Riemann

1
Riemann
2016 2 13 () ∗
http://www.math.tohoku.ac.jp/˜kuroki/LaTeX/20160213ReciprocitiesForRiemannSurfaces.pdf
0 2
1 Heisenberg 2
1.1 Riemann . . . . . . . . . . . . . . . . . . . . . 2
1.2 Heisenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 tame Weil 3
2.1 tame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 tame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 tame Weil . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Steinberg . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 tame Steinberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Contou-Carrère 8
3.1 Laurent . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Contou-Carrère . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Contou-Carrère . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Contou-Carrère . . . . . . . . . . . . . . . . . . 11
3.5 Contou-Carrère Steinberg . . . . . . . . . . . . . . . . . . . . . . 13
4 13
5 : 13
∗ 2016 2 16 () Ver.1.0.

2 1. Heisenberg
0
Riemann tame Contou-Carrère
.
twitter .
: https://twitter.com/genkuroki/status/694780107794182144
1 Heisenberg
1.1 Riemann
X Riemann , X K = C(X)
. x ∈ X , K x ̂K x . ,
z(x) = 0 x z , ̂K x Laurent
C((z)) . x Laurent K ⊂ ̂K x
. C[[z]] ⊂ K x Ôx .
∏ ̂K x∈X x A X :
{
A X = (f x ) x∈X ∈ ∏ }
∣ x f x ∈ Ôx .
x∈X
A X Riemann X . K → A X , f ↦→ (f) x∈X
K ⊂ A X .
1.2 Heisenberg
C Lie A x C Lie
h X = A x ⊕ C :
[(f x ) x∈X , (g x ) x∈X ] = ∑ x∈X
Res x (f x dg x )
(
(fx ) x∈X , (g x ) x∈X ∈ A X
)
.
Res x x . x f x , g x ∈ Ôx
. h X Heisenberg .
Heisenberg 1 .
1 , , [11] .
Heisenberg [11] 2.1 . Heisenberg h x = C((z)) ⊕ C
[z m , z n ] = mδ m+n,0 [11] 2.1 a m z m .
Lie C((z)) ( x ) Lie
. C[[z]] z , z −1 C[z −1 ] x
.
z −1 C[z −1 ] . Riemann
.

1.3. 3
1.3
,
∑
Res x (f dg) = 0
x∈X
(f, g ∈ K = C(X))
K = C(X) .
Heisenberg h A Lie A X ,
K = C(X) .
.
, 2 .
2 tame Weil
Riemann .
[4], [7] .
2.1 tame
x Laurent f ∈ ̂K x = C((z)) v x (f)
. f = a 0 z r + a 1 z r+1 + a 2 z r+2 · · · , r ∈ Z, a i ∈ C, a 0 ≠ 0 v x (f) = r
, v x (0) = ∞ . f ∈ ̂K x v x (f) ≧ 0 f ∈ Ôx . f ∈ Ôx
f = a 0 + a 1 z + a 2 z 2 + · · · , a i ∈ C , f x f(x) = a 0 ∈ C
(z(x) = 0 ).
f, g ∈ ̂K × x tame [f, g] x ∈ C× (tame symbol) :
[
[f, g] x
= (−1) v x(f)v x (g) f
v x
]
(g)
(x).
g v x(f)
, v x (f v x(g) /g v x(f) ) = 0 , f v x(g) /g v x(f) x [f v x(g) /g v x(f) ](x)
, 0 .
Riemann tame (Weil
) 3 .
2 [11] pp.210–211 .
. x i ∈ X z i z i (x i ) = 0
. x i X 1-form ω , x i
g i = g i (z i )dz i ∈ C((z))dz , , x i X f
, ∑ i Res z i=0(f(z i )g i (z i )dz i ) = ∑ x∈X (fω) = 0 . , g i = g i ((z i ))dz i ∈ C((z i ))dz i
x i X 1-form
. , x i X f
∑
i Res z i =0(f(z i )g i (z i )dz i ) = 0 . ([11] p.210
) .
Riemann .
, P SL 2 (Z)
Riemann 2 .
(
) .
3 . 0 (resultant) .

4 2. tame Weil
2.2 tame
tame tame
.
x 0 x 0
C . 4 :
1
2πi
∫
(log z · d log z − d log z · log x 0 )
C
= 1 ∫
d ( (log z) 2) − log x 0
4πi C
= 1 (
(log x0 + 2πi) 2 − (log x 0 ) 2) − log x 0 = πi.
4πi
exponential −1 . tame
.
f, g ∈ K × = C(X) × . f, g X
S .
Riemann X g 5 , X , X 4g
6 . , S
4g . 4g x 0 , x 0
S .
x ∈ S , x 0 x x 0
4g C x . ,
[ ∫ 1 (
[f, g] x
= exp log f · d log g − d log f · log g(x0 ) )]
(∗)
2πi C x
. . φ(f, g) .
log(ab) = log a + log b φ(fg, h) = φ(f, h)φ(g, h), φ(f, gh) = φ(f, g)φ(f, h)
. φ bimultiplicative 7 .
0 .
[12] Weil .
.
4 [4] p.153 .
.
5 g g .
. ( 1 ) . 4g
α 1 , β 1 , α 1, ′ β 1, ′ . . . , α g , β g , α g, ′ β g ′ , α′ i α−1 i , β i ′ β−1
, g ( g ) .
6 Riemann 4g .
7 G, H, K , φ : G × H → K bimultiplicative
φ(gg ′ , h) = φ(g, h)φ(g ′ , h), φ(g, hh ′ ) = φ(g, h)φ(g, h ′ ) (g, g ′ ∈ G, h, h ′ ∈ H)
. G, H, K Abel , bimultiplicative φ : G × H → K G ⊗ Z H → K,
g ⊗ h ↦→ φ(g, h) . G ⊗ Z H G, H Z Z
.

2.2. tame 5
φ(f, g) = φ(g, f) −1 :
∫
1 (
log f · d log g − d log f · log g(x0 ) )
2πi C x
= 1 (
(log f(x0 ) + 2πi v x (f))(log g(x 0 ) + 2πi v x (g)) − log f(x 0 ) · log g(x 0 )
2πi ∫
− d log f · log g − 2πi v x (f) log g(x 0 ) )
C x
= log f(x 0 ) · v x (g) + 2πi v x (f)v x (g) − 1 ∫
d log f · log g
2πi C x
= − 1 ∫
(
d log f · log g − log f(x0 ) · d log g ) + 2πi v x (f)v x (g).
2πi C x
exponential φ(f, g) = φ(g, f) −1 .
x f, g 0 , Cauchy φ(f, g) = 1
.
f g f = z vx(f) f 0 , g = z vxg g 0 , v x (f 0 ) = 0, v x (g 0 ) = 0 .
[ ∫
]
1
φ(z, z) = exp (log z · d log z − d log z · log x 0 ) = exp[πi] = −1,
2πi C
[ ∫
x
]
1
φ(f 0 , z) = exp (log f 0 · d log z − d log f 0 · log x 0 ) = exp(log f 0 (x)) = f 0 (x),
2πi C x
φ(z, g 0 ) = φ(g 0 , z) −1 = g 0 (x) −1 ,
φ(f 0 , g 0 ) = 1.
(∗) .
(∗) (iterated integral, [3]) .
1-forms ω 1 , . . . , ω r γ(t) (0 ≦ t ≦ 1) iterated integral ∫ ω γ r ◦ · · · ◦ ω 1
: γ ∗ ω i = f i (t) dt ,
∫
∫ ∫
ω r ◦ · · · ◦ ω 1 = · · ·
f r (t r ) · · · f 1 (t 1 ) dt r · · · dt 1 .
γ
0

6 2. tame Weil
, (∗)
[f, g] x
= exp
[ ∫ 1 (
d log f ◦ d log g + log f(x0 ) · d log g − d log f · log g(x 0 ) )]
2πi C x
(∗∗)
.
2.3 tame Weil
2.1 (Weil ). Riemann X 0 f, g
,
∏
[f, g] x
= 1
x∈X
. ( S x .)
. X 4g x 0
x 0 Γ , (∗∗) Cauchy
∏
[ ∫
]
1
[f, g] x
= exp d log f ◦ d log g .
2πi Γ
x∈X
Γ Riemann α i , β i Γ = α 1 β 1 α1 −1 β1 −1 · · · α g β g αg
−1
∫
ω r ◦ · · · ◦ ω 1 = (−1)
∫γ r ω 1 ◦ · · · ◦ ω r ,
−1
∫
r∑
∫
ω 2 ◦ ω 1 = ω 2 ◦ ω 1 +
γ 1···γ r i=1 γ i
∫
∫ ∫ ∫
ω 2 ◦ ω 1 = ω 2 ω 1 −
αβα −1 β −1
β
γ
α
∑
1≦

2.5. tame Steinberg 7
k K K 2 (k) k × ⊗ Z k × { f ⊗ (1 − f) | f, 1 − f ∈ k × }
(). Steinberg
Abel K 2 (k) → G .
Steinberg φ φ(f, −f) = 1 :
φ(f, g)φ(g, f) = 1 :
1 = φ(f, 1 − f) = φ(f, −f)φ(f, 1 − f −1 )
= φ(f, −f)φ(f −1 , 1 − f −1 ) −1 = φ(f, −f).
1 = φ(fg, −fg) = φ(f, −f)φ(f, g)φ(g, f)φ(g, −g) = φ(f, g)φ(g, f).
φ(f, f) = φ(f, −1) :
1 = φ(f, −f) = φ(f, f)φ(f, −1)
, φ bimultiplicative φ(f, −1) 2 = 1 , φ(a, a) = φ(f, −1)
. , φ(f, 1 − f) = 1 (f, 1 − f ∈ k × ) φ(f/g, g − f) = φ(−f, g)
(f, g, g − f ∈ k × ) : h = g − f 1 = (g − f)h −1 = (−fh −1 ) + gh −1
1 = φ(−fh −1 , gh −1 ) = φ(−f, g)φ(−f, h) −1 φ(h, g) −1 φ(h, h)
= φ(−f, g)φ(−f, h) −1 φ(h, g) −1 φ(h, −1) = φ(−f, g)φ(f/g, h) −1 .
φ(f/g, g − f) = φ(f/g, h) = φ(−f, g).
2.5 tame Steinberg
tame [ , ] x
: ̂K x × × ̂K x
× → C × Steinberg . f, g ∈ ̂K x
×
f + g = 1 . v x (f) > 0 v x (g) = 0 g(x) = 1 tame
[f, g] x
= 1/g(x) vx(f) = 1 . v x (f) = 0 v x (g) ≧ 0 . v x (f) = 0
v x (g) > 0 [f, g] x
= [g, f] −1
x
= 1. v x (f) = v x (g) = 0
tame [f, g] x
= 1 . tame [h, −h] x
= 1 (h ∈ ̂K x )
. v x (f) < 0 , h = f −1
,
[f, g] x
= [f, 1 − f] x
= [ f −1 , 1 − f ] −1
x
= [ f −1 , 1 − f ] −1 [
x f −1 , −f −1] −1
x
= [ f −1 , −(1 − f)f −1] −1
= [ f −1 , 1 − f −1] = 1.
x x
v x (f −1 ) > 0 .
(∗)( (∗∗)) , tame f, 1 − f ∈ K × = C(X) × , [f, g] x
= 1
. .
dilogarithm 9 . w y 0 ≠ 0, 1
, δ 0 ( δ 1 ) y 0 0 ( 1) y 0
9 (dilogarithm) Li 2 (z) Li 2 (z) = − ∫ z
0 d log w · log(1 − w) , Li 2(z) = ∑ ∞
n=1 zn /n 2
(|z| < 1) Taylor . polylogarithm Li r (z) Li r (z) = ∫ z
0 d log w · Li r−1(w)
, Li r (z) = ∑ ∞
n=1 zn /n r (|z| < 1) Taylor . 5 .

8 3. Contou-Carrère
.
∫
1
[log w · d log(1 − w) − d log w · log(1 − y 0 ))
2πi δ 0
= 1
2πi [(log y 0 + 2πi) log(1 − y 0 ) − log y 0 · log(1 − y 0 )]
− 1 ∫
d log w · log(1 − w) − log(1 − y 0 )
2πi δ 0
= − 1 ∫
d log w · log(1 − w) = − log 1 ∈ 2πiZ,
2πi δ 0
∫
1
(log w · d log(1 − w) − d log w · log(1 − y 0 )) = log 1 ∈ 2πiZ.
2πi δ 1
f, 1 − f ∈ K × = C(X) × , γ Riemann X x 0 x 0
f, 1 − f . γ f δ = f ◦ γ
y 0 = f(x 0 ) y 0 0, 1
. , ,
1
2πi
∫
γ
= 1
2πi
(
log f · d log(1 − f) − d log f · log(1 − f(x0 )) )
∫
δ
(
log w · d log(1 − w) − d log w · log(1 − y0 ) ) ∈ 2πiZ.
γ = C x [f, 1 − f] x
= 1 .
3 Contou-Carrère
Riemann . , X
Riemann , K , x ∈ X K
̂K x . z(x) = 0 x z ̂K x = C((z))
.
[7], [9], [8], [6] .
A = C[ε]/(ε N+1 ) (A ε N ≠ 0, ε N+1 = 0), m = εA 10 .
3.1 Laurent
K A = K ⊗ A = K[ε]/(ε N+1 ) , x ∈ X ̂K A,x = A((z)) =
̂K x [ε]/(ε N+1 ) . x Laurent ̂K A ⊂ ̂K A,x .
R , f ∈ R[ε]/(ε N+1 ) R = R[ε]/(ε) f ε=0 = f mod ε
. R[ε]/(ε N+1 ) :
(
R[ε]/(ε N+1 ) ) ×
= { f ∈ R[ε]/(ε N+1 ) | f ε=0 ∈ R × }.
10 (A, m) C Artin .

3.1. Laurent 9
A((z)) × =
(
̂Kx [ε]/(ε N+1 )) ×
:
A((z)) × = ̂K x [ε]/(ε N+1 )
= { f 0 + f 1 ε + f 2 ε 2 + · · · + f N ε N ∈ ̂K x [ε]/(ε N+1 ) | f 0 ∈ ̂K × x , f 1 , . . . , f N ∈ ̂K x }.
.
Laurent f ∈ A((z)) × :
∏
f = z wx(f) a 0 (1 − a i z i ).
0≠i∈Z
w x (f) ∈ Z, a 0 ∈ A × , a i ∈ A, a −i ∈ m (i > 0) , i
a −i = 0 .
.
A((z)) × z Z G :
z Z = { z m | a ∈ m ∈ Z },
G = { g ∈ A((z)) × | g ε=0 ∈ C[[z]] × }
, A((z)) , z Z × H ∼ = A((z)) × .
ϕ ∈ C((z)) × z v x (ϕ) . f ∈ A((z)) ×
m = w x (f) = v x (f ε=0 ) , f g ∈ G f = z m g
. z Z × G ∼ = A((z)) × .
G G ± :
G + = A[[z]] × , G − = 1 + εz −1 A[z −1 ].
, A((z)) , G + ×G −
∼ = G . g ∈ G
g ± ∈ G ± :
,
g = a 0 + a 1 ε + · · · + a N ε N ,
g + = b 0 + b 1 ε + · · · + b N ε N ,
a 0 ∈ C[[z]] × , a 1 , . . . , a N ∈ C((z)),
b 0 ∈ C[[z]] × , b 1 , . . . , b N ∈ C[[z]],
g − = 1 + c 1 ε + · · · + c N ε N , c 1 , . . . , c N ∈ εz −1 C[z −1 ].
g + g − = b 0 + (b 0 c 1 + b 1 )ε + (b 0 c 2 + b 1 c 1 + b 2 )ε + (b 0 c 2 + b 1 c 2 + b 2 c 1 + b 3 ) + · · ·
b i , c i g + g − = g b 0 = a 0
. G + × G −
∼ = G
.
A((z))
( ) ×
A × z Z × G + × G −
∼ = A((z)) × = ̂Kx [ε]/(ε N+1 )
.

10 3. Contou-Carrère
g + ∈ G + = A[[z]] × :
∏ ∞
g + = a 0 (1 − a i z i ), a 0 ∈ A × , a 1 , a 2 , . . . ∈ A
i=1
a 0 − a 0 a 1 z − a 0 a 2 z 2 − · · · − a 0 (a n + (a 1 , . . . , a n−1 ))z n − · · ·
. g + :
g + = c 0 + c 1 z + c 2 z 2 + · · · , c 0 ∈ A × , c 1 , c 2 , . . . ∈ A.
, a i g + = ()
. g + .
g − ∈ G − = 1 + εz −1 A[z −1 ] :
g − =
∏
(1 − a i z i ), a i ∈ εA = m.
i 0) ,
i a −i = 0, b −i = 0 . f, g Contou-Carrère ⟨f, g⟩ x
(Contou-Carrère symbol, CC )
⟨f, g⟩ x
= (−1) wx(f)wx(g) aw x(g)
0
b w x(f)
0
∏ ∞
i,j=1
(1 − a j/(i,j)
i
∏ ∞
i,j=1
(1 − a j/(i,j)
−i
b i/(i,j)
−j
b i/(i,j)
j
) (i,j)
) (i,j)
∈ A ×
. (i, j) i, j . i, j a −i = 0,
b −j = 0 , i > 0 a −i , b −j ∈ m = εA m a m −i = 0,
b m −j = 0 , .
CC z ,
. , CC , tame
. .

3.3. Contou-Carrère 11
3.3 Contou-Carrère
A = C (ε = 0) , CC tame . , CC
tame .
ε 2 ≠ 0 . f, g ∈ A((z)) , w x (f) = w x (g) = 0,
a 0 ≡ 1 + εα 0 (mod ε 2 ), b 0 ≡ 1 + εβ 0 (mod ε 2 ),
a i ≡ −εα i (mod ε 2 ), b i ≡ −εβ i (mod ε 2 ) (i ≠ 0),
α i , β i ∈ C
:
f ≡ ∏ i∈Z(1 + εα i z i ) ≡ 1 + εϕ (mod ε 2 ),
g ≡ ∏ i∈Z(1 + εβ i z i ) ≡ 1 + εψ (mod ε 2 ).
ϕ, ψ ∈ C((z)) :
ϕ = ∑ i
α i z i ,
ψ = ∑ i
β i z i .
i, j > 0 modulo ε 3 a j/(i,j)
i
, mod ε 3 ,
⟨f, g⟩ x
≡ 1 − ∑ i>0
ia i b −i + ∑ j>0
b i/(i,j)
−j
, a j/(i,j)
−i b i/(i,j)
j i = j = (i, j)
ja −j b j ≡ 1 − ε 2 ∑ i∈Z
iα i β −i ≡ 1 + ε 2 Res x (ϕ · dψ)
. , CC (ϕ, ψ) Res x (ϕ · dψ)
.
3.4 Contou-Carrère
3.1 (CC ). , f, g ∈ K × A ,
∏
⟨f, g⟩ x
= 1
x∈X
. .
, Weil
. Weil .
: f, g ∈ K × A ,
⟨f, g⟩ x
[ ∫ 1 (
= exp log f · d log g − d log f · log g(x0 ) )] ($)
2πi C
[ ∫
x
1 (
= exp d log f ◦ d log g + log f(x0 ) · d log g − d log f · log g(x 0 ) )] ($$)
2πi C x

12 3. Contou-Carrère
($),($$) (∗),(∗∗) ,
. CC
.
K A = K[ε]/(ε N+1 ) f ∈ K × A f = f 0(1 − ϕ), f 0 ∈ K × , ϕ ∈ εK A .
ϕ N+1 = 0 . log f :
log f = log f 0 + log(1 − ϕ) = log f 0 −
, log f X f 0 Riemann
log f 0 εK[ε]/(ε N+1 ) . ε
.
($),($$) . ,
N∑
n=1
ϕ n
n .
exp
f = 1 − az k , g = 1 − bz −l , a ∈ A, b ∈ m, k, l ∈ Z >0
[ ∫ 1 (
log f · d log g − d log f · log g(x0 ) )] = ( 1 − a l/(k,l) b k/(k,l)) (k,l)
2πi C x
. , 1
, 2 :
exp [Res x (log f · d log g)] = ( 1 − a l/(k,l) b k/(k,l)) (k,l)
. ($$$)
k, l
. .
. Taylor log(1 − X) = − ∑ ∞
i=1 Xi /i ,
log f = log(1 − az k ) = −
log g = log(1 − bz −l ) = −
∞∑
i=1
∞∑
j=1
log f · d log g = −l
a i
i zik ,
b j
j z−jl ,
∞∑
i,j=1
d log g = l
a i b j
z ik−jl−1 dz.
i
∞∑
b j z −jl−1 dz
ik = jl i, j −la i b j /i . ik = jl
ik/(k, l) = jl/(k, l) , m i = ml/(k, l),
j = mk/(k, l) .
∞∑ a ml/(k,l) b mk/(k,l)
Res x (log f · d log g) = −l
ml/(k, l)
m=1
( ∞∑ a l/(k,l) b k/(k,l)) m
= −(k, l)
= (k, l) log ( 1 − a l/(k,l) b k/(k,l)) .
m
($$$) .
m=1
j=1

3.5. Contou-Carrère Steinberg 13
3.5 Contou-Carrère Steinberg
tame , f, 1 − f ∈ K × A
CC Steinberg
⟨f, 1 − f⟩ x
= 1 ($$) .
f, g ∈ ̂K × A,x = A((z))× ⟨f, g⟩ x
, f, 1 − f ∈ K × A
CC Steinberg f, 1 − f ∈ ̂K × A,x
Steinberg .
4
, “symbols” Tate [10] .
, “symbols” , [2], [1] . tame
CC Grassmann Lie GL ∞
([1]). 1 Lie
.
5 :
(polylogarithm) [5] .
A(t) 11 , U(t)
dU(t)
dt
= U(t)A(t), U(t 0 ) = E = ()
.
U(t) = E +
∫ t
t 0
U(s)A(s) ds
. , t 0 ≦ t .
:
U(t) = E +
= E +
∫ t
t 0
A(t 1 ) dt 1 +
∫ t
∫ t
t 0
∫ t1
t 0
A(t 1 ) dt 1 + · · · +
t 0
U(t 2 )A(t 2 ) dt 2 · A(t 1 ) dt 1 = · · ·
∫ t
t 0
∫ t1
t 0
· · ·
∫ tn−1
t 0
U(t n )A(t n ) dt n · · · A(t 2 ) dt 2 · A(t 1 ) dt 1 .
(Picard , ),
:
U(t) =
∞∑
∫
n=0
∫
· · ·
A(t n ) dt n · · · A(t 2 ) dt 2 · A(t 1 ) dt 1 .
t 0

14 5. :
,
. ,
∞∑
∫
1 t ∫ t
U(t) = · · · T [A(t n ) · · · A(t 1 )] dt n · · · dt 1 .
n! t 0 t 0
n=0
[
∑ ∞ (∫
1 t
) n
] [ (∫ t
)]
U(t) = T
A(s) ds = T exp A(s) ds .
n! t 0 t 0
n=0
, .
(time-ordered exponential) . ,
(path-ordered product) ,
(path-ordered exponential) 12 .
2.2 (iterated integral) ,
: 1-form A A = A(t) dt ,
∫ t
∞∑
U(t) = A} ◦ ·{{ · · ◦ A}
,
n=0 t 0 n times
∫ t
∫ ∫
A} ◦ ·{{ · · ◦ A}
= · · ·
t 0 n times
t 0

15
A = A(t) dt
⎡
0 ω 1
0 ω 0
0 ω 0
A =
0 ω 0
⎢
⎣
0
⎤
. .. ⎥
⎦
. ..
, dU(t) = U(t)A, U(0) = E , U(z) (1, r + 1)
Li r (z) . [5] .
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