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。
ϨϜχεέʔτ͔Βପԁؔക࡚@unaoya͢͏͕͘ͿΜ͔MATHPOWER2018 10/6
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ΞʔϕϧͱପԁੵwikipediaΑΓΞʔϕϧֶ͕ʹ֮Ίͯ200
ΨεͱϨϜχεέʔτੵ
ࢉज़زԿฏۉaͱbͷࢉज़ฏۉa + b2aͱbͷزԿฏۉ√ab
ࢉज़زԿฏۉa0= 1, b0=1√2= 0.7071 · · · ͔ΒॳΊͯ࣍ʑ܁Γฦ͢ɻa1=a0+ b02= 0.853553 · · ·b1=√a0b0= 0.840896 · · ·
ࢉज़زԿฏۉa2=a1+ b12= 0.847224 · · ·b2=√a1b1= 0.847201 · · ·a3=a2+ b22= 0.847213 · · ·b3=√a2b2= 0.847213 · · ·
ϨϜχεέʔτr2 = cos 2θOP
ϨϜχεέʔτੵPQ RPQ2 + QR2 = PR2√(dr)2 + (rdθ)2 = ds
ϨϜχεέʔτੵr2 = cos 2θ2rdr = −2 sin 2θdθ4r2(dr)2 = 4 sin2 2θ(dθ)2= 4(1 − cos2 2θ)(dθ)2 = 4(1 − r4)(dθ)2
ϨϜχεέʔτੵr41 − r4(dr)2 = r2(dθ)2∫ √(rdθ)2 + (dr)2 =∫ √11 − r4dr
ϨϜχεέʔτੵs(t) =∫PO1√1 − r4drOP
ପԁੵͷඪ४ܗr2 = 1 − sin2 θrdr = −2 cos θ sin θdθdr =−2 cos θ sin θdθ√1 − sin2 θ
ପԁੵͷඪ४ܗdr√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
∫10dr√1 − r4=12∫π/20dθ√1 − (1/√2)2 sin2 θ)K(k) =∫π/20dθ√1 − k2 sin2 θ
ϥϯσϯมͱࢉज़زԿฏۉkn=bnan, kn+1=bn+1an+1ʹରͯ͠1anK(kn) =1an+1K(kn+1)
ϧδϟϯυϧͷؔࣜE(k) =∫π/20√1 − k2 sin2 θdθk′2 + k2 = 1E(k)K(k′) + E(k′)K(k) − K(k)K(k′) =π2
ϧδϟϯυϧͷؔࣜಛʹk =1√2ͷ࣌2E(1√2)K(1√2) − K(1√2)2 =π2
·ͱΊ▶ ϨϜχεέʔτੵପԁੵK(1√2)▶ ࢉज़زԿฏۉͱପԁੵͷؔʢϥϯσϯมʣ▶ ପԁੵͱԁपͷؔʢϧδϟϯυϧͷؔࣜʣ
ڏ๏ପԁੵͷؔࣜ∫it01√1 − r4dr =∫t01√1 − (ir′)4d(ir′)= i∫t01√1 − r′4dr′
ڏ๏ପԁੵs(t) =∫t01√1 − r4drڏ๏ͱ͍͏ؔࣜΛຬͨ͢s(it) = is(t)
ڏ๏ପԁੵK(k) =∫π/20dθ√1 − k2 sin2 θk ͝ͱʹ৭ʑଘࡏ͢ΔɻͦͷதͰϨϜχεέʔτੵK(1√2)ಛผͳରশੑΛ࣋ͭɻ
Ξʔϕϧͱؔͷੵ
ϨϜχεέʔτੵs(t) =∫t01√1 − r4drʹ͍ͭͯϑΝχϟʔϊΦΠϥʔͷݚڀ
ΦΠϥʔͷՃ๏ఆཧx =y −√1 − z4 + z√1 − y41 + y2z2ͷͱ͖∫x01√1 − r4dr =∫y01√1 − r4dr +∫z01√1 − r4dr
ΞʔϕϧͷҰൠԽ·ͣପԁੵ∫dx√x3 + ax2 + bx + cΛߟ͑Δɻ
ΞʔϕϧͷҰൠԽr =√−xdr = −dx2√−x∫dr√1 − r4= −12∫dx√(1 − x2)(−x)
ͦͷલʹԁͷހ∫dx√1 − x2x = sin t ͱஔੵ
ࡾ֯ؔͷՃ๏ఆཧC : x2 + y2 = 1L(t) : y = t1x + t2P1(t)P2(t)O
ΞʔϕϧC ͱL(t)ͷަP1(t), P2(t)∫dxy=∫dx√1 − x2u(t) =∫P1(t)Odxy+∫P2(t)Odxy
t2Λಈ͔͢P1(t)P2(t)O∂u(t)∂t2= 0
t1Λಈ͔͢P1(t)P2(t)O∂u(t)∂t1= −2(arctan t1)′
͜ͷ͜ͱ͔Βɺu(t) = −2 arctan t1= arcsin(−2t11 + t21)
Ұํɺx1, x2͕x2 + (t1x + t2)2 = 1ͷղͳͷͰx1x2=t22− 1t21+ 1, x1+ x2=−2t1t2t21+ 1Ͱ͋Δ͜ͱ͔Βɺx1y2+ x2y1= x1(t1x2+ t2) + x2(t1x1+ t2)= 2t1x1x2+ (x1+ x2)t2=−2t11 + t21
ͭ·Γɺu(P1(t)) + u(P2(t)) = u(t)∫(x1,y1)(0,1)dxy+∫(x2,y2)(0,1)dxy=∫x1y2+x2y1(0,1)dxyͱͳΔɻ
ٯؔu(s) =∫s0dxyͷٯؔu =∫s(u)0dxyࠓͷ߹͜Ε͕ࡾ֯ؔ
Ճ๏ఆཧu(t)ͷٯؔΛt = sin(u)ͱ͔͘ͱɺsin(u(P1) + u(P2)) = x1y2+ x2y1= cos u(P1) sin u(P2) + sin u(P2) cos u(P1)
·ͱΊ1. u =∫s0dx√1 − x2ͷٯ͕ؔsin u2. x2 + y2 = 1ͷΞʔϕϧ∫P1(t)Odxy+∫P2(t)Odxy3. ࡾ֯ؔͷՃ๏ఆཧ
ପԁੵy2 = x3 + ax2 + bx + c∫POdx√x3 + ax2 + bx + c=∫POdxy
ΞʔϕϧͷՃ๏ఆཧC ͱL(t)ͷަP1(t), P2(t), P3(t)P1(t)P2(t)P3(t)
ΞʔϕϧͷՃ๏ఆཧC ͱL(t)ͷަP1(t), P2(t), P3(t)C ͷΞʔϕϧu(t) =∫P1(t)Odxy+∫P2(t)Odxy+∫P3(t)Odxy
ΞʔϕϧͷՃ๏ఆཧC ͱL(t)ͷަP1(t), P2(t), P3(t)Ξʔϕϧͷఆཧu(t) =∫P1(t)Odxy+∫P2(t)Odxy+∫P3(t)Odxy= 0
ପԁؔͷՃ๏ఆཧପԁੵͷٯؔu =∫s(u)OdxyΛΈͨ͢s(u)͕ପԁؔ
ପԁؔͷՃ๏ఆཧP3ͷ࠲ඪy = t1x + t2ͱy2 = x3 + ax2 + bx + c ͔Βతʹٻ·ΔP1P2 P3
ପԁؔͷՃ๏ఆཧ∫P1Odxy+∫P2Odxy+∫P3Odxy= 0u1+ u2+ u3= 0s(u1+ u2) = s(−u3) = P1ͱP2ͷతͳࣜ
·ͱΊ1. u =∫s0dx√x3 + ax2 + bx + cͷٯ͕ؔପԁؔ2. y2 = x3 + ax2 + bx + c ͷΞʔϕϧ∫P1(t)Odxy+∫P2(t)Odxy+∫P3(t)Odxy= 03. ପԁؔͷՃ๏ఆཧ
ΞʔϕϧੵP1(t), . . . , Pn(t)ΛC ͱDtͷަͱ͢Δu(t) =n∑i=0∫Pi(t)P0r(x, y)dx͜͜Ͱr(x, y)dx dxyͷΑ͏ͳ༗ཧࣜ
Ξʔϕϧͷఆཧu(t) =n∑i=0∫Pi(t)P0r(x, y)dxu(t) = R(t) +∑logiSi(t)͜͜ͰɺR(t), S(t)t ͷ༗ཧؔ
Ξʔϕϧͷఆཧω =pdxfy∂u(t)∂t1=−x2p(x, t1x + t2)f (x, t1x + t2)ͷఆ߲∂u(t)∂t2=−xp(x, t1x + t2)f (x, t1x + t2)ͷఆ߲
Ξʔϕϧͷఆཧp ͷ͕࣍খ͚͞Εu(t)ఆ∫P1P0ω +∫P2P0ω + · · ·∫PnP0ω = 0
Ξʔϕϧͷఆཧͱपظ
ΞʔϕϧͷఆཧͷٯΞʔϕϧͷఆཧP1, P2, P3͕Ұઢ্ͷͱ͖u(P1) + u(P2) + u(P3) = 0ΞʔϕϧͷఆཧͷٯC ্ͷP1, P2, P3ʹର͠u(P1) + u(P2) + u(P3) = 0ͳΒP1, P2, P3Ұઢ্ɻ
ΞʔϕϧͷఆཧͷٯC ͕n࣍ۂઢf (x, y) = 0ͷͱ͖P1, . . . , PgͱQ1, . . . , Qg͔Β∑iu(Pi) +∑iu(Qi) +∑iu(Ri) = 0ΛΈͨ͢R1, . . . , Rg͕ܾ·Δɻ
पظੵͷ࣮Ұͭʹܾ·Βͣɺੵܦ࿏ʹґଘ͢ΔɻPQO
पظίʔγʔͷੵఆཧಛҟΛճΒͳ͚ΕੵͷมΘΒͳ͍पظಛҟͷपΓΛҰपճͬͨੵͨͪ∫γω
ϗϞϩδʔɺίϗϞϩδʔຊ࣭తʹҟͳΔܦ࿏͕ͲΕ͙Β͍͋Δ͔ʁγ ∈ H1(C, Z)पظ֨ࢠΛ(C) = {(∫γωi) | γ ∈ H1(C, Z)} ⊂ C
ΞʔϕϧϠίϏͷఆཧC ͕ࡾ࣍ࣜͷ࣌C → C/Λ(C)ಉҰࢹΛ༩͑Δɻ
ϗοδཧඃੵؔω ͱੵܦ࿏γ ͷؔ∫γω͕ۂઢf (x, y) = 0ʹґଘͨ͠ྔΛ༩͑Δɻ
ϞδϡϥΠ▶ ପԁؔͰҟͳΔͷ͕ͲΕ͙Β͍͋Δ͔ʁ▶ पظ͕ͲΕ͙Β͍͋Δ͔ʁ͜ΕΒΛूΊͯҰͭͷزԿֶతରͱͯ͠ѻͬͨͷ͕ϞδϡϥΠۭؒ
ࢀߟจݙ▶ פޫɺશପԁੵͱΨεɾϧδϟϯυϧ๏ʹΑΔπ ͷܭࢉ▶ Phillip Griffiths, The legacy of Abel inalgebraic geometry▶ Phillip Griffiths, Variations on a Theorem ofAbel