Principe de Hasse Table des matières

∼
K
F(X1, . . . , Xn)KOK
x KnOn
K
F(x) = 0
Z
p p Fn
p
ZF((x) = 0 mod p
Fn
pp2p3pk
pZp
Zp= lim
←− Z/pkZ.
xZp(αk)k>1αk∈ {0, . . . , pk−1}
k>1αk+1 ≡αkmod pkZp
Z
Zn
pZ
Q
pQp= (Zp)
Qpp
p
p|.|pQpa/b
a/b =pv.a0/b0a0b0p
|a/b|p=p−v.
QpQ|.|p
Qp
Q|.|p|x+y|p6max{|x|p,|y|p}
0

p
fi(x1, ..., xn) = 0,16i6r
Z Zp
pZ/pZ
Zp
∂fi
∂xji,j
Zpp
Zp
p2a b c
p ax2+by2=cZp
p
ax2+by2=c,
a b c Z/pZ
Z/pZ Z/pZ
p+1
2Z/pZax2c−by2
(0,0)
n d
n > d P (X1, . . . , Xn) = Pn
k=1 akXd
kd
Zn > d p -dQak
P= 0 Zp
KOK
pOk

Kpp p
|.|pK KpK
|.|pR Q
K
σ:K →C|x|σ=|σ(x)| |.|Cσ
K MKOK
K MK
KQK
Kvv∈K K
F(X)
V K
V(K)V K
(V(K)6=∅)⇒(∀v∈MK, V (Kv)6=∅).
V/K
(∀v∈MK, V (Kv)6=∅)⇒(V(K)6=∅).
Q
q(x1, . . . , xn)
K q(x1, . . . , xn) = 0

Kn
Kvv∈ MKK
n= 2
K2
ax2+bxy +cy2b2−4ac
K
Q
n= 3
q
Z3R3q
q= 0
n>4
n= 4
q q(x1, x2, x3, x4) = P4
i=1 aix2
i
p q = 0 a∈Q∗
Qg(x1, x2) = a1x2
1+a2x2 2h(x3, x4) = −a3x2 3−a4x2 4
q
6
q q = 0
x= (x1, . . . , xn)Qxi6= 0
K L K
x∈K∗y∈L∗
v K w L K x Kv
yw∈L∗
w
(e1, ..., en)K L
NL/K n
X
i=1
Xiei!=a.