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楕円曲線の有理点と BSD 予想

8554778f1060f77c8eec4e88a30369ac?s=47 Naoya Umezaki
October 06, 2018
610

楕円曲線の有理点と BSD 予想

MATHPOWER2018での講演スライド。 BSD予想についての解説。

8554778f1060f77c8eec4e88a30369ac?s=128

Naoya Umezaki

October 06, 2018
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  1. ପԁۂઢͷ༗ཧ఺ͱBSD༧૝ ക࡚௚໵@unaoya ͢͏͕͘ͿΜ͔ɺ਺ཧۭؒ τ ´ oπoζ MATHPOWER2018 10/6

  2. ฏํ਺ͱཱํ਺ ฏํ਺ 1, 4, 9, 16, 25, 36, 49, 64,

    . . . ཱํ਺ 1, 8, 27, 64, 125, 216, 343, 512, . . . ฏํ਺ͱཱํ਺ͷ͕ࠩ1 ฏํ਺ͱཱํ਺ʹڬ·Εͨ།Ұͷ਺26
  3. ପԁۂઢ y2 = x3 + 1, (x, y) = (2,

    3) y2 = x3 − 2, (x, y) = (3, 5) ༗ཧ఺ x, y ࠲ඪ͕༗ཧ਺ͳ఺
  4. ༗ཧ఺ͷ܈ P Q R P+Q P, Q ͕༗ཧ఺ ௚ઢPQ ͸༗ཧ਺܎਺

    R ΋༗ཧ఺ P + Q ΋༗ཧ఺
  5. ༗ཧ఺ͷ܈ P Q 2P P ͕༗ཧ఺ ઀ઢ͸༗ཧ਺܎਺ Q ΋༗ཧ఺ 2P

    ΋༗ཧ఺
  6. y2 = x3 + 1 P Q R P+Q P

    = (−1, 0), Q = (0, 1) PQ : y = x + 1 (x + 1)2 = x3 + 1 x = −1, 0, 2 R = (2, 3), P + Q = (2, −3)
  7. y2 = x3 + 1 P Q 2P P =

    (2, 3) yy′ = 3x2 ઀ઢ͸ y = 2(x − 2) + 3 = 2x − 1 (2x − 1)2 = x3 + 1 x = 0, 2 Q = (0, −1), 2P = (0, 1)
  8. y2 = x3 + 1 P Q R P +

    Q y2 = x3 + 1ͷ༗ཧ఺͸ (−1, 0), (0, ±1), (2, ±3), O ͷ6ݸɻ
  9. y2 = x3 − 2 P = (3, 5) 2P

    = (129/100, −383/1000) 3P = (164323/29241, −66234835/5000211) 4P = (2340922881/58675600, 113259286337279/44945509600) ༗ཧ఺͸nP ͷΈ
  10. y2 = x3 − 17x P = (−1, 4) 2P

    = (1089/16, −35871/64) 3P = (−4169764/1329409, 7264943878/1532808577) 4P = (1416749814529/82350633024, − 1637173839697065089/23631996457631232)
  11. y2 = x3 − 17x Q = (−4, 2) 2Q

    = (81/16, 423/64) 3Q = (−36481/9409, −2520436/912673) 4Q = (119093569/11451456, − 1193164200991/38751727104)
  12. y2 = x3 − 17x R = (0, 0) 2R

    = O ༗ཧ఺͸nP + mQ, nP + mQ + R Ͱશͯɻ
  13. ϞʔσϧϰΣΠϢ֊਺ ༗ཧ఺ͷʢແݶ෦෼ͷʣ࠷খͷੜ੒ݩͷݸ਺ 1. y2 = x3 + 1͸ϞʔσϧϰΣΠϢ֊਺͸0 2. y2

    = x3 − 2͸nP ͷܗͳͷͰϞʔσϧ ϰΣΠϢ֊਺1 3. y2 = x3 − 17x ͸nP + mQ ͷܗͳͷͰ ϞʔσϧϰΣΠϢ֊਺2
  14. mod pͷ఺ͷݸ਺ ପԁۂઢE ͷ mod p ͷ఺ͷݸ਺Np (E)Λ਺ ͑Δɻ

  15. E : y2 = x3 + 1 N3 (E)͸ mod

    3Ͱ (x, y) = (0, 0), (1, 0), (0, 1), (1, 1) 02 ̸= 03 + 1 02 = 13 + 1 12 = 03 + 1 12 ̸= 13 + 1
  16. E : y2 = x3 + 1 N3 (E)͸ mod

    2Ͱx = 0, 1, 2, y = 0, 1, 2 12 = 03 + 1, 22 = 03 + 1, 02 = 23 + 1 ͷ3ͭʹແݶԕ఺ΛՃ͑ͯ N3 (E) = 4
  17. E : y2 = x3 + 1 ∏ p Np

    (E) p Λߟ͑Δɻ N2 (E) 2 , N2 (E) 2 N3 (E) 3 , N2 (E) 2 N3 (E) 3 N5 (E) 5 , . . .
  18. E : y2 = x3 + 1

  19. E : y2 = x3 − 2

  20. E : y2 = x3 − 17x

  21. ∏ Np(E)/p

  22. Lؔ਺ L(s, E) = ∏ p 1 1 − (1

    + p − Np (E))p−s + p1−2s ϦʔϚϯθʔλؔ਺ͷପԁۂઢ൛ ζ(s) = ∏ p 1 1 − p−s
  23. Lؔ਺ L(1, E) = ∏ p 1 1 − (1

    + p − Np (E))p−1 + p1−2 = ∏ p 1 1 − p−1 − 1 + Np (E)p−1 + p−1 = ∏ p 1 Np (E)/p
  24. Birch and Swinnerton-Dyer༧૝ ▶ L(s, E)ͷs = 1Ͱͷॏෳ౓ͱ E ͷϞʔσϧϰΣΠϢ֊਺͕౳͍͠

    ▶ L(1, E) ̸= 0 ⇐⇒ ༗ཧ఺͕༗ݶ ෦෼తղܾ͋Γɻ ׬શʹղ͍ͨΒ100ສυϧ.ɻ
  25. ࢀߟจݙ 1. ాޱ༤Ұ࿠, ༗ཧ఺ͷ੔਺࿦ 2. Birch and Swinnerton-Dyer, Notes on

    elliptic curves. II.