Jan 21, 1985 - ${\rm Log}((a,\tilde{k}/k)/q)$ which permits us to describe $G_{r}$ and its subgroups such as the inertia and higher ramification groups and to ...
TOKYO J. MATH. VOL. 9, No. 1, 1986
Decomposition and Inertia Groups in
-Extensions
$Z_{p}$
Georges GRAS University
of Franche-Comt\’e-Besan\caon
(Communicated
by K. Katase)
Introduction
In this paper, we shall give a canonical description of the decomposition and inertia groups, in a -extension of a number field , for any prime ideal of ; when this prime ideal divides , and ramifies in the extension, all the higher ramification groups are also described. This description gives immediately a numerical knowledge of the previous groups, as soon as the p-class group and the group of units of are numerically known. Of course, if $K/k$ is any abelian extension, the law of decomposition of prime ideals of is known if and only if the Artin group of $K/k$ is given; but in practice we have the opposite situation: the extension $K/k$ is specified by mean of some property (for instance $K/k$ is a -extension...) and the problem is to determine its Artin group. The results obtained in [2] give a general method for this kind of problem, via the use of a logarithm function, $Log$ , which induces a canonical isomorphism between of all -extensions of , and an the Galois group $G$ of the compositum explicit -module attached to and depending (numerically) on ideal in classes and units. It is well known that the decomposition group , of any prime ideal of , is the closure in $G$ of the image of ; then it is sufficient to by the Hasse norm residue symbol compute we obtain an explicit formula for for any ${\rm Log}((a,\tilde{k}/k)/q)$ and its subgroups such as which permits us to describe the inertia and higher ramification groups and to give some properties of jumps of ramification (\S 2). Finally, to illustrate this study, we consider (in \S 3) the case of imaginary quadratic fields and give all details for a numerical utilization. Some basic tools used in this paper may be compared with some ones $k$
$Z_{p}$
$k$
$p$
$Z_{p^{-}}$
$k$
$k$
$Z_{p}$
$\tilde{k}$
$Z_{p}$
$k$
$k$
$Z_{p}$
$G_{r}$
$\tilde{k}/k$
$\mathfrak{q}$
$k$
$k^{\times}$
$(( , \tilde{k}/k)/\mathfrak{q})$
${\rm Log}((a,\tilde{k}/k)/q)$
$a\in k^{\times};$
$G_{r}$
Received January 21, 1985 Revised October 28, 1985
42
GEORGES GRAS
of T. Kubota [6] who first gave important insights in S-ramification aspects of class-field theory (see also some developments of Kubota’s ideas in [8]). However, we think that the technics introduced in [2] and used here will give a new and more powerful approach for these questions. I am pleased to thank Professor H. Miki for his remarks and suggestions about this paper, and Professor S. Iyanaga for his help conceming its publication.
\S 1. Canonical description of
$G$
.
We fixe a prime number . the compositum of all For any number field , we denote by $G$ extensions of , and by the Galois group . If $K$ is any extension of , contained in , we know that the decomposition and inertia groups in $K/k$ of a prime ideal of , and more generally all its higher ramification groups (with upper numbering), are obtained by taking the restrictions of the corresponding groups in to $K$ (see [1], chap. 11); then it is equivalent to solve the problem for . any prime For ideal of we call: , the decomposition group of in , the inertia group of in , the higher ramification groups of (with upper in $p$
$\tilde{k}$
$k$
$Z_{p^{-}}$
$Ga1(\tilde{k}/k)$
$k$
$\tilde{k}$
$k$
$k$
$\tilde{k}/k$
$\tilde{k}$
$k$
$q$
$G_{r}$
$q$
$\tilde{k}/k$
$G_{Q}^{0}$
$G_{r}^{i},$
$\tilde{k}/k$
$\mathfrak{q}$
$i\geqq 0$
$\tilde{k}/k$
$q$
numbering).
-modules, they will be written additively. As these groups are free $S$ prime dividing $p$ . the set of We call ideals of To recall the main result which gives a canonical description of $G$ , we use the following notations concerning the field : $P,$ (i) are respectively the group of ideals, of principal ideals, of ideals prime to $S$ , of principal ideals prime to $S$ , $E$ are respectively the ring of integers and the group of (ii) $Z_{p}$
$k$
$k$
$I,$
$I_{s},$
$P_{s}$
$Z_{k},$
units,
, are respectively the product of com. corresponding product of groups of local and the at is identified with its diagonal embedding in , (iv) is the usual p-adic logarithm (we have log $=(log,),\in S$ $log,:U,\rightarrow k$ , is the classical extension of the p-adic logarithm on where (iii)
$C_{s}=\prod,eSk,,$
pletions of units; here,
$k$
$U_{s}=\prod,eSU$
$\mathfrak{p}\in S$
$k$
$C_{s}$
$log:U_{s}\rightarrow C_{s}$
’
$U,)$
,
, where $V_{s}=Q_{p}$ log $E$ is the -subspace of by the logarithms of units of , (vi) is the function defined by (v)
$B_{s}=C_{s}/V_{s}$
$Q_{p}$
$k$
$Log:I_{s}\rightarrow B_{s}$
$C_{s}$
generated
$Z_{p}$
Log
43
-EXTENSIONS
log $a+V_{s}$ ,
$\mathfrak{a}=\frac{1}{n}$
, is any integer such that (vii) . is the Artin map We know that Log is trivial on Ker , and then may be dePned on , and finally on (See [2], \S 2); the image of , by Log is the closure of Log in :
where
$n$
$\mathfrak{a}^{*}=aZ_{k},$
$a\in k^{\times}$
$(\alpha(\mathfrak{a})=((\tilde{k}/k)/\mathfrak{a}))$
$\alpha:I_{s}\rightarrow G$
$\alpha$
$G$
$\alpha(I_{s})$
$G$
$B_{s}$
$I_{s}$
The continuous map
THEOREM 1.1. dense subgroup of onto
$\alpha(I_{s})$
$G$
$\overline{{\rm Log} I_{s}}$
of
$G$
by Log
$Log:G\rightarrow\overline{{\rm Log} I_{s},}$
$\alpha(\mathfrak{a})={\rm Log} \mathfrak{a}$
, is
defined on the
a canonical isomorphism
$\overline{{\rm Log} I_{s}.}$
Now we identify $G$ with and we will describe any subgroup $G$ for instance, if of in terms of the corresponding subgroup of $G$ , , $k=Q$ the corresponding group where $q=p$ (resp. 4) if $p\neq 2$ is $\overline{{\rm Log} I_{s}}$
$\overline{{\rm Log} I_{s};}$
$qZ_{p}$
(resp. $p=2$ ).
is any group of automorphisms of , the groups $G$ , are canonically $Z_{p}[H]$ -modules and Log an isomorphism of
Note that if $C_{s},$
$V_{s},$
$U_{s},$
$Z_{p}[H]$
$B_{S}$
$H$
$k$
-modules.
\S 2. Canonical description of
$G_{r}$
.
2.1. The tame case. If the prime $q=I$ is not in $S$ , then we have immediately the result is a topological because I does not ramify and the Artin symbol generator of : $\alpha(I)$
$G_{\iota}$
THEOREM 2.1. the decomposition
If I is a prime ideal of group is of I in $\tilde{k}/k$
$G_{t}$
$k$
which does not divide Log I.
$p$
,
$G_{\iota}=Z_{p}$
REMARK. As I does not ramify, it is known that the Hasse norm , where , is given by residue symbol is the I-adic valuation on then we have, with the Log function: $((a,\tilde{k}/k)/I),$
$((\tilde{k}/k)/I)^{-v_{I}\{a)}$
$a\in k^{x}$
$v_{\iota}:k^{x}\rightarrow>Z$
$k^{\times};$
${\rm Log}(\frac{a,\tilde{k}/k}{I})=-v_{I}(a){\rm Log}$
then,
as the image of
$G_{\iota}=Z_{p}$
$k^{\times}$
by
$(( \tilde{k}/k)/I)$
I;
is dense in
$G_{\iota}$
,
we find again that
Log 1.
The wild case. , then , and the description of $G,,$ and ramifies in $G,$ , higher groups requires Log the an extension of the ramification
2.2. If
$q=\mathfrak{p}\in S$
$\tilde{k}/k$
$\mathfrak{p}$
$G_{l}^{0}$
44
GEORGES GRAS
function. Extension of the Log function to . ; if , we will put Log $\mathfrak{a}=(1/n)\log a+V_{s}$ , in the Let a same way as for elements of ; then we must define log on : the more natural and efficient definition is that of Iwasawa ([5], \S 4.2): Let and log $=(log,)_{es}$ (in the usual sense); if we consider in then there exists integers $e>0$ such that $u\in U,$ , and we put: a)
$I$
$\mathfrak{a}eI$
$\mathfrak{a}\cdot=aZ_{k},$
$ek^{x}$
$k^{x}$
$I_{s}$
$a\in k^{x}$
$a$
$k^{\times}$
$a=p^{\lambda}u,$
$log,$ $a=\frac{1}{e}log,$
(equivalently
we put
$u$
). for each The properties of Iwasawa’s log function show that homomorphism of groups. $log,p=0$
$\lambda\in Z,$
$\mathfrak{p}\in S$
$Log:I\rightarrow B_{s}$
is an
REMARK. It must be noted that for any me the value Logm is defined in but is not, in general, in . b) Class field theory. As the Artin map is not defined in , we use Hasse norm for residue symbol, whose main properties are the following ones: (i) the image of by is a dense subgroup of $G,$ ; (ii) more generally, if , is the subgroup , where is the usual filtration of the group $U,$ , then the image of by is a dense subgroup of the i-th higher ramification group , upper (in numbering). Of course, as $G$ is a pro-p-group, we have $G^{0}=G^{1}$, for all . Then the problem is reduced to the computation of , for . and c) Computation of Let be an element of and let . Let , a prime to ; we recall that may be approximated by a suitable Artin symbol in the following manner: , and let Let be any large enough integer $(n>v,(a)$ for instance), and let be such that the two following multiplicative congruences are satisfied: $\langle S\rangle$
$B_{s}$
$\overline{{\rm Log} I_{s}}$
$\tilde{k}/k$
$\mathfrak{p}\in S$
$(( \tilde{k}/k)/\mathfrak{p})$
$k^{\times}$
$k_{(’)}^{l},$
$i\geqq 0$
$k^{x}\cap U^{l}$
$\{U_{1}^{l}\}_{\iota\geq 0}$
$k_{(’)}^{\iota}$
$(( \tilde{k}/k)/\mathfrak{p})$
$G^{\iota}$
$\mathfrak{p}\in S$
$((a,\tilde{k}/k)/\mathfrak{p})$
$a\in k^{x}$
$\mathfrak{p}\in S$
$((a,\tilde{k}/k)/\mathfrak{p})$
.
$k^{x}$
$a$
$\mathfrak{p}\in S$
$aZ_{k}=\mathfrak{p}^{v(l)}a,$
$\mathfrak{a}\in I$
$((a,\tilde{k}/k)/\mathfrak{p})$
$\mathfrak{p}$
$m=\mathfrak{p}^{-1}\prod_{\mathfrak{g}eS}q$
$n$
$b_{n}\in k^{\times}$
$b.\equiv amod^{\times}\mathfrak{p}$
.
$ b.\equiv 1mod^{\times}m\cdot$
then if we put
$b.Z_{k}=\mathfrak{p}^{v_{1}(b)}b_{*},$
$\mathfrak{y}$
.
,
$\in I$
;
we see that
$v,(b.)=v,(a)$
and that
$Z_{p}$
.
45
-EXTENSIONS
Then where is $mod (\prod_{qeS}q)^{n}$ $G$ class field ; then in as . We have which is approximated by Log ; using the extension of Log to , described in \S a, we have Log , and we see the following facts: (i) if , then is close to ; (ii) if , then is close to $log,$ ; then is $(log, a, 0, \cdots, 0)+V_{s}$ ; then finally we obtain: $b_{n}\in I_{s}$
$\sigma_{1*}=((a,\tilde{k}/k)/\mathfrak{p})((\tilde{k}/k)/\mathfrak{b}_{r})^{-1}\in Ga1(\tilde{k}/k^{(n)})$
$\sigma_{n}\rightarrow 1$
$k^{(n)}$
the ray
$ n\rightarrow\infty$
${\rm Log}((a,\tilde{k}/k)/\mathfrak{p})$
$\mathfrak{b}_{n}$
$I$
but, by
Log
$\mathfrak{b}_{n}=$
$b_{n}-v,(a){\rm Log} \mathfrak{p}$
$\mathfrak{q}\in S,$
$0$
$\log_{r}b_{n}$
$q\neq \mathfrak{p}$
$\log_{\mathfrak{g}}b_{*}$
$q=\mathfrak{p}$
THEOREM 2.2. For any and in is given by: $a\in k^{\times}$
$((a,\tilde{k}/k)/\mathfrak{p})\in G$
${\rm Log} b$
$a$
$\mathfrak{p}\in S$
REMARK.
close to
, the canonical image
of
$\overline{{\rm Log} I_{s}}$
${\rm Log}(\frac{a,\tilde{k}/k}{\mathfrak{p}})=(log, a, 0, \cdots, 0)-v,(a){\rm Log} \mathfrak{p}$
c
.
Let
$a\in k^{\times}$
.
.
Then we have (See \S 2.1)
$\sum_{q}{\rm Log}(\frac{a,\tilde{k}/k}{q})=\sum_{qeS}{\rm Log}(\frac{a,\tilde{k}/k}{q})+\sum_{q\not\in S}{\rm Log}(\frac{a,\tilde{k}/k}{q})$
$=(log, a),eS-\sum_{qeS}v_{0}(a){\rm Log} q-\sum_{rs}v_{\mathfrak{g}}(a){\rm Log} q$
$={\rm Log} a-{\rm Log} a=0$
.
Of course this phenomena means the product formula. d) The decomposition group $G,$ . To obtain $G$, it is sufficient to compute the $sub- Z_{p}$ -module generated by the elements $(log, a, 0, \cdots, 0)-v,(a){\rm Log} \mathfrak{p}$ , when in varies in we obtain: $B_{s}$
$a$
THEOREM 2.3.
The decomposition group
$G$
, of
$\mathfrak{p}\in S$
in
$k^{\times};$
$\tilde{k}/k$
is:
$G,=((log, \pi,, 0, \cdots, 0)-{\rm Log} \mathfrak{p})Z_{p}+(log, U_{p})\times\{0\}\times\cdots x\{0\}+V_{s}/V_{s}$
where
$\pi$
, is any
prime element in
PROOF. We can choose Let
$a\in k^{\times};$
$\pi$
, in
$k^{\times}$
considering
$a$
$k,$
,
.
. in
$k_{\mathfrak{p}}^{x}$
,
we have
$a=\pi^{\lambda},u,$
$\lambda\in Z,$
$u\in k^{\times}\cap U,$
, and
${\rm Log}((a,\tilde{k}/k)/\mathfrak{p})=(\log_{\mathfrak{p}}a, 0, \cdots, 0)-v,(a){\rm Log} \mathfrak{p}=(x\log_{\mathfrak{p}}\pi,+log, u, 0, \cdots, 0)-$
. Conversely, as may be choosen independently, and by the fact that ) is dense in (resp. $U,$ ), the result follows easily.
$x{\rm Log} \mathfrak{p}=x((log, \pi_{\mathfrak{p}}, 0, \cdots, 0)-{\rm Log} \mathfrak{p})+(log, u, 0, \cdots, 0)+V_{s}$ $\lambda\in Z$
$Z$
and
(resp.
,
$u\in k^{\times}\cap U$
$k^{\times}\cap U,$
COROLLARY. Let
$Z_{p}$
$\tilde{H}^{s}$
the maximal subextension
of
$\tilde{k}$
which is unrami-
46
GEORGES GRAS
fied on
and such that $S$ splits completely (i.e. in which all completely). Then the field fixed by the subgroup $k$
$\mathfrak{p}\in S$
split
$\tilde{H}^{s}i\epsilon$
$({\rm Log}\langle S\rangle+{\rm Log} k^{\times})\cap\overline{{\rm Log} I_{s}}$
of
$\overline{{\rm Log} I_{s}}$
.
is fixed by the subgroup of Of course ; then it is the subgroup: $\tilde{H}^{s}$
$G$
generated by all the
$G,$
,
$\mathfrak{p}\in S$
$\sum_{eS}((0, \cdots, 0, log, \pi,, 0, \cdots, O)-{\rm Log} \mathfrak{p})Z_{p}+{\rm Log} U_{s}+V_{S}/V_{s}$
.
It is not difficult to prove the equality. REMARK 1. This result is consistent with the following situation of be and let class field theory: Let $H$ be the Hilbert p-class field of $S$ splits completely, $H$ , where and put the maximal subfield of . Then in the canonical isomorphism $Ga1(H/k)\simeq I/P$ (the class group $C$ ) we have $Ga1(H/H^{s})\simeq\{mP, m\in\langle S\rangle\}$ (denoted by $C(S)$ ) and ). therefore $Ga1(H^{s}/k)\simeq I/\langle S\rangle P\simeq C/C(S)$ (the S-class group $H^{s}$
$k$
$\tilde{H}=\tilde{k}\cap H$
$\tilde{H}^{s}=\tilde{k}\cap H^{s}$
$C^{s}$
, , and REMARK 2. Suppose that we write $w\in U,$ ; then a representative of Log is $(1/h)\log\omega=\log\pi,+(1/h)\log w$ $(log,\pi,, 0, \cdots, 0)-{\rm Log} \mathfrak{p}$ is represented by $(log, \pi,, 0, \cdots, 0)$ -log ,and $\mathfrak{p}^{h}=\omega Z_{k},$
$\omega ek^{x}$
$\omega=\pi_{l}^{h}w$
$\mathfrak{p}$
$\pi$
$(1/h)\log w=(-(1/h)log, w, \cdots, -\log_{r}\pi_{l}-(1/h)\log_{0}w, \cdots)_{r\neq},$
.
This shows that we obtain a kind of local-global computation involving the class group (via the number h) and the local units via $log,$ ; for instance, $(1/h)log,$ $w$ is not necessarly in $\log_{l}U,$ . $\pi,$
and, Consider $k=Q(\sqrt{-15})$ and $p=2$ ; we have $\mathfrak{p}^{2}=((1+\sqrt{-15})/2)Z_{k}$ ; if we take $\pi,=2$ , we have, as $C\simeq Z/2Z$, we see that $w=(1+\sqrt{-15})/8\in U,$ ; as $\omega=(1+\sqrt{-15)}/2=4w$ , where in $-25$ mod , we have $w\equiv-3$ mod 16, then $(1/2)log,$ $w=2u,$ we have . Then also $(1/2)\log_{r}w=2v,$
EXAMPLE.
$S=\{\mathfrak{p}, q\}$
$\sqrt{-15}\equiv$
$k_{l},$
$u\in Z_{l}^{*};$
$\mathfrak{p}^{\tau}$
$v\in Z_{2}^{*}$
$G,=(2,2)Z_{2}+(log, U,)\times\{0\}=(2,2)Z_{2}\oplus(4,0)Z_{2}$
As
$G=\overline{{\rm Log} I_{s}}=\langle{\rm Log} I_{8}\rangle+{\rm Log} U_{s}$
$G=(2,2)Z_{2}\oplus(4,0)Z_{2}$
e)
The higher ramification groups. We have recall in \S b that $G^{l},,$
(where
( $=G$ ,
$i\geqq 0$
$I_{8}|3$
and
.
$I_{s}^{2}=3Z_{k}$
)
we obtain
in this example).
, is the
$Z_{p}$
-module generated by
$Z_{p}$
the
47
-EXTENSIONS
, and the general computation of this symbol gives immediately (of course, as $G$ is a pro-p-group, we have \S c)
${\rm Log}((a,\tilde{k}/k)/\mathfrak{p}),$
(See
$G^{0},=G^{1},)$
$a\in k_{(’)}^{\iota}$
:
THEOREM 2.4. For any numbering) of in is:
$i\geqq 0$
, the
ramification group
$G_{l}^{l}=(log, U^{l})\times\{0\}\times\cdots\times\{0\}+V_{s}/V_{s}$
.
(in
upper
$\tilde{k}/k$
$\mathfrak{p}\in S$
$\tilde{k}/k$
,
$G^{l}$
.
Now we give some remarks concerning the jumps of ramification in Let
, be the absolute index of ramification of
.
We know that if then is an isomorphism (where fi denotes the closure of in the ring of integers of ). Thus we deduce, from the previous description of , the following results: $e$
$i>e,/(p-1)$ ,
$\mathfrak{p}$
$log,:U^{t}\rightarrow(\overline{\mathfrak{p}})^{l}$
$k,$
$\mathfrak{p}$
$G^{l},$
COROLLARY 1. For $i>e,/(p-1),$ , (hence if $i>e,/(p-1)$ is a jump of ramificat,bon of is also a jump of ramification).
$G^{l},=(\overline{\mathfrak{p}})^{\iota}\times\{0\}\times\cdots\times\{0\}+V_{s}/V_{s}$
$G^{l+\cdot \mathfrak{p}}=pG^{l}$
The case of a
$Z_{p}$
, and
$G,$
$i+e$
,
-extension is interesting and yields the following
corollary (cf. Wyman [9]):
COROLLARY 2. Let
be a -extension of and let be its Galois group. Then, for $i>e,/(p-1)$ , the set of jumps of ramification of , for , is $\{i_{0}+xe,, x\in N\}$ , where is the first $jump>e_{\mathfrak{p}}/(p-1)$ . $K/k$
$k$
$Z_{p}$
$\Gamma$
$\Gamma$
$\mathfrak{p}\in S$
PROOF.
$i_{0}$
Let
([1], chap. 11,
$H=Ga1(\tilde{k}/K)$
, and
.
$\Gamma=G/H$
We know that
$\Gamma^{l},=G^{l},H/H$
\S 2).
By the previous Corollary 1, if $i>e,/(p-1)$ is a jump of $i+e$ , is also a iump because $\Gamma^{l+\epsilon}=G^{l+l}’ H/H=pG^{l},H/H=p\Gamma^{i},$ . Let $m>n>e,/(p-1)$ be two consecutive jumps of ramification of We prove now that , if and . As , $i>e,/(p-1)$ , we have ; but we have the surjection; $\Gamma,$
$\Gamma$
$\Gamma^{n+1}=p\Gamma^{n}$
$\Gamma^{\iota+1}=p^{\alpha}\Gamma_{l}^{i},$
$\Gamma^{m+1}=p\Gamma^{m}$
$\Gamma,\simeq Z_{p}$
.
$\Gamma^{l+1}\neq\Gamma^{l},$
$\alpha\geqq 1$
$G^{i},/G^{t+1}\rightarrow G^{l},H/G^{i+1}H=\Gamma^{i},/\Gamma^{l+1}$
,
and must be a jump of $G$ , but, as $G^{l+}’=pG^{i},$ , this proves that is of exponent and then . Therefore we have ; then and has the exponent at least; but as $G^{n+}=$ $pGC$ , it is necessary to have $m+1>n+e,$ , then $m-n\geqq e,$ ; we obtain $m=$ $n+e$ , as desired. To compare these results with the local case, see [7] and the bibliogra$i$
.
$G^{i},/G^{l+1}$
$p$
$\Gamma_{\mathfrak{p}}^{m+1}=p^{2}\Gamma^{n}$
$\Gamma^{l+1}=p\Gamma^{l},$
$G^{n}/G_{l}^{m+1}$
$\Gamma^{n+1}=p\Gamma_{l}^{n}=\Gamma\phi$
$p^{2}$
48
GEORGES GRAS
phy of this paper. f) Some particular
cases. , we have Log (i) If does not split in $k/Q$ , then as $(1/e,){\rm Log} p=0$ ; but for a prime element of , log is not necessarily (except for instance if is principal) (in this direction, we observe that ). In this case , and then Log if we have: $pZ_{k}=\mathfrak{p}^{e_{p}}$
$p$
$\pi$
$0$
$k,$
$\mathfrak{p}=$
$\pi$
$\mathfrak{p}$
$\pi\in k^{x},$
$\pi Z_{k}=\infty,$
$\pi={\rm Log} \mathfrak{a}\in\overline{{\rm Log} I_{s}}$
$\mathfrak{a}\in I_{s}$
log $\pi+\log U_{s}+V_{s}/V_{s}$ ,
$G,=Z_{p}$
$G^{l},=\log U_{s}^{l}+V_{s}/V_{s}$
does not ramify in then $log,$ $\pi,=0$ ; but (except if Log is not necessarily . In this case we have: (ii)
If
$p$
,
for all
for any we may take $\pi,=p$ , does not split, then is inert in $k/Q$ )
$k/Q$ , $p$
.
$i\geqq 0$
$\mathfrak{p}\in S$
$0$
$\mathfrak{p}$
,
$G,=Z_{p}{\rm Log} \mathfrak{p}+(log, U,)\times\{0\}\times\cdots\times\{0\}+V_{s}/V_{s}$
,
$G:=(log, U^{l})\times\{0\}\times\cdots\times\{0\}+V_{s}/V_{s}$
for all . g) Local norms. contained in , we obtain an effective If $K$ is a finite extension of computation of $((a, K/k)/q)$ and a criterion for the condition is a local norm at in $K/k’$ . For this we must know the Artin group ([2], corollary ; then of to Theorem 2.1.), and the map: $i\geqq 0$
$\tilde{k}$
$k$
$a$ $ek^{x}$
$A\subset I_{s}$
$q$
$Ga1(K/k)\simeq\overline{{\rm Log} I_{s}}/\overline{{\rm Log} A}$
$K(A=\{\mathfrak{a}\in I_{s}, ((K/k)/\mathfrak{a})=1\})$
$k^{\times}\rightarrow\overline{{\rm Log} I}_{s}/\overline{{\rm Log} A}$
, $a\rightarrow\dagger_{(\log_{1}a,0},$
$-v_{r}(a){\rm Log} q+-v_{q}(a){\rm Log} q+\frac{{\rm Log} A}{{\rm Log} A}$
$\cdots,$
$0$
$ifif$
$q\not\in Sq\in S$
)
is, essentially, the Hasse norm residue symbol at (cf. \S 2, c). gives the local norms at
$q$
in
$K/k$ ,
and its kernel
$q$
which are local norms at For instance this gives the elements , in any finite subfield of This gives also (only for subfields of k) a new approach concerning explicit reciprocity laws. $a\in k^{\times}$
$\tilde{k}$
$q$
.
\S 3. The case of imaginary quadratic fields. Such a situation offers many possibilities of numerical computations.
$Z_{p}$
Let
49
-EXTENSIONS
square free, be an imaginary quadratic field, and put $H=\{1, s\}=Ga1(k-/Q)$ . Such a field has two fundamental -extensions which are normal over $Q$ (See [2], \S 3, or [4], \S 3): the cyclotomic one, , for which the law of decomposition of prime ideals is rather trivial (because is abelian), and the prodiedral one, $F$; we have , but, for $p=2,$ is of degree 1 or 2 over . We call the Galois group $Ga1(F/k)$ (then $H$ acts on via the , for all relation ). As log $E=0$ , we have $B_{s}=C_{s}$ and then Log $=log$ . We can then describe the subgroups of $G$ corresponding to and $F$ (See [4], \S 3): $k=Q(\sqrt{-m}),$
$m$
$Z_{p}$
$k_{\infty}=kQ_{\infty}$
$k_{\infty}/Q$
$k_{\infty}F=\tilde{k}$
$k_{\infty}\cap F$
$k$
$\Gamma\simeq Z_{p}$
$\Gamma$
$ s\gamma=-\gamma$
$\gamma\in\Gamma$
$k_{\infty}$
$Ga1(\tilde{k}/k_{\infty})=G^{*}=\overline{\log I_{s}}^{*}$
the kernel of the trace map
$C_{s}\rightarrow Q_{p}$
restricted to
$Ga1(\tilde{k}/F)=G^{H}=\overline{\log I_{S}}^{H}$
Of course, if $p\neq 2,$
$k_{\infty}$
and
$F$
, $\overline{\log I_{s}}$
,
.
are linearly disjoint over and ; if $p=2$ , we recall that $k$
$((1-\underline{s)/2)\overline{lo}}gI_{s}, \overline{\log I_{S}}^{H}=((1+s)/2\overline{)\log I}_{s}$
$\overline{l\log I_{s^{*}}}=$
$[k_{\infty}\cap F:k]=$
or 1 (See [4], Theorem 3.1 and its corollary). Here we give only the results concerning the law of decomposition of prime ideals of , for the various in $F/k$ (the general case, in extensions, is obtained easily from the corresponding information in and, mainly, in $F/k$ ). , all subfields of $F$ are characterised by their degree over As . For this we suppose that is numerically known (then , and are known). $(2Z_{2}:\log I_{s}^{H})=2^{\chi},$
$x=0$ $k$
$\tilde{k}/k$
$Z_{p}-$
$k_{\infty}/k$
$\Gamma\simeq Z_{p}$
$k$
$\overline{\log I_{s}}$
$\overline{\log I_{s}}^{H}$
$\overline{l\log I_{s^{*}}}$
$\chi$
(resp. ) be the intersection of REMARK. Let with the Hilbert $F$ $H$ p-class field (resp. ); if of is the unique subfield of $F$ ( degree of : log ); if $p=2$ , we have shown in [4] (See \S 2, pp. 13 (of course and 14) how to find and or 2). We distinguish the tame and wild cases. a) Tame case. ; the problem is to determine Let which is given by the index $(G:G_{\iota}+G^{H})$ . We know (See \S 2) that log 1; then we have the following cases: (i) if I does not split in $k/Q,$ $I=I$ and , therefore I splits completely in $F$. (ii) if I splits in $k/Q$ , we have two cases: -if $p\neq 2,$ $G/Z_{p}$ log $I+G^{H}\simeq G^{*}\oplus G^{H}/Z_{p}$ log $I+G^{H}\simeq G^{*}/Z_{p}\log(I^{1-})$ , and $\tilde{H}$
$\tilde{F}$
$p\neq 2,\tilde{H}=\tilde{F}$
$k$
$\overline{\log I_{s}}$
$\tilde{k}$
$U_{s}$
$\tilde{H}$
$\tilde{F}$
$[\tilde{H}:\tilde{F}]=1$
$\mathfrak{q}=I\not\in S$
$(\Gamma;\Gamma_{t})$
$G_{\iota}=$
$Z_{p}$
$G_{I}\subset G^{H}$
( -if $p=2$ ,
$(\Gamma:\Gamma_{1})=$
). log we have the exact sequence:
$\overline{\log I_{s^{*}}}:Z_{p}$
$I^{1-}$
50
GEORGES GRAS $ 1\rightarrow G^{*}/G^{*}\cap$
(
$Z_{2}$
log $I+G^{H}$)
$\rightarrow G/Z_{2}$
log $I+G^{H}$
$\rightarrow^{Tr}4Z_{2}/4\times 2(l)Z_{2}+4\times 2^{\chi}Z_{2}\rightarrow 1$
,
odd. where $lZ_{k}=I\cap Z$, and $\pm l=1+4\times 2(l)u,$ $\sigma=a\log I+\sigma_{0}$ ; then $0=$ , , be such that Let , and $\sigma_{0}=-(1/2)a$ log . As $G^{H}=2^{x+1}Z_{2}$ and log $l=2(l)+2v,$ odd, log ; this is sufficient , therefore we must have log . ( log $I+G^{H}$) and gives : Then we have the following values for $u$
$a\in Z_{2}$
$\sigma\in G^{*}$
$a$
$\sigma_{0}\in G^{H}$
$v$
$l$
$l+2\sigma_{0}$
$a\in 2^{\chi-\cdot(t)}Z_{2}\cap Z_{2}$
$2^{1+\cdot(l)}a\in 2^{1+\chi}Z_{2}$
$ G^{*}\cap$
$Z_{2}$
$=2^{{\rm Max}(\chi-*(l),0)-1}Z_{2}$
$I^{1-}$
$(\Gamma:\Gamma_{\iota})$
The result does not depend on the choice of . ; here the prime ramifies in $F/k$ and b) Wild case. Let , (which do not depend on the choice we must determine the groups ); the decomposition field is contained in , and the inertia field is of which is known ([3], array III, p. 14). Then it remains to compute for in $F/k$ ; we see that instance which is the residual degree of $(\Gamma,:\Gamma^{0},)=(G,+G^{H}:G^{0},+G^{H})=(G,:G_{l}^{0}+G_{1}^{H})$ and we have the following cases: (i) if is inert in $k/Q$ , then (See case (ii) in \S 2, f) we have $G,=$ $G^{0},=\log U_{s}$ , and therefore the decomposition field is also the inertia field (ii) if ramifies in $k/Q$ (See case (i) in \S 2, f) we have the following results: -if $p\neq 2$ , then $p|m$ and we can take $\pi=\sqrt{-m}$ ; then $G,=$ $Z_{p}\log(\sqrt{-m})+\log U_{s}=\log U_{s}$ ; therefore the decomposition field is also . $\pi=\sqrt{-m}$ $\pi+\log U_{s}$ if $G,=Z_{2}$ where log -if $p=2$ , we have $\pi=1+\sqrt{-m}$ computation of log and that 4, The 4. mod if mod given in [3] (See array I, p. 10) give the following results conof log $G,$ , and the index $(G,:G^{0},+G^{H})=$ ( $G,$ : log $U_{S}+G^{H}$): cerning $I|l$
$q=\mathfrak{p}\in S$
$\Gamma,,$
$\mathfrak{p}$
$\Gamma^{0}$
$\tilde{F}$
$\tilde{F}$
$\mathfrak{p}\in S$
$\mathfrak{p}$
$(\Gamma,:\Gamma_{l}^{0})$
$\mathfrak{p}$
.
$\tilde{F}$
$\mathfrak{p}$
$\tilde{F}$
$ m\equiv$
$m\equiv 1$
$2$
$U_{s}$
$\pi$
$Z_{p}$
51
-EXTENSIONS
In this array, means . (iii) if splits in $k/Q$ (See case (ii) in \S 2, f), then $G,=Z$, log $G^{0},=\log_{l}U,$ : $p\neq 2$ , -if we recall that the inertia and decomposition fields of in $F/k$ (resp. ) do not depend on the choice of ; but and is fixed $log,$ by , and is fixed by log $\{0\}+G^{H}=Z_{p}$ log ; therefore ( log : log ) $=$ $(a)$
$aZ_{p}$
$\mathfrak{p}$
$\mathfrak{p}+$
$\log_{\mathfrak{p}}U_{l}\times\{0\},$
$\times\{0\}$
$\mathfrak{p}$
$\tilde{H}$
$\tilde{H}^{s}$
$\tilde{H}$
$\mathfrak{p}$
$U_{\mathfrak{p}}\times\{0\}+G^{H}=\log U_{s}$ $\mathfrak{p}+\log U_{s}$
(
$Z_{p}$
$\tilde{H}^{s}$
$Z_{p}$
$(\Gamma,:\Gamma^{0},)=$
$\mathfrak{p}+\log_{l}U,$
$\mathfrak{p}+\log U_{s}$
$Z_{p}$
$\times$
$U_{s}$
log ). -if $p=2$ , the result of [4] (array III of Theorem 2.2) shows that is cyclic, and the final result depends on ; if $\chi=0$ , , then log $\mathfrak{p}+(4,0)Z_{2}+(2,2)Z_{2}$ : 0)Z_{2}+(2,2)Z_{2})$ ; . if $\chi=1,\tilde{H}=\tilde{F}=k$ , and in this case, is totally ramified $\mathfrak{p}+pZ_{p}\times pZ_{p}:pZ_{p}\times pZ_{p}$
$\tilde{H}/k$
.
$(4,
$x$
$[\tilde{H}:\tilde{F}]=2$
$(\Gamma,:\Gamma^{0},)=(Z_{2}$
$\Gamma,=\Gamma^{0},=\Gamma(\mathfrak{p}$
in
$F/k$ ).
References [11 [21
[3]
[41
E. ARTIN and J. TATE, Class Field Theory, W. A. Benjamin, New-York, 1967. G. GRAS, Logarithme p-adique et groupes de Galois, J. Reine Angew. Math., 343 (1983), 64-80. G. GRAS, Sur les $Z_{2^{-}}extensions$ d’un corps quadratique imaginaire, Ann. Inst. Fourier, vol. 33, no. 4 (1983), 1-18. G. GRAS, Logarithme p-adique, P-ramificatIon ab\’elienne et , S\’eminaire de Th\’eorie des Nombres, Bordeaux, Ann\’ee 1982-1983, . K. IWASAWA, Lectures on p-adic L-functions, Ann. of Math. Stud. 74, Princeton University Press, Princeton, 1972. T. KUBOTA, Galois group of the maximal abelian extension of an algebraic number field, Nagoya Math. J., 12 (1957), 177-189. H. MIKI, On the ramification numbers of cyclic p-extensions over local fields, J. Reine Angew. Math., 328 (1981), 99-115. H. MIKI, On the maximal abelian l-extension of a finite algebraic number field with given ramification, Nagoya Math. J., 7O (1978), 183-202. B. F. WYMAN, Wildly ramified gamma extensions, Amer. J. Math., 91 (1969), 135-152. $K_{2}$
$n^{o}12$
[51 [61
[71 [81
[91
Present Address: UNIVERSIT\’E DE ET
FRANCHE-CoMTB-BESANQON
C. N. R. S.
FACULT\’E DES
SCIENCES
MATH\’EMATIQUES U. A. 741 F-25030 BESANqON CEDEX FRANCE
