Le Théorème de Poitou-Tate

p
p
p
p

k ksk Gk
(ks/k)k
p
Z
Q Z
f∈Z[X1, ..., Xn]
f(X1, ..., Xn) = 0 Z
•R
f(x1, ..., xn)<0f(x1, ..., xn)>0
•m f(x1, ..., xn) = 0
Z/mZ
p m f(x1, ..., xn)=0
Z/pmZ
p f(X1, ..., Xn)=0 Z/pmZm > 0
Zp= lim
←−mZ/pmZ
p
f(X1, ..., Xn)=0 Z
R Zpp
Z
f(X1, ..., Xn)∈Z[X1, ..., Xn]f(x1, ..., xn)=0
Q R
QpZpp
Q
Q R
Qp
Qp
Qp
Zp

FpQ Qp
p
Qp
p dp(x, y) = p−vp(x−y)
QvppQ R
Q QpQ
Qp
Q
Q
K
|.|:K→R+
∀x∈K, |x|= 0 ⇔x= 0
∀(x, y)∈K2,|xy|=|x||y|
∀(x, y)∈K2,|x+y| ≤ |x|+|y|
K K
d(x, y) = |x−y|
K K K
K|.|K K|.|
K|.|
K K
K|.|
Q
kOkk
kP Ok
(C/R)corps(k, C)R
corps(k, C)/(C/R)
P ∪ R→ { }
p∈ P 7→ (|.|p:x7→ e−vp(x))
σ∈R7→ (|.|σ:x7→ |σ(x)|)
vp(x)v x ∈pvp∈ P k|.|p
pQpp∈pσ∈R k|.|σ
R(σ)⊆R C
|.|p|.|s,p:x7→ s−vp(x)
s > 1s=e
Ok/p
k=Qk

k v
k((kv)s/kv) (ks/k)
v
k
k
k
kΩk k
Ak=((xv)∈Y
v∈Ω
kv/ v k |xv|v≤1)
|.|vv kvk Ik=A×
k
k×IkCk=Ik/k×
I= lim
−→LILC= lim −→LCLL k
•
•
X
k k
X k k
X
K
Pn(K)Kn+1 n
K
Kn+1
K Ks
Pn(K)Pn(Ks)
GK= (Ks/K) (Ks)n+1 Pn(Ks)
Pn(Ks)GK=Pn(K)Pn(K)⊆Pn(Ks)GKu∈(Ks)n+1
Ksu∈Pn(Ks)GKs∈GKφ(s)∈Ks×
s·u=φ(s)u φ
φ(s0s1) = φ(s0)s0(φ(s1)) λ∈Ks×λu ∈Kn+1
µ∈Ks×φ(s) = s(µ)µ−1s∈GKKsu∈Pn(K)

φ
H1(GK, Ks×) := {f:GK→Ks×/∀(s0, s1)∈G2
K, f(s0s1) = f(s0)s0(f(s1))}
{GK→Ks×, s 7→ s(µ)µ−1/µ ∈Ks×}
Pn(Ks)GK=Pn(K)
H1(G, M) = {f:G→M/∀(s0, s1)∈G2, f (s0s1) = f(s0) + s0·f(s1)}
{G→M, s 7→ s·m−m/m ∈M}
G M G
s·(m1+m2) = s·m1+s·m2s∈G m1, m2∈M
G
G M G s ·(m1+
m2) = s·m1+s·m2s∈G m1, m2∈M
M G
F(G0, M)d0
//F(G1, M)d1 //F(G2, M)d2 //...
G0={1} F(Gi, M)GiM
dif(s0, ..., si) = s0f(s1, ..., si) +
i
X
k=1
(−1)kf(s0, ..., sk−1sk, ..., si)+(−1)i+1f(s0, ..., si−1)
f∈ F(Gi, M)di+1 ◦di= 0
(di)⊆(di+1)r M Hr(G, M) =
(dr)/(dr−1)
G
Hr(G, −)r
M→MGG
G
G= (K/k)G(L/k)
L k K
G
G G M
G×M→M, (g, m)7→ gm
M G r Hr(G, M)M
F(Gi, M)
GiM r
M7→ MGG