Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.                                       Upcoming SlideShare
×

# 「ガロア表現」を使って素数の分解法則を考える #mathmoring

3,960 views

Published on

https://connpass.com/event/82142/

で発表したスライドです。

tsujimotter
http://tsujimotter.info

Published in: Science
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here • Be the first to comment

### 「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