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2018.04.14 #mathmoring

• tsujimotter
•
•
•

Def.
GK : K
V : E –
etc.`C
⇢ : GK
cont. hom.
! AutEV
(⇢, V ) V
etc.

K
L
Gal(L/K)
Gal(L/L) = 1
K = LGal(L/K)

K
L
K
K
K
L/K

K
L
Gal(K/K) = GK
K
Gal(K/L)
Gal(K/K)
K
K

K
K
K
L
Gal(K/K) = GK
K
Gal(K/L)
Gal(K/K)
' GK/Gal(K/L)
L/K

K
L
K
' GK/Gal(K/L)
K
K
K
K
' GK/Gal(K/L0
)
' GK/Gal(K/L00
)
L0
L00
( )K GK

GK y K
VGK
GK y V
V GK
GK

E
n
E
E ℓ
ℓ
E
ℓ
GLn(E)
V GK

p = x2
+ y2
p :
= (x + y
p
1)(x y
p
1) () p ⌘ 1 (mod 4)
2
pQ(
p
1)
13 = 22
+ 32
29 = 22
+ 52

(1 +
p
1)2
(
p
1)
2 p (⌘ 1 (mod 4)) (⌘ 3 (mod 4))p
(x + y
p
1)(x y
p
1) pK
Q
= Q(
p
1)

Q
K = Q(
p
1)K
Q
Gal(Q/Q) = GQ
Gal(Q/K)
Gal(Q/Q)
2
2
p Frobp

1
◆⇤
p : Gal(Qp/Qp) ! GQ
◆p : Q ,! Qp
Gal(Qp/Qp) ⇢ GQ
p

Gal(Qp/Qp) ⇣ Gal(Fp/Fp)

:
0 ! Ip ,! Gal(Qp/Qp) ⇣ Gal(Fp/Fp) ! 0
K
p
Ip
K/Q

:
0 ! Ip ,! Gal(Qp/Qp) ⇣ Gal(Fp/Fp) ! 0
2
(x 7! xp
)
( )
K
p
Ip
K/Q

:
0 ! Ip ,! Gal(Qp/Qp) ⇣ Gal(Fp/Fp) ! 0
2
2
(x 7! xp
)
( )
7!Frobp
p
Frobp|K
K/Q
K
p
Ip
K/Q
Gal(K/Q)

K/QFrobp|K = idK () p
K/QpProp.

1◆p : Q ,! Qp Frobp
◆p, ◆0
p : Q ,! Qp Frobp, Frob0
p
2 Gal(K/Q) Frob0
p|K = Frobp|K
1
Frobp, Frob0
p
Frob0
p|K ⇠ Frobp|K
Def. Frob0
p|K ⇠ Frobp|KGal(K/Q)

1. 1K/Q
Frob0
p|K ⇠ Frobp|K
() Frob0
p|K = Frobp|K
1
= ( 1
) Frobp|K
= Frobp|K
OK
2. 1K/Q

Q
Q
Gal(Q/Q) = GQ
Gal(Q/Q)
K 0 ! Ker ⇢ ,! GQ ⇣ GLn(E) ! 0
⇢ : GQ ! GLn(E)
0 ! Ker ⇢ ,! GQ ⇣ GLn(E) ! 0K/Q
Prop.
K/Q
Prop.
K/Q
⇢(Ip) = { 1 } () p
⇢(Frobp) = 1 () p
K/Q
K := Q
Ker ⇢

K := Q
Ker ⇢
g K
() p K/Q
Prop.
K/Q⇢(Ip) = { 1 } () p
Ip ⇢ Ker ⇢ g 2 Ip K
⇢(Ip) = { 1 }
() Ip|K = {idK}

Prop.
K/Q⇢(Frobp) = 1 () p
g 2 GQ
K := Q
Ker ⇢
⇢(g) = 1 g K
g|K = idK
⇢(Frobp) = 1
() Frobp|K = idK
() p K/Q

Frob0
p|K = Frobp|K
1
⇢(Frob0
p) = ⇢( )⇢(Frobp)⇢( ) 1
B = PAP 1
Tr ⇢(Frob0
p) = Tr ⇢(Frobp)
well-defined

K = Q(
p
1)
⇢ : GQ ⇣ Gal(K/Q) ' {±1} ,! C⇥
= GL1(C)
2
2
idK 7! +1
17!
⇢(g) = 1 () g|K = idK Q
Ker ⇢
= K
K/Q

K = Q(
p
1)
⇢ : GQ ⇣ Gal(K/Q) ' {±1} ,! C⇥
= GL1(C)
Gal(Q(⇣N )/Q) ' (Z/NZ)⇥
GQ ! Gal(K/Q) ! C⇥
'
⇢ :
(Z/4Z)⇥
!
: (Z/4Z)⇥
! C⇥

⇢ : GQ ! Gal(Q(
p
1)/Q) ' (Z/4Z)⇥
2
2
Frobp p
! GL1(C)
(p)
2
p
1 7! (
p
1)p
7! 7!
2
⇢(Frobp) = (p)

{ Q 1 ⇢ } ! { }

{ Q 1 ⇢ } ! { }
{ Q 2 ` ⇢ } ! { f }
2

f =
P1
n=1 anqn
Q
{ Q 2 ` ⇢f,` } { f }
⇢f,` : Gal(Q/Q) ! GL2(E)
Tr(⇢f,`(Frob 1
p )) = ap
k 2

f =
P1
n=1 anqn
Q
{ Q 2 ` ⇢f,` } { f }
⇢f,` : Gal(Q/Q) ! GL2(E)
Tr(⇢f,`(Frob 1
p )) = ap

f =
P1
n=1 anqn 2 ℓ
⇢f,` : Gal(Q/Q) ! GL2(E)
Prop.
=)
p K/Q
p K/Q
character table
ap = 2
Ker ⇢f,` K/Q
ap = Tr ⇢f,`(Frob 1
p ) = 2⇢f,`(Frob 1
p ) =
✓
1 0
0 1
◆
=)

Q(
p
1)/Q
K/Q
f = q
1Y
n=1
(1 qn
)(1 q23n
) =
1X
n=1
anqn
K X3
X2
+ 1
ap = 2 () p
p ⌘ 1 (mod 4) () p

•
•
•
•
• ( ) ︎ mod N
• (ℓ ) ︎

• 2009 l
•
• http://tsujimotter.hatenablog.com/entry/2018-april

sage: M = NumberField(x^2 + 23, 'a’); M
Number Field in a with defining polynomial x^2 + 23
sage: K = M.hilbert_class_field('b’); K
Number Field in b with defining polynomial x^3 - x^2 + 1 over its base
field
sage: I = K.ideal(59); I
Fractional ideal (59)
sage: I.factor()
(Fractional ideal ((6/23*a + 1)*b^2 - 2/23*a*b + 5/46*a + 1/2))
* (Fractional ideal ((-1/46*a - 3/2)*b^2 + (4/23*a + 1)*b - 5/23*a + 1))
* (Fractional ideal ((-6/23*a + 1)*b^2 + 2/23*a*b - 5/46*a + 1/2))
* (Fractional ideal ((-13/46*a + 1/2)*b^2 + (6/23*a - 1)*b + 4/23*a + 1))
* (Fractional ideal ((1/46*a - 3/2)*b^2 + (-4/23*a + 1)*b + 5/23*a + 1))
* (Fractional ideal ((-13/46*a - 1/2)*b^2 + (6/23*a + 1)*b + 4/23*a - 1))

K/Q
K
ap = 2 () p
X5
X4
+ X3
+ X2
2X + 1
f = q
1Y
n=1
(1 qn
)(1 q23n
) =
1X
n=1
anqn
f(⌧) = ✓A(⌧)
1 +
p
5
2
!
✓B(⌧)
1
p
5
2
!
✓C(⌧) 2 S1
✓
0(47),
✓
47
⇤
◆◆
1
p
5
2
!
✓C(⌧) 2 S1
✓
0(47),
✓
47
⇤
◆◆
✓A(⌧) =
X
m,n2Z
qm2
+mn+12n2
✓B(⌧) =
X
m,n2Z
q3m2
+mn+4n2
✓C(⌧) =
X
m,n2Z
q2m2
+mn+6n2