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tsujimotter
tsujimotter.info
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
...
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
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)⇥
!...
⇢ : 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
...
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_fiel...
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
...
「ガロア表現」を使って素数の分解法則を考える #mathmoring
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「ガロア表現」を使って素数の分解法則を考える #mathmoring

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松森さん歓迎&数理学院立ち上げ記念セミナー
https://connpass.com/event/82142/

で発表したスライドです。

tsujimotter
http://tsujimotter.info

Published in: Science
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「ガロア表現」を使って素数の分解法則を考える #mathmoring

  1. 1. tsujimotter tsujimotter.info 2018.04.14 #mathmoring
  2. 2. •  tsujimotter •  •  • 
  3. 3. Def. GK : K V : E – etc.`C ⇢ : GK cont. hom. ! AutEV (⇢, V ) V etc.
  4. 4. K L Gal(L/K) Gal(L/L) = 1 K = LGal(L/K)
  5. 5. K L K K K L/K
  6. 6. K L Gal(K/K) = GK K Gal(K/L) Gal(K/K) K K
  7. 7. K K K L Gal(K/K) = GK K Gal(K/L) Gal(K/K) ' GK/Gal(K/L) L/K
  8. 8. K L K ' GK/Gal(K/L) K K K K ' GK/Gal(K/L0 ) ' GK/Gal(K/L00 ) L0 L00 ( )K GK
  9. 9. GK y K VGK GK y V V GK GK
  10. 10. E n E E ℓ ℓ E ℓ GLn(E) V GK
  11. 11. 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
  12. 12. (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)
  13. 13. Q K = Q( p 1)K Q Gal(Q/Q) = GQ Gal(Q/K) Gal(Q/Q) 2 2 p Frobp
  14. 14. 1 ◆⇤ p : Gal(Qp/Qp) ! GQ ◆p : Q ,! Qp Gal(Qp/Qp) ⇢ GQ p
  15. 15. Gal(Qp/Qp) ⇣ Gal(Fp/Fp)
  16. 16. : 0 ! Ip ,! Gal(Qp/Qp) ⇣ Gal(Fp/Fp) ! 0 K p Ip K/Q
  17. 17. : 0 ! Ip ,! Gal(Qp/Qp) ⇣ Gal(Fp/Fp) ! 0 2 (x 7! xp ) ( ) K p Ip K/Q
  18. 18. : 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)
  19. 19. K/QFrobp|K = idK () p K/QpProp.
  20. 20. 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)
  21. 21. 1. 1K/Q Frob0 p|K ⇠ Frobp|K () Frob0 p|K = Frobp|K 1 = ( 1 ) Frobp|K = Frobp|K OK 2. 1K/Q
  22. 22. 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 ⇢
  23. 23. 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}
  24. 24. 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
  25. 25. Frob0 p|K = Frobp|K 1 ⇢(Frob0 p) = ⇢( )⇢(Frobp)⇢( ) 1 B = PAP 1 Tr ⇢(Frob0 p) = Tr ⇢(Frobp) well-defined
  26. 26. 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
  27. 27. 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⇥
  28. 28. ⇢ : 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)
  29. 29. { Q 1 ⇢ } ! { }
  30. 30. { Q 1 ⇢ } ! { } { Q 2 ` ⇢ } ! { f } 2
  31. 31. f = P1 n=1 anqn Q { Q 2 ` ⇢f,` } { f } ⇢f,` : Gal(Q/Q) ! GL2(E) Tr(⇢f,`(Frob 1 p )) = ap k 2
  32. 32. f = P1 n=1 anqn Q { Q 2 ` ⇢f,` } { f } ⇢f,` : Gal(Q/Q) ! GL2(E) Tr(⇢f,`(Frob 1 p )) = ap
  33. 33. 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 ◆ =)
  34. 34. 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
  35. 35. •  •  •  •  •  ( ) ︎ mod N •  (ℓ ) ︎
  36. 36. •  2009 l •  •  http://tsujimotter.hatenablog.com/entry/2018-april
  37. 37. 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))
  38. 38. 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

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