਺࿦زԿͱ෼ذ ക࡚ ௚໵ ౦ژେֶ਺ཧՊֶݚڀՊ August 8, 2014 ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

਺࿦زԿͱ͸ʁ ݚڀର৅ɿํఔࣜͷղશମͷͳ͢ਤܗɻ ྫ y = x2, x2 + y2 = 1, y2 = x(x − 1)(x − 2) ͍Ζ͍ΖͳՃݮ৐Λ΋ͭ਺ͷू߹ʢ੔਺શମɺ༗ཧ਺શମɺ࣮਺ શମɺෳૉ਺શମͳͲʣͰํఔࣜΛߟ͑Δɻ ྫɻϑΣϧϚʔ༧૝ xn + yn = 1 ͷ༗ཧ਺ղɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

ෳૉۂઢͷ෼ذ ෳૉۂઢͷྫ P1ɿෳૉฏ໘ʹҰ఺ແݶԕ఺Λ͚ͭՃ͑ͨ΋ͷɻٿ໘ͱಉ͡ ͔ͨͪɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

ෳૉۂઢͷ෼ذ y2 = x(x − 1)(x − 2) Ұൠʹ͸ x ͷ஋ΛܾΊΔͱ y ͷ஋͕;͖ͨͭ·Δɻٿ໘;ͨ ͭͱ͍͍ͩͨಉ͡ɻ ͱ͜Ζ͕ x = 0, 1, t, ∞ Ͱ͸ॏղΛ΋ͭɺͭ·Γ y ͷ஋͸ͻ ͱͭɻ͜ͷΑ͏ͳ఺Λ෼ذ఺ͱ͍͏ɻ ͜ͷۂઢͷղશମʢʹແݶԕ఺Λ͚ͭՃ͑ͨ΋ͷʣͷ͔ͨͪ ͸ʁٿ໘ೋͭΛ૊Έ߹ΘͤͯɺυʔφπܕΛͭ͘Δɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

छ਺ ۂઢͷෆมྔɿ݀ͷ਺ʢछ਺ gʣʹΑ͓͓ͬͯ·͔ʹ෼ྨ͢Δɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

Hurwitz ͷެࣜ ೋͭͷۂઢ Y → X ͷؒͷछ਺ͷެࣜ 2g(Y) − 2 = d(2g(X) − 2) + ∑ P (eP − 1) g ͕݀ͷ਺ɺ2g − 2 ΛΦΠϥʔ਺ʢߴ࣍ݩͷਤܗʹ͍ͨͯ͠ ΋ఆٛͰ͖ΔʣͱΑͿɻd ͕Ұൠతͳ఺ͷ্ʹ͋Δ఺ͷ਺ɺ P ͸෼ذ఺ɺeP ͸෼ذͷେ͖͞ʢղͷॏෳ౓ʣ ɻ લͷྫͰ͸ɺd = 2, P = 0, 1, 2, ∞, eP = 2, g(X) = 0 ͳͷͰ g(Y) = 1 ͱͳΔɻ ͱ͘ʹɺ͜Ε͔Β P1 ্ෆ෼ذɺҰ఺Ͱ෼ذ͢Δඃ෴͸ଘࡏ͠ͳ ͍͜ͱ͕Θ͔Δɻ2g − 2 = −2d + 1 ͱ͢Δͱ g = −d + 3 2 < 0 ͱ ͳΔͷͰɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

੔਺ͷ෼ذ ૉ਺શମʹۂઢ ༗ཧ਺ʹํఔࣜͷղΛ͚ͭՃ֦͑ͯେ͢Δʢ࣮਺͔Βෳૉ਺Λͭ ͘ΔΑ͏ʹʣ ෼ذΛݟΔ͜ͱͰ్தʹ͋Δ֦େΛ͠Δ͜ͱ͕Ͱ͖Δɻ ్தͰ෼ذͯͨ͠Βɺ্·Ͱ͍ͬͯ΋෼ذɻ ྫɺQ(ζ5)ɿ༗ཧ਺શମʹ x5 = 1 ͷղΛ͚ͭՃ͑ͨମɻ͜͜ͰͲ Μͳೋ࣍ํఔ͕ࣜղ͚Δ͔ʁQ(ζ5) Ͱ͸ 5 ͚ͩ෼ذɺx2 = n ͸ n ͕ 5 ͰΘΕͳ͚Ε͹ղ͚ͳ͍ʂ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

༗ݶମ ੔਺Λૉ਺ p ͰΘͬͨ͋·Γͷͳ͢ू߹ Fp Λߟ͑Δɻ͜Ε͸Ճ ݮ৐আͰด͡Δɻ F3 = {0, 1, 2}, F5 = {0, 1, 2, 3, 4} F3 Ͱ͸ 2 × 2 = 1 ͱͳΓɺ1/2 = 2 ͱͳΔɻ ͞ΒʹҰม਺ํఔࣜͷղʢͨͱ͑͹ x2 = −1 ͷղͳͲʣΛ͢΂ͯ ͚ͭ͘Θ͑ͨ΋ͷΛΛ ¯ Fp ͱ͔͘ɻ͜Ε΋ p ͝ͱʹଘࡏɻෳૉ਺ ͷྨࣅɻ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

༗ݶମ্ͷۂઢͷ෼ذ ༗ݶମ্ͷۂઢͷྫɻ P1ɿ ¯ Fp શମͱແݶԕ఺ʢٿ໘ͷྨࣅʣ yp − y = x x Λ P1 ͷ࠲ඪͱΈͯɺͦͷ্ͷඃ෴ͱߟ͑Δɻ ͨͱ͑͹ x = 0 ͩͱ y = 0, 1, 2, . . . , p − 1 ͕ղɻ ෼ذ͢Δ͔ʁ ॏղ͕ଘࡏ͢ΔͳΒ͹ɺඍ෼ͱͷڞ௨Ҽࢠ͋Δɻඍ෼͢Δͱ pyp−1 − 1 = −1 ͰɺͲ͜΋ফ͑ͳ͍ɻͭ·Γ x = ∞ Ҏ֎Ͱ͸෼ ذ͠ͳ͍ɻ P1 ্Ұ఺Ͱ෼ذ͢Δඃ෴͕ଘࡏɻHurwitz ͷެ͕ࣜͳΓͨͨͳ͍ʂ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

Grothendieck-Ogg-Shafarevich ެࣜ ༗ݶମ্ͷۂઢͰ͸෼ذͷ༷ࢠΛΑΓਂ͘ଊ͑Δඞཁ͕͋Δɻ ෼ذͷΑ͏͢Λ͋ΒΘ͋ͨ͢Β͍͠ෆมྔɿSwan ಋख SwP ʢSerreʣΛఆٛɻ Grothendieck-Ogg-Shafarevich ެࣜ χc(U, F) = rankFχc(U, Q ) − ∑ P SwPF F ͕ඃ෴ɺχc(U, F) ͕ΦΠϥʔ਺ɻ ͞Βʹ͜ΕΒͷߴ࣍ݩԽɻ ʢม਺΍ํఔࣜͷ਺Λ૿΍ͯ͠ਤܗΛ ߟ͑Δɻ ʣ ߴ࣍ݩͷਤܗʹମ͢Δ Swan ಋखͷఆٛɺGOS ެࣜɻ ʢՃ౻-ࡈ౻ʣ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ

ݱࡏͷݚڀ ෳૉ਺ͷઢܗඍ෼ํఔࣜʢD Ճ܈ʣͷෆ֬ఆಛҟ఺ͱ༗ݶମ্ͷ ෼ذͷྨࣅɻ D Ճ܈ͷΦΠϥʔ਺ʹղͷ࣍ݩ ྫɻexp z ͸ෳૉฏ໘্ਖ਼ଇͰ z = ∞ Ͱ͸ෆ֬ఆಛҟ఺Λ΋ͭ D Ճ܈ʹ͓͍ͯಛੑαΠΫϧ͕ॏཁͳෆมྔɻ ͜ͷྨࣅΛ༗ݶମ΍੔਺܎਺ͷํఔࣜͷͳ͢ਤܗʹରͯ͠ఆٛ͠ ͍ͨɻͦΕΛ࢖ͬͯΦΠϥʔ਺ͷܭࢉͳͲΛߦ͏ɻ ʢݱࡏਐߦதʣ ക࡚ ௚໵ ਺࿦زԿͱ෼ذ