レムニスケートから楕円関数へ

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

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Ξʔϕϧͱପԁੵ෼ wikipediaΑΓ Ξʔϕϧ͕਺ֶʹ໨֮Ίͯ200೥

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Ψ΢εͱϨϜχεέʔτੵ෼

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ࢉज़زԿฏۉ aͱbͷࢉज़ฏۉ a + b 2 aͱbͷزԿฏۉ √ ab

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

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

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ϨϜχεέʔτ r2 = cos 2θ O P

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

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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

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

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

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

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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

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∫ 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

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

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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

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

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

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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′

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

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

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Ξʔϕϧͱ୅਺ؔ਺ͷੵ෼

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

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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

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

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

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

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

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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

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t2 Λಈ͔͢ P1 (t) P2 (t) O ∂u(t) ∂t2 = 0

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

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

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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

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ͭ·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

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

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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 )

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·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

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

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

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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

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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

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

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

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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

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·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

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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

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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਺

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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਺߲

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

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Ξʔϕϧͷఆཧͱपظ

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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

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

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पظ ੵ෼ͷ஋͸࣮͸Ұͭʹܾ·Βͣɺੵ෼ܦ࿏ʹ ґଘ͢Δɻ P Q O

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

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

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

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

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

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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