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パーフェクトイド空間とコホモロジー

8554778f1060f77c8eec4e88a30369ac?s=47 Naoya Umezaki
October 06, 2018
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 パーフェクトイド空間とコホモロジー

MATHPOWER2018での講演。フィールズ賞受賞者Peter Scholzeの業績紹介。

8554778f1060f77c8eec4e88a30369ac?s=128

Naoya Umezaki

October 06, 2018
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  1. ύʔϑΣΫτΠυۭؒͱ ίϗϞϩδʔ Peter Scholzeͷۀ੷঺հ ക࡚௚໵@unaoya ͢͏͕͘ͿΜ͔ MATHPOWER2018 10/6

  2. ड৆ཧ༝ p ਐ਺Ͱͷ୅਺زԿͷݚڀ ▶ ύʔϑΣΫτΠυۭؒͷཧ࿦ ▶ ϥϯάϥϯζରԠ΁ͷԠ༻ ▶ ৽͍͠ίϗϞϩδʔཧ࿦

  3. pਐ਺ ༗ཧ਺͔Β࣮਺΁3.14159265 · · · ༗ཧ਺͔Βp ਐ਺΁ · · ·

    245123 = 3+2p+1p2 +5p3 +4p4 +2p5 +· · ·
  4. pਐ਺ ▶ ࣮਺Ͱ͸0.9999 · · · = 1 ▶ p

    = 2ͷͱ͖ɺpਐ਺Ͱ͸· · · 111111 = −1
  5. ୅਺زԿֶ ଟ߲ࣜΛߟ͑Δͱਤܗ͕ܾ·Δɻ ▶ ԁx2 + y2 = 1 ▶ ପԁۂઢy2

    = x3 + x ▶ ϑΣϧϚʔۂઢxn + yn = 1
  6. ίϗϞϩδʔ ݀ͷ਺Λ਺͑Δɻਤܗͷ෼ྨ͕Ͱ͖Δɻ H1 sing (X) = H1 dR (X) =

    R2
  7. ίϗϞϩδʔ ༷ʑͳίϗϞϩδʔ͕͋Δɻ υϥʔϜ ίϗϞϩδʔ ಛҟ ίϗϞϩδʔ ؔ਺΍ ඍ෼ํఔࣜ ۭؒͷதͷ ਤܗͷมܗ

  8. ίϗϞϩδʔͷൺֱ υϥʔϜ ίϗϞϩδʔ ಛҟ ίϗϞϩδʔ ϗοδ ίϗϞϩδʔ ίϗϞϩδʔͷൺֱ͔Βपظ͕ग़ͯ͘Δɻ

  9. ੔਺࿦ͱίϗϞϩδʔ ▶ ੲ͔Βߟ͑ΒΕ͍͍ͯͨΖΜͳ໰୊͕ί ϗϞϩδʔΛ࢖ͬͯදݱͰ͖Δɻ ▶ ੔਺࿦ʹ΋Ԡ༻ʢϦʔϚϯ༧૝ͷྨࣅʣ ▶ ίϗϞϩδʔΛௐ΂Ε͹৭ʑΘ͔Δʂ

  10. ύʔϑΣΫτΠυۭؒ ▶ ୅਺زԿ͸ଟ߲ࣜx, x2 + ax + b, . .

    . ▶ ղੳزԿ͸ऩଋႈڃ਺x + px + p2x2 + · · · ▶ ύʔϑΣΫτΠυۭؒ͸ 1/x + p + p2x + · · · , 1/xp + 1 + px + · · · , 1/xp2 + 1/xp + · · · , . . .
  11. ύʔϑΣΫτΠυۭؒ ύʔϑΣΫτΠυۭؒΛ࢖͏ͱίϗϞϩδʔ ͕ௐ΂΍͘͢ͳΔɻ

  12. pਐHodgeཧ࿦ ίϗϞϩδʔͷൺֱఆཧ Hi ´ et (X, Fp ) ⊗ OC

    /p ∼ = Hi ´ et (X, O+ X /p) Hi ´ et (X, Qp ) ⊗Qp BdR ∼ = Hi dR (X0 ) ⊗k BdR
  13. pਐपظࣸ૾ ϗοδཧ࿦ͷp ਐ൛ πHT : S∗ Kp → F ପԁۂઢͷ

    ϞδϡϥΠ ίϗϞϩδʔ ͷൺֱ
  14. LanglandsରԠ ΨϩΞදݱ อܕදݱ ପԁۂઢ อܕܗࣜ ▶ ࠨ޲͖ɿΨϩΞදݱͷߏ੒ ΞΠώϥʔ-ࢤଜ etc ▶

    ӈ޲͖ɿΨϩΞදݱͷอܕੑ ςΠϥʔ-ϫΠϧζ etc
  15. LanglandsରԠ ΨϩΞදݱ อܕදݱ 1. ίϗϞϩδʔͷൺֱఆཧ 2. ہॴରশۭؒͷp-torsionίϗϞϩδʔ͕ ௐ΂ΒΕΔ 3. ΑΓ޿͍อܕදݱ͔ΒΨϩΞදݱͷߏ੒

    4. ΨϩΞදݱͷอܕੑʹ΋Ԡ༻
  16. ৽͍͠ίϗϞϩδʔཧ࿦ ίϗϞϩδʔΛ౷Ұతʹѻ͍͍ͨ ʁ Τλʔϧ ΫϦε λϦϯ υϥʔϜ

  17. ৽͍͠ίϗϞϩδʔཧ࿦ ϥϯάϥϯζରԠͷݚڀ͔Β γτΡΧʁ Τλʔϧ ΫϦε λϦϯ υϥʔϜ