Densité d`un sous-groupe du groupe de Morava

S2
2 3
S2
W

p > 3
Homc(Sl0
2,Fp2)
p
p= 2
R K
A K R
A R L A R
A K
R
A
R=Z
ˆ
XY
X Y X
I
A Spec (A/I)Spec A
A I
ˆ
A= lim
n∈NA/In.
Spec k[X1, . . . , Xn] = An
k
0 (X1, . . . , Xn)Spec k[[X1, . . . , Xn]]
C
0
+ : C×C→C
ˆ
C C 0
z=−x/y w =−1/y
y2+a1xy +a3y=x3+a2x2+a4x+a6

w=f(z, w)
f=X3+a1XY +a2X2Y+a3Y2+a4XY 2+a6Y3∈Z[X, Y ].
W∈Z[a1, . . . , a6][[Z]]
W=f(Z, W ).
x=Z/W y =−1/W P (Z)
F P (Z1) +
CP(Z2) = P(F(Z1, Z2))
+
CC
F
p
X= (X0, X1, . . . , Xn, . . . )Y= (Y0, Y1, . . . , Yn, . . . )
Wn=
n
X
i=0
piTpn−i
i∈Z[T0, . . . , Tm, . . . ].
ϕ∈Z[X, Y ]
Φ = (φ0, φ1, . . . , φn, . . . )∈Z[X,Y]N
Wn(Φ) = ϕ(Wn(X), Wn(Y))
n∈N
S P ϕ=X+Y
ϕ=XY A W (A)
AN+. a +b=S(a, b)a.b =P(a, b)
A W (A)
A
A

WiW(A)
ANp A
W(Fp)pZp
A p W (A)
(p)A
q p x W (Fq)
x=X
i∈N
aipi
aq
i=aii
W(Fq)Fq
aip
E a 7→
¯a T r(a) = a+ ¯a N(a) = a¯a a ∈E
A E u u = ¯u
N(ab) = N(a)N(b)
γ A F A
E < s > E
s s2−γ sa−¯as a ∈E
E2 1 s
s F
(a+bs)(a0+b0s) = aa0+γb¯ b0+ (¯a0b+ab0)s.
F A
T r(u) = u¯u N(u) = u¯u a +bs = ¯a−bs ;
T r(a+bs) = T r(a)N(a+bs) = N(a)−γN (b).
uv = ¯v¯u N(uv) = N(u)N(v)
A F F
0
E
A E =A[t]/(t2−α)
¯
t=−t

n∈N∗Fn
nFpn[p]Fn=Xpn
F i i F
[i]F(X) = X+
F. . . + FX i
2
End (Fn)b Onb O Op
Qp
b
Dnb D Dpn2
1/n
n= 2
SnAut (Fn) = b O×
Fn
p n2
n= 2
W=W(Fp2)W
Fp2Wu7→ ¯u
b
OW
γ=p
b
Db O ⊗
Z
Qb Ob D
b
O=b O ⊗
Z
Z→b O ⊗
Z
Q
b
D
Zp=W(Fp)→W(Fp2) = W Q
Qp→W⊗
Z
Q= Frac(W)Qp
b
D
CFp2
OFp
Cˆ C C
2p aXp2ˆ C
1 2 F2
C