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数論幾何と分岐
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Naoya Umezaki
June 26, 2018
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数論幾何と分岐
ある企業の研究者の方に自分の研究の概要を説明したものです。
Naoya Umezaki
June 26, 2018
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Transcript
زԿͱذ ക࡚ ౦ژେֶཧՊֶݚڀՊ August 8, 2014 ക࡚ زԿͱذ
زԿͱʁ ݚڀରɿํఔࣜͷղશମͷͳ͢ਤܗɻ ྫ y = x2, x2 + y2 =
1, y2 = x(x − 1)(x − 2) ͍Ζ͍ΖͳՃݮΛͭͷू߹ʢશମɺ༗ཧશମɺ࣮ શମɺෳૉશମͳͲʣͰํఔࣜΛߟ͑Δɻ ྫɻϑΣϧϚʔ༧ xn + yn = 1 ͷ༗ཧղɻ ക࡚ زԿͱذ
ෳૉۂઢͷذ ෳૉۂઢͷྫ P1ɿෳૉฏ໘ʹҰແݶԕΛ͚ͭՃ͑ͨͷɻٿ໘ͱಉ͡ ͔ͨͪɻ ക࡚ زԿͱذ
ෳૉۂઢͷذ y2 = x(x − 1)(x − 2) Ұൠʹ x
ͷΛܾΊΔͱ y ͷ͕;͖ͨͭ·Δɻٿ໘;ͨ ͭͱ͍͍ͩͨಉ͡ɻ ͱ͜Ζ͕ x = 0, 1, t, ∞ ͰॏղΛͭɺͭ·Γ y ͷͻ ͱͭɻ͜ͷΑ͏ͳΛذͱ͍͏ɻ ͜ͷۂઢͷղશମʢʹແݶԕΛ͚ͭՃ͑ͨͷʣͷ͔ͨͪ ʁٿ໘ೋͭΛΈ߹ΘͤͯɺυʔφπܕΛͭ͘Δɻ ക࡚ زԿͱذ
ക࡚ زԿͱذ
छ ۂઢͷෆมྔɿ݀ͷʢछ gʣʹΑ͓͓ͬͯ·͔ʹྨ͢Δɻ ക࡚ زԿͱذ
Hurwitz ͷެࣜ ೋͭͷۂઢ Y → X ͷؒͷछͷެࣜ 2g(Y) − 2
= d(2g(X) − 2) + ∑ P (eP − 1) g ͕݀ͷɺ2g − 2 ΛΦΠϥʔʢߴ࣍ݩͷਤܗʹ͍ͨͯ͠ ఆٛͰ͖ΔʣͱΑͿɻd ͕Ұൠతͳͷ্ʹ͋Δͷɺ P ذɺeP ذͷେ͖͞ʢղͷॏෳʣ ɻ લͷྫͰɺd = 2, P = 0, 1, 2, ∞, eP = 2, g(X) = 0 ͳͷͰ g(Y) = 1 ͱͳΔɻ ͱ͘ʹɺ͜Ε͔Β P1 ্ෆذɺҰͰذ͢Δඃ෴ଘࡏ͠ͳ ͍͜ͱ͕Θ͔Δɻ2g − 2 = −2d + 1 ͱ͢Δͱ g = −d + 3 2 < 0 ͱ ͳΔͷͰɻ ക࡚ زԿͱذ
ͷذ ૉશମʹۂઢ ༗ཧʹํఔࣜͷղΛ͚ͭՃ֦͑ͯେ͢Δʢ࣮͔ΒෳૉΛͭ ͘ΔΑ͏ʹʣ ذΛݟΔ͜ͱͰ్தʹ͋Δ֦େΛ͠Δ͜ͱ͕Ͱ͖Δɻ ్தͰذͯͨ͠Βɺ্·Ͱ͍ͬͯذɻ ྫɺQ(ζ5)ɿ༗ཧશମʹ x5 = 1
ͷղΛ͚ͭՃ͑ͨମɻ͜͜ͰͲ Μͳೋ࣍ํఔ͕ࣜղ͚Δ͔ʁQ(ζ5) Ͱ 5 ͚ͩذɺx2 = n n ͕ 5 ͰΘΕͳ͚Εղ͚ͳ͍ʂ ക࡚ زԿͱذ
༗ݶମ Λૉ p ͰΘͬͨ͋·Γͷͳ͢ू߹ Fp Λߟ͑Δɻ͜ΕՃ ݮআͰด͡Δɻ F3 = {0,
1, 2}, F5 = {0, 1, 2, 3, 4} F3 Ͱ 2 × 2 = 1 ͱͳΓɺ1/2 = 2 ͱͳΔɻ ͞ΒʹҰมํఔࣜͷղʢͨͱ͑ x2 = −1 ͷղͳͲʣΛͯ͢ ͚ͭ͘Θ͑ͨͷΛΛ ¯ Fp ͱ͔͘ɻ͜Ε p ͝ͱʹଘࡏɻෳૉ ͷྨࣅɻ ക࡚ زԿͱذ
༗ݶମ্ͷۂઢͷذ ༗ݶମ্ͷۂઢͷྫɻ P1ɿ ¯ Fp શମͱແݶԕʢٿ໘ͷྨࣅʣ yp − y =
x x Λ P1 ͷ࠲ඪͱΈͯɺͦͷ্ͷඃ෴ͱߟ͑Δɻ ͨͱ͑ x = 0 ͩͱ y = 0, 1, 2, . . . , p − 1 ͕ղɻ ذ͢Δ͔ʁ ॏղ͕ଘࡏ͢ΔͳΒɺඍͱͷڞ௨Ҽࢠ͋Δɻඍ͢Δͱ pyp−1 − 1 = −1 ͰɺͲ͜ফ͑ͳ͍ɻͭ·Γ x = ∞ Ҏ֎Ͱ ذ͠ͳ͍ɻ P1 ্ҰͰذ͢Δඃ෴͕ଘࡏɻHurwitz ͷެ͕ࣜͳΓͨͨͳ͍ʂ ക࡚ زԿͱذ
Grothendieck-Ogg-Shafarevich ެࣜ ༗ݶମ্ͷۂઢͰذͷ༷ࢠΛΑΓਂ͘ଊ͑Δඞཁ͕͋Δɻ ذͷΑ͏͢Λ͋ΒΘ͋ͨ͢Β͍͠ෆมྔɿSwan ಋख SwP ʢSerreʣΛఆٛɻ Grothendieck-Ogg-Shafarevich ެࣜ χc(U,
F) = rankFχc(U, Q ) − ∑ P SwPF F ͕ඃ෴ɺχc(U, F) ͕ΦΠϥʔɻ ͞Βʹ͜ΕΒͷߴ࣍ݩԽɻ ʢมํఔࣜͷΛ૿ͯ͠ਤܗΛ ߟ͑Δɻ ʣ ߴ࣍ݩͷਤܗʹମ͢Δ Swan ಋखͷఆٛɺGOS ެࣜɻ ʢՃ౻-ࡈ౻ʣ ക࡚ زԿͱذ
ݱࡏͷݚڀ ෳૉͷઢܗඍํఔࣜʢD Ճ܈ʣͷෆ֬ఆಛҟͱ༗ݶମ্ͷ ذͷྨࣅɻ D Ճ܈ͷΦΠϥʔʹղͷ࣍ݩ ྫɻexp z ෳૉฏ໘্ਖ਼ଇͰ z
= ∞ Ͱෆ֬ఆಛҟΛͭ D Ճ܈ʹ͓͍ͯಛੑαΠΫϧ͕ॏཁͳෆมྔɻ ͜ͷྨࣅΛ༗ݶମͷํఔࣜͷͳ͢ਤܗʹରͯ͠ఆٛ͠ ͍ͨɻͦΕΛͬͯΦΠϥʔͷܭࢉͳͲΛߦ͏ɻ ʢݱࡏਐߦதʣ ക࡚ زԿͱذ