Classes logarithmiques des corps totalement réels

∗
`
`
K
K r σvc
σv
−
σvr+c
L∞|K×−→ ⊕
v|∞ R=Rr+c
x K×
L∞(x) = (log∞|x|v)v|∞ = (log∞NKv/R(x))v|∞,
|x|v=|σv(x)|v|x|v=|σv(x)|2=NKv/R(σv(x)) v
log∞R×
R×
+log∞(xy) = log∞x+ log∞y
K l ` σl
` l ` `
L`|K×−→ ⊕
l|`Q`=Ql
`
x K×`
L`(x) = (log`|x|l)l|`= (log`NKl/Q`(x))l|`
∗

|x|l=|σl(x)|NKl/Q`(x) = NKl/Q`(σl(x))
U1
`Q×
`=µ◦
`U1 ``Zlog`
Q×
``log``= 1
log`(xy) = log`x+ log`y.
`
E0
K={x∈K×|vp(x) = 0 ∀p-`}
`
EK={x∈K×|vp(x) = 0 ∀p-∞}
R
L∞(EK)EKRr+cH∞=
{(x1, . . . , xr+c)∈Rr+c|Pxi= 0}`
Q`
L`(E0
K)Ql
`` E0
K
K l ` H`={(x1, . . . , xl)∈Ql
`|Pxi = 0}
` K
Z`f D`K
Z`D`K=⊕pZ`p
K δ =Ppnppdeg δ=
Ppnpdeg p= 0 f
P`K
f
D`KRK=Z`⊗ZK×K×
`
` `
D`K=JK/Qpe Up`JK=Qres
pRp
KQpe Up
e
JK/Qpe Up,e JK
JKZ`Kc
Kf P`KD`KRK
f
C`K=f D`K/f P`K'e JK/Qpe UpRK
Gal(Klc/Kc)KcZ`
K Klc `
K
Kc

f
C`K
`C0
∞=Gal(C0 ∞/Kc)
` C0
∞Kc
f
C`K'ΓC0
∞=C0 ∞/C0γ−1 ∞,
Γ = Gal(Kc/K) = γZ`Z`
KcKc=K∞=∪n∈NKn
`
C0
∞=∪n∈NC0 n`
C0
nKn`
`
Gal(C0
n/Kn)` `
C`0
nKn`
`
Gal(C0
∞/K∞
C0
∞= lim
←− C`0
n.
Klc C0
n
Kf C`K=Gal(Klc/Kc)
ΓC0
∞C0 ∞Γ
` `
K
f
C`KlK `
evl=log | |l
deg ll
`f C`KK
(evl)l|`E0
K=Z`⊗ZE0 K`
KZ`e ⊕l|`Z`l
`
K `
§
E0
K−→ e
⊕l|`Z`l−→ f C`K−→ C`0
K
` ` C`0
K

K `
D`G=
Gal(K/Q)
H=KG
Q`[G/D`]Mn(D)n > 2
K
` ` f C`KK
H
K H `
Hlc H
Z`HcKHlc
` K
Kc=KHcf C`K=Gal(Klc/Kc)
H `
`
K
` ` Q`
Q`[Gal(K/Q)]
Mn(D)n > 2
L ` K
K=H L
K ` K
H`(L) = H`(K)Q`
E0
KE0 L
`
Q`[G]
` G
2

K G
`
K
E+
K
Z[G]χaug =χreg −1K
` E0
K`
χreg +χaug = 2χreg −1
Z`[G]−
L(E0
K)K`=⊕l|`Kl=⊕l|`Q`
E0
KχL(E0)=χreg −1
χL(E0)≤χaug L`(E0)
`E0
K=Z`⊗ZE0 K` E0 K
H`Q`[G]χaug
χL(E0)≥χreg −1,
Q`ϕ χaug
χL(E0).
1 2
ϕ χaug
∈E+
K
Z[G]E+
Kϕ=Στ∈Gϕ(τ)τ−1E0
K
Z`[G]ϕ
L`(ϕ)K`ϕ
ZϕE0
KL`(ϕ)6= 0
ϕ χaug χE0rϕ= 4
ϕ χreg dϕ= 2
k= [rϕ/dϕ] 2
χL(E0)
ρϕ≥rϕdϕ/(rϕ+dϕ) = 8/6>1ϕ
χE0
ϕ=nϕPφ|ϕφ