数理解析研究所講究録
971 巻 1996 年 116-124

116

On Demu\v{s}kin Groups
Maurice Arrigoni*
Tokyo Metropolitan University
Abstract
Let be an algebraic number field, an odd prime, and,whereis the maximal pro-rof unramified outside the set ofprimes dividing . We consider situations whereis a Demu\v{s}kin group, especially when is totally real or of CM type.
$k$$G_{S}=\mathrm{G}\mathrm{a}1(ks/k)$$p$$k_{S}$$\mathrm{e}\mathrm{X}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$

$S$

$k$

$G_{S}$$p$$k$

Several authors gave conditions in order to express the Galois grouppro-p-prOduCt:$G_{S}$

as a free$G_{S}=(v\in T\star G_{v})\star F$

where $F$ is a free
is a subset of andbeing
the maximal pro-p-extension of the localization of at . However these conditions
are rather restrictive, implying in particular that the decomposition groupof anextension of $v\in T$ tois equal to the whole local group. Keeping this in mind, weare all the more interested in the cases whereis a Demu\v{s}kin group without being
equal tofor any $v\in S$ .
When is totally real, we can give a criterion for
to be a Demu\v{s}kin groupinvolving Iwasawa theory, see Proposition 5. Using numerical data, we obtain many
examples when $p=3$ and is a real quadratic field.
We classify the Demu\v{s}kin groups
in two types, depending on the existence of a
$v\in
S$
place
such that the natural map$S$$\tau$$\mathrm{p}\mathrm{r}\mathrm{o}-p- \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{P}},$$G_{v}=\mathrm{G}\mathrm{a}1(k_{v}(p)/k_{v}),$$k$$k_{v}(p)$$v$$G_{S}^{v}$$k_{S}$$G_{v}$$G_{S}$$G_{v}$$k$$G_{S}$$k$$G_{S}$$H^{2}(G_{S}, \mathbb{Z}/p\mathbb{Z})arrow H^{2}(G_{v}, \mathbb{Z}/p\mathbb{Z})$is an isomorphism. If such a exists, we say thatis a Demu\v{s}kin group of local type.
When is totally real or of CM type, we give necessary and sufficient conditionsforto be a Demu\v{s}kin group of local type, involving only the arithmetic of thebase field , see Theorems 1 and 2. Examples are given in the case $p=3$ and is
an imaginary -quadratic field. In these examples we have $G_{S}=G_{S}^{v}$ , where is theunique place of dividing 3 and $G_{S}\neq G_{v}$ ; this was the situation we were interested
in, without allowing the case is totally real where the Proposition 5 gives an easy
characterization.
$G_{S}$$v$$k$$G_{S}$$k$$k$$\mathrm{b}\mathrm{i}$

$v$

$k$

$k$

*This research was partly supported by-Aid for Scientific Research, Ministry of Education,
Science, Sports and Culture, Japanese Government, No. 06094224.
$\mathrm{c}_{\mathrm{r}\mathrm{a}\mathrm{n}}\mathrm{t}_{-\mathrm{i}}\mathrm{n}$

117

1

Introduction

Let us introduce some notations associated with the field$k$and the prime . We denote$p$$-r_{1}\mathrm{b}\mathrm{y}$(resp. ) the number of real (resp. complex) archimedean places of$-k_{S}$ the maximal pro-p-extension ofunramified outsideandthe cyclotomic -extension of-primary part of$-\mu_{p}$ the group of p-th roots of unity, and for a field $F,$ $\mu(F)$ the$F$the group of roots of unity contained in$-k_{v}$ the localization of(resp.
) the group of units (resp.
at a prime
the local degree, whereprincipal units) in the ring of integers of
anddivides the prime ,$-k_{v}(p)$ the maximal pro-p-extension of$k$$r_{2}$$S$$k$$-k_{\infty}$$k(k_{\infty}\subset k_{S}),$$\mathbb{Z}_{p}$$\Gamma=\mathrm{G}\mathrm{a}1(k_{\infty}/k)$$\Lambda=\mathbb{Z}_{\mathrm{P}}[[\mathrm{r}]]$

$p$

$k$

$U_{v}^{(1)}$$U_{v}$$v,$$n_{v}=(k_{v} :$k_{v}$\mathbb{Q}_{\ell})$$\ell$$v$$k_{v}$$-G_{S}=\mathrm{G}\mathrm{a}1(k_{S}/k),$ $G_{v}=\mathrm{G}\mathrm{a}1(k_{v}(p)/k_{v})$the decomposition group of an extension of)
automorphism of$-G_{S}^{v}$

$v$

to

$k_{S}(G_{S}^{v}$

is defined up to an inner$G_{S}$$-k^{J}=k(\mu_{p}),$$\triangle=\mathrm{G}\mathrm{a}1(k’/k)$

1 if
1 if
otherwiseotherwise$-E$ the unit group of the ring of integers of) the -Sylow of the class group (resp. -class group) of–Cl (resp.
$-V_{S}=$ {. By comparing class
for
for
}
field theory and Kummer theory, we obtain an isomorphism$\mu_{p}\subset k$$\mu_{p}\subset k_{v}$$-\delta=\{$$\delta_{v}=\{$

$0$

$0$

$k$

$\mathrm{C}1_{S}$

$S$

$p$

$a\in k^{\cross}|a\in k_{v}^{\cross p}$$k$$v\not\in S$$v\in S;a\in U_{v}k_{v}^{\cross \mathrm{p}}$$/k^{\cross p}$(1)$Vs\simeq \mathrm{H}\mathrm{o}\mathrm{m}\Delta(\mathrm{c}1_{s}(k’), \mu p)$

When there is more than one algebraic number field in consideration, we will append
the name of the field to the above objects.
Let be a. For every integer , let $H^{n}(G)=H^{n}(G, \mathbb{Z}/p\mathbb{Z}),$ acting trivially on. $d(G):=\dim H^{1}(G)$ is the generator rank of , and $r(G):=\dim H^{2}(c)$is the relation rank of .
A motivation for our problem is given by the following local result
$G$

$G$

$n$

$\mathrm{p}\mathrm{r}\mathrm{o}-p- \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{P}}$$G$$\mathbb{Z}/p\mathbb{Z}$

$G$

1. If
2. If

$\mu_{p}\not\subset k_{v}$, then$G_{v}$

is a free$\mu_{p}\subset k_{v}$, then$G_{v}$

is a Demu\v{s}kin group.

Definition 1
cup-product$G$.

$\mathrm{p}\mathrm{r}\mathrm{o}-p- \mathrm{g}_{\Gamma \mathrm{O}}\mathrm{u}_{\mathrm{P}}$is defined to be a Demu\v{s}kin group if $d(G)$ is finite, $r(G)=1$ , and the$H^{1}(G)\cross H^{1}(G)arrow H^{2}(G)\dot{i}S$a perfect pairing.

Let us now consider the global situation. The Galois groupwell-known property:$\mathrm{c}\mathrm{d}(Gs)$(the cohomological dimension of$G_{S}$) is$\leq 2$

$G_{S}$

has the following, and $d(G_{s)}$ is finite.

118

is a free proand this property holds also for a Demu\v{s}kin group. After the caseis a Demu\v{s}kin group might be-group (or equivalently $cd(c_{s)}=1)$ , the caseconsidered as the easiest situation to handle in restricted ramification Theory.
, the Poitou-Tate duality gives the
In order to evaluate the relation rank offollowing exact sequence$G_{S}$$G_{S}$$p$$G_{S}$(2)$0 arrow\mu_{p}(k)arrow\prod_{v\in S}\mu p(kv)arrow H^{2}(G_{S})^{*}arrow V_{S}arrow 0$

denotes the Pontrjagin dual of a locally compact abelian group. In theand the map
is in duality withsequence (2), the local group.
correspto the composition
rank and the relation rank
generatortheWe will use the following relation between
giving the value of its Euler-Poincar\’e characteristic:
of

where

$A^{*}$$\mu_{p}(k_{v})arrow$$H^{2}(G_{v})$$\mu_{p}(k_{v})$$G_{v}arrow G_{S}^{v}arrow G_{S}$$H^{2}(G_{s)^{*}}$$G_{S}$(3)$\chi(G_{s)}:=1-d(c_{s^{)+}}r(c_{s})=-r_{2}$is a Demu\v{s}kin group. Then the relation rank of
Let us suppose thatto one, so there are only two possibilities:$G_{S}$$G_{S}$

is equal

is a Demu\v{s}kin group of local
will say that
1. $V_{S}=0$ andtype. Then the natural map $H^{2}(G_{S})arrow H^{2}(G_{v})$ is an isomorphism, for every$v\in S$ such that.
$G_{S}$$\delta+1=\sum_{v\in s}\delta_{v}-\mathrm{w}\mathrm{e}$$\delta_{v}=1$

is a Demu\v{s}kin group of global
will say that
2. $\dim V_{S}=1$ andtype. Then, $H^{2}(G_{s)}arrow H^{2}(G_{v})$ is the zero map for every $v\in S$ .
$G_{S}$$\delta=\sum_{v\in S}\delta_{v}-\mathrm{w}\mathrm{e}$. Furthermore such
In any case there are at most two primes $v\in S$ such that
otherwise there would exist a finite extension $K$ in suchprimes can not split in$G_{S}(K)$ , having finite index in
, is
that
) $\geq p>2$ , which is absurd because
also a Demu\v{s}kin group. So we have proved the following$\delta_{v}=1$$k$$k_{s;}$$G_{S}$$\Sigma_{v\in S(K}$Proposition 1 Suppose that
. Then $G_{S}=G_{S}^{v}$ .

$G_{S}$

is a Demu\v{s}kin group and there exists $v\in S$ such that$\delta_{v}=1$2Known Results

We quote here some results relevant to our problem.
is a Demu\v{s}kin group. Then, by Proposiand
Suppose first that contains
isand we have $|S|=1$ (resp. $|S|=2$ ) if
tion 1, the places of do not split ina Demu\v{s}kin group of global (resp. local) type. Conversely, if contains , using a
result of Kuz’min, see [4, Fondamental Theorem, A&B(a)], we have$k$

$G_{S}$$\mu_{p}$$S$$G_{S}$$k_{S}$$k$

1. if $|S(k_{S})|=1$ , then

$G_{S}$

is a free$\mathrm{p}\mathrm{r}\mathrm{o}-p- \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{P}}$.

$\mu_{p}$119

2. if $|S(k_{S})|=2$ , then $G_{S}=G_{v_{1}}=G_{v_{2}}$ , where $S(k)=\{v_{1}, v_{2}\}$ , hence
Demu\v{s}kin group.

$G_{S}$

is a

Hence, we deduce the following (essentially due to Kuz’min). Thenis a Demu\v{s}kin group
and
Proposition 2 Suppose that contains$\dot{i}=1,2$. Moreover, we haveand only if $S=\{v_{1}, v_{2}\}$ and does not split infor$G_{S}=G_{v_{1}}=G_{v_{2}}$ , andis a Demu\v{s}kin group of local type.
$k$$\dot{i}f$$G_{S}$$S=S_{\mathrm{p}}$$\mu_{p}$$k_{S}$$v_{i}$$G_{S}$

Ku’zmin gave the following example: Let $p=3$ and$k=\mathbb{Q}(\sqrt{-3}, \sqrt{15})$. Then$G_{S}$

is a

Demu\v{s}kin group.

In [6], Wingberg gives necessary and sufficient conditions fora free pro-p-group $F$ , to the free pro-p-product of the local groups
is a maximal subset with the property $(+)$ :where$G_{S}$

to be equal, up to,for$v\in(S\backslash S_{0})$$G_{v}$$S_{0}\subset S$

with $(+)$ :$G_{S}=(\star G_{v})\star Fv\in(s\backslash s_{)}0$$\sum_{w\in s_{0}}\delta_{w}=\delta$

In the particular case of Demu\v{s}kin groups , here are the conditions (seeorem (i) and Corollary b):
Proposition 3 Let $v\in S$ such thatgroup of local type) if and only$\delta_{v}=1$. Then $c_{s=}G_{v}$ (hence$G_{S}$$1\mathrm{o}\mathrm{c}$.

$\mathrm{c}\mathrm{i}\mathrm{t}$. The-

is a Demu\v{s}kin$\dot{i}f$$S_{0}=S\backslash \{v\}$where$V_{S_{0}}^{S}=${has the property $(+),$ $r_{2}=n_{vf}$ and$a\in k^{\cross}|$$a$$\in k_{v}^{\cross p}$

for$v\in S_{0};a\in U_{v}k_{v}^{\cross p}$$V_{S_{0}}^{S}=0$

for,$v\not\in S$}$/k^{\cross p}$.

Remark. Except the case $\delta=1$ studied by Kuz’min, there is no example of fieldsatisfying Proposition 3.

$k$Totally real fields

3

In order to tackle the totally real case, we will use the following result, see [1, Prop. 1]
Proposition 4 Let
be a pro-p-group having a closed normal subgroup $H$ such thatthe quotient $\Gamma=G/H$ is one-generator free pro-p-group. Then the following conditions are equivalent
$G$

$a$

1.

$G$

is a two-generator Demu\v{s}kin group.

2. $H$ is$a$

one-generator free pro-p-group.

We suppose now that the field is totally real. Then by the equality (3), we have$d(G_{s)}=1+r(Gs)$ . Hence, by the previous Proposition,
is a Demu\v{s}kin group
if and only if the groupis isomorphic to , and from a property
isof pro-p-groups, this is true if and only if the maximal abelian pro-p-quotientisomorphic to . We take the following notations of Iwasawa’s theory:$k$$G_{S}$$H=\mathrm{G}\mathrm{a}1(k_{S}/k_{\infty})$$\mathbb{Z}_{p}$$H^{ab}$$\mathbb{Z}_{p}$120$\bullet$$H=\mathrm{G}\mathrm{a}1(k_{s}/k_{\infty})$$\bullet$$k_{n}$$\bullet$,$\mathcal{X}=H^{ab}$the maximal A-module unramified outside

the n-th layer of the cyclotomic extension$S$$k_{\infty}/k$, the projective limits being taken via thenorm maps. By class field theory, $X$ (resp. $X’$ ) is isomorphic to the Galois
group of the maximal abelian pro-p-extension ofwhich is unramified (resp.
split totally).
unramffied and where the places of$X= \lim_{arrow}\mathrm{C}1(k_{n})$,$X’= \lim_{arrow}\mathrm{C}1s(k_{n})$$k_{\infty}$$S(k_{\infty})$$\bullet$$\trianglearrow\omega \mathbb{Z}_{p}^{\cross}$

For a

the Teichm\"uller describing the action of on-module $M$ , we denote by
the -component of $M$ :$M_{\omega}$$\mathbb{Z}_{p}[\triangle]$$\lambda(M)$.

$\omega${ $m\in M|\delta.m=\omega(a)m$ for every$M_{\omega}=$$\bullet$$\zeta\in\mu_{p}:\sigma(\zeta)=\zeta^{\omega(\sigma)}$$\triangle$$\delta\in\triangle$}

and $\mu(M)$ the Iwasawa invariants of a Noetherian A-module $M$ .

We use the following standard results of Iwasawa’s theory, see for instance [3]andare Noetherian torsion A-modules having the sameinvariants, and X has no finite submodule other than .
$X(k/)_{\omega}$$\mathcal{X}(k)$$\mu$$\lambda$

and

$0$

This implies that-rank of is equal to$\mu(\mathcal{X})=0$$\mathcal{X}$$\mathbb{Z}_{p}$if and only if is a free
. Hence, we obtain:$\mathcal{X}$$\mathbb{Z}_{p}$-module, and in that case the$\lambda(\mathcal{X})$Proposition 5 Let be a totally real field. Then
if $\mu(X(k’)\omega)=0$ and $\lambda(X(k’)_{\omega})=1$ .
$k$$G_{S}$

is a Demu\v{s}kin group if and only

Example. Let
be the maximal real subfield of. Then $\mu(X(k;))$ isequal to by the Theorem of Ferrero-Washington, and for $p<$ 125000,
isequal to the irregularity index
of , see [5, Remark following Corollary 10.3]. So
for $p<125000,$is a Demu\v{s}kin group if and only if $i(p)=1$ . In this case, the uniqueplace $v\in S$ is totally ramified in , because $\mathrm{C}1(k)=0$ ( $Vandiver^{;}S$ Conjecture) is true
for $p<125000$ . In particular we have $G_{S}=G_{S}^{v}$ , although
is a Demu\v{s}kin group of
global type.
We will now consider the case whereis a Demu\v{s}kin group of local type for a
totally real field . The discussion at the end of the introduction shows that we must
suppose that there exists a unique $v\in S$ such thatand does not split in any-ramified extension of . We have the following criterion using only the arithmetic of, see [1, Th. 1]:$k=\mathbb{Q}(\mu_{p})^{+}$$\mathbb{Q}(\mu_{p})$$0$$\lambda(X(k’)_{\omega}$$\dot{i}(p)$$p$$c_{S}$$k_{S}$$G_{S}$$G_{S}$$k$$\delta_{v}=1$

$S$

$v$

$k$

$k$

Theorem 1 Let be a totally real field, having a unique $v\in S$ such that, andsuppose that does not split in. Then the following conditions are equivalent.
$k$$\delta_{v}=1$$k_{\infty}/k$

$v$

1.

$G_{S}$$\cdot$is a Demu\v{s}kin group (of local type).

2. Let $w(F)$ be the cardinal of $\mu(F)$ for a field , and let
be respectively
the class number, the -adic regulator, and the discriminant of . Then
$F$

$p$

$h,$

$R_{p},$

$D$
$k$

121

$\frac{w(k’)hRp}{w(k_{v})\sqrt{D}}\prod_{w\in S}(1-N_{or}m_{k_{w}/\mathbb{Q}_{p}}w-1)$

is a -adic$p$$un\dot{i}t$.

3. $x^{J}(k’)\omega 0=$

4. The map$E/E^{p} arrow\prod_{w\in S}U_{w}/U_{w}p$

is injective, and the -Hilbert class field of

$k$

$p$

maximal abelian unramified pro-p-extension of ) is contained in$k$$k_{\infty}$

(the

.

Example.
is a real quadratic field, and $p=3$
We suppose that $d>1$ is square free integer. Let $k^{*}=\mathbb{Q}(\sqrt{-m})$ the “mirror field”,
where $m=d/3$ if 3 divides , and $m=3d$ otherwise. In standard way, we have$X(k’)\omega=X(k^{*})$ , and also similar results for $X’(k*),$By the Theorem of Ferrero-Washington, we have $\mu(X(k*))=0$ . Hence by Proposition 5, we obtain:$k=\mathbb{Q}(\sqrt{d})$$d$$\mathrm{C}1(k^{*}),$$G_{S}(k)$$\mathrm{C}1_{S}(k^{*})\ldots$

is a Demu\v{s}kin group $\Leftrightarrow\lambda(X(k*))=1$

We use the tables of [2] giving the -invariant of $X(k^{*})$ for $0<m<$ 100000. The
case
happens quite often (23489 times), giving us a good provision of Demu\v{s}kingroups. We give examples in two cases, just to illustrate different situations which werediscussed: $d\equiv-3$ (mod 9), and(mod 3).
$d\equiv-3$(mod 9) In this case, 3 is ramified in , and, where $\{v\}=S$ .
are the first ten values of $m\equiv-1$ (mod 3) for which $\lambda(X(k*))=1:m=2,5$ ,11, 17, 23, 26, 29, 38, 53, 59. Hence$\lambda$$\lambda=1$$d\equiv 2$$k$$\mu_{3}\subset k_{v}$$\mathrm{H}\mathrm{e}\mathrm{r}\overline{\mathrm{e}}$is a Demu\v{s}kin group of local type for $d=6,15,33,51,69,78$ ,87, 114, 159, 177$G_{S}(\mathbb{Q}\sqrt{d})$$\ldots$

It is also possible to use the last condition of Theorem 1: Here,is totally ramified
at , hence the 3-Hilbert class field of is disjoint to. So we obtain:$k_{\infty}/k$

$k$

$v$

$k_{\infty}$

Proposition 6 Let, where is a positive square free integer such that $d\equiv-3$(mod 9), and let $p=3$ . Then $G_{S}(k)$ is a Demu\v{s}kin group (of local type) if and only
if $Cl(k)=0$ and the fundamental unit of is not a cube in .
$k=\mathbb{Q}\sqrt{d}$$d$$\epsilon$$k$$U_{v}$(mod 3) In this case, 3 remains prime in , and for every integer , theideal of dividing 3 is principal, hence, which implies $X(k)=$$X’(k)$ . Here are the first values ofsuch thatis a Demu\v{s}kin group:$d\equiv 2$$k$$\mathrm{u}\mathrm{n}\mathrm{i}\overline{\mathrm{q}\mathrm{u}\mathrm{e}}$$k_{n}$$n$$\mathrm{C}1(k_{n})=\mathrm{C}1_{s(k_{n})}$$d$$G_{S}$

is a Demu\v{s}kin group of global type for $d=29,74,113,122,131$ ,137, 173, 182, 206, 251, 254, 257. . .

$G_{S}(\mathbb{Q}\sqrt{d})$122

CM fields

4

In this section, we generalize a process used by Kuz’min to construct Demu\v{s}kin groups,
see [4, Proposition 5.4], where he considered an extension $k/k^{+}$ of CM type, the fieldcontaining . In order to find new examples , we make the following weaker hypothesis:$k$$\mu_{p}$$k/k^{+}$

is an extension of $CM$ type, with Galois group$J=\mathrm{G}\mathrm{a}1(k/k^{+})$.

respectively the invariant and antiand-module $M$ , we denote by
For ainvariant part of $M$ under the action of .
Taking the inflation followed by the restriction, for every integer , we have isomorphisms$M^{+}$$\mathbb{Z}_{p}[J]$$M^{-}$$J$$\dot{i}$$H^{i}(G_{S}(k+)) \inf_{arrow H^{i}(}\mathrm{G}\mathrm{a}1(k_{S}/k+))arrow H^{i}(\mathrm{r}\mathrm{e}\mathrm{s}Gs(k))^{+}$

for the following reasons: the order of is prime to , so the restriction maps arebecauseisomorphisms. The isomorphisms hold trivially for $i=0$ , and also foris an
the cohomology groups vanish. The inflation. At last,is the maximal pro-p-quotient ofisomorphism, becauseby the sequence (2), $H^{2}(c_{S}(k+))$ andfrom the description of
isomorphic. In the case of Demu\v{s}kin groups, we can make a more precise statement,see [1, Prop. 10]:
$J$

$p$

$\dot{i}\geq 3$$H^{1}(Gs(k+))arrow H^{1}(\mathrm{G}\mathrm{a}1(k_{S}/k^{+}))$$\mathrm{G}\mathrm{a}1(k_{S}/k_{S}^{+})$$G_{S}(k^{+})$$H^{2}(G_{S}(k))+\mathrm{a}\mathrm{r}\mathrm{e}$$H^{2}(c_{s})$Proposition 7 If $G_{S}(k)$ is a Demu\v{s}kin group, then $G_{S}(k+)$ is also a Demu\v{s}kin group,and$H^{2}(G_{S}(k))=H^{2}(c_{s())^{+}}k$.

If we restrict to Demu\v{s}kin groups of local type, the previous proposition has the following converse, see [1, Th. 2]:
is a Demu\v{s}kin
Theorem 2 Let $k/k^{+}$ be an extension of CM type, such that
. Let us also denotegroup , and suppose that there exists $v\in S(k)$ such that
$G_{S}(k)$Demu\v{s}kin
groupand only if theisadividing . Thenby the place offollowing conditions hold:$G_{S}(k^{+})$$\mu_{p}\subset k_{v}$

$v$

(i)

$k^{+}$$v$$\dot{i}f$$\mu_{p}\subset k_{v}^{+}$(ii) $S(k^{+})=\{v\}$ and $|S(k)|=1+\delta$(iii) $X’(k)-=0$

In this case, $G_{S}(k)$ and $c_{s}(k^{+})$ are both Demu\v{s}kin groups of local type.

.
We now derive some consequences of Theorem 2. We suppose first that$X’(k)-=X’(k)\omega$
7,
Proposition
weobtainthe. Hence, by using Theorem 1 andThenfollowing result, see [1, Cor. 1]:$\mu_{p}\subset k$

123

Corollary 1 (compare with , Proposition 5.4]) Let $k/k^{+}$ be an extension of CM
type, such that contains . Then the following conditions are equivalent:$l\mathit{4}$$k$$\mu_{p}$1. $G_{S}(k)$ is a Demu\v{s}kin group.

2.
3.

$G_{S}(k^{+})$$0S.(k^{+})$

is a Demu\v{s}kin group of local type, and $|S(k^{+})|=1$ .

has only one element$v_{f}\mu_{p}\subset k^{+},$In this case we have $G_{S}=G_{S}^{w}=G_{w}$ , where

$v$

$w$

does not split in$k_{\infty}^{+}/k^{+}$

and $x’(k)^{-}=$

is one of the two places ofdividing .

$k$

$v$

Example. Let $p=3$ , letbe a real quadratic field, and.
In the previous section, we gave the first values of such that $G_{S}(k+)$ is a Demu\v{s}kingroup of local type. We obtain:$k^{+}=\mathbb{Q}\sqrt{d}$$k=\mathbb{Q}(\sqrt{-3}, \sqrt{d})$$d$$Gs_{3}(\mathbb{Q}(\sqrt{-3}, \sqrt{d}))$114, 159, 177

is a Demu\v{s}kin group for $d=6,15,33,51,69,78,87$ ,$\ldots$

We suppose now that
. If $G_{S}(k+)$ is a Demu\v{s}kin group, and if the condition(ii) of the Theorem holds, then we have $|S(k_{\infty})|=1$ . And under this hypothesis, theconditions $X’(k\infty)^{-}=0$ andare equivalent, because,see [5, Lemma 13.15] for an analogous result. Hence we obtain the following
Corollary 2 Let $k/k^{+}$ be an extension of CM type such that does not contain ,and suppose that there exists $v\in S(k)$ such that
contains . Then the following
conditions are equivalent:$\mu_{p}\not\subset k$$\mathrm{C}1_{S}(k)^{-}=0$$\mathrm{C}1_{S}(k)^{-}=X’(k_{\infty})_{\Gamma}^{-}$$k$$\mu_{p}$$k_{v}$$\mu_{p}$1. $G_{S}(k)$ is a Demu\v{s}kin group.

2. $S(k)=\{v\},$$\mu_{p}\subset k_{v\mathrm{z}}^{+}G_{S}(k^{+})$

is a Demu\v{s}kin group and $Cl_{S()^{-}}k=0$ .

In this case $G_{S}(k)=G_{S}^{v}(k)$ , but $G_{S}(k)\neq G_{v}(k)$ .
The last assertion of the Corollary comes from the fact that the Euler-Poincar\’e characteristic $-r_{2}(k)$ of $G_{S}(k)$ and the Euler-Poincar\’e characteristic $-n_{v}(k)$ of $G_{v}(k)$ aredifferent.
Example. Let $p=3$ , and let us find the -quadratic fields satisfying Corollary
2. The real maximal subfield, where is a square free integer, must be oftype given by Proposition 6. must have a unique place dividing 3, and the inertia
group of in
is non trivial (because
is ramified) and cyclic, by class field
theory. Hence the inertia field is an imaginary (otherwise would be real) quadratic
field:, where is a square free integer congruent to 1 (mod 3), because
3 remains prime in . Let
be the other quadratic subfield of . We havecanonically$k$$\mathrm{b}\mathrm{i}$$k^{+}=\mathbb{Q}\sqrt{d}$

$d$

$k$

$v$

$k/\mathbb{Q}$$v$$k^{+}/\mathbb{Q}$$k^{v}$$k^{v}=\mathbb{Q}\sqrt{-d’}$

$k$

$d$

$k^{v}$$\tilde{k}=\mathbb{Q}\sqrt{-dd’}$$k$$\mathrm{C}1_{S()^{-}}k\simeq \mathrm{C}1_{S}(k^{v})\oplus \mathrm{C}1_{S}(\tilde{k})$Furthermore
because the prime 3 is principal in
because the prime ideal dividing 3 in has order 1 or 2 intotally ramified at 3. Hence we obtain the following$\mathrm{C}1_{S(k^{v})}=\mathrm{C}1(k^{v})$$\mathrm{C}1(\tilde{k})$$k^{v}$, and$\mathrm{C}1_{S}(\tilde{k})=$$\tilde{k}$$\mathrm{C}1(\tilde{k}),\tilde{k}/\mathbb{Q}$

being

124

and
, whereProposition 8 Let $p=3$ and$d’\equiv
1$(mod 3). Thenintegers such that $d\equiv-3$ (mod 9) andgroup if and only if the following conditions hold:$k=\mathbb{Q}(\sqrt{d}, \sqrt{-d’})$1.

$Gs(\mathbb{Q}\sqrt{d})$

$d$

$d’$

$G_{S}$

are square free
is a Demu\v{s}kin

is a Demu\v{s}kin group.

2. $Cl(\mathbb{Q}\sqrt{-d’})=Cl(\mathbb{Q}\sqrt{-dd’})=0$ .
. The following table gives the
Let $m$ be the square free integer such thatvalue of $m$ when $m<500$ . .The first line and the first column give respectively the
first values d’ and satisfying the congruences of Proposition 8, such that 3 does not
is a Demu\v{s}kin group (See the list
anddivide the class number ofmeans that 3 divides the class number of . The
given in the previous section).
, except the cases
table shows that $G_{S}(k)$ is a Demu\v{s}kin group, with
$(7,
33)$
$(d’, d)=(37,6)$ or.
$\tilde{k}=\mathbb{Q}\sqrt{-m}$$d$$\mathbb{Q}\sqrt{-d’}$$G_{S}(\mathbb{Q}\sqrt{d})$$\tilde{k}$$\gamma h$$k=\mathbb{Q}(\sqrt{d}, \sqrt{-d’})$$171013192234374346$$d’$$d\downarrowarrow$$64215781143351222 258 69$

$6$

$15$

$15 105 6 195 285 330$$33$

$33 2\beta 1 330 429 6$

$51$

$69$
$78$

$51 357 6$$69
6$$78 195 6 483
429$

References[1] M. Arrigoni, Representation of Demu\v{s}kin groups, Tokyo Metropolitan Univ. Math.
Preprint Series, 1995
9.
$\mathrm{n}^{\mathrm{O}}$[2] T. Rkuda, Iwasawa -invariants of imaginary quadratic fields, J. of the Collegeof Industrial Technology, Nihon Univ. 27 (1994), 35-88, Corrigendum to appear
in ibid.
$\lambda$[3] K. Iwasawa, On(1973), 243-326.

$\mathbb{Z}_{l}$-extensions of algebraic number fields, Annals of Math. 98[4] L. V. Kuz’min, Local extensions associated with -extensions with given ramification, Math. USSR Izvestija 9-4 (1975), 693-726.
$\ell$[5] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math.
83, Springer, 1982.

[6] K. Wingberg, On Galois groups of -closed algebraic number fields with restricted
ramification II, J. reine. angew. Math. 416 $(1991),187-194$ .
$p$