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。

Transcript

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