha$$\alpha|F..=\mathrm{i}\mathrm{d}$$\alpha|F\neq \mathrm{i}\mathrm{d}$,,.

$K=\mathrm{Q}(\sqrt{\frac{13+\sqrt{5}}{2}}\dot{i})$$L(\mathrm{O}, x)=2,$

if
if. Then we have$h_{F}=h_{K}=1,$ $|C_{\mathrm{f}}|=2$$\epsilon=\frac{3+\sqrt{5}}{2}$,. Since $h_{0}=1$ , we may takeexist and are
and. Then for even positive integers and. Takeby (6.13). By a numerical experiment, we findanduniquely determined. Define. (The numerical value coincide to the 45-th decimal place. This quantity isequal to $ABQ$ in the notation of [Y2], \S 4, Example 1.) We also find$\alpha_{1}=(1)$

$b,$

$a$

$\mathrm{q}=(1)$$Q_{1}$$\lambda_{a,b}^{(2)}$$\lambda_{a,b}^{(1)}$$Q_{2}$$Q_{1}Q_{2}= \frac{25}{9}$$\epsilon^{-}Q_{1}22/4122/41Q+\epsilon 2=\frac{490}{3\cdot 41}$.

Solving these equations and comparing with numerical values to determine the sign ofsquare roots, we get$Q_{1}=\epsilon^{22}/41$.

$\frac{245+60^{\sqrt{5}}}{3\cdot 41}$,

$Q_{2}= \epsilon^{-22/}\frac{245-60\sqrt{5}}{3\cdot 41}41.$.

The numerical coincidence is to the 45-th decimal place.4 In this example, is a 41-st root
. Here the involvement of the fractional power ofof unity given by
is curious and this was the main difficulty of the numerical experiment. Letclass of (6). We have $|R(\epsilon, C_{i})|=20,\dot{i}=1$ ,is the identity. We may take
where
2 and
$\zeta_{1}$$(\epsilon^{22/41})^{\alpha}=\zeta_{1}((\epsilon^{\alpha})^{22}/41)$$C_{\mathrm{f}}=\{c_{1}, c_{2}\}$$\epsilon$$c_{2}=\mathrm{t}\mathrm{h}\mathrm{e}$$c_{1}$$V(c_{1})=( \epsilon’-\epsilon)\log\epsilon\cdot\frac{19}{2\cdot 3\cdot 41}$,

$V(C_{2})=( \epsilon-’)\epsilon\log\epsilon\cdot\frac{-2.9}{2\cdot 341}$,$V(_{C_{1})(C_{2})\frac{8}{41}\sqrt{5}}-V=- \log\epsilon, W(c_{1})=-\frac{1}{2}\log(41)$$W(c_{2})= \frac{1}{2}$,log(41).

Thus we have$\pi^{2}g_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})^{4}=\frac{1}{41}\epsilon^{-}8\sqrt{5}/41(\prod_{)z\in R(\epsilon C1},\frac{\Gamma_{2}(_{Z},(1,\epsilon))}{\rho_{2}((1,\epsilon))})(\prod_{(z\in R\epsilon C_{2})},\frac{\Gamma_{2}(_{\mathcal{Z}},(1,\epsilon))}{\rho_{2}((1,\epsilon))})^{-1}$.

is a transcendental number by the Gelfond-Schneider theorem. It is
We note thatcan be “explained” by $22-8\sqrt{5}\equiv 0$ mod . Similar
remarkable that the power
phenomena will be noticed in the following examples.
$\epsilon^{\sqrt{5}}$$\epsilon^{22/41}$

4For the sake of completeness, we carried out the second computation withproved the numerical coincidence to the 112-th decimal place.

$\mathrm{f}’$$\mathrm{h}\mathrm{i}.\mathrm{g}$her precision and im-

189
Example 2. Let$F=\mathrm{Q}(\sqrt{41}),$$K=\mathrm{Q}(\sqrt{13+2\sqrt{41}}\dot{i})$$h_{F}--h_{K}=1,$$\mathrm{f}=(13+2\sqrt{41}),$ $N(\mathrm{f})=5,$ $L(\mathrm{O}, \chi)=2,$. Then we have $\epsilon=2049+320\sqrt{41}$,.
. We take $a_{1}=(1),$$\mathrm{q}=(1)$$|C_{\mathrm{f}}|=2$

Then we find$Q_{1}=\epsilon^{4/25_{*\frac{546718+85383\sqrt{41}}{2^{2}\cdot 3\cdot 5}}}$,$Q_{2}=\epsilon^{-4}/25$.

$\frac{546718-85383\sqrt{41}}{2^{2}\cdot 3\cdot 5}$

The numerical coincidence is to the 42-nd decimal place. Let$C_{\mathrm{f}}=\{c_{1}, c_{2}\},$.

$c_{1}=1$. Then1280, $\dot{i}=1,2$ . We have $V(c_{1})-V(c_{2})=- \frac{1}{25}\sqrt{41}\log\epsilon$ and $4-\sqrt{41}\equiv 0$
appears in the denominator of $V(c_{1})-$modholds. (The “reason” of square is that
$V(c_{2})$ . Compare with Example 11.)$|R(\epsilon, c_{i})|=$$5^{2}$$\mathrm{f}^{\prime 2}$

Example 3. Let $F=\mathrm{Q}(\sqrt{29}),$$\mathrm{f}=(\frac{9+\sqrt{29}}{2}),$$N(\mathrm{f})=\mathrm{i}3,$$K=\mathrm{Q}(\sqrt{\frac{9+\sqrt{29}}{2}}\dot{i})$$h_{F}=h_{K}--1,$$L(\mathrm{O}, \chi)=2,$. Then we have$|C_{\mathrm{f}}|=2$

Then we find$Q_{1}= \epsilon^{9/13}\cdot\frac{2935+503^{\sqrt{29}}}{2^{4}\cdot 3\cdot 13}$,$\epsilon=\frac{27+5\sqrt{29}}{2}$. We take $a_{1}=(1),$..

$Q_{2}= \epsilon^{-9/}\frac{2935-503\sqrt{29}}{2^{4}\cdot 3\cdot 13}13.$$C_{\mathrm{f}}=\{c_{1}, c_{2}\},$

have $V(c_{1})-V(c2)--- \frac{1}{13}\sqrt{29}\log\epsilon$ and $9-\sqrt{29}\equiv 0$ mod

holds.

Example 4. Let $F=\mathrm{Q}(\sqrt{13}),$ $K=\mathrm{Q}(\sqrt{9+2\sqrt{13}}\dot{i})$ . Then we have$\mathrm{f}=(9+2^{\sqrt{13}),N(\mathrm{f}})=29,$$L(0, \chi)=2,$ $h_{F}=h_{K}=1,$$c_{1}=1$. We$\epsilon=\frac{11+3\sqrt{13}}{2}$. We take $a_{1}=(1),$$|C_{\mathrm{f}}|=2$.

.

The numerical coincidence is to the 29-th decimal place. Let$\mathrm{f}’$$\mathrm{q}=(1)$,$\mathrm{q}=(1)$,.

Then we find$Q_{1}= \epsilon^{28/29}\cdot\frac{5669+274^{\sqrt{13}}}{2^{3}\cdot 3^{2}\cdot 29}$,

$Q_{2}= \epsilon^{-28/}\frac{5669-274\sqrt{13}}{2^{3}\cdot 3^{2}\cdot 29}29.$

The numerical coincidence is to the 45-th decimal place. Let
have $V(c_{1})-V(c_{2})=- \frac{3}{29}\sqrt{13}\log\epsilon$ and $28-3\sqrt{13}\equiv 0$ mod,$C_{\mathrm{f}}=\{c_{1}, c_{2}\},$$\mathrm{f}’$$c_{1}=1$. We

holds.

Example 5. Let $F=\mathrm{Q}(\sqrt{17}),$ $K=\mathrm{Q}(\sqrt{9+2\sqrt{17}}\dot{i})$ . Then we have $\epsilon=33+8\sqrt{17}$ ,$\mathrm{f}=(9+2^{\sqrt{17}),N(\mathrm{f}})=13,$ $L(0, \chi)=2,$ $h_{F}=h_{K}=1,$.
. We take $a_{1}=(1),$Then we find$\mathrm{q}=(1)$$|C_{\mathrm{f}}|=2$$Q_{1}= \epsilon^{11/1\mathrm{s}}\cdot\frac{285+.6\sqrt{17}}{2^{2}13}$,$Q_{2}= \epsilon^{-11/}\frac{285-.6\sqrt{17}}{2^{2}13}13.$

The numerical coincidence is to the 29-th deciinal place.

.

$\mathrm{L}\mathrm{e}\mathrm{t}\prime C_{\mathrm{f}}=\{c_{1}, c_{2}\},$

have $V(c_{1})-V(c_{2})=- \frac{1}{13}\sqrt{17}\log\epsilon$ and $11-\sqrt{17}\equiv 0$ mod$\mathrm{f}’$holds.

$c_{1}=1$

. We

190

Example 6. Let $F=\mathrm{Q}(\sqrt{13}),$$\mathrm{f}=(\frac{9+\sqrt{13}}{2}),$$K=\mathrm{Q}(\sqrt{\frac{9+\sqrt{13}}{2}}i)$$N(\mathrm{f})=17,$ $L(\mathrm{O}, \chi)=2,$$h_{F}=h_{K}=1,$. Then we have$\epsilon=\frac{11+3\sqrt{13}}{2}$. We take $a_{1}=(1),$$|C_{\mathrm{f}}|=2$$\mathrm{q}=(1)$,.

Then we find$Q_{1}= \epsilon^{1/17}\cdot\frac{5522+1519\sqrt{13}}{3^{2}\cdot 17}$,

$Q_{2}=\epsilon^{-1}/17$.

$\frac{5522-1519\sqrt{13}}{3^{2}\cdot 17}$

The numerical coincidence is to the 44-th decimal place. Let
have $V(c_{1})-V(c_{2})=- \frac{2}{17}\sqrt{13}\log\epsilon$ and $1-2\sqrt{13}\equiv 0$ mod$\mathrm{f}’$.
$c_{1}=1$$C_{\mathrm{f}}=\{c_{1}, c_{2}\},$. We

holds.

Example 7. Let $F=\mathrm{Q}(\sqrt{5}),$ $K=\mathrm{Q}(\sqrt{7+2\sqrt{5}}\dot{i})$ . Then we have$\epsilon=\frac{3+\sqrt{5}}{2}$,. We take,. The prime ideal (2) of ramifies in . Put (2). Let $i=1$ or 2. For every
positive even integers and , there exists two Gr\"ossencharactersof conductor (1).
They are determined by) $=1$ or-l. First determine
takingfor$(6.14)$$h_{K}=2,$
everyby (6.13). Thoughand . Define
andstill follows(6.12). We find$\mathrm{f}=(4)(7+2\sqrt{5}),$ $N(\mathrm{f})=2429,$ $L(\mathrm{O}, x)=4,$$h_{F}=1,$ $h_{K}=2,$
$K$

$F$

$\mathrm{q}=(1)$

$b$

$a$

$\lambda_{a,b}^{(i)}$$\lambda_{a,b(\mathfrak{P}2}^{(i)}$

$a,$

$b$

$i$

$Q_{1}$$\lambda_{a,b}^{(i)}$taking$\lambda_{a,b}^{(i)}(\mathfrak{P}2)=1$$\lambda_{a,b}^{(i)}$$Q_{2}$$\mathrm{h}\mathrm{o}\mathrm{m}$$Q_{1}= \epsilon^{19/29}\cdot\frac{49675-4569\sqrt{5}}{2\cdot 3\cdot 13\cdot 29}$

Next determine$\alpha_{1}=(1)$$|C_{\mathrm{f}}|=16$$=\mathfrak{P}_{2}^{2}$$\lambda_{a,b(\mathfrak{P}2}^{(i)}$,$Q_{2}= \epsilon^{-}\frac{49675+4569\sqrt{5}}{2\cdot 3\cdot 13\cdot 29}19/29.$) $=-1$ for every$Q_{1}= \epsilon^{19/29}\cdot\frac{24645-823\sqrt{5}}{2\cdot 5^{2}\cdot 29}$

,

$a,$

$b$

.

and . Then we find$\dot{i}$$Q_{2}= \epsilon^{-19}\frac{24645+823\sqrt{5}}{2\cdot 5^{2}\cdot 29}/29.$.

The numerical coincidence is to the 44-th decimal place. Let $C_{\mathrm{f}}=\{c_{i}|1\leq i\leq 16\}$ . We

have$\frac{1}{2}\sum_{i=1}^{16}x(Ci)V(C_{i})=-\frac{15}{58}\sqrt{5}\log\epsilon$

and$38-15\sqrt{5}\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} (7-2\sqrt{5})$holds.

\S 7. Numerical examples II: Quartic extensions of real quadratic fields
We keep the notation in the previous section and proceed to consider the normal closureof $K$ over Q. We have $[L:\mathrm{Q}]=8$ . Defineby$L=\mathrm{Q}(\xi, \xi’)$$\sigma.’\tau\in \mathrm{G}\mathrm{a}1(L/\mathrm{Q})$a $:(\xi, \xi’)arrow(\xi’, -\xi)$ ,(Previous
is extended.) Then
which are subject to the relations$\sigma$$\mathrm{G}\mathrm{a}1(L/\mathrm{Q})$$\sigma^{4}=\tau^{2}=1$,$\tau:(\xi, \xi’)arrow(\xi’, \xi)$.

is the dihedral group generated by$\tau\sigma=\sigma^{\mathrm{s}}\tau$,$\sigma^{2}=\rho$.

$\sigma$and$\tau$

191
The reflex field $K’$ ofis contained in and given by $K’=\mathrm{Q}(\sqrt{2(x+\sqrt{d’}}\dot{i})$ ,$d’=x^{2}-y^{2}d$ . The maximal real subfield of $K’$ is. We have$L$$(K, \{\mathrm{i}\mathrm{d}, \sigma\})$$F’=\mathrm{Q}(\sqrt{d’})$$\mathrm{G}\mathrm{a}1(L/K)=\{1, \sigma\tau\}$,$\mathrm{G}\mathrm{a}1(L/K’)=\{1, \tau\}$.

for allin terms of the mulLet us consider the problem to give,
tiple gamma function.
has five conjugacy classes. Therefore, in view of Theorem Sl, (3), our previous Conjecture 1 can give
. We haveessentially only$\alpha\in \mathrm{G}\mathrm{a}1(L/\mathrm{Q})$$p_{L}(\mathrm{i}\mathrm{d}, \alpha)$$\mathrm{G}\mathrm{a}1(L/\mathrm{Q})$$\{\rho\},$$\{\mathrm{i}\mathrm{d}\},$$\{\sigma, \sigma\rho\},$$\{\mathcal{T}, \mathcal{T}\beta\}$$\{\sigma\tau, \sigma\tau\rho\}$$p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})$,$p_{K}(\sigma, \sigma)\sim p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})p_{L}(\mathrm{i}\mathrm{d}, \sigma \mathcal{T})^{-}1$$p_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})\sim p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})pL(\mathrm{i}\mathrm{d}, \sigma \mathcal{T})$,$p_{K’}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})\sim p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})p_{L}(\mathrm{i}\mathrm{d}, \tau)$,$p_{K’}(\sigma\tau, \sigma \mathcal{T})\sim p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})pL(\mathrm{i}\mathrm{d}, \mathcal{T})^{-1}$and these quantities were examined in examples discussed in \S 6. In other words, we. Whatand forexamined the validity of our conjectures for$F_{*}=\mathrm{Q}(\sqrt{dd’})\subset L$. Put. We haveremains to be considered is. By Theorems Sl and S2, we can easily derive the relation$p_{L}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d}),$$p_{L}(\mathrm{i}\mathrm{d}, \tau)$$p_{L}(\mathrm{i}\mathrm{d}, \sigma \mathcal{T})$$\mathrm{G}\mathrm{a}1(L/F_{*})=$$p_{L}(\mathrm{i}\mathrm{d}, \sigma)$$\langle\sigma\rangle\cong \mathrm{Z}/4\mathrm{Z}$$\frac{L_{K()}1,\lambda_{4,2}(1)(1,\lambda(2)L_{K})2,4}{L_{K()}1,\lambda_{2,4}(1)(1,\lambda(2)L_{K})4,,2}\sim p_{L()^{8}}\mathrm{i}\mathrm{d},$$\sigma$.

Hence we put
,

(7.1)

$Q=’ \frac{L_{K}(1,\lambda_{4}^{(1)}2)LK(1,\lambda_{2,4}(2))LK(1,\lambda_{2,4}(1))^{-}1LK(1,\lambda_{4}(2))2-1}{g_{L}(\mathrm{i}\mathrm{d},\sigma)^{8}}$and are going to examine the algebraicity of . Hereplace of ) and (6.4), being an abelian extension of .
Letbe an integral ideal of ’. For.integers
such thatcharacters of$Q$$g_{L}(\mathrm{i}\mathrm{d}, \sigma)$$L$$\tau$

is defined by (4.7) (in$F_{*}$$\mathrm{q}’$$\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}}$

$K|$

$a$

and , let$b$$\mu_{a,b}^{(1)}$$I_{\mathrm{q}’}(K’)$$\mu_{a,b}^{(1)}((\alpha))$$=( \frac{\alpha^{\rho}}{|\alpha|})^{a}(\frac{\alpha^{\sigma\tau\rho}}{|\alpha^{\sigma 7^{\sim}}|})^{b}$,$\alpha\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{q}’$,.

$\mu_{a,b}^{(2)}((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})^{a}(\frac{\alpha^{\sigma\tau}}{|\alpha^{\sigma\tau\rho}|})^{b}$,$\alpha\underline{=}1\mathrm{m}\mathrm{o}\mathrm{d}^{\cross}\mathrm{q}’$.

Then we have$\sigma$$\frac{L_{K’}(1,\mu_{2,4}(1))LK\prime(1,\mu_{4,2}(2))}{L_{K’}(1,\mu_{4,2}(1))LKJ(1,\mu_{2,4}(2))}\sim p_{L()^{8}}\mathrm{i}\mathrm{d},$.

We put(7.2)$\sigma$.

$R=’ \frac{L_{K}\prime(1,\mu_{2}4)(1)LK^{J}(1,\mu_{4,2}^{(})L2)(1,\mu_{4,2}(K^{J}))-1L_{K}J(1\mu 1,)(2)2,4-1}{g_{L}(\mathrm{i}\mathrm{d},\sigma)^{8}}$and$\mu_{a,b}^{(2)}$

be

192

Example 8. Let $F=\mathrm{Q}(\sqrt{5}),$$\mathrm{Q}(\sqrt{13+2\sqrt{41}}\dot{i})$$\mathrm{Q}(\sqrt{225})$$K=\mathrm{Q}(\sqrt{\frac{13+\sqrt{5}}{2}}\dot{i})$. Then $F’=\mathrm{Q}(\sqrt{41}),$ $K’=$. These fields are considered in Examples 1 and 2. We have $F_{*}=$. The fundamental unit$\epsilon_{0}$of$F_{*}$

is$\frac{43+3\sqrt{205}}{2}$, which is totally positive. Set$13+\sqrt{5}$$\epsilon=\epsilon_{0}$. We also have $h_{F_{*}}=2,$ $h_{0}=4$ . We have $D(K/F)=(\overline{2})$ , $D(K’/F’)=$. Hence $D(L/\mathrm{Q})=N_{K/\mathrm{Q}}(D(L/K))\cdot(5^{4}\cdot 41^{2})=NK’/\mathrm{Q}(D(L/KJ))\cdot(5^{2}\cdot 41^{4})$.
Since, only prime factors of (41) (resp. (5)) can ramify in$L/K$ (resp. $L/K’$ ). Here V denotes the operation to make the composite field. Then weeasily get. Since $D(L/\mathrm{Q})=N_{F_{*}/\mathrm{Q}(}D(L/F*))D(F*/\mathrm{Q})^{4}$ , we must have
$D(L/F_{*})=(1)$ . Therefore
is unramified over . Hence is the maximal ray class fieldof conductor (1)of . Set$(13+2\sqrt{41})$$L=K\vee \mathrm{Q}(\sqrt{41})=K’\vee \mathrm{Q}(\sqrt{5})$$D(L/\mathrm{Q})=5^{4}\cdot 41^{4}$$L$$\mathfrak{p}_{3}=3\mathrm{Z}+\frac{1+\sqrt{205}}{2}\mathrm{Z}$,$\mathfrak{p}_{3}’=3\mathrm{Z}+\frac{1-\sqrt{205}}{2}$andare prime ideals of
and we have (3)narrow ideal class group of , we can takeThen

$F_{*}$$\mathfrak{p}_{3}’$$\mathfrak{p}_{3}$$L$$F_{*}$$F_{*}$$\infty_{1}\infty_{2}$Z.

. As representatives of the$=\mathfrak{p}_{3}\mathfrak{p}_{3}’$$F_{*}$$a_{1}=(1)$We have$( \frac{L/F_{*}}{\mathfrak{p}_{3}})=\sigma$,$\alpha_{2}=\mathfrak{p}_{3}$. For$|R(\epsilon, c_{1})|=3$,$1\leq\dot{i}\leq 4$,$a_{3}=\mathfrak{p}_{\mathrm{s}}^{2}=(14-\sqrt{205})$, let$c_{i}$

be the class of$a_{i}$,$\alpha_{4}=\mathfrak{p}_{3}’$.

. We have.

$|R(\epsilon, c_{2})|=|R(\epsilon, c_{4})|=9$,$|R(\epsilon, c_{3})|=27$$\zeta_{F_{*}}(0, C2)=\zeta_{F}*(\mathrm{o}, c_{4})=0$,$\zeta_{F_{*}}(0, c3)=-1$

By (6.5), we get$\zeta_{F_{*}}(0, c1^{\cdot})=1$

,

.

Hence
$g_{L}( \mathrm{i}\mathrm{d}, \sigma)=\exp(\frac{1}{4}(x(c_{4})-x(_{C_{2})}))$.

Defineas in Example 1 and let us consider the quantity defined by (7.1). Conjecture
does not apply directly tosince
is not a Gr\"ossencharacter of . To reduce tothe situation of Conjecture , regard
as a Hecke character ofand let be the
Hecke character ofwhich corresponds to the quadratic extension $L/K$ . We have
. Set$\lambda_{a,b}^{(i)}$

$Q$

$Q$

$\mathrm{B}$$\mathrm{B}$$L$$\lambda_{a,b}^{(i)}$$\lambda_{a,b}^{(i)}$$K_{A}^{\cross}$$K_{A}^{\cross}$$L_{L}(s, \lambda_{a,b^{\circ}}^{()}iN_{L}K)/=L_{K}(s, \lambda(i))a,bLK(s, \lambda_{a,b}(i))\omega$$\Lambda=\frac{L_{K}(1,\lambda^{(}))4,2LK(1,\lambda^{()})12,42}{L_{K()}1,\lambda_{2,4}(1)L_{K}(1,\lambda(2))4}$,$\Lambda’=\frac{L_{K}(1,\lambda_{4,2}(1)_{\omega})L_{K}(1,\lambda(2))2,4\omega}{L_{K}(1,\lambda_{2,4}^{(}1)\omega)LK(1,\lambda_{4,2}\omega)(2)}$$\omega$

193
and for$\alpha\in \mathrm{G}\mathrm{a}1(L/\mathrm{Q})$, set$\Lambda_{\alpha}=\frac{L_{K}(1,(\lambda^{(1})4,2))\alpha LK(1,(\lambda^{(})^{\alpha}2,4)2)}{L_{K}(1,(\lambda^{(1}))2}$,.

$\Lambda_{\alpha}’=\frac{L_{K}(1,(\lambda_{4,2}\omega)^{\alpha}(1))L_{K}(1,(\lambda_{2,4}(2)\omega)^{\alpha})}{L_{K}(1,(\lambda_{2,4}(1)_{\omega})\alpha)LK(1,(\lambda_{4,2}(2)\omega)^{\alpha})}$We can easily verify that$\Lambda_{\sigma}=\Lambda$

Put$Q_{*}= \frac{\Lambda\Lambda’}{g_{L}(\mathrm{i}\mathrm{d},\sigma)^{16}}$,$\Lambda_{\tau}=\Lambda^{-1}$. Conjecture$\mathrm{B}$,$\Lambda_{\sigma}’=\Lambda^{J}$

to

and

$\zeta_{\tau}$$\Lambda_{\tau}’=\Lambda^{\prime-1}$

can be applied to. We use the same letter. The conjecture implies that$g_{L}(\mathrm{i}\mathrm{d}, \sigma\rho)=g_{L}(\mathrm{i}\mathrm{d}, \sigma)-1$$\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$,$\sigma$(resp.

$Q_{*}^{\sigma}=\zeta_{\sigma}Q_{*},$. Note that

$Q_{*}$$\tau$.
$g_{L}(\mathrm{i}\mathrm{d}, \mathcal{T}-1\sigma\tau)=$) for any extension of (resp. )with roots of unity$Q_{*}^{\tau}=\zeta_{\tau}Q_{*}^{-1}$$\zeta_{\sigma}$. On the other hand, by Proposition 7.3 of [Y2], we immediately get$( \frac{\Lambda’}{\Lambda})^{\tau}=\pm\frac{\Lambda}{\Lambda},$$\cdot$Since$Q^{2}=Q_{*} \cdot\frac{\Lambda’}{\Lambda}$$\tau$$\sigma$$( \frac{\Lambda’}{\Lambda})^{\sigma}=\pm\frac{\Lambda’}{\Lambda}$,, we conclude that

$Q^{\sigma}=\zeta_{\sigma}’Q$,$Q^{\tau}=\zeta_{7}’.Q^{-1}$with roots of unityand . We searchso thatis close toa rational number with a simple denominator.5 We find $a=6$ fits this scheme and$\zeta_{\sigma}’$$\zeta_{\tau}’$$a\in \mathrm{Z}$$(\epsilon^{a/27}Q)^{2}+(\epsilon^{a/27}Q)^{-2}$$( \epsilon^{2/9}Q)^{2}+\frac{1}{(\epsilon^{2/9}Q)^{2}}=\frac{367}{3^{4}}$.

Solving this equation and comparing with the numerical value, we find

$Q= \epsilon^{-2/9}\cdot\frac{1}{3^{2}}\sqrt{\frac{367+23\sqrt{205}}{2}}$.

The numerical coincidence is to the 47-th decimal place. Similarly, taking
we find$R= \epsilon^{-2/9}\cdot\frac{\sqrt{637530967.-18592639^{\sqrt{205}}}}{3^{2}31\cdot 61}$$\mathrm{q}’=(1)$for$\mu_{a,b}^{(i)}$,.

The numerical coincidence is to the 48-th decimal place.
Example 9. Let $F=\mathrm{Q}(\sqrt{17}),$ $K=\mathrm{Q}(\sqrt{9+2\sqrt{17}}\dot{i})$ . Then $F’=\mathrm{Q}(\sqrt{13}),$ $K’=$$\mathrm{Q}(\sqrt{\frac{9+\sqrt{13}}{2}}\dot{i})$. These fields are considered in Examples 5 and 6. We have

$F_{*}=\mathrm{Q}(\sqrt{221})$

5Practically we can do such test using the expansion of a real number into the continued fraction.

.

194
. Set
is
of
The fundamental unit
is the maximal ray class field of conductor (1)$\epsilon_{0}$$F_{*}$$\frac{15+\sqrt{221}}{2}$$\infty_{1}\infty_{2}$$\mathfrak{p}_{5}=5\mathrm{Z}+\frac{1+\sqrt{221}}{2}\mathrm{Z}$Then we have (5)of$F_{*}$$=\mathfrak{p}_{5}\mathfrak{p}_{5}’,$$( \frac{L/F_{*}}{\mathfrak{p}_{5}})=\sigma$,. We have $h_{F_{*}}=2,$ $h_{0}=4$ andof . Set$\epsilon=\epsilon_{0}$$L$$F_{*}$$\mathfrak{p}_{5}’=5\mathrm{Z}+\frac{1-\sqrt{221}}{2}$Z.

. As representatives of the narrow ideal class group, we take$a_{1}=(1)$Define$\lambda_{a,b}^{(i)}$(resp.

,$\mu_{a,b}^{(i)}$$\alpha_{2}=\mathfrak{p}_{5}$) taking,$\alpha_{3}=\mathfrak{p}_{5}^{2}=(14-\sqrt{221})$$\mathrm{q}=(1)$(resp.

$\mathrm{q}’=(1)$,$a_{4}=\mathfrak{p}_{5}’$.

). Then we findThe numerical coincidence is to the 30-th decimal place.
$R= \epsilon^{-1/5}\cdot\frac{\sqrt{64769+3600^{\sqrt{221}}}}{191}$.

The numerical coincidence is to the 47-th decimal place.
Example 10. Let $F=\mathrm{Q}(\sqrt{29}),$$K=\mathrm{Q}(\sqrt{\frac{9+\sqrt{29}}{2}}\dot{i})$. Then $F’=\mathrm{Q}(\sqrt{13}),$ $K’=$. These fields are considered in Examples 3 and 4. We have $F_{*}=$. We have $h_{F_{*}}=2$ ,is $233+12\sqrt{377}$ . Set. The fundamental unit of$h_{0}=4$ andof . Set
is the maximal ray class field of conductor (1)$\mathrm{Q}(\sqrt{9+2\sqrt{13}}\dot{i})$$\mathrm{Q}(\sqrt{377})$$\epsilon_{0}$$F_{*}$$\epsilon=\epsilon_{0}$$L$$\mathfrak{p}_{2}=2\mathrm{Z}+\frac{1+\sqrt{377}}{2}\mathrm{Z}$Then we have (2)of$F_{*}$$F_{*}$$\infty_{1}\infty_{2}$$=\mathfrak{p}_{2}\mathfrak{p}_{2}’,$$( \frac{L/F_{*}}{\mathfrak{p}_{2}})=\sigma$,$\mathfrak{p}_{2}’=2\mathrm{Z}+\frac{1-\sqrt{377}}{2}$Z.

. As representatives of the narrow ideal class group, we take$a_{1}=(1)$Define$\lambda_{a,b}^{(i)}$(resp.

,$\mu_{a,b}^{(i)}$$\alpha_{2}=\mathfrak{p}_{2}$) taking,$\mathrm{q}=(1)$$a_{3}= \mathfrak{p}_{2}^{2}=(\frac{19-\sqrt{377}}{2})$(resp.

$\mathrm{q}’=(1)$,$\alpha_{4}=\mathfrak{p}_{2}’$). Then we find

$Q= \epsilon^{-1/4}\cdot\frac{1}{2^{2}\cdot 1889}\sqrt\frac{490432649-24564411\sqrt{377}}{2}$

.

.

195
The numerical coincidence is to the 32-nd decimal place.

.

$R= \epsilon^{-1/4}\cdot\frac{1}{2^{2}\cdot 23\cdot 179}\sqrt{\frac{2459444473-123549195\sqrt{377}}{2}}$The numerical coincidence is to the 49-th decimal place.
Remark. There exist intricate relations between quantities considered in \S 6 and \S 7. Let us
in Example 5. The ideals (2) and (53) decompose completely inconsider the quantity; so let (2)(53)
be the decompositions into prime ideals. Then
we have the factorization into prime ideals:$Q_{1}$$\mathrm{Q}(\sqrt{17})$$=\mathfrak{p}_{53}\mathfrak{p}’53$$=\mathfrak{p}_{2}\mathfrak{p}_{2}’,$$(*)$$(\epsilon^{-11/1}13Q1)=\mathfrak{p}_{2}^{-}2\mathfrak{p}^{\prime-}22(3)\mathfrak{p}_{53}\mathrm{f};\mathrm{f}-$always appears in the factorizations of
where $\mathrm{f}=(9+2\sqrt{17})$ . We can check that$1\sim7$. (In Example 7, is not prime, but a similar fact
similar quantities in Examples
in Example
We have the factorization into prime
holds.)
er the$\mathrm{f}’\mathrm{f}^{-1}$$\mathrm{f}$Consi..d$\mathrm{N}\mathrm{o}..\mathrm{W}$.

ideals.: .,$9..\cdot$$\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}..\mathrm{t}..\mathrm{y}Q$.

$(\epsilon^{1/52}Q)=\mathrm{q}_{5\mathrm{s}(}^{2}\mathrm{q}’5\mathrm{s})^{-2}$:,. The appearance of 53 in the denominator ofinwhere (53)$(.*)$. Similar relations exi.st for al.l the otherbe explain.ed by$=\mathrm{q}_{53}\mathrm{q}_{5}^{J}3$$\mathrm{Q}(\sqrt{221})$$Q$

seems to$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{m}_{\mathrm{P}^{\mathrm{l}\mathrm{e}\mathrm{S}}}..\cdot$\S 8. Numerical examples III: Quadratic extension of a totally real cubic field
Let $F$ be a totally real cubic field and let $K$ be a totally imaginary quadratic extension. Let
to an element ofwithand extend eachof $F$ . Let$K$
be charactersand , letbe an integral ideal of . For positive integers
ofsuch that$J_{F}=\{\sigma_{1}, \sigma_{2}, \sigma_{3}\}$$\sigma_{1}=\mathrm{i}\mathrm{d}$$J_{K}$$\sigma_{i}$

$a,$

$b$

$\lambda_{a,b}^{(i)},$$c$$1\leq\dot{i}\leq 4$$I_{\mathrm{q}}(K)$$\lambda_{a,b}^{(1)b\mathrm{C}},C((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})a(\frac{\alpha^{\sigma_{2}\rho}}{|\alpha^{\sigma_{2}}|})(\frac{\alpha^{\sigma_{3}\rho}}{|\alpha^{\sigma_{3}}|})$,,$\lambda_{a,b_{C}}^{(2)},((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})a(\frac{\alpha^{\sigma_{2}\rho}}{|\alpha^{\sigma_{2}}|})^{b}(\frac{\alpha^{\sigma_{3}}}{|\alpha^{\sigma_{3}\rho}|})^{c}$,$\lambda_{a,b}^{(3)b},C((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})a(\frac{\alpha^{\sigma_{2}}}{|\alpha^{\sigma_{2}\rho}|})(\frac{\alpha^{\sigma \mathrm{s}\rho}}{|\alpha^{\sigma_{3}}|})^{c}$.

$\lambda_{a,b}^{(4)b},C((\alpha))=(\frac{\alpha^{\rho}}{|\alpha|})a(\frac{\alpha^{\sigma_{2}}}{|\alpha^{\sigma_{2}\rho}|})(\frac{\alpha^{\sigma_{3}}}{|\alpha^{\sigma_{3}\rho}|})^{c}$

Since$\prod_{i=1}^{4}\frac{L_{K}(1,\lambda_{4,2,2}^{(i}))}{L_{K}(1,\lambda_{2,2,2}^{(i}))}\sim\pi^{4}p_{K}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})^{8}$

holds (independently of the extensions$\sigma_{i}$), Conjecture$\mathrm{B}$states that$( \frac{\prod_{i=1}^{4}L_{K}(1,\lambda^{(i}4,)2,2)L_{K}(1,\lambda)(2,2,2i)-1}{\pi^{4}g_{K}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{8}})^{\alpha}.\cdot$(8.1)$= \zeta\cdot\frac{\prod_{i=1}^{4}L_{K}(1,(\lambda_{4,2,2}(i))\alpha)L_{K}(1,(\lambda(i))^{\alpha}2,2,2)-1}{\pi^{4}g_{K^{\alpha}}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{8}}$$\mathrm{q}$196
for every$\alpha\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$with a root of unity . Put$\zeta$$Q_{1}= \frac{\prod_{i=1}^{4}LK(1,\lambda^{(i}4,)2,2)L_{K}(1,\lambda)(2,2,2i)-1}{\pi^{4}g_{K}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{8}}$

(8.2)

,

$Q_{2}= \frac{\prod_{i=1}^{4}LK(1,\lambda^{(i}2,)4,2)L_{K}(1,\lambda)(2,2,2i)-1}{\pi^{4}g_{K^{\sigma_{2}}}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{8}})$$Q_{3}= \frac{\prod_{i=1}^{4}LK(1,\lambda^{(i}2,)2,4)L_{K}(1,\lambda)(2,2,2i)-1}{\pi^{4}g_{K^{\sigma}3}(\mathrm{i}\mathrm{d},\mathrm{i}\mathrm{d})^{8}}$Example 11. Let$F=\mathrm{Q}(\zeta_{7}+\zeta_{7}^{-1})$$\omega_{1}=2\cos\frac{2}{7}\pi=\zeta_{7}+\zeta_{7}-1$

We have,. Set$\zeta_{7}=e^{2\pi i/7}$. Set$\omega_{2}=2\cos\frac{4}{7}\pi=\zeta 7^{+\zeta}27-2$$\mathrm{J}\supset_{F}=\mathrm{Z}+\mathrm{Z}\omega_{1}+\mathrm{Z}\omega_{2}$$\omega_{1}^{\sigma_{3}}=\omega_{3}$, where. Define$\sigma_{1},$$\epsilon=(1+\omega_{1})2\omega-21=2+\omega_{1},$$\sigma_{2},$.

,$\omega_{3}=2\cos\frac{8}{7}\pi=\zeta_{7}^{4}+\zeta_{7^{-4}}$,by. Then it is known that $h_{F}=1$$\sigma_{3}\in \mathrm{G}\mathrm{a}1(F/\mathrm{Q}\rangle$$\eta=\omega_{1}^{2}=2+\omega_{2}$and that$E_{F}=\langle-1, \omega_{1},1+\omega_{1}\rangle$,.

$E_{F}^{+}=\langle\epsilon, \eta\rangle$$\sigma_{1}=\mathrm{i}\mathrm{d},$$\omega_{1}^{\sigma_{2}}=\omega_{2}$.

has an arbitrary signature distribution, the class number of $F$ in the narrowSince
sense is also 1. By Shintani [Shl], p. 415,
holds by taking (cf.
$E_{F}$$\mathrm{R}_{+}^{3}=\bigcup_{u\in E_{F}^{+^{u(C)}}}\bigcup_{j1j}^{6}=$also Thomas and Vasquez [TV], Theorem 1)$C_{1}=C(1, \epsilon, \epsilon\eta)$$C_{4}=C(1, \eta)$

Now put
We have,,$C_{2}=C(1, \eta, \eta\epsilon)$$C_{5}=C(1, \epsilon\eta)$

in

$F$

$C_{3}=C(1, \epsilon)$,$C_{6}=C(1)$ .

,is totally positive and $N(\delta)=167$ .
. We specify bywith conjugate prime ideals$\delta=6+\omega_{1}-\omega_{2},$ $K=F(\sqrt{\delta}\dot{i})$$(167)=\mathfrak{p}\mathfrak{p}^{r}\mathfrak{p}^{J\prime}$,. We see that$\delta$$\mathfrak{p}’’$$\mathfrak{p}’,$$\mathfrak{p},$$\mathfrak{p}$$\omega_{1}\equiv 19$. Then we can verify that.
. We check thatis integral over
Therefore we see that $K$ is a CM-field, is the only prime ideal which ramifies in $K/F$ and$\mathrm{m}\mathrm{o}\mathrm{d}$$\frac{\sqrt{\delta}\dot{i}+1+\omega_{2}}{2}$$\mathfrak{p}=(\delta)$$\mathfrak{p}$$0_{F}$$\mathfrak{p}$. We can also
thatfield of conductorof $F$ . We haveclass of (5). We see that$\mathrm{D}_{K}=\mathrm{J}\supset_{F}\frac{\sqrt{\delta}\dot{i}+1+\omega_{2}}{2}\oplus \mathrm{J}\supset_{F}$$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\Phi$that $K$ is the maximal ray class
where $c_{1}=$ the class of (1),$C_{\mathfrak{p}}=\{c_{1}, c_{2}\}$$\mathfrak{p}\infty_{1}\infty_{2}\infty_{3}$$c_{2}=\mathrm{t}\mathrm{h}\mathrm{e}$$|R(c_{1}, c_{1})|=|R(C_{1}, C_{2})|=83$,$|R(C_{2}, C_{1})|=|R(C_{2}, C_{2})|=166$

and $R(C_{j}, c_{1})=R(C_{j}, c_{2})=\emptyset$ if $j\geq 3$ . We compute$\zeta_{F}(0, c1)=2$,

by (2.16) and obtain$\zeta_{F}(0, c_{i})$$\zeta_{F}(\mathrm{o}, C2)=-2$.

which corresponds to the quadratic extension $K/F$ .
Let be the Hecke character of. Since $E_{K}=E_{F}$ , we get $h_{K}=1$ (cf. [Y2], \S 3). We have
We have$F_{A}^{\cross}$$\chi$$L(\mathrm{O}, \chi)=4$.

$g_{K^{\sigma}}( \mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})=\pi^{-1/\sigma}2\exp(\frac{1}{8}(x(C_{1})-X(C_{2}^{\sigma})), \sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$197
Here to defineto$\mathrm{G}\mathrm{a}1(F/\mathrm{Q})$$g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})$$\sigma_{1},$$(^{\sqrt{\delta}}\dot{i})^{\sigma_{2}}=\sqrt{\delta^{(2)}}\dot{i}$, we useby$\sigma_{3}\in J_{K}$$\sigma_{2},$,and$\{C_{j}^{\sigma}\}_{1\leq j}\leq 6$$\sigma_{1}=\mathrm{i}\mathrm{d}$$a_{1}^{\sigma}=(1)$. We extend$\sigma_{1},$$\sigma_{3}\in$$\sigma_{2},$,$\delta^{(2)}=6+\omega_{2}-\omega_{3}$,$(^{\sqrt{\delta}}\dot{i})^{\sigma_{3}}=\sqrt{\delta^{(3)}}\dot{i}$,$\delta^{(3)}=6+\omega_{3}-\omega_{1}$.

We computeby the asymptotic expansion of the multiple gamma function given
in Barnes [Ba2], p. 424 and by the difference equation (2.1). We takeand consider
the quantities
defined by (8.2). (Note that for every positive even integers
and, there exists uniqueof conductor (1) . Then (8.1) implies$g_{K^{\sigma}}(\mathrm{i}\mathrm{d}, \mathrm{i}\mathrm{d})$$\mathrm{q}=(1)$$Q_{i}$$a,$$1\leq\dot{i}\leq 4$$\lambda_{a,b,c}^{(i)}$$(Q_{1})^{\alpha}=\zeta_{1}\cdot Q_{1}$

(8.3)

$\{$

$(Q_{1})^{\alpha}=\zeta_{1}\cdot Q_{2}$$(Q_{1})^{\alpha}=\zeta_{1Q_{3}}.$

,

$c$

$)$

$(Q_{2})^{\alpha}=\zeta_{2Q_{2}}.$

,
,

$b,$

$(Q_{2})^{\alpha}=\zeta_{2}\cdot Q_{3}$$(Q_{2})^{\alpha}=\zeta_{2}\cdot Q_{1}$,,,$(Q_{3})^{\alpha}=\zeta_{\mathrm{s}}\cdot Q_{3}$,$(Q_{3})^{\alpha}=\zeta_{3Q_{1}},$,$(Q_{3})^{\alpha}=\zeta_{\mathrm{s}}\cdot Q_{2}$

,

if
if
if

$\alpha|F=\mathrm{i}\mathrm{d}$,$\alpha|F=\sigma_{2}$,$\alpha|F=\sigma_{3}$with roots of unity
and which depend on . We computeby Shimura’s
method explained in [Y2], \S 3. We obtain (listing to the 20-th decimal place)$\zeta_{1},$$\zeta_{2}$$\zeta_{3}$$\alpha$$Q_{1}=19.59569210489183464935\cdots$ ,$L_{K}(1, \lambda_{a,b_{C}}^{(i}),)$$Q_{2}=0.04269484846107960286\cdots$,$Q_{3}=4.61030480431463779839\cdots$

We find easily that
. The coincidence is to the 33-rd decimal place. We
have $V(c_{2})=-V(c_{1}),$ $W(c_{2})=-W(c_{1})$ ,$Q_{1}Q_{2}Q_{3}= \frac{27}{7}$$V(c_{1})-V(c_{2})= \frac{1}{2^{2}\cdot 3^{2}\cdot 167^{2}}\{(-85174-389193\omega_{1}+133671\omega_{2})\log\epsilon$(8.4)$+(-313883-283680\omega_{1}-657969\omega 2)\log\eta\}$ ,$W(c_{1})-W(c2)=- \frac{4}{3}\log 167$.

To identify
with algebraic numbers is morethan the case of quadratic fields.
We proceed in the following way. Note that (7). $=(2-\omega_{1})^{3}$ in $F$ . Rom the experience inthe quadratic field case, we suppose that$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{C}\mathrm{u}}\mathrm{i}\mathrm{t}$$Q_{i}’ \mathrm{s}$$(*)$$167^{-4/3}Q_{1}= \epsilon^{a}\eta^{b}\cdot\frac{3}{2-\omega_{1}}(\delta^{(1}))^{\alpha}(\delta^{(2)})^{\beta}(\delta^{()}3)^{\gamma}$

6Barnes did not estimate the remainder term explicitly. Probably this is the reason for that Shintani[Sh2] gave another as ymptotic expansion ofwith an explicit estimate of the remainder term.
Theor finds an elementary method to estimate the remainder term of Barnes’ asymptotic expansionofif
for all 1 $\leq i<j\leq r$ . This result includes classical estimate of theremainder term of Stirling’s series ([WW], p. 252) as a special case and is quite practical for our present
purpose. The details will be exposed elsewhere.
$\mathrm{r}_{2}(z,\omega)$$\log\overline{\rho_{2}(\omega)}$$\mathrm{a}\mathrm{u}\mathrm{t}\acute{\mathrm{h}}$$\log\frac{\Gamma_{r}(z,\omega)}{\rho_{r}(\omega)}$$\omega_{i}/\omega_{j}\in\overline{\mathrm{Q}}-\mathrm{Q}$

198
where
action of$\alpha,$$\beta,$$\sigma_{2}$$\gamma\in \mathrm{Z}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\theta\alpha+\beta+\gamma=-4$, we would have$167^{-4/3}Q_{2}=\eta(a\epsilon\eta)-b$$(**)$

.

an.d

$a,$

$b\in(2^{2}\cdot 3^{2}\cdot 167^{2})^{-1}\mathrm{z}$$\frac{3}{2-\omega_{2}}(\delta^{(}2))\alpha(\delta^{(}3))\beta(\delta^{()}1)^{\gamma}$. Then, by the.

and are given, we can solve $(*)$ and $(**)$ and obtain and . We search ,is close to an integer for the solution $(a, b)$ . This procedureso that$\beta=\gamma=0$$\alpha=-4,$
and we findworks well for

When$\beta,$$\alpha,$$\beta$$b$$a$$\gamma$$\alpha$$2^{2}\cdot 3^{2}\cdot 167^{2}a$$\gamma\in \mathrm{Z}$$a= \frac{901097}{2\cdot 3^{2}\cdot 167^{2}}$,$b= \frac{927649}{2\cdot 3^{2}\cdot 167^{2}}$.

Thus we obtain identifications$Q_{1}=167^{4/3}\cdot\epsilon^{901097}\eta^{9}/2\cdot \mathrm{s}12.67272649/2\cdot 3162.7^{2}$$=\sqrt[3]{167}$.

.

$\epsilon^{901097}\eta^{927649/2\cdot 3}\frac{23395929-8832486\omega_{1}+13899651\omega_{2}}{7\cdot 167^{3}}/2\cdot 3^{2}\cdot 167^{2}2.1672$$Q_{2}=167^{4/\mathrm{s}}\cdot\epsilon^{-9276}\eta^{-}49/2\cdot \mathrm{s}^{2}\cdot 167226552/2\cdot 3^{2}\cdot 167^{2}$$=\sqrt[3]{167}$.

$\frac{3}{2-\omega_{1}}\frac{1}{\delta^{4}}$.

,$\frac{3}{2-\omega_{2}}\frac{1}{(\delta^{(2}))^{4}}$$\epsilon^{-92764}9/2\cdot \mathrm{s}1672.2265\eta^{-}\frac{9496278-13899651\omega_{1}-22732137\omega 2}{7\cdot 167^{3}}52/2\cdot 3^{2}\cdot 167^{2}$,$Q_{\mathrm{s}=}167^{4/\mathrm{s}} \cdot\epsilon^{26552}\eta\frac{3}{2-\omega_{3}}/2\cdot 3^{2}\cdot 167^{2}-901097/2\cdot 3^{2}\cdot 167^{2}.\frac{1}{(\delta^{(3)})^{4}}$$=\sqrt[3]{167}$.

$\epsilon^{26552}\eta\frac{32228415+22732137\omega_{1}+8832486\omega_{2}}{7\cdot 167^{3}}/2\cdot 3^{2}\cdot 167^{2}-901097/2\cdot 32.1672$.

The numerical values coincide with these algebraic numbers to the 33-rd decimal place.
Concerning the exponents of and , we find (cf. (8.4))$\epsilon$$\eta$

2, $901097+(-85174-389193\omega_{1}+133671\omega_{2})\equiv 0$

2 $\cdot 927649+(-313883-283680\omega_{1}-657969\omega 2)\equiv 0$
whereespecially to Example 2.
$\mathfrak{p}’=(\delta^{(2)}),$$\mathfrak{p}’’=(\delta^{(3)})$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}^{\prime 2}\mathfrak{p}’\prime 2$$\mathrm{m}\mathrm{o}\mathrm{d}$,$\mathfrak{p}^{;2\prime\prime 2}\mathfrak{p}$,. This phenomenon is quite similar to Examples $1\sim 7$ ,References[A] G. W. Anderson, Logarithmic derivatives of Dirichlet
abelian varieties, Comp. Math. 45(1982), 315-332.

$L$-functions and the periods of[Bal] E. W. Barnes, The theory of the double gamma function, Philos. Rans. Roy. Soc. (A),196 (1901), 265-388.

199[Ba2] E. W. Barnes, On the theory of the multiple gamma function,
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$r_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{s}}$. Cambridge Philos.

[B1] D. Blasius, On the critical values of Hecke -series, Ann. of Math. 124(1986), 23-63.
$L$[C] P. Colmez, P\’eriods des vari\’et\’es ab\’eliennes \‘a multiplication complexe, Ann. of Math.
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[SC] A. Selberg and S. Chowla, On Epstein’s zeta-function, Jour. f\"ur reine und angew. Math.
227(1967), 86-110. (Papers of A. Selberg, No. 34)$=\mathrm{C}_{0}11\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{e}\mathrm{d}$[S1] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. 102(1975), 491-515.

[S2] G. Shimura, Automorphic forms and periods of abelian
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varieties.’ J. Math. Soc. Japan[S3] G. Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. 111(1980), 313-375.
[S4] G. Shimura, Algebraic relations between critical values of zeta functions and innerproducts, Amer. J. Math. 104(1983), 253-285.

[S5] G. Shimura, Abelian varieties with complex multiplication and modular functions, toappear from Princeton University Press.
[Shl] T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at
non-positive integers, J. Fac. Sci. Univ. Tokyo 23(1976), 393-417.
[Sh2] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ.
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[Sh3] T. Shintani, On values at s $=1$ of certain L functions of totally real algebraic number
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[Sh4] T. Shintani, On certain ray class30(1978), 139-167.

invari.ants of real quadratic fields, J. Math. Soc. Japan[Sh5] T. Shintani, On special values of(1977), 204-216 (in Japanese).

$L$-functions of algebraic number fields, Sugaku 29[ST] G. Shimura and Y. Taniyama, Complex multiplications of abelian varieties and its
applications to number theory, Publ. Math. Soc. Japan No. 6, 1961.
[TV] E. Thomas and A. T. Vasquez, On the resolution of cusp singularities and the Shintani
decomposition in totally real cubic number fields, Math. Ann. 247(1980), 1-20.
[WW] E. T. Whittaker and G. N. Watson, A course of Modern Analysis, fourth edition, Cambridge University Press, 1927.
[Y1] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular
forms, Duke Math. J. 75(1994), 121-191.

200[Y2] H. Yoshida, On absolute CM-periods, to appear in Symposia Pure Math. (Shimura
honorary volume).
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-01, Japanemail address: yoshida@kusm.kyoto-u.ac.jp