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/

Transcript

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

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide