Radical hilbertien et tour localement cyclotomique

∗
`
K
` K
K(rK, cK)
KcZ`K
 Kc
∞K
 K Kc
 K∞=∪n∈NK(n)
K Kc
∞= (K∞)cK∞
[K∞:K]Kc
∞=K∞= (K∞)cK∞
K 
KH
K∞
 KH
 K∞
K 
K∞ K K∞/K
G= Gal(Kc
∞/K)KZ`Kc ∞
K∞H2(G, F`)
ΛK/NK∞/K (e EK∞)R`
KRK=Z`⊗K× K×
ΛK={x∈ RK/x ∈e J`
Ke
UK}Ke EK∞
K∞
Gal(K∞/K)
Gal(K∞/K)
Gal(K∞/K)
d`f CK+rK+cK+δ d`f CKf CKδ1
K  0
∗

 K
pK Kp
RpK×
pRp= lim
←− K×
p/K× p
`n
e
UpRp
Z`Kpp
e Up
Kp p e Up
 Kp
Z`[Kp:Q`]
pe UpUpRpp|
pe UpRp
JkQres Rp
(xp)p
Up
e
UKe Up
JKRK=Z`⊗K×
e
EK=RK∩e UK
K
Z`rK+cK
rKK cK
Z`KcK
e
JKe JKJK
Klc  K
Klc
pKlc pZ`
Kc
pKpKlc 
K Kc
Klc RKe Uk
Gal(Klc/Kc)'e JK/RKe UK
f
CK=e JK/e UKRK
 K
K
ΛK=RK∩e UKJ`
K=RK∩e
UKe J`
KΛK/R` K
K 
RK/R`
K
ζ, x
p
K0=K Kn+1 
Knf CKn
Klc
nγ.c γ
Gal(Klc
n/Kn)Gal(Kc n/Kn)'Z`c
f
CKn'Gal(Klc
n/Kc n)K∞=∪n∈NKn
[K∞:K]
K

F F
K 
KZ`
Kc
K/k PK
eeP(K/k) = [KP:KP∩kpc Qp]
P|p|pc Qpˆ Z QpZq
Qpq
PK/k K/k
Gal(KP/kp∩kpc Qp)
PK/k e IP(K/k)e Ip(K/k)
Gal(K/k)
e
fP(K/k) = [KP∩kpc Qp:kp] = [KP:kp]/eeP(K/k)
eeP(K/k)e fP(K/k) [KP:kp]
e
Up
Up
e
Ip(K/k)
[e Ip(K/k),Gal(Kp/kp)] .
k K/k
pk Kpkp
K k DpKp/kpe Ip
[Dp, Dp] = [Dp,e Ip]
Dp/e Ip
¯ K K
¯
K  K ¯ G= Gal( ¯ K/K)H
¯
Gp
¯
Ke Ip(¯ K/K)¯ Kp/Kc
p
p¯ K/K
H¯ G H
Klc
∞=Kc ∞
L=Klc
∞=Kc ∞
G= Gal(L/K)F¯ K[¯ G, H]
F/K∞

K
Kc
K∞
L¯
K
•
•
•
• •
•
•
[¯ G, H]
H
F
G
¯
G
ΛK
1−→ e
EK
e
E`
KNK∞/K (e
EK∞)−→ ΛK R`
KNK∞/K (e
EK∞)−→ `f
CK−→ 1.
x=uy`∈ΛKu∈e Uky∈e JKy
`f CKf CK
1−→ Z−→ Z−→ F`−→ 1
1−→ H2(G, Z)`−→ H2(G, F`)−→ H1(G, Z)[]−→ 1.
`H1(G, Z) = `Gab '`Gal(Klc/K)'`f
CK.
H2(G, Z)
F
Z`FcFnFc/F Fn
FcnFΓnFn/F
ΓFc/F NFc/F (e EFc)e EF
Fc/F
NFc/F (e EFc) = \
n
NFn/F (e EFn)

F
e
EFF F c/F
e
EF=NFc/F (e EFc)
n > 0 Γn
e
EFnΓnFn/F
|ˆ H0(Γn,e EFn)|=|H1(Γn,e EFn)|
F FnFn/F
H1(Γn,e EFn)
e
EF/NFn/F e EFn=ˆ H0(Γn,e EFn) = 1
−1
F
K Gn= Gal(Fn/K)G= Gal(Fc/K)
lim
←−
n
ˆ
H−1(Gn,e CFn)'lim
←−
n
ˆ
H−1(Gn,CFn),
CFn=JFn/RFne CFn=e JFn/RFn
Fn
lim
←−
n
CFn'lim
←−
ne
CFn.
CFn/e CFn'Gal(Fc/Fn)CFn+1 /e CFn+1
CFn/e CFnGal(Fc/Fn+1)→Gal(Fc/Fn)Gn
e
CFnCFn
1−→ lim
←−
ne
CFn−→ lim
←−
n
CFn−→ lim
←−
n
CFn
e
CFn
= 1
CFce CFcCFne CFn
ˆ
H−1(G, e CFc)
ˆ
H−1(Gn,e CFc) := lim
←− ˆ
H−1(Gn,e CFn) = lim
←− NFn/F e
CFn/IGn(e CFn).
ˆ
H−1(Gn,CFc)CFce CFcG
ˆ
H−1(Γ,CFc) = lim
←− ˆ
H−1(Gn, Nn(CFc)),
Nn(CFc) := Tm≥nNFm/Fn(CFm)
ˆ
H−1(Γ,e CFc) = lim
←− ˆ
H−1(Gn, Nn(e CFc)).
lim
←− ˆ
H−1(Gn, NnCFc)'lim
←− NFn/F NFc/Fn(CFc)/lim ←− IGnNFc/Fn(CFc).