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Chowla-Selberg公式

Naoya Umezaki
September 23, 2018
8.4k

 Chowla-Selberg公式

数学について話す会

https://unaoya.github.io/event.html

Naoya Umezaki

September 23, 2018
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  1. Chowla-Selberg ͷެࣜ
    ക࡚௚໵@unaoya
    September 24, 2018

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  2. Chowla-Selberg ͷެࣜ

    a∈Cl(k)
    ∆(a)∆(a−1) =
    (

    d
    )12h ∏
    a∈(Z/dZ)×
    Γ
    ( a
    d
    )6wϵ(a)
    ▶ k = Q(

    −d) ͸ڏೋ࣍ମͰ ok
    Λͦͷ੔਺؀
    ▶ Cl(k) ͸ΠσΞϧྨ܈Ͱ h = |Cl(k)|, w = |o×
    k
    |
    ▶ ϵ ͸ೋ࣍ମ k ʹରԠ͢Δ Dirichlet ࢦඪʢฏํ৒༨ه߸ʣ
    ▶ ∆ ͸ weight 12 ͷ cusp form

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  3. ղੳతͳূ໌
    Chowla-Selberg ʹΑΔ

    a∈Cl(k)
    ∆(a)∆(a−1) =
    (

    d
    )12h ∏
    a∈(Z/dZ)×
    Γ
    ( a
    d
    )6wϵ(a)
    ํ਑
    ζ′
    k
    ζk
    Λ;ͨ௨Γͷํ๏Ͱܭࢉɻ
    1. Kronecker limit formula Λ࢖͏ͱࠨล͕ग़ͯ͘Δ
    2. Lerch ͷެࣜͱ Dirichlet ྨ਺ެࣜΛ࢖͏ͱӈล͕ग़ͯ͘Δ

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  4. ୅਺زԿతͳূ໌
    Gross ʹΑΔ

    a∈Cl(k)
    ∆(a)∆(a−1) ∼
    (

    d
    )12h ∏
    a∈(Z/dZ)×
    Γ
    ( a
    d
    )6wϵ(a)
    ҧ͍
    1. ΑΓҰൠͷ৔߹ʹ΋ެࣜ
    2. ୅਺త਺ഒͷͣΕ͸ಛఆͰ͖ͳ͍
    ํ਑
    1. ྆ลΛ͋ΔΞʔϕϧଟ༷ମͷपظͱͯ͠ղऍ
    2. ͜ΕΒͷΞʔϕϧଟ༷ମΛ͏·͘࿈ଓతʹมܗ͢Δ
    3. มܗͨ࣌͠ʹपظ͕୅਺త਺ഒͷͣΕͰ͋Δ͜ͱΛࣔ͢

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  5. discriminant
    ∆(z) ͸ Ez = C/(Z + zZ) ͷ discriminant Ͱ͋Γɺ͜Ε͸ z ͷؔ਺ͱͯ͠ weight 12 Ͱ
    level SL(2, Z) ͷอܕܗࣜͰ͋Δɻ
    Ez : y2 = 4x3 − g2(τ)x − g3(τ)
    ʹ͍ͨ͠
    ∆(τ) = g2(τ)3 − 27g3(τ)2

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  6. पظ
    ൺֱఆཧ
    ඍ෼ܗࣜΛੵ෼͢Δ͜ͱͰίϗϞϩδʔͷಉܕ͕ಘΒΕΔɻ
    Hi
    dR
    (X/k) ⊗ C → Hi (X(C), C)
    ω → (γ →

    γ
    ω)
    Hodge ෼ղ
    H1
    dR
    (X/C) = H0(X, Ω1) ⊕ H1(X, OX )
    ପԁۂઢ X = E ͷ৔߹ ωE
    ͕ H0(E, Ω1) ͷجఈʢਖ਼ଇඍ෼ܗࣜʣ

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  7. ପԁۂઢͷੵ
    ପԁۂઢ n = ϕ(d) ݸͷੵ A=E1 × · · · × En Ͱ k Ͱڏ਺৐๏Λ࣋ͭ΋ͷΛ࡞Δɻ
    ▶ ֤ Ei ͕ k Ͱڏ਺৐๏Λ࣋ͭ΋ͷͱ͢Δ
    ▶ ͜ΕΒ͸શͯಉछͰ͋Γɺपظ͸୅਺త਺ഒͷͣΕ
    ▶ E ͕ڏ਺৐๏Λ࣋ͭͱ͖ɺ∆(τ) ͱपظ ω12
    E
    ͷͣΕ͸୅਺త਺ɻ
    ʢWeil ͷຊʣ
    A ͷपظͱͯ͠ࠨล͕Ͱͯ͘Δɻ

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  8. Fermat ۂઢ
    B ؔ਺
    B(
    a
    d
    ,
    b
    d
    ) =

    1
    0
    t a
    d
    −1(1 − t)b
    d
    −1dt
    =

    1
    0
    xa−1yb−d dx mod Q×
    ͜͜Ͱ y = d

    1 − xd ͱ͢Δɻ͜Ε͸ Fermat ۂઢ F(d) : xd + yd = 1 ͷपظɻ

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  9. Jacobian
    Jacobian
    ۂઢ C ʹରͯ͠ఆ·ΔΞʔϕϧଟ༷ମ JC
    ɻ
    ▶ H1 ͸ C ͱҰக
    ▶ dim JC = g(C)
    C = F(d) : xd + yd = 1 ͷ৔߹ɺJC
    ͸ n ࣍ݩͷ঎Λ࣋ͭɻ·ͨ µd
    ͷ࡞༻͔Βڏ਺৐
    ๏Λ࣋ͭɻ

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  10. ପԁۂઢͷ଒
    ૬ର 1 ܗࣜ
    τ ∈ H ʹରͯ͠ Eτ = C/(Z ⊗ τZ) ͸ y2 = 4x3 − g2(τ)x − g3(τ) ͱॻ͚ͯ
    ωτ =
    dx
    y
    =
    dx

    4x3 − g2(τ)x − g3(τ)
    Ϟδϡϥʔۂઢ
    L = {(τ, x), x ∈ Z ⊕ τZ ⊂ C} ͱ͠ɺπ : h × C/L → h Λ SL2(Z) ͰΘΔɻ
    ڏ਺৐๏
    ڏೋ࣍ମʹରԠ͢Δ఺Λ h ͷ෦෼ू߹ͱࢥ͍ɺSL2(Z) ͕࡞༻ɻ͜ΕʹପԁۂઢΛҾ
    ͖໭ׂͯ͠Δɻ

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  11. ૬ରతͳঢ়گ
    π : A → S ΛΞʔϕϧଟ༷ମͷ଒ͱ͢ΔɻHn
    dR
    (A/S) ͸ Hn
    dR
    (As/s) Λ·ͱΊͨ S ্ͷ
    ૚ɻRnπ∗C ͸ Hn(As, C) Λ·ͱΊͨ S ্ͷ૚ɺOS
    ͸ਖ਼ଇؔ਺ͷ૚ɻ
    ૬ର൛ൺֱఆཧ
    Hn
    dR
    (A/S) → Rnπ∗C ⊗C OS
    HdR(A/S) ͷ੾அ ω ʹ͍ͭͯɺ֤఺ s ∈ S ͝ͱʹ ωs ͷपظ͕ఆ·Δɻω ͕ఆ਺पظΛ
    ࣋ͭͱ͸͜ͷपظ͕ҰఆͰ͋Δ͜ͱɻ

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  12. ࢤଜଟ༷ମ
    k = Q(

    −d) ͱ͠ɺok
    Ͱڏ਺৐๏Λ࣋ͭ n ࣍ݩΞʔϕϧଟ༷ମΛશ෦ूΊΔɻ
    ʢภۃ
    ͱϨϕϧߏ଄΋͚ͭΔʣ
    ࢤଜଟ༷ମ
    ීวతͳΞʔϕϧଟ༷ମͷ଒
    A → S
    ͕Ͱ͖ͯɺS ͸୅਺ମ্ͷ୅਺ଟ༷ମʹͳΔ

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  13. େҬ੾அ
    ڏ਺৐๏Λ࣋ͭ͜ͱΛ࢖ͬͯɺ֤఺Ͱͷपظ͕ Q ഒͷͣΕͰ͋Δ͜ͱΛূ໌Ͱ͖Δɻ
    ▶ ڏ਺৐๏ʹΑΔ k ͷ Hn
    dR
    (A/S) ΁ͷ࡞༻Λ෼ղͯ͠ɺS ্ͷେҬ੾அ ω Λ࡞Δɻ
    ▶ ҰํͰ Rnπ∗C ʹ΋ τn Ͱ࡞༻͢Δ෦෼ۭ͕ؒଘࡏ͠ɺ͜ΕΒ͸ൺֱಉܕͰରԠ
    ͢Δɻ
    ▶ Hn
    dR
    (A/S) ʹ͸ Gauss-Manin ઀ଓͱ͍͏୅਺తͳඍ෼ํఔ͕ࣜఆ·͍ͬͯΔɻ͜
    ΕΛ༻͍ͯɺ্ͷେҬ੾அ͕୅਺తͰɺ͞Βʹ Q×
    ্ఆٛ͞ΕΔ͜ͱ͕Θ͔Δɻ
    ▶ ·ͨ͜ͷେҬ੾அͷ࣍ݩ͸ίϯύΫτԽΛ༻͍ͯܭࢉ͢Δͱ 1 Ͱ͋Δ͜ͱ͕Θ
    ͔Δɻ

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  14. Hodge ༧૝
    Deligne ͷίϝϯτ
    I began by saying that I had found a new proof of the period implication of the Chowla
    Selberg formula, using some techniques from algebraic geometry. Deligne immediately
    asked, in all seriousness, if I had proved the Hodge conjecture. I replied that I would be
    delighted to hear that I had done so, as I was still looking for a thesis topic (and felt
    that a proof of the Hodge conjecture would probably be sufficient).

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  15. ࢀߟจݙ
    1. Andre Weil, ΞΠθϯγϡλΠϯͱΫϩωοΧʔʹΑΔପԁؔ਺࿦
    2. Benedict H. Gross, On the Periods of Abelian Integrals and a Formula of Chowla
    and Selberg
    3. Benedict H. Gross, On the periods of abelian varietiese

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