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# クンマーの合同式とゼータ関数の左側 - 数学カフェ #mathcafe_height

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### クンマーの合同式とゼータ関数の左側 - 数学カフェ #mathcafe_height

1. 1. @tsujimotter AQ(⇣691) ' Z/691Z Z/691Z ⇣( 11)
2. 2. tsujimotter
3. 3. •  •  •
4. 4. tsujimotter
5. 5. 1810 – 1893
6. 6. ⇣(s) = 1X n=1 1 ns (Re s > 1)
7. 7. s s (s)
8. 8. ⇣(1 s) = 21 s ⇡ s sin ✓ ⇡(1 s) 2 ◆ (s)⇣(s)
9. 9. ⇣(1 r) = Br r r Br r
10. 10. 13 + 23 + 33 + · · · + n3 12 + 22 + 32 + · · · + n2 = 1 2 n2 + 1 2 n12 + 22 + 32 + · · · + n2 = 1 3 n3 + 1 2 n2 + 1 6 n = 1 4 n4 + 1 2 n3 + 1 4 n2 + 0 · n 14 + 24 + 34 + · · · + n4 = 1 5 n5 + 1 2 n4 + 1 3 n3 + 0 · n2 - 1 30 n n1
11. 11. B0 = 1, B1 = 1 2 , B2 = 1 6 , B3 B0 = 1, B1 = 1 2 , B2 = 1 6 , B3 = 0 B0 = 1, B1 = 1 2 , B2 = 1 6 , B3 = 0, B4 0 = 1, B1 = 1 2 , B2 = 1 6 , B3 = 0, B4 = - , B1 = 1 2 , B2 = 1 6 , B3 = 0, B4 = - 1 30 ,
12. 12. x ex 1 = 1X n=0 Bn n! xn = 1 1 2 · x + 1 6 · 2! · x2 + 0 · x3 1 30 · 4! · x4 + 0 · x5 + · · ·
13. 13. ⇣( 1) = 1 12 = 1 22 · 3 ⇣( 3) = 1 120 = 1 23 · 3 · 5 ⇣( 5) = 1 252 = 1 22 · 32 · 7 ⇣( 7) = 1 240 = 1 24 · 3 · 5 ⇣( 9) = 1 132 = 1 22 · 3 · 11 ⇣( 11) = 691 32760 = 691 23 · 32 · 5 · 7 · 13
14. 14. ⇣( 11) = 691 32760 = 691 23 · 32 · 5 · 7 · 13 von-Staudt Clausen
15. 15. von Staudt–Clausen Dm = Y (p 1)|m p Bm
16. 16. 1, 2, 3, 4, 6, 8, 12, 2424 ! ⇣(1 24) = B24 24 2, 3, 4, 5, 7, 9, 13, 25 D24 = Y (p 1)|24 p = 2 · 3 · 5 · 7 · 13 = 24 · 2 · 3 · 5 · 7 · 13⇣(1 24) = B24 24
17. 17. •  ⇣( 23) = 236364091 65520 = 103 · 2294797 24 · 32 · 5 · 7 · 13
18. 18. ⇣(1 r1) ⌘ ⇣(1 r2) (mod p) r1 ⌘ r2 (mod p 1) p r1, r2 r1 - p 1 A ⌘ B (mod p) A B p
19. 19. 1 321 68 ⇣( 31) = 37 · 683 · 305065927/26 · 3 · 5 · 17 ⇣( 67) = 37 · 101 · 123143 · 1822329343 · 5525473366510930028227481/23 · 3 · 5 r2 r1 r2 r1 ⌘ 0 (mod p 1) p = 37 ⇣(1 r2) ⇣(1 r1) ⌘ 0 (mod p)
20. 20. p = 37 36363636 ⇣( 31)⇣( 67)⇣( 103)⇣( 139)⇣( 175) 36 37
21. 21. -1 -199 37
22. 22. ⇣( 23) ⇣( 11) = 103 · 2294797 24 · 32 · 5 · 7 · 13 1 22 · 3 = 103 · 2294797 + 22 · 3 · 5 · 7 · 13 24 · 32 · 5 · 7 · 13 = 11 · 21488141 24 · 32 · 5 · 7 · 13 ⇣( 23) ⇣( 11) = 103 · 2294797 24 · 32 · 5 · 7 · 13 1 22 · 3 = 103 · 2294797 + 22 · 3 · 5 · 7 · 13 24 · 32 · 5 · 7 · 13 = 11 · 21488141 24 · 32 · 5 · 7 · 13 ⇣( 23) ⇣( 11) = 103 · 2294797 24 · 32 · 5 · 7 · 13 1 22 · 3 = 103 · 2294797 + 22 · 3 · 5 · 7 · 13 24 · 32 · 5 · 7 · 13 = 11 · 21488141 24 · 32 · 5 · 7 · 13 p = 11 10 ⇣( 13)
23. 23. mod p mod pn n = 1
24. 24. p r1, r2 r1 - p 1 (1 1/p1 r1 )⇣(1 r1) ⌘ (1 1/p1 r2 )⇣(1 r2) (mod pn ) r1 ⌘ r2 (mod (p 1)pn 1 ) n 1
25. 25. p ⇣p(1 r) := (1 1/p1 r )⇣(1 r)
26. 26. p r1, r2 r1 - p 1 n 1 ⇣p(1 r1) ⌘ ⇣p(1 r2) (mod pn ) r1 ⌘ r2 (mod (p 1)pn 1 )
27. 27. r1 ⌘ r2 (mod (p 1)pn 1 ) ⇣p(1 r1) ⌘ ⇣p(1 r2) (mod pn ) (p 1)pn 1 pn
28. 28. y = f(x) x y
29. 29. pn-1 pn
30. 30. p |x|p := p vp(x) p x p –1 –11 5 –1 –251 5 | 11 ( 1)|5 = | 10|5 = 1 5 = 1 5 | 251 ( 1)|5 = | 250|5 = 1 53
31. 31. –1 –251 –11 –1 –11 5 –1 –251 5 | 11 ( 1)|5 = | 10|5 = 1 5 = 1 5 | 251 ( 1)|5 = | 250|5 = 1 53 = 1 53 = 1 5
32. 32. p r0 r1 r2 ⇣p(1 r0) ⇣p(1 r1) ⇣p(1 r2)
33. 33. •  •  p
34. 34. •  •  •
35. 35. xn + yn = zn n 3 (xyz 6= 0) (x, y, z) FLT(n)
36. 36. FLT(4) FLT(3) FLT(5) FLT(7) FLT(14)
37. 37. p FLT(p)
38. 38. FLT(p) FLT(3) FLT(5) FLT(7) FLT(11) FLT(13) FLT(17) FLT(19) FLT(23) FLT(29) FLT(31) FLT(37) FLT(41) FLT(43) FLT(47) FLT(53) FLT(59) FLT(61) FLT(67) FLT(71) FLT(73) FLT(79) FLT(83) FLT(89) FLT(97)
39. 39. Q(⇣p) Q(⇣p) Z[⇣p] Z[⇣p] aq q0 + q1⇣p + · · · + qp 2⇣p 2 p a0 + a1⇣p + · · · + ap 2⇣p 2 p
40. 40. ⇣p
41. 41. FLT xp + yp = zp (x + y)(x + ⇣py)(x + ⇣2 p y) · · · (x + ⇣p 1 p y) = zp Q(⇣p) z = ✏pe1 1 · · · peg g = (✏pe1 1 · · · peg g )p x + ⇣k p y = ✏0 ↵p (x ⇣k p y), (x ⇣k0 p y)
42. 42. ( )( ) = z2 = (P1P2)2 z = P1P2 = (P1P2)(P1P2) = P2 1 P2 2
43. 43. ( )( ) = z2 z = P1P2 = Q1Q2 = (P1P2)(Q1Q2) = (P1Q1)(P2Q2)
44. 44. p = 23 6 = 2 · 3 = ⇠1 · ⇠2 ⇠1 = ⇣23 + ⇣4 23 + ⇣9 23 + ⇣16 23 + ⇣2 23 + ⇣13 23 + ⇣3 23 + ⇣18 23 + ⇣12 23 + ⇣8 23 + ⇣6 23 ⇠2 = ⇣22 23 + ⇣19 23 + ⇣14 23 + ⇣7 23 + ⇣21 23 + ⇣10 23 + ⇣20 23 + ⇣5 23 + ⇣11 23 + ⇣15 23 + ⇣17 23
45. 45. 6 = 2 · 3 = ⇠1 · ⇠2 A, B, C, D
46. 46. Z “ ” 3Z = (3) 3Z + 5Z = (3, 5) “ ” “ ”
47. 47. Z[⇣p] 2Z[⇣p] = (2) 3Z[⇣p] = (3) ⇠1Z[⇣p] = (⇠1) ⇠2Z[⇣p] = (⇠2) 3Z[⇣p] + ⇠2Z[⇣p] = (3, ⇠2) 3Z[⇣p] + ⇠1Z[⇣p] = (3, ⇠1) 2Z[⇣p] + ⇠1Z[⇣p] = (2, ⇠1) 2Z[⇣p] + ⇠2Z[⇣p] = (2, ⇠2)
48. 48. (2)(3) = (6) (⇠1)(⇠2) = (⇠1⇠2) (2, ⇠1)(3, ⇠2) = (2 · 3, 2⇠2, 3⇠1, ⇠1⇠2) (3)(2, ⇠1) = (6, 3⇠1)
49. 49. 積 (2, ⇠1)(2, ⇠2) = (22 , 2⇠1, 2⇠2, ⇠1⇠2) = (22 , 2⇠1, 2⇠2, 6) = (2)(1) 2, 3 = 1 = (2)(2, ⇠1, ⇠2, 3) 2 = (2)(1)
50. 50. (2) = (2, ⇠1)(2, ⇠2)
51. 51. (6) = (2)(3) = (⇠1)(⇠2) (6) = (2, ⇠1) (2, ⇠2) (3, ⇠1) (3, ⇠2)
52. 52. (↵)↵ Q(⇣p) Q(⇣p) Q(⇣p) A A2 ⇥A A3 ⇥A
53. 53. •  •  •  •  JK PK Cl(K) := JK PK PK ⇢ JK #Cl(K) K
54. 54. Cl(Q(⇣p)) Cl(Q(⇣p))
55. 55. xp + yp = zp (x + y)(x + ⇣py)(x + ⇣2 p y) · · · (x + ⇣p 1 p y) = zp Q(⇣p) (z) = pe1 1 · · · peg g = pe1 1 · · · peg g p (x + y)(x + ⇣py)(x + ⇣2 p y) · · · (x + ⇣p 1 p y) = (z)p (x + ⇣k p y) = Ap (x ⇣k p y), (x ⇣k0 p y)
56. 56. A = (↵) (x + ⇣k p y) = (↵p ) x + ⇣k p y = ✏0 ↵p A = (↵) (x + ⇣k p y) = Ap
57. 57. A (x + ⇣k p y) = Ap #Cl (Q(⇣p)) pA = (↵) p A = (↵) A = (↵) p p 1 1
58. 58. #Cl(Q(⇣p)) p #Cl(Q(⇣p)) p FLT p FLT p
59. 59. #Cl (Q(⇣23)) = 3 #Cl(Q(⇣7)) = 1 #Cl(Q(⇣11)) = 1 #Cl(Q(⇣13)) = 1 #Cl(Q(⇣17)) = 1 #Cl(Q(⇣19)) = 1 #Cl(Q(⇣29)) = 8
60. 60. #Cl(Q(⇣p)) = hp h+ p p p
61. 61. h7 = 1 h11 = 1 h13 = 1 h17 = 1 h19 = 1 h23 = 3 h29 = 8 h31 = 9 h37 = 37 37
62. 62. 100 h37 = 371 h59 = 591 · 699 h67 = 671 · 12739 Remark
63. 63. 200 h37 = 371 h59 = 591 · 699 h67 = 671 · 12739 h101 = 1011 · 35122815625 h103 = 1031 · 88049462555 h131 = 1311 · 217529616253985775 h149 = 1491 · 4616697044880367249149 h157 = 1572 · 2281404020463379154005
64. 64. •  •  ex. ( ) hp p •
65. 65. •  •  •
66. 66. p p
67. 67. h37 37 ⇣( 31) 37
68. 68. h103 103 103⇣( 23)
69. 69. p p (1) p hp p (2) p ⇣( 1), ⇣( 3), ⇣( 5), ⇣( 7), . . .
70. 70. 691
71. 71. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3)) (2) p (1) p hp p
72. 72. 31 –1 –27 31
73. 73. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3)) (2) p (1) p hp p
74. 74. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3)) (2) p
75. 75. p- p- p Cl(Q(⇣p)) = AQ(⇣p) A0 Q(⇣p) A!i Q(⇣p) = {x 2 AQ(⇣p) | 8 2 , (x) = x!( )i } = Gal(Q(⇣p)/Q) !: ⇠ ! (Z/pZ)⇥ AQ(⇣p) = p 2M i=0 A!i Q(⇣p)
76. 76. (1) (2) (1) (2) p ⇣(1 r) 2 (x) = x!( )1 r p xCl(Q(⇣p)) A!1 r Q(⇣p) 6= {e}
77. 77. p = 37 AQ(⇣37) ' Z/37Z ⇣( 31) = 7709321041217 16320 = 37 · 683 · 305065927 26 · 3 · 5 · 17 37 (x) = x!( ) 31 p xk ⇠ ! k. mod 37
78. 78. p = 691 AQ(⇣691) ' Z/691Z Z/691Z (x) = x!( ) 11 (y) = y!( ) 199 xk · yl ⇠ ! (k. mod 691, l. mod 691) p 691⇣( 11), ⇣( 199)
79. 79. 691
80. 80. •  •  p ó p •
81. 81. [1] [2] [3] [1] [2] [3]
82. 82. > zeta199.numerator => 49838404942833341476492863214039966210849588745720667496 80558226172636696215236875688658023022109991326014126976 13279391058654527145340515840099290478026350382802884371 712359337984274122861159800280019110197888555893671151 > zeta199.numerator % 691 => 0 ⇣( 199) ⌘ 0 (mod 691)
83. 83. ↵A ↵(1) = (↵) ↵ 2 K⇥ A ⇢ OK
84. 84. AB = (n) A B n ↵A ⇥ 1 ↵n B = ↵ n↵ AB = 1 n (n) = (1)
85. 85. Sagemath x = k.ideal(6) x.factor() (Fractional ideal (2, z^11 + z^10 + z^6 + z^5 + z^4 + z^2 + 1)) * (Fractional ideal (2, z^11 + z^9 + z^7 + z^6 + z^5 + z + 1)) * (Fractional ideal (3, z^11 + z^10 + z^9 - z^8 - z^7 + z^5 + z^3 - 1)) * (Fractional ideal (3, z^11 - z^8 - z^6 + z^4 + z^3 - z^2 - z - 1)) a = k.ideal(2, z^11 + z^10 + z^6 + z^5 + z^4 + z^2 + 1) b = k.ideal(2, z^11 + z^9 + z^7 + z^6 + z^5 + z + 1) c = k.ideal(3, z^11 + z^10 + z^9 - z^8 - z^7 + z^5 + z^3 - 1) d = k.ideal(3, z^11 - z^8 - z^6 + z^4 + z^3 - z^2 - z – 1) x = k.ideal(23) x.factor() z1 = z + z^4 + z^9 + z^16 + z^2 + z^13 + z^3 + z^18 + z^12 + z^8 + z^6 z2 = z^22 + z^19 + z^14 + z^7 + z^21 + z^10 + z^20 + z^5 + z^11 + z^15 + z^17 k.ideal(z1) k.ideal(z1).reduce_equiv() k.ideal(z2) k.ideal(z2).reduce_equiv() z1*z2