Naoya Umezaki
October 20, 2019
670

# 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/

October 20, 2019

## Transcript

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