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レムニスケートから楕円関数へ

Naoya Umezaki
October 06, 2018
1.4k

 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。

Naoya Umezaki

October 06, 2018
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  1. ࢉज़زԿฏۉ a0 = 1, b0 = 1 √ 2 =

    0.7071 · · · ͔ΒॳΊͯ࣍ʑ ܁Γฦ͢ɻ a1 = a0 + b0 2 = 0.853553 · · · b1 = √ a0 b0 = 0.840896 · · ·
  2. ࢉज़زԿฏۉ a2 = a1 + b1 2 = 0.847224 ·

    · · b2 = √ a1 b1 = 0.847201 · · · a3 = a2 + b2 2 = 0.847213 · · · b3 = √ a2 b2 = 0.847213 · · ·
  3. 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
  4. ϨϜχεέʔτੵ෼ r4 1 − r4 (dr)2 = r2(dθ)2 ∫ √

    (rdθ)2 + (dr)2 = ∫ √ 1 1 − r4 dr
  5. ପԁੵ෼ͷඪ४ܗ r2 = 1 − sin2 θ rdr = −2

    cos θ sin θdθ dr = −2 cos θ sin θdθ √ 1 − sin2 θ
  6. 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
  7. ∫ 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
  8. ϥϯσϯม׵ͱࢉज़زԿฏۉ kn = bn an , kn+1 = bn+1 an+1

    ʹରͯ͠ 1 an K(kn ) = 1 an+1 K(kn+1 )
  9. 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
  10. ϧδϟϯυϧͷؔ܎ࣜ ಛʹk = 1 √ 2 ͷ࣌ 2E( 1 √

    2 )K( 1 √ 2 ) − K( 1 √ 2 )2 = π 2
  11. 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′
  12. ڏ਺৐๏ ପԁੵ෼ s(t) = ∫ t 0 1 √ 1

    − r4 dr ͸ڏ਺৐๏ͱ͍͏ؔ܎ࣜΛຬͨ͢ s(it) = is(t)
  13. ڏ਺৐๏ ପԁੵ෼ K(k) = ∫ π/2 0 dθ √ 1

    − k2 sin2 θ ͸k ͝ͱʹ৭ʑଘࡏ͢ΔɻͦͷதͰϨϜχε έʔτੵ෼K( 1 √ 2 )͸ಛผͳରশੑΛ࣋ͭɻ
  14. ϨϜχεέʔτੵ෼ s(t) = ∫ t 0 1 √ 1 −

    r4 dr ʹ͍ͭͯϑΝχϟʔϊ΍ΦΠϥʔͷݚڀ
  15. 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
  16. ΞʔϕϧͷҰൠԽ r = √ −x dr = − dx 2

    √ −x ∫ dr √ 1 − r4 = − 1 2 ∫ dx √ (1 − x2)(−x)
  17. ࡾ֯ؔ਺ͷՃ๏ఆཧ C : x2 + y2 = 1 L(t) :

    y = t1 x + t2 P1 (t) P2 (t) O
  18. 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
  19. 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
  20. ͭ·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
  21. ٯؔ਺ u(s) = ∫ s 0 dx y ͷٯؔ਺ u

    = ∫ s(u) 0 dx y ࠓͷ৔߹͸͜Ε͕ࡾ֯ؔ਺
  22. 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 )
  23. ·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
  24. ପԁੵ෼ y2 = x3 + ax2 + bx + c

    ∫ P O dx √ x3 + ax2 + bx + c = ∫ P O dx y
  25. 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
  26. 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
  27. ପԁؔ਺ͷՃ๏ఆཧ P3 ͷ࠲ඪ͸y = t1 x + t2 ͱ y2

    = x3 + ax2 + bx + c ͔Β୅਺తʹٻ·Δ P1 P2 P3
  28. 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
  29. ·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
  30. 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
  31. 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਺
  32. 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਺߲
  33. 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
  34. Ξʔϕϧͷఆཧͷٯ C ͕n࣍ۂઢf (x, y) = 0ͷͱ͖ P1 , .

    . . , Pg ͱQ1 , . . . , Qg ͔Β ∑ i u(Pi ) + ∑ i u(Qi ) + ∑ i u(Ri ) = 0 ΛΈͨ͢R1 , . . . , Rg ͕ܾ·Δɻ
  35. 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