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レムニスケートから楕円関数へ
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Naoya Umezaki
October 06, 2018
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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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Transcript
ϨϜχεέʔτ͔Β ପԁؔ ക࡚@unaoya ͢͏͕͘ͿΜ͔ MATHPOWER2018 10/6
Ξʔϕϧͱପԁੵ wikipediaΑΓ Ξʔϕϧֶ͕ʹ֮Ίͯ200
ΨεͱϨϜχεέʔτੵ
ࢉज़زԿฏۉ aͱbͷࢉज़ฏۉ a + b 2 aͱbͷزԿฏۉ √ ab
ࢉज़زԿฏۉ a0 = 1, b0 = 1 √ 2 =
0.7071 · · · ͔ΒॳΊͯ࣍ʑ ܁Γฦ͢ɻ a1 = a0 + b0 2 = 0.853553 · · · b1 = √ a0 b0 = 0.840896 · · ·
ࢉज़زԿฏۉ a2 = a1 + b1 2 = 0.847224 ·
· · b2 = √ a1 b1 = 0.847201 · · · a3 = a2 + b2 2 = 0.847213 · · · b3 = √ a2 b2 = 0.847213 · · ·
ϨϜχεέʔτ r2 = cos 2θ O P
ϨϜχεέʔτੵ P Q R PQ2 + QR2 = PR2 √
(dr)2 + (rdθ)2 = ds
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
ϨϜχεέʔτੵ r4 1 − r4 (dr)2 = r2(dθ)2 ∫ √
(rdθ)2 + (dr)2 = ∫ √ 1 1 − r4 dr
ϨϜχεέʔτੵ s(t) = ∫ P O 1 √ 1 −
r4 dr O P
ପԁੵͷඪ४ܗ r2 = 1 − sin2 θ rdr = −2
cos θ sin θdθ dr = −2 cos θ sin θdθ √ 1 − sin2 θ
ପԁੵͷඪ४ܗ 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
∫ 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
ϥϯσϯมͱࢉज़زԿฏۉ kn = bn an , kn+1 = bn+1 an+1
ʹରͯ͠ 1 an K(kn ) = 1 an+1 K(kn+1 )
ϧδϟϯυϧͷؔࣜ 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
ϧδϟϯυϧͷؔࣜ ಛʹk = 1 √ 2 ͷ࣌ 2E( 1 √
2 )K( 1 √ 2 ) − K( 1 √ 2 )2 = π 2
·ͱΊ ▶ ϨϜχεέʔτੵପԁੵK( 1 √ 2 ) ▶ ࢉज़زԿฏۉͱପԁੵͷؔ ʢϥϯσϯมʣ
▶ ପԁੵͱԁपͷؔ ʢϧδϟϯυϧͷؔࣜʣ
ڏ๏ ପԁੵͷؔࣜ ∫ it 0 1 √ 1 − r4
dr = ∫ t 0 1 √ 1 − (ir′)4 d(ir′) = i ∫ t 0 1 √ 1 − r′4 dr′
ڏ๏ ପԁੵ s(t) = ∫ t 0 1 √ 1
− r4 dr ڏ๏ͱ͍͏ؔࣜΛຬͨ͢ s(it) = is(t)
ڏ๏ ପԁੵ K(k) = ∫ π/2 0 dθ √ 1
− k2 sin2 θ k ͝ͱʹ৭ʑଘࡏ͢ΔɻͦͷதͰϨϜχε έʔτੵK( 1 √ 2 )ಛผͳରশੑΛ࣋ͭɻ
Ξʔϕϧͱؔͷੵ
ϨϜχεέʔτੵ s(t) = ∫ t 0 1 √ 1 −
r4 dr ʹ͍ͭͯϑΝχϟʔϊΦΠϥʔͷݚڀ
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
ΞʔϕϧͷҰൠԽ ·ͣପԁੵ ∫ dx √ x3 + ax2 + bx
+ c Λߟ͑Δɻ
ΞʔϕϧͷҰൠԽ r = √ −x dr = − dx 2
√ −x ∫ dr √ 1 − r4 = − 1 2 ∫ dx √ (1 − x2)(−x)
ͦͷલʹ ԁͷހ ∫ dx √ 1 − x2 x =
sin t ͱஔੵ
ࡾ֯ؔͷՃ๏ఆཧ C : x2 + y2 = 1 L(t) :
y = t1 x + t2 P1 (t) P2 (t) O
Ξʔϕϧ C ͱL(t)ͷަP1 (t), P2 (t) ∫ dx y =
∫ dx √ 1 − x2 u(t) = ∫ P1(t) O dx y + ∫ P2(t) O dx y
t2 Λಈ͔͢ P1 (t) P2 (t) O ∂u(t) ∂t2 =
0
t1 Λಈ͔͢ P1 (t) P2 (t) O ∂u(t) ∂t1 =
−2(arctan t1 )′
͜ͷ͜ͱ͔Βɺ u(t) = −2 arctan t1 = arcsin( −2t1 1
+ t2 1 )
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
ͭ·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
ٯؔ u(s) = ∫ s 0 dx y ͷٯؔ u
= ∫ s(u) 0 dx y ࠓͷ߹͜Ε͕ࡾ֯ؔ
Ճ๏ఆཧ u(t)ͷٯؔΛt = sin(u)ͱ͔͘ͱɺ sin(u(P1 ) + u(P2 )) =
x1 y2 + x2 y1 = cos u(P1 ) sin u(P2 ) + sin u(P2 ) cos u(P1 )
·ͱΊ 1. u = ∫ s 0 dx √ 1
− x2 ͷٯ͕ؔsin u 2. x2 + y2 = 1ͷΞʔϕϧ ∫ P1(t) O dx y + ∫ P2(t) O dx y 3. ࡾ֯ؔͷՃ๏ఆཧ
ପԁੵ y2 = x3 + ax2 + bx + c
∫ P O dx √ x3 + ax2 + bx + c = ∫ P O dx y
ΞʔϕϧͷՃ๏ఆཧ C ͱL(t)ͷަP1 (t), P2 (t), P3 (t) P1 (t)
P2 (t) P3 (t)
ΞʔϕϧͷՃ๏ఆཧ C ͱL(t)ͷަP1 (t), P2 (t), P3 (t) C ͷΞʔϕϧ
u(t) = ∫ P1(t) O dx y + ∫ P2(t) O dx y + ∫ P3(t) O dx y
ΞʔϕϧͷՃ๏ఆཧ C ͱL(t)ͷަP1 (t), P2 (t), P3 (t) Ξʔϕϧͷఆཧ u(t)
= ∫ P1(t) O dx y + ∫ P2(t) O dx y + ∫ P3(t) O dx y = 0
ପԁؔͷՃ๏ఆཧ ପԁੵͷٯؔ u = ∫ s(u) O dx y ΛΈͨ͢s(u)͕ପԁؔ
ପԁؔͷՃ๏ఆཧ P3 ͷ࠲ඪy = t1 x + t2 ͱ y2
= x3 + ax2 + bx + c ͔Βతʹٻ·Δ P1 P2 P3
ପԁؔͷՃ๏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 ͷతͳࣜ
·ͱΊ 1. u = ∫ s 0 dx √ x3
+ ax2 + bx + c ͷٯ͕ؔ 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. ପԁؔͷՃ๏ఆཧ
Ξʔϕϧੵ P1 (t), . . . , Pn (t)ΛC ͱDt
ͷަͱ͢Δ u(t) = n ∑ i=0 ∫ Pi(t) P0 r(x, y)dx ͜͜Ͱr(x, y)dx dx y ͷΑ͏ͳ༗ཧࣜ
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
Ξʔϕϧͷఆཧ ω = pdx fy ∂u(t) ∂t1 = −x2p(x, t1
x + t2 ) f (x, t1 x + t2 ) ͷఆ߲ ∂u(t) ∂t2 = −xp(x, t1 x + t2 ) f (x, t1 x + t2 ) ͷఆ߲
Ξʔϕϧͷఆཧ p ͷ͕࣍খ͚͞Εu(t)ఆ ∫ P1 P0 ω + ∫ P2
P0 ω + · · · ∫ Pn P0 ω = 0
Ξʔϕϧͷఆཧͱपظ
Ξʔϕϧͷఆཧͷٯ Ξʔϕϧͷఆཧ P1 , P2 , P3 ͕Ұઢ্ͷͱ͖ u(P1 )
+ u(P2 ) + u(P3 ) = 0 Ξʔϕϧͷఆཧͷٯ C ্ͷP1 , P2 , P3 ʹର͠ u(P1 ) + u(P2 ) + u(P3 ) = 0ͳΒP1 , P2 , P3 Ұ ઢ্ɻ
Ξʔϕϧͷఆཧͷٯ C ͕n࣍ۂઢf (x, y) = 0ͷͱ͖ P1 , .
. . , Pg ͱQ1 , . . . , Qg ͔Β ∑ i u(Pi ) + ∑ i u(Qi ) + ∑ i u(Ri ) = 0 ΛΈͨ͢R1 , . . . , Rg ͕ܾ·Δɻ
पظ ੵͷ࣮Ұͭʹܾ·Βͣɺੵܦ࿏ʹ ґଘ͢Δɻ P Q O
पظ ίʔγʔͷੵఆཧ ಛҟΛճΒͳ͚ΕੵͷมΘΒͳ͍ पظ ಛҟͷपΓΛҰपճͬͨੵͨͪ ∫ γ ω
ϗϞϩδʔɺίϗϞϩδʔ ຊ࣭తʹҟͳΔܦ࿏͕ͲΕ͙Β͍͋Δ͔ʁ γ ∈ H1 (C, Z) पظ֨ࢠ Λ(C) =
{( ∫ γ ωi ) | γ ∈ H1 (C, Z)} ⊂ C
ΞʔϕϧϠίϏͷఆཧ C ͕ࡾ࣍ࣜͷ࣌ C → C/Λ(C) ಉҰࢹΛ༩͑Δɻ
ϗοδཧ ඃੵؔω ͱੵܦ࿏γ ͷؔ ∫ γ ω ͕ۂઢf (x, y)
= 0ʹґଘͨ͠ྔΛ༩͑Δɻ
ϞδϡϥΠ ▶ ପԁؔͰҟͳΔͷ͕ͲΕ͙Β͍͋ Δ͔ʁ ▶ पظ͕ͲΕ͙Β͍͋Δ͔ʁ ͜ΕΒΛूΊͯҰͭͷزԿֶతରͱͯ͠ ѻͬͨͷ͕ϞδϡϥΠۭؒ
ࢀߟจݙ ▶ פޫɺશପԁੵͱΨεɾϧδϟ ϯυϧ๏ʹΑΔπ ͷܭࢉ ▶ Phillip Griffiths, The legacy
of Abel in algebraic geometry ▶ Phillip Griffiths, Variations on a Theorem of Abel