Naoya Umezaki
September 23, 2018
8.4k

# Chowla-Selberg公式

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

## Naoya Umezaki

September 23, 2018

## Transcript

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

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

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

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

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

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

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

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

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

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

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

12. ࢤଜଟ༷ମ
k = Q(

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

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

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).

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