GT
GT
Gal(Q/Q)
ΦKZ
C
:C ×C C a: (UV)WU(VW)
c:UVVU
((UV)W)X
aUV,W,X
ttiiiiiiiiiiiiiiiiaU,V,W IdX
**
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
(UV)(WX)
aU,V W X
(U(VW)) X
aU,V W,X
U(V(WX)) U((VW)X)
IdUaV,W,X
oo
U(VW)cU,V W
//(VW)U
aV,W,U
((
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
(UV)W
aU,V,W
66
m
m
m
m
m
m
m
m
m
m
m
m
m
cU,V IdW
((
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
V(WU)
(VU)WaV,U,W//V(UW)
IdVcU,W
66
m
m
m
m
m
m
m
m
m
m
m
m
m
(UV)WcUV,W
//W(UV)
a1
W,U,V
((
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
U(VW)
a1
U,V,W
66
m
m
m
m
m
m
m
m
m
m
m
m
m
IdUcV,W
((
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
(WU)V
U(WV)a1
U,W,V//(UW)V
cU,W IdV
66
m
m
m
m
m
m
m
m
m
m
m
m
m
(A, , ε, Φ) A
k∆ : AAA
εa k∆(1) = 1 ε(1) = 1 Φ
AAA
(Id ∆)(∆(a)) = Φ.(∆ Id(∆(a)).Φ1
(Id Id ∆)(Φ).(∆ Id Id)(Φ) = (1 Φ).(Id Id)(Φ).1)
(εId) = Id = (Id ε)
(Id εId)(Φ) = 1
(A, , ε, Φ, R) (A, , ε, Φ)
R A A
0(a) = R∆(a)R1
(∆ Id)(R)=Φ312R13132)1R23Φ
(Id ∆)(R) = (Φ231)1R13Φ213R12Φ1
0=σσ R =Paibi
R12 =Paibi1R13 =Pai1biR23 =P1aibiΦ = Pxiyizi
Φ312 =Pyizixi
A A
R12 6=R1
F A A(A, , ε, Φ, R)
(Id ε)(F)=1=(εId)
˜
∆(a) = F∆(a)F1
˜
Φ = F23(Id ∆)(F)Φ(∆ Id)(F1)(F12)1
˜
R=R21RF 1
(A, ˜
, ε, ˜
Φ,˜
R)F(A, δ, ε, Φ, R)
k
k[[h]] (A, , ε, Φ, R)A/hA
A
k[[h]] V[[h]] V k
Φ1 mod h R 1 mod h F 1 mod h
gk t gg g
A=Ughg
R=eht/2ΦAAA
(A, , ε, Φ, R)
QΦτ=ht
PQΦ
exp(P(ht12, ht23))
ΦKZ
GT
C
(V1V2)V3
V1(V2V3)
(V1V2)V3(VV)V V
CB3σ1
cId σ2a1(Id c)a α B3
(V1V2)V3)
(Vi1Vi2)Vi3(i1, i2, i3)
α(V1V2)V3K3= ker(B3S3)ϕK3
ψK2
ψK2σ2m
λ= 2m+ 1 ϕK3f(σ2
1, σ2
2)(σ1σ2)3nnZ
f(X, Y ) (σ1σ2)3= (σ2σ1)3
B3
(λ, f, n)n= 0
(λ, f)
f(Y, X) = f(X, Y )1
f(X3, X1)Xm
3f(X2, X3)Xm
2f(X1, X2)Xm
1= 1 X1X2X3= 1, m = (λ1)/2
f(x12, x23x24)f(x13x23, x34) = f(x23x34)f(x12x13, x24x34)fx12, x23),
16i < j 6n xij = (σj1· · · σi)σ2
i(σj1· · · σi)1Kn
(λ, f)λ1 + 2Z
C
C0F:C1→ C2F
C0
1C0
2
(λ1, f1).(λ2, f2) =
(λ, f)
λ=λ1λ2,
f(X, Y ) = f1(f2(X, Y )Xλ2f2(X, Y )1, , Y λ2).f2(X, Y )
(A, , ε, Φ, R)
A
(λ, f) Φ R
R=R(R21R)m= (RR21)mR m = (λ1)/2
Φ = Φf(R21R12,Φ1R32R23Φ)
=fR21R12Φ1, R32R23
(λ, f)
f(X, Y )
X Y λ =±1f(X, Y ) = YrXr
k
λk f(X, Y )k
f(X, Y ) exp(F(ln X, ln Y))
F k
GT(k) (λ, f) GT(k)
(λ, f)GT(k) (A, , ε, Φ, R)
k[[h]]
GT(k)
Ug g k[[h]]
R=eht/2Φ = exp(P(ht12, ht23)) p
RΦR=eλht/2Φ = exp(P(ht12, ht23))
P k
Gal(Q/Q)
ΦKZ
(A, , ε, Φ, R) Φ
A
A
g g k =C
ΦA R =eαt t
g g g alpha h
G0(z) = 1
2u0
z+u11zG(z)
GC C << u0, u1>>
C
D=C\(] − ∞; 0] [1; [)
G0D G0(z)
z0zu0
G1D G1(z)
z1 (1 zu1
zu0= exp(uoln(z)) f
ag fg1
a1a
ΦKZ =G0G1
1C<< u0;u1>>
z
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