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パーフェクトイド空間とコホモロジー
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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。
Naoya Umezaki
October 06, 2018
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Transcript
ύʔϑΣΫτΠυۭؒͱ ίϗϞϩδʔ Peter Scholzeͷۀհ ക࡚@unaoya ͢͏͕͘ͿΜ͔ MATHPOWER2018 10/6
डཧ༝ p ਐͰͷزԿͷݚڀ ▶ ύʔϑΣΫτΠυۭؒͷཧ ▶ ϥϯάϥϯζରԠͷԠ༻ ▶ ৽͍͠ίϗϞϩδʔཧ
pਐ ༗ཧ͔Β࣮3.14159265 · · · ༗ཧ͔Βp ਐ · · ·
245123 = 3+2p+1p2 +5p3 +4p4 +2p5 +· · ·
pਐ ▶ ࣮Ͱ0.9999 · · · = 1 ▶ p
= 2ͷͱ͖ɺpਐͰ· · · 111111 = −1
زԿֶ ଟ߲ࣜΛߟ͑Δͱਤܗ͕ܾ·Δɻ ▶ ԁx2 + y2 = 1 ▶ ପԁۂઢy2
= x3 + x ▶ ϑΣϧϚʔۂઢxn + yn = 1
ίϗϞϩδʔ ݀ͷΛ͑Δɻਤܗͷྨ͕Ͱ͖Δɻ H1 sing (X) = H1 dR (X) =
R2
ίϗϞϩδʔ ༷ʑͳίϗϞϩδʔ͕͋Δɻ υϥʔϜ ίϗϞϩδʔ ಛҟ ίϗϞϩδʔ ؔ ඍํఔࣜ ۭؒͷதͷ ਤܗͷมܗ
ίϗϞϩδʔͷൺֱ υϥʔϜ ίϗϞϩδʔ ಛҟ ίϗϞϩδʔ ϗοδ ίϗϞϩδʔ ίϗϞϩδʔͷൺֱ͔Βपظ͕ग़ͯ͘Δɻ
ͱίϗϞϩδʔ ▶ ੲ͔Βߟ͑ΒΕ͍͍ͯͨΖΜͳ͕ί ϗϞϩδʔΛͬͯදݱͰ͖Δɻ ▶ ʹԠ༻ʢϦʔϚϯ༧ͷྨࣅʣ ▶ ίϗϞϩδʔΛௐΕ৭ʑΘ͔Δʂ
ύʔϑΣΫτΠυۭؒ ▶ زԿଟ߲ࣜx, x2 + ax + b, . .
. ▶ ղੳزԿऩଋႈڃx + px + p2x2 + · · · ▶ ύʔϑΣΫτΠυۭؒ 1/x + p + p2x + · · · , 1/xp + 1 + px + · · · , 1/xp2 + 1/xp + · · · , . . .
ύʔϑΣΫτΠυۭؒ ύʔϑΣΫτΠυۭؒΛ͏ͱίϗϞϩδʔ ͕ௐ͘͢ͳΔɻ
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
pਐपظࣸ૾ ϗοδཧͷp ਐ൛ πHT : S∗ Kp → F ପԁۂઢͷ
ϞδϡϥΠ ίϗϞϩδʔ ͷൺֱ
LanglandsରԠ ΨϩΞදݱ อܕදݱ ପԁۂઢ อܕܗࣜ ▶ ࠨ͖ɿΨϩΞදݱͷߏ ΞΠώϥʔ-ࢤଜ etc ▶
ӈ͖ɿΨϩΞදݱͷอܕੑ ςΠϥʔ-ϫΠϧζ etc
LanglandsରԠ ΨϩΞදݱ อܕදݱ 1. ίϗϞϩδʔͷൺֱఆཧ 2. ہॴରশۭؒͷp-torsionίϗϞϩδʔ͕ ௐΒΕΔ 3. ΑΓ͍อܕදݱ͔ΒΨϩΞදݱͷߏ
4. ΨϩΞදݱͷอܕੑʹԠ༻
৽͍͠ίϗϞϩδʔཧ ίϗϞϩδʔΛ౷Ұతʹѻ͍͍ͨ ʁ Τλʔϧ ΫϦε λϦϯ υϥʔϜ
৽͍͠ίϗϞϩδʔཧ ϥϯάϥϯζରԠͷݚڀ͔Β γτΡΧʁ Τλʔϧ ΫϦε λϦϯ υϥʔϜ