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関数等式と双対性

8554778f1060f77c8eec4e88a30369ac?s=47 Naoya Umezaki
October 20, 2019

 関数等式と双対性

ロマンティック数学ナイトプライム@ゼータでの発表

https://mathparty.localinfo.jp/

8554778f1060f77c8eec4e88a30369ac?s=128

Naoya Umezaki

October 20, 2019
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  1. ؔ਺౳ࣜͱ૒ରੑ ക࡚௚໵@unaoya 2019 ೥ 10 ݄ 20 ೔ϩϚϯςΟοΫ਺ֶφΠτϓϥΠϜˏθʔλ 1

  2. Riemann ζ ζ(s) = ∞ n=1 n−s = p (1

    − p−s)−1 ˆ ζ(s) = π−s/2Γ( s 2 )ζ(s) ͱ͓͘ͱɺؔ਺౳ࣜ ˆ ζ(s) = ˆ ζ(1 − s) ͕੒ཱɻFourier ม׵ʢPoisson ࿨ެࣜʣΛ༻͍ͯࣔͤΔɻ 2
  3. Dirichlet L ಋख f ͷ Dirhchlet ࢦඪ χ : Z

    → C χ(nm) = χ(n)χ(m)ɺn ͕ f ͱޓ͍ʹૉͳΒ χ(n) = 0ɻ Legendre ه߸ͳͲ͕ྫɻ L(χ, s) = ∞ n=1 χ(n)n−s = p (1 − χ(p)p−s)−1 શͯͷ n Ͱ χ(n) = 1 ͱ͢Δͱ Riemann ζ L(1, s) = ∞ n=1 n−s = p (1 − p−s)−1 3
  4. ؔ਺౳ࣜ ˆ L(χ, s) = f s/2 χ Γ(χ, s)L(s,

    χ) ͱ͢Δɻf1 = 1, Γ(s, 1) = π−s/2Γ(s) Ͱ͋Δɻ ˆ L(χ, 1 − s) = W (χ)ˆ L(χ, s) ิਖ਼߲ W (χ) ͕ଘࡏ͢ΔɻFourier ม׵ʢPoisson ࿨ެࣜʣΛ༻͍ ͯࣔͤΔɻ 4
  5. Dedekind ζ ୅਺ମ K ʹରͯ͠ɺ ζK (s) = a (NK/Qa)−s

    = p (1 − (NK/Qp)−s)−1 K = Q ͷ࣌ɺNQ/Q(p) = p ͳͷͰ ζK (s) = ζ(s) ͱͳΔɻ 5
  6. ؔ਺౳ࣜ ˆ ζK (s) = |DK |s/2ΓK (s)ζK (x) ͱ͢ΔɻDK

    ͸ K ͷ൑ผࣜͰ DQ = 1ɻΓQ(s) = π−s/2Γ( s 2 ) Ͱ ͋Δɻ ˆ ζK (s) = ˆ ζK (1 − s) 6
  7. Hecke L ಋख f ͷ Hecke ࢦඪ χ : AK

    → C×ɻ͜Εͷಛผͳ৔߹͕ Dirichlet ࢦඪɻ L(χ, s) = p (1 − χ(πp)N(p)−s)−1 ʢѱ͍ૉ఺Ͱ͸मਖ਼͢Δɻ ʣ 7
  8. ؔ਺౳ࣜ ˆ L(χ, s) = |DK |s/2f s/2 χ Γ(χ,

    s)L(χ, s) ͱ͢Δͱɺؔ਺౳ࣜ ˆ L(χ, s) = W (χ)ˆ L(χ, 1 − s) Λຬͨ͢ɻΞσʔϧ্ͷ Fourier ม׵Λ༻͍ͯࣔ͢ɻ 8
  9. ߹ಉ ζ ༗ݶମ্ͷଟ༷ମ X/Fq ͸͍͍ͩͨଟ߲ࣜ f = 0 Ͱఆ·Δਤܗɻ ͜Εͷղͷݸ਺

    |X(Fqm )| Λ਺͑Δ͜ͱͰɺ Z(X, t) = exp ∞ m=1 |X(Fqm )|tm m ΛఆΊΔɻ d dt log(Z(X, t)) = m |X(Fqm )|tm Ͱ͋Δɻ ζX (s) = x∈X (1 − (Nx)−s)−1 = Z(X, q−s) ͱදࣔͰ͖Δɻ 9
  10. ؔ਺౳ࣜ X ͷίϗϞϩδʔ Hi (X) ͷ Lefschetz ੻ެࣜʹΑΓɺFrobenius ࡞ ༻ͷݻ༗ଟ߲ࣜΛ༻͍ͯ

    Z(X, t) Λهड़Ͱ͖Δɻ Z(X, t) = det(1 − Frobt | H1(X)) · · · det(1 − Frobt | H2n−1(X)) det(1 − Frobt | H0(X)) · · · det(1 − Frobt | H2n(X)) ؔ਺౳ࣜ Z(X, 1 qnt ) = ±qnχ(X)/2tχ(X)Z(X, t) ζX (n − s) = ±qnχ(X)/2−χ(X)sζX (s) ͕੒ཱɻίϗϞϩδʔͷ Poincare ૒ରੑɻ 10
  11. Hasse-Weil ζ ୅਺ମ K ্ͷଟ༷ମ X ʹର͠ɺͦͷ i ࣍෦෼ Hi

    (X) ʹରͯ͠ L(Hi (X), s) = p det(1 − Frobpp−s | Hi (X))−1 ʢѱ͍ૉ఺Ͱ͸मਖ਼͢Δɻ ʣ ˆ L(Hi (X), s) = Ns/2Γ(Hi (X), s)L(Hi (X), s) 11
  12. ؔ਺౳ࣜ ؔ਺౳ࣜʢ༧૝ʣ ˆ L(Hi (X), s) = ±ˆ L(Hi (X),

    i + 1 − s) Q ্ͷପԁۂઢ E Ͱ͸ Wiles ͳͲʹΑΓূ໌͞Εͨɻ อܕܗࣜ fE Ͱ͋ͬͯ L ؔ਺͕Ұக͢Δ΋ͷΛ࡞Δɻอܕܗࣜ fE ͷ L ؔ਺ͷؔ਺౳ࣜ͸ Hecke ͳͲʹΑΓ Fourier ม׵ͳͲΛ༻͍ ͯূ໌͞Ε͍ͯͨɻ 12
  13. ℓ ਐ૚ͷ L X ͕༗ݶମ্ͷଟ༷ମɺF Λ ℓ ਐ૚ͱ͢Δɻ L(X, F,

    t) = x det(1 − tdeg(x)Fx , F¯ x )−1 = det(1 − Frobt | H1(X, F)) · · · det(1 − Frobt | H2n−1(X, F)) det(1 − Frobt | H0(X, F)) · · · det(1 − Frobt | H2n(X, F)) F ͕ఆ਺૚ Λ ͷͱ͖ɺ߹ಉθʔλɻ ۂઢ X ্ͷ଒ f : Y → X ʹରͯ͠ɺF = Hi (Yx ) ΋ ℓ ਐ૚ͷྫɻ ؔ਺౳ࣜ L(X, F, t) = ε(X, F)t−χ(X,F)L(X, D(F), t−1) 13
  14. ෼ذͱ ε Ҽࢠ ѱ͍ૉ఺Ͱͷ༷ࢠɺ൑ผࣜɺಋखɺؔ਺౳ࣜʹݱΕΔิਖ਼߲ͳͲ ͷ৘ใ͕ॏཁɻ ʢෆมྔͱͯ͠΋ڧྗɻ ʣ ෼ذͷزԿతͳෆมྔͱͯ͠ಛੑαΠΫϧͱ͍͏΋ͷ͕͋Δɻಛ ੑαΠΫϧ͸ݩʑ͸ඍ෼ํఔࣜʢD Ճ܈ʣͷཧ࿦Ͱߟ͑ΒΕͨ΋

    ͷͰɺ෼ذͷ༷ࢠΛهड़͢Δɻ ؔ਺౳ࣜͷ ε(X, F) ͱಛੑαΠΫϧͷؔ܎ ఆཧ (U.-Yang-Zhao) det ρ(−ccX F) = ε(X, F ⊗ ρ) ε(X, F)dim ρ 14