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関数等式と双対性
Naoya Umezaki
October 20, 2019
Science
1
540
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/
Naoya Umezaki
October 20, 2019
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Transcript
ؔࣜͱରੑ ക࡚@unaoya 2019 10 ݄ 20 ϩϚϯςΟοΫֶφΠτϓϥΠϜˏθʔλ 1
Riemann ζ ζ(s) = ∞ n=1 n−s = p (1
− p−s)−1 ˆ ζ(s) = π−s/2Γ( s 2 )ζ(s) ͱ͓͘ͱɺؔࣜ ˆ ζ(s) = ˆ ζ(1 − s) ཱ͕ɻFourier มʢPoisson ެࣜʣΛ༻͍ͯࣔͤΔɻ 2
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
ؔࣜ 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 มʢPoisson ެࣜʣΛ༻͍ ͯࣔͤΔɻ 4
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
ؔࣜ ˆ ζK (s) = |DK |s/2ΓK (s)ζK (x) ͱ͢ΔɻDK
K ͷผࣜͰ DQ = 1ɻΓQ(s) = π−s/2Γ( s 2 ) Ͱ ͋Δɻ ˆ ζK (s) = ˆ ζK (1 − s) 6
Hecke L ಋख f ͷ Hecke ࢦඪ χ : AK
→ C×ɻ͜Εͷಛผͳ߹͕ Dirichlet ࢦඪɻ L(χ, s) = p (1 − χ(πp)N(p)−s)−1 ʢѱ͍ૉͰमਖ਼͢Δɻ ʣ 7
ؔࣜ ˆ L(χ, s) = |DK |s/2f s/2 χ Γ(χ,
s)L(χ, s) ͱ͢Δͱɺؔࣜ ˆ L(χ, s) = W (χ)ˆ L(χ, 1 − s) Λຬͨ͢ɻΞσʔϧ্ͷ Fourier มΛ༻͍ͯࣔ͢ɻ 8
߹ಉ ζ ༗ݶମ্ͷଟ༷ମ X/Fq ͍͍ͩͨଟ߲ࣜ f = 0 Ͱఆ·Δਤܗɻ ͜Εͷղͷݸ
|X(Fqm )| Λ͑Δ͜ͱͰɺ 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
ؔࣜ 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
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 ʢѱ͍ૉͰमਖ਼͢Δɻ ʣ ˆ L(Hi (X), s) = Ns/2Γ(Hi (X), s)L(Hi (X), s) 11
ؔࣜ ؔࣜʢ༧ʣ ˆ L(Hi (X), s) = ±ˆ L(Hi (X),
i + 1 − s) Q ্ͷପԁۂઢ E Ͱ Wiles ͳͲʹΑΓূ໌͞Εͨɻ อܕܗࣜ fE Ͱ͋ͬͯ L ͕ؔҰக͢ΔͷΛ࡞Δɻอܕܗࣜ fE ͷ L ؔͷؔࣜ Hecke ͳͲʹΑΓ Fourier มͳͲΛ༻͍ ͯূ໌͞Ε͍ͯͨɻ 12
ℓ ਐͷ L X ͕༗ݶମ্ͷଟ༷ମɺF Λ ℓ ਐͱ͢Δɻ 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 ਐͷྫɻ ؔࣜ L(X, F, t) = ε(X, F)t−χ(X,F)L(X, D(F), t−1) 13
ذͱ ε Ҽࢠ ѱ͍ૉͰͷ༷ࢠɺผࣜɺಋखɺؔࣜʹݱΕΔิਖ਼߲ͳͲ ͷใ͕ॏཁɻ ʢෆมྔͱͯ͠ڧྗɻ ʣ ذͷزԿతͳෆมྔͱͯ͠ಛੑαΠΫϧͱ͍͏ͷ͕͋Δɻಛ ੑαΠΫϧݩʑඍํఔࣜʢD Ճ܈ʣͷཧͰߟ͑ΒΕͨ
ͷͰɺذͷ༷ࢠΛهड़͢Δɻ ؔࣜͷ ε(X, F) ͱಛੑαΠΫϧͷؔ ఆཧ (U.-Yang-Zhao) det ρ(−ccX F) = ε(X, F ⊗ ρ) ε(X, F)dim ρ 14