レムニスケートから楕円関数へ

MATHPOWER2018での講演スライド。レムニスケートと楕円関数に関わるアーベルの業績について解説。

Transcript

1. ϨϜχεέʔτ͔Β
   ପԁؔ਺΁
   ക࡚௚໵@unaoya
   ͢͏͕͘ͿΜ͔
   MATHPOWER2018 10/6
   

   View Slide

2. Ξʔϕϧͱପԁੵ෼
   wikipediaΑΓ
   Ξʔϕϧ͕਺ֶʹ໨֮Ίͯ200೥
   

   View Slide

3. Ψ΢εͱϨϜχεέʔτੵ෼
   

   View Slide

4. ࢉज़زԿฏۉ
   aͱbͷࢉज़ฏۉ
   a + b
   2
   aͱbͷزԿฏۉ
   √
   ab
   

   View Slide

5. ࢉज़زԿฏۉ
   a0
   = 1, b0
   =
   1
   √
   2
   = 0.7071 · · · ͔ΒॳΊͯ࣍ʑ
   ܁Γฦ͢ɻ
   a1
   =
   a0
   + b0
   2
   = 0.853553 · · ·
   b1
   =
   √
   a0
   b0
   = 0.840896 · · ·
   

   View Slide

6. ࢉज़زԿฏۉ
   a2
   =
   a1
   + b1
   2
   = 0.847224 · · ·
   b2
   =
   √
   a1
   b1
   = 0.847201 · · ·
   a3
   =
   a2
   + b2
   2
   = 0.847213 · · ·
   b3
   =
   √
   a2
   b2
   = 0.847213 · · ·
   

   View Slide

7. ϨϜχεέʔτ
   r2 = cos 2θ
   O
   P
   

   View Slide

8. ϨϜχεέʔτੵ෼
   P
   Q R
   PQ2 + QR2 = PR2
   √
   (dr)2 + (rdθ)2 = ds
   

   View Slide

9. ϨϜχεέʔτੵ෼
   r2 = cos 2θ
   2rdr = −2 sin 2θdθ
   4r2(dr)2 = 4 sin2 2θ(dθ)2
   = 4(1 − cos2 2θ)(dθ)2 = 4(1 − r4)(dθ)2
   

   View Slide

10. ϨϜχεέʔτੵ෼
    r4
    1 − r4
    (dr)2 = r2(dθ)2
    ∫ √
    (rdθ)2 + (dr)2 =
    ∫ √
    1
    1 − r4
    dr
    

    View Slide

11. ϨϜχεέʔτੵ෼
    s(t) =
    ∫
    P
    O
    1
    √
    1 − r4
    dr
    O
    P
    

    View Slide

12. ପԁੵ෼ͷඪ४ܗ
    r2 = 1 − sin2 θ
    rdr = −2 cos θ sin θdθ
    dr =
    −2 cos θ sin θdθ
    √
    1 − sin2 θ
    

    View Slide

13. ପԁੵ෼ͷඪ४ܗ
    dr
    √
    1 − r4
    =
    −2 cos θ sin θdθ
    √
    1 − (1 − sin2 θ)2
    √
    1 − sin2 θ
    =
    −2 sin θdθ
    √
    1 − (1 − sin2 θ)2
    =
    −2 sin θdθ
    √
    2 sin2 θ − sin4 θ
    =
    −2dθ
    √
    2 − sin2 θ
    

    View Slide

14. ∫
    1
    0
    dr
    √
    1 − r4
    =
    1
    2
    ∫
    π/2
    0
    dθ
    √
    1 − (1/
    √
    2)2 sin2 θ)
    K(k) =
    ∫
    π/2
    0
    dθ
    √
    1 − k2 sin2 θ
    

    View Slide

15. ϥϯσϯม׵ͱࢉज़زԿฏۉ
    kn
    =
    bn
    an
    , kn+1
    =
    bn+1
    an+1
    ʹରͯ͠
    1
    an
    K(kn
    ) =
    1
    an+1
    K(kn+1
    )
    

    View Slide

16. ϧδϟϯυϧͷؔ܎ࣜ
    E(k) =
    ∫
    π/2
    0
    √
    1 − k2 sin2 θdθ
    k′2 + k2 = 1
    E(k)K(k′) + E(k′)K(k) − K(k)K(k′) =
    π
    2
    

    View Slide

17. ϧδϟϯυϧͷؔ܎ࣜ
    ಛʹk =
    1
    √
    2
    ͷ࣌
    2E(
    1
    √
    2
    )K(
    1
    √
    2
    ) − K(
    1
    √
    2
    )2 =
    π
    2
    

    View Slide

18. ·ͱΊ
    ▶ ϨϜχεέʔτੵ෼͸ପԁੵ෼K(
    1
    √
    2
    )
    ▶ ࢉज़زԿฏۉͱପԁੵ෼ͷؔ܎
    ʢϥϯσϯม׵ʣ
    ▶ ପԁੵ෼ͱԁप཰ͷؔ܎
    ʢϧδϟϯυϧͷؔ܎ࣜʣ
    

    View Slide

19. ڏ਺৐๏
    ପԁੵ෼ͷؔ܎ࣜ
    ∫
    it
    0
    1
    √
    1 − r4
    dr =
    ∫
    t
    0
    1
    √
    1 − (ir′)4
    d(ir′)
    = i
    ∫
    t
    0
    1
    √
    1 − r′4
    dr′
    

    View Slide

20. ڏ਺৐๏
    ପԁੵ෼
    s(t) =
    ∫
    t
    0
    1
    √
    1 − r4
    dr
    ͸ڏ਺৐๏ͱ͍͏ؔ܎ࣜΛຬͨ͢
    s(it) = is(t)
    

    View Slide

21. ڏ਺৐๏
    ପԁੵ෼
    K(k) =
    ∫
    π/2
    0
    dθ
    √
    1 − k2 sin2 θ
    ͸k ͝ͱʹ৭ʑଘࡏ͢ΔɻͦͷதͰϨϜχε
    έʔτੵ෼K(
    1
    √
    2
    )͸ಛผͳରশੑΛ࣋ͭɻ
    

    View Slide

22. Ξʔϕϧͱ୅਺ؔ਺ͷੵ෼
    

    View Slide

23. ϨϜχεέʔτੵ෼
    s(t) =
    ∫
    t
    0
    1
    √
    1 − r4
    dr
    ʹ͍ͭͯϑΝχϟʔϊ΍ΦΠϥʔͷݚڀ
    

    View Slide

24. ΦΠϥʔͷՃ๏ఆཧ
    x =
    y −
    √
    1 − z4 + z
    √
    1 − y4
    1 + y2z2
    ͷͱ͖
    ∫
    x
    0
    1
    √
    1 − r4
    dr =
    ∫
    y
    0
    1
    √
    1 − r4
    dr +
    ∫
    z
    0
    1
    √
    1 − r4
    dr
    

    View Slide

25. ΞʔϕϧͷҰൠԽ
    ·ͣପԁੵ෼
    ∫
    dx
    √
    x3 + ax2 + bx + c
    Λߟ͑Δɻ
    

    View Slide

26. ΞʔϕϧͷҰൠԽ
    r =
    √
    −x
    dr = −
    dx
    2
    √
    −x
    ∫
    dr
    √
    1 − r4
    = −
    1
    2
    ∫
    dx
    √
    (1 − x2)(−x)
    

    View Slide

27. ͦͷલʹ
    ԁͷހ௕
    ∫
    dx
    √
    1 − x2
    x = sin t ͱஔ׵ੵ෼
    

    View Slide

28. ࡾ֯ؔ਺ͷՃ๏ఆཧ
    C : x2 + y2 = 1
    L(t) : y = t1
    x + t2
    P1
    (t)
    P2
    (t)
    O
    

    View Slide

29. Ξʔϕϧ࿨
    C ͱL(t)ͷަ఺P1
    (t), P2
    (t)
    ∫
    dx
    y
    =
    ∫
    dx
    √
    1 − x2
    u(t) =
    ∫
    P1(t)
    O
    dx
    y
    +
    ∫
    P2(t)
    O
    dx
    y
    

    View Slide

30. t2
    Λಈ͔͢
    P1
    (t)
    P2
    (t)
    O
    ∂u(t)
    ∂t2
    = 0
    

    View Slide

31. t1
    Λಈ͔͢
    P1
    (t)
    P2
    (t)
    O
    ∂u(t)
    ∂t1
    = −2(arctan t1
    )′
    

    View Slide

32. ͜ͷ͜ͱ͔Βɺ
    u(t) = −2 arctan t1
    = arcsin(
    −2t1
    1 + t2
    1
    )
    

    View Slide

33. Ұํɺx1
    , x2
    ͕x2 + (t1
    x + t2
    )2 = 1ͷղͳͷͰ
    x1
    x2
    =
    t2
    2
    − 1
    t2
    1
    + 1
    , x1
    + x2
    =
    −2t1
    t2
    t2
    1
    + 1
    Ͱ͋Δ͜ͱ͔Βɺ
    x1
    y2
    + x2
    y1
    = x1
    (t1
    x2
    + t2
    ) + x2
    (t1
    x1
    + t2
    )
    = 2t1
    x1
    x2
    + (x1
    + x2
    )t2
    =
    −2t1
    1 + t2
    1
    

    View Slide

34. ͭ·Γɺ
    u(P1
    (t)) + u(P2
    (t)) = u(t)
    ∫
    (x1,y1)
    (0,1)
    dx
    y
    +
    ∫
    (x2,y2)
    (0,1)
    dx
    y
    =
    ∫
    x1y2+x2y1
    (0,1)
    dx
    y
    ͱͳΔɻ
    

    View Slide

35. ٯؔ਺
    u(s) =
    ∫
    s
    0
    dx
    y
    ͷٯؔ਺
    u =
    ∫
    s(u)
    0
    dx
    y
    ࠓͷ৔߹͸͜Ε͕ࡾ֯ؔ਺
    

    View Slide

36. Ճ๏ఆཧ
    u(t)ͷٯؔ਺Λt = sin(u)ͱ͔͘ͱɺ
    sin(u(P1
    ) + u(P2
    )) = x1
    y2
    + x2
    y1
    = cos u(P1
    ) sin u(P2
    ) + sin u(P2
    ) cos u(P1
    )
    

    View Slide

37. ·ͱΊ
    1. u =
    ∫
    s
    0
    dx
    √
    1 − x2
    ͷٯؔ਺͕sin u
    2. x2 + y2 = 1ͷΞʔϕϧ࿨
    ∫
    P1(t)
    O
    dx
    y
    +
    ∫
    P2(t)
    O
    dx
    y
    3. ࡾ֯ؔ਺ͷՃ๏ఆཧ
    

    View Slide

38. ପԁੵ෼
    y2 = x3 + ax2 + bx + c
    ∫
    P
    O
    dx
    √
    x3 + ax2 + bx + c
    =
    ∫
    P
    O
    dx
    y
    

    View Slide

39. ΞʔϕϧͷՃ๏ఆཧ
    C ͱL(t)ͷަ఺P1
    (t), P2
    (t), P3
    (t)
    P1
    (t)
    P2
    (t)
    P3
    (t)
    

    View Slide

40. ΞʔϕϧͷՃ๏ఆཧ
    C ͱL(t)ͷަ఺P1
    (t), P2
    (t), P3
    (t)
    C ͷΞʔϕϧ࿨
    u(t) =
    ∫
    P1(t)
    O
    dx
    y
    +
    ∫
    P2(t)
    O
    dx
    y
    +
    ∫
    P3(t)
    O
    dx
    y
    

    View Slide

41. ΞʔϕϧͷՃ๏ఆཧ
    C ͱL(t)ͷަ఺P1
    (t), P2
    (t), P3
    (t)
    Ξʔϕϧͷఆཧ
    u(t) =
    ∫
    P1(t)
    O
    dx
    y
    +
    ∫
    P2(t)
    O
    dx
    y
    +
    ∫
    P3(t)
    O
    dx
    y
    = 0
    

    View Slide

42. ପԁؔ਺ͷՃ๏ఆཧ
    ପԁੵ෼ͷٯؔ਺
    u =
    ∫
    s(u)
    O
    dx
    y
    ΛΈͨ͢s(u)͕ପԁؔ਺
    

    View Slide

43. ପԁؔ਺ͷՃ๏ఆཧ
    P3
    ͷ࠲ඪ͸y = t1
    x + t2
    ͱ
    y2 = x3 + ax2 + bx + c ͔Β୅਺తʹٻ·Δ
    P1
    P2 P3
    

    View Slide

44. ପԁؔ਺ͷՃ๏ఆཧ
    ∫
    P1
    O
    dx
    y
    +
    ∫
    P2
    O
    dx
    y
    +
    ∫
    P3
    O
    dx
    y
    = 0
    u1
    + u2
    + u3
    = 0
    s(u1
    + u2
    ) = s(−u3
    ) = P1
    ͱP2
    ͷ୅਺తͳࣜ
    

    View Slide

45. ·ͱΊ
    1. u =
    ∫
    s
    0
    dx
    √
    x3 + ax2 + bx + c
    ͷٯؔ਺͕
    ପԁؔ਺
    2. y2 = x3 + ax2 + bx + c ͷΞʔϕϧ࿨
    ∫
    P1(t)
    O
    dx
    y
    +
    ∫
    P2(t)
    O
    dx
    y
    +
    ∫
    P3(t)
    O
    dx
    y
    = 0
    3. ପԁؔ਺ͷՃ๏ఆཧ
    

    View Slide

46. Ξʔϕϧੵ෼
    P1
    (t), . . . , Pn
    (t)ΛC ͱDt
    ͷަ఺ͱ͢Δ
    u(t) =
    n
    ∑
    i=0
    ∫
    Pi(t)
    P0
    r(x, y)dx
    ͜͜Ͱr(x, y)dx ͸
    dx
    y
    ͷΑ͏ͳ༗ཧࣜ
    

    View Slide

47. Ξʔϕϧͷఆཧ
    u(t) =
    n
    ∑
    i=0
    ∫
    Pi(t)
    P0
    r(x, y)dx
    ͸
    u(t) = R(t) +
    ∑
    logi
    Si
    (t)
    ͜͜ͰɺR(t), S(t)͸t ͷ༗ཧؔ਺
    

    View Slide

48. Ξʔϕϧͷఆཧ
    ω =
    pdx
    fy
    ∂u(t)
    ∂t1
    =
    −x2p(x, t1
    x + t2
    )
    f (x, t1
    x + t2
    )
    ͷఆ਺߲
    ∂u(t)
    ∂t2
    =
    −xp(x, t1
    x + t2
    )
    f (x, t1
    x + t2
    )
    ͷఆ਺߲
    

    View Slide

49. Ξʔϕϧͷఆཧ
    p ͷ࣍਺͕খ͚͞Ε͹u(t)͸ఆ਺
    ∫
    P1
    P0
    ω +
    ∫
    P2
    P0
    ω + · · ·
    ∫
    Pn
    P0
    ω = 0
    

    View Slide

50. Ξʔϕϧͷఆཧͱपظ
    

    View Slide

51. Ξʔϕϧͷఆཧͷٯ
    Ξʔϕϧͷఆཧ
    P1
    , P2
    , P3
    ͕Ұ௚ઢ্ͷͱ͖
    u(P1
    ) + u(P2
    ) + u(P3
    ) = 0
    Ξʔϕϧͷఆཧͷٯ
    C ্ͷP1
    , P2
    , P3
    ʹର͠
    u(P1
    ) + u(P2
    ) + u(P3
    ) = 0ͳΒP1
    , P2
    , P3
    ͸Ұ
    ௚ઢ্ɻ
    

    View Slide

52. Ξʔϕϧͷఆཧͷٯ
    C ͕n࣍ۂઢf (x, y) = 0ͷͱ͖
    P1
    , . . . , Pg
    ͱQ1
    , . . . , Qg
    ͔Β
    ∑
    i
    u(Pi
    ) +
    ∑
    i
    u(Qi
    ) +
    ∑
    i
    u(Ri
    ) = 0
    ΛΈͨ͢R1
    , . . . , Rg
    ͕ܾ·Δɻ
    

    View Slide

53. पظ
    ੵ෼ͷ஋͸࣮͸Ұͭʹܾ·Βͣɺੵ෼ܦ࿏ʹ
    ґଘ͢Δɻ
    P
    Q
    O
    

    View Slide

54. पظ
    ίʔγʔͷੵ෼ఆཧ
    ಛҟ఺ΛճΒͳ͚Ε͹ੵ෼ͷ஋͸มΘΒͳ͍
    पظ
    ಛҟ఺ͷपΓΛҰपճͬͨੵ෼஋ͨͪ
    ∫
    γ
    ω
    

    View Slide

55. ϗϞϩδʔɺίϗϞϩδʔ
    ຊ࣭తʹҟͳΔܦ࿏͕ͲΕ͙Β͍͋Δ͔ʁ
    γ ∈ H1
    (C, Z)
    पظ֨ࢠ
    Λ(C) = {(
    ∫
    γ
    ωi
    ) | γ ∈ H1
    (C, Z)} ⊂ C
    

    View Slide

56. ΞʔϕϧϠίϏͷఆཧ
    C ͕ࡾ࣍ࣜͷ࣌
    C → C/Λ(C)
    ͸ಉҰࢹΛ༩͑Δɻ
    

    View Slide

57. ϗοδཧ࿦
    ඃੵ෼ؔ਺ω ͱੵ෼ܦ࿏γ ͷؔ܎
    ∫
    γ
    ω
    ͕ۂઢf (x, y) = 0ʹґଘͨ͠ྔΛ༩͑Δɻ
    

    View Slide

58. ϞδϡϥΠ
    ▶ ପԁؔ਺ͰҟͳΔ΋ͷ͕ͲΕ͙Β͍͋
    Δ͔ʁ
    ▶ पظ͕ͲΕ͙Β͍͋Δ͔ʁ
    ͜ΕΒΛूΊͯҰͭͷزԿֶతର৅ͱͯ͠
    ѻͬͨ΋ͷ͕ϞδϡϥΠۭؒ
    

    View Slide

59. ࢀߟจݙ
    ▶ פ઒ޫɺ׬શପԁੵ෼ͱΨ΢εɾϧδϟ
    ϯυϧ๏ʹΑΔπ ͷܭࢉ
    ▶ Phillip Griffiths, The legacy of Abel in
    algebraic geometry
    ▶ Phillip Griffiths, Variations on a Theorem of
    Abel
    

    View Slide