Introduction Grothendieck ʖे KᲴஊྸૠ˳ Q ɥஊᨂဃ঺Ƴ˳ ӑ୺ႎ ˊૠ୺ዴ /K → Ტ࠹˴ႎᲣؕஜ፭+ ٳ Galois ˺ဇ

ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ Grothendieck ^{ʖेƴƭƍƯ}

ɶ஭ ҦଯᲢᣃᇌٻྸᲣ

ྚ߷ܤᬱဏᲢʮٻૠྸᄂᲣ ஓஉ ૼɟᲢʮٻૠྸᄂᲣ 1. Introduction

1.1. ˊૠٶಮ˳Ʒؕஜ፭ǁƷ ٳ Galois _˺ဇ

1.2. ˊૠٶಮ˳Ʒૠᛯႎؕஜ፭ 2. Grothendieck _ʖे

3. _{ɶ஭ƷˁʙƷኰʼ} 4. _{ྚ߷ƷˁʙƷኰʼ} 5. _{ஓஉƷˁʙƷኰʼ}

1

1. Introduction Grothendieck _ʖे

K_{Ჴஊྸૠ˳} Q ^{ɥஊᨂဃ঺Ƴ˳}

ӑ୺ႎ

ˊૠ୺ዴ /K → ^{Ტ࠹˴ႎᲣؕஜ፭}+ _ٳ Galois _˺ဇ

·||·

ૠᛯႎؕஜ፭ [_{࠹˴ႎݣᝋ}] [_{ˊૠႎݣᝋ}]

←−−−−−−−−

ܦμƴൿܭ ᲢࣄΨᲣ

2

1.1. ˊૠٶಮ˳Ʒؕஜ፭ǁƷ ٳ Galois ^˺ဇ

ˊૠٶಮ˳ƷᲢˮႻႎᲣؕஜ፭ X_{Ჴˊૠٶಮ˳} /C

Ტᙐእٶಮ˳ƱƠƯƷˮႻƴǑǓˮႻᆰ᧓Უ x ∈ X_Ჴؕໜ

^{ᲢˮႻႎᲣؕஜ፭} π₁^top(X, x) π₁^top(X) X _{Ʒᘮᙴμ˳ǛወСᲴ}

(f : Y → X) → f⁻¹(x)

(_ᘮᙴ/X) →^∼

π₁^top(X, x) - _˺ဇ

˄Ɩᨼӳ

∪ ∪

ஊᨂഏ ᘮᙴ /X

→∼

π₁^top(X, x) - _˺ဇ

˄Ɩஊᨂᨼӳ

3

profinite _ܦͳ҄

π₁(X) ^def= π₁^top(X) ƜƜưŴ፭ Γ _ƴݣƠƯ

Γ ^def= lim

←−(Γ _{ƷƢǂƯƷஊᨂՠ}) ᲢdzȳȑǯȈμɧᡲኽˮႻ፭Უ ᐯ໱Ƴݧ Γ → Γ^{Ʒ΂Ƹᆓ݅ưƋǓŴ}

Γ - _˺ဇ

˄Ɩஊᨂᨼӳ

=

Γ- _ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

ཎƴ

ஊᨂഏ ᘮᙴ /X

→∼

π₁(X) - _ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

4

Galois _˺ဇ

X _ƕ K(⊂ C) _{ɥܭ፯ƞǕƯƍǔئӳ}

Aut(C/K) X

ƩƕŴ X Ʒᙐእٶಮ˳ƱƠƯƷˮႻƴ᧙ƠƯ ᡲዓưƳƍƷư

Aut(C/K) π₁^top(X) ƸࡽƖឪƜƞǕƳƍ

Riemann ^Ʒ܍נܭྸ

X ƷஊᨂഏᘮᙴƸžˊૠႎſ i.e.

∀ f : Y → X _{ஊᨂഏᘮᙴ}

∃₁ Y ƷɥƷˊૠٶಮ˳Ʒನᡯ s.t. f _{Ƹˊૠٶಮ˳Ʒݧ}

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σ ∈ Aut(C/K) _ƴݣƠ

σ(f : Y → X) = (f^σ : Y ^σ → X^σ = X) Ʊܭ፯ƢǔƜƱƴǑǓ

Aut(C/K) (_{ஊᨂഏᘮᙴ} /X)

π₁(X) - _ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

Aut(C/K) π₁(X)

ദᄩƴƸŴ π₁(X, x)

∼σ

→π₁(X, σ(x)) ɧܭࣱ

π₁(X, x)

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ƠƨƕƬƯ

Aut(C/K) → Out(π₁(X))

Aut(π₁(X))/ Inn(π₁(X))

ƜǕƸ

Aut(C/K) Gal(K) ^def= Gal(K/K) Ǜኺဌ

ᲢRiemann Ʒ܍נܭྸưӲஊᨂഏᘮᙴƴ

λǔˊૠٶಮ˳ƷನᡯƸŴܱƸ K _ɥܭ፯ ƞǕǔᲣ

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1.2. ˊૠٶಮ˳Ʒૠᛯႎؕஜ፭ K_Ჴ˓ॖƷ˳

X_{Ჴˊૠٶಮ˳} /K finite ´etale _ᘮᙴ

X ƷžஊᨂഏᘮᙴſǛኝˊૠႎƴܭ፯

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̊ X = A¹_K − {0} = Spec(K[T, T ⁻¹])

̊ 1 A¹_K − {0} → A¹_K − {0} = X K[S, S⁻¹] ← K[T, T ⁻¹]

K = K

S^N ← T

̊ 2 A¹_K − {0} → A¹_K − {0} = X K[S, S⁻¹] ← K[T, T ⁻¹]

K = K

aS^N ← T

(a ∈ K^×)

̊ 3 A¹_K − {0} → A¹_K − {0} = X K[T, T ⁻¹] ← K[T, T ⁻¹]

K ⊃ K

T ← T

ᲢK_Ჴ K _{ƷஊᨂഏЎᩉਘٻᲣ}

̊ 4 _{ƜǕǒƷฆӳ}

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ؕஜ፭

π₁(X)_Ჴ profinite _ˮႻ፭ s.t.

finite ´etale ᘮᙴ /X

∼

→

π₁(X) - _ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

X _{ƕ᩼ཎီƳǒƹ}

Gal(K(X)) π₁(X) ཎƴ

Gal(K) = π₁(Spec(K))

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ഏƷܦμЗƕؕஜႎᲢX → Spec(K) _ƴݣƢ ǔ fiber homotopy _ЗᲣ

1 → π₁(X_K) → π₁(X) ^pr→^X Gal(K) → 1 π₁(X)_{Ჴૠᛯႎؕஜ፭}

π₁(X_K)_{Ჴ࠹˴ႎؕஜ፭}

ρ_X : Gal(K) → Out(π₁(X_K))

ද 1 K ⊂ C ^Ʒ଺Ƹ π₁(X_K) = π₁(X_C) _ư ρ_{X Ƹ} 1.1 _{ƷᘙྵƱɟᐲ}

ද 2 π₁(X_K) _{Ʒɶ࣎ƕᐯଢƳǒƹ}

(π₁(X), pr_X) (π₁(X_K), ρ_X)

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pro-l _ؕஜ፭Ტl_ᲴእૠᲣ (pro-l _፭) = lim

←−(_ஊᨂ l - _፭) Π_Ჴ profinite _፭

Π^l_Ჴ Π _Ʒஇٻ pro-l _ՠ

def= lim

←−(Π _{ƷƢǂƯƷˮૠ} l _ǂƖஊᨂՠ) π₁^(l)(X_K) ^def= π₁(X_K)^l

π₁^(l)(X) ^def=

π₁(X)/ Ker(π₁(X_K) → π₁^(l)(X_K))

1 → π₁^(l)(X_K) → π₁^(l)(X) ^pr

(l)

→X Gal(K) → 1 ρ⁽_X^l) : Gal(K) → Out(π₁^(l)(X_K))

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2. Grothendieck ^ʖे

ʖे

K_Ჴ Q ^{ɥஊᨂဃ঺Ƴ˳}

C_{Ჴӑ୺ႎˊૠ୺ዴ} /K

S_Ჴ smooth _ˊૠٶಮ˳ /K ƜƷ଺Ŵ

Hom^dom_K (S, C)

→∼ Hom ext^open

Gal(K)(π₁(S), π₁(C)) C^∗_Ჴ C _{ƷdzȳȑǯȈ҄}, g_Ჴ C^∗ _Ʒᆔૠ Σ = C^∗ − C, ν = (Σ(K))

C_Ჴӑ୺ႎ ⇐⇒ χ(C) ^def= 2 − 2g − ν < 0

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⎛

⎜⎝

S → C _Ჴ dominant _ݧ

Spec(K)

⎞

⎟⎠

↓

⎛

⎜⎝

π₁(S) → π₁(C) _{Ჴᡲዓ᧏፭แӷ׹}

Gal(K)

⎞

⎟⎠ modulo Inn(π₁(C_K))

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ॖԛ

žK ɥƷӑ୺ႎˊૠ୺ዴƸƦƷૠᛯႎؕஜ፭ ƔǒܦμƴࣄΨƞǕǔſ

Ტᙸ૾Უ

(1) S = C _{Ტӑ୺ႎˊૠ୺ዴᲣ} ӑ୺ႎˊૠ

୺ዴ /K

_dom

π₁

→

profinite _፭ / Gal(K)

_open

ext

ƕܱࣙΪ฼ŵ ཎƴ

žૠᛯႎؕஜ፭ƕӷ׹Ƴ K _{ɥƷӑ୺ႎˊૠ}

୺ዴƸӷ׹ſ

(2) _߼ᡀ = C _Ʒ S - _{ஊྸໜᨼӳ}

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ද

(i) _{Ǣȸșȫٶಮ˳Ʒ} Tate _ʖेᲢFaltings _Ʒ ܭྸᲣƱ᫏˩

Hom_K(A, A) ⊗_Z Zˆ

→∼ Hom_Gal(K)(π₁(A_K), π₁(A_K))

(ii) ᭗ഏΨٶಮ˳ǁƷɟᑍ҄ᲢசᚐൿᲣ

ʖेžanabelian Ƴˊૠٶಮ˳Ƹૠᛯႎؕஜ

፭ƔǒܦμƴࣄΨƞǕǔſ

“anabelian” _ƱƸᲹ

சܭ፯ŵžabelian ƔǒDŽƲᢒƍؕஜ፭ƴǑƬ ƯƦƷ࠹˴ƕወСƞǕǔǑƏƳٶಮ˳ſ

̊Ჴӑ୺ႎ୺ዴƷ successive fibration ӑ୺ႎ୺ዴƷȢǸȥȩǤᆰ᧓

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(iii) Section _{ʖेᲢசᚐൿᲣ}

ቇҥƷƨNJ C_{ᲴݧࢨႎᲢ} ⇐⇒ C = C^∗_Უ ʖे

Hom_K(S, C)

→∼ Hom ext_Gal(K)(π₁(S), π₁(C))

ཎƴ S = Spec(K) _ƱƢǕƹ

C(K) → {^∼ pr_C : π₁(C) → Gal(K)

ƷᡲዓŴ፭ᛯႎ sections}_ext

ᲢΣ = ∅ ^{Ʒ଺Ƹ̲ദᙲ} “tangential sections”_Უ

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3. ^ɶ஭Ʒˁʙ

K_Ჴ Q ^{ɥஊᨂဃ঺Ƴ˳}

C_Ჴ g = 0 _Ტν ≥ 3_Უ ቇҥƷƨNJ

C = P¹_K − Σ, P¹(K) ⊃ Σ ⊃ {0, 1, ∞}

ܭྸᲢAnderson - _˙ҾᲣ

ρ⁽_C^l) : Gal(K) → Out(π₁^(l)(C_K))

ƷఋƴݣࣖƢǔ K _{ƷᲢ໯ᨂഏ} Galois_Უਘٻ

˳ K_C^(l) _ƸŴ Σ _{ƔǒžᙐൔſƱž}l _ʈఌſǛጮ ǓᡉƠƯဃơǔžૠſǛ K _{ƴชьƠƨ˳}

̊ Q⁽_P^l)1

Q−{0,1,2,∞} = Q⁽_P^l)1

Q−{0,1,3,∞}(l = 3)

૾ᤆᲴ Σ Ʒࡈ೅ǛܦμƴࣄΨƢǔDŽƲƷऴإ ǛਤƬƨ K _Ʒਘٻ˳Ǜ C _{Ʒૠᛯႎؕஜ፭Ɣ} ǒ፭ᛯႎƴЏǓЈƤƳƍƔᲹ

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C_{Ჴӑ୺ႎ୺ዴᲢ}g_ᲴɟᑍᲣ, x ∈ Σ

π₁(C) ⊃ D_x

ᲢЎᚐ፭Უ ⊃ I_x

Ტॄࣱ፭Უ Zˆ 1→ I_x → D_x → Gal(κ(x))→1

∩ ∩ ∩open

1→π₁(C_K)→π₁(C)^pr→^C Gal(K) →1 LJƨ D_x = N_π1(C)(I_x)

ᙀ᫆ (ॄࣱ፭Ʒ፭ᛯႎཎࣉ˄ƚ)

߹ׅᢿЎ፭ J ⊂ π₁(C_K) _ƴݣƠ J = I_x(∃x ∈ Σ) ⇐⇒

ž∀H ⊂

open π₁(C_K) _ƴݣƠƯ J ∩ H _ƕ H^ab Ʒ ‘_όЎႎᢿЎ’ ƴλǔſNjƷƷɶưಊٻ

Riemann-Weil ʖेƴǑǓࢸƷவˑƸ፭ᛯႎ 19

C = P¹_K − {0, 1, ∞, λ} ^Ტλ ∈ K − {0, 1}^Უ Ტν > 4 _{Ʒ଺Ƹᙀ᫆Ǜ̅ƬƯ} ν = 4 _ƴ࠙ბᲣ Step1 π₁(C) K(ζ_n, λ ⁿ¹ ) (n ∈ N)

0, ∞ ƷLjưᲢܦμᲣЎޟƢǔ P¹_K(ζn) ^Ʒ n _ഏ

߹ׅᘮᙴƴƓƚǔ 1, λ _Ʒ fibers _Ʒй˷˳Ʒ ӳ঺˳ǛᎋƑŴƦƷǑƏƳᘮᙴμƯǛឥƬƯ ƦƷσᡫᢿЎǛӕǔᲢᙀ᫆ƴǑǓž፭ᛯႎſᲣ Step2 {K(ζ_n, λ ⁿ¹ )}ⁿ_∈N λ

Kummer ྸᛯŴҥૠ፭Ʒஊᨂဃ঺ࣱ

Step3

P¹_K − {0, 1, ∞, λ} λ

|

P¹_K − {0, 1, ∞, 1 − λ} 1 − λ

|

P¹_K − {0, 1, ∞, _λ−1^λ } _λ−1^λ

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

λ

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pro-l _᫏˩

Aut_K(C)→^∼ Aut ext_Gal(K)(π₁^(l)(C)) ƱƳǔ̊ǛɨƑƨ

৖ඥᲴ pro-l _{ϙ΂᫏፭Ტ}⊂ Out(π₁^(l)(C_K))_Უ Ʒ weight filtration _ǛဇƍƯ pro-l _ٳ Galois ᘙྵƴƓƚǔ Galois _΂Ǜᚡᡓ

᭗ᆔૠƷ୺ዴǍƦƷᣐፗᆰ᧓ƷئӳƴNjᄂᆮ Ტᚌႏܨȷ᭗ރݵനƱσӷᲣ

21

4. ^ྚ߷Ʒˁʙ

Grothendieck _ʖे / _ஊᨂ˳

k_Ჴஊᨂ˳

C_{Ჴӑ୺ႎ୺ዴ} /k, ν > 0 Ტν = 0 _{ƸசᚐൿᲣ}

ܭྸᲢϋဋᲣ

k(C) _Ƹ Gal(k(C)) _{ƔǒࣄΨƞǕǔ} ᚰଢƷ૾ᤆᲴ

Step1 C^∗ ƷӲໜƴݣƢǔЎᚐ፭Ʒ፭ᛯႎ ཎࣉ˄ƚ

Step2 _ʈඥ፭ k(C)^× _ƷࣄΨ

Step3 k(C) = k(C)^× ∪ {0} ^{ƷɥƷьඥನᡯ} ƷࣄΨ

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Step1

ቇҥƷƨNJ x ∈ C {1}

^αx

1→ I_x → D_x →^∼ Gal(κ(x))→1

∩ ∩ ∩open

1→π₁(C_K)→π₁(C)^pr→^C Gal(k) →1 x ∈ C(k) _Ƴǒƹ

α_x : Gal(k) → π₁(C), pr_C ◦ α_x = id_Gal(k)

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ᡞƴ

α : Gal(k) → π₁(C), pr_C ◦ α = id_Gal(k) ƴݣƠƯ

α = α_x(∃x ∈ C(k))

⇐⇒ α(Gal(k)) _ƴݣࣖƢǔ C _ƷᲢpro -_Უ ᘮᙴƕ k - _{ஊྸໜǛਤƭ}

⇐⇒ α(Gal(k)) ⊂ ∀H ⊂

open π₁(C)

H _{ƴݣࣖƢǔᘮᙴƕ} k - _{ஊྸໜǛਤƭ}

Lefschetz ួπࡸƴǑǓஇࢸƷவˑƸ፭ᛯႎ

Step2 k(C) _{Ʒ᫏˳ᛯƷႻʝࢷᲴ} k(C)^× = Artin _ϙ΂Ʒఋ Step3 _ဦ

24

π₁^tame/ _ஊᨂ˳Ʒ Grothendieck _ʖे

π₁/ _{ஊᨂဃ঺˳ /Q Ʒ}⇓ Grothendieck _ʖे

ද ࠹˴ႎؕஜ፭ƴƭƍƯ C_Ჴ୺ዴ / _೅ૠ 0 _Ʒˊૠ᧍˳

=⇒ π₁(C) _Ƹ (g, ν) _{ƷLjưൿLJǔ} ܭྸ

୺ዴ C/F^p, g = 0 ƷᲢǹǭȸȠƱƠƯƷᲣ ӷ׹᫏Ƹ π₁(C) ƴǑƬƯܦμƴൿܭƞǕǔ Ტg_{ᲴɟᑍƸசᚐൿᲣ}

25

5. ^ஓஉƷˁʙ

Grothendieck _ʖे /p _ᡶ˳

ኽௐ

K_Ჴ Q^p ɥஊᨂဃ঺Ƴ˳ƷᢿЎ˳

̊ K_Ჴ Q ^{ɥஊᨂဃ঺}

K_Ჴ Q^{p ɥஊᨂဃ঺}

K = ∪

[K:Q]≤NK

26

ܭྸᲫ

C_{Ჴӑ୺ႎˊૠ୺ዴ} /K

S_Ჴ smooth _ˊૠٶಮ˳ /K ƜƷ଺Ŵ

Hom^dom_K (S, C)

→∼ Hom ext^open

Gal(K)(π₁(S), π₁(C))

→∼ Hom ext^open

Gal(K)(π₁^(p)(S), π₁^(p)(C)) ܭྸᲬ

L, MᲴᲢ˓ॖഏΨᲣˊૠٶಮ˳ /K _Ʒ᧙ૠ˳

ƜƷ଺Ŵ

Hom_K(M, L)

→∼ Hom ext^open

Gal(K)(Gal(L), Gal(M))

27

ද

(i) _{Ǣȸșȫٶಮ˳Ʒ} Tate _ʖेƸ p _ᡶ˳ɥ ưƸ঺ᇌƠƳƍŵ

(ii) K _ƕ Q ɥஊᨂဃ঺Ʒ଺ŴܭྸᲬƷ Isom

༿Ƹ Pop _Ʒܭྸ

ᚰଢᲴ p _ᡶ Hodge _{ྸᛯƷࣖဇ} Ტ[K : Q^p] < ∞^Უ

28

Grothendieck _ʖे

ٶಮ˳Ʒݧ ൔ᠋

←→ ^{ؕஜ፭ᲢƷݧᲣ}

ž᧙ૠſႎ

ˮႻႎ

´etale _ႎ

Galois _ᘙྵႎ p _ᡶ Hodge _ྸᛯ

p ᡶ˳ɥƷˊૠٶಮ˳ƴݣƠƯ de Rham

(crystalline) cohomology

←→ൔ᠋ p _ᡶ ´etale cohomology

ࣇЎь፭ႎ ž᧙ૠſႎ

ˮႻႎ

´etale _ႎ

Galois _ᘙྵႎ

29

ቇҥƷƨNJ C = C^∗, non-hyperelliptic π₁^(p)(C_K)^ab T_p(J_K)

ᲢJ_Ჴ C _Ʒ Jacobi _ٶಮ˳Უ

Hodge-Tate _ЎᚐᲢC^p = (K)^ᲣᲴ π₁^(p)(C_K)^ab ⊗

ZpC^p {Γ(C, Ω¹_C/K)⊗

KC^p} ⊕ {H¹(C, O^C)⊗

KC^p(1)} ǏƑƴ

(π₁^(p)(C_K)^ab ⊗

ZpC^p)^Gal(K)

= Γ(C, Ω¹_C/K) = Kω₁ ⊕ · · · ⊕ Kω_g canonical embedding_Ჴ C → P^g_K⁻¹

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΂ƷࣄΨƷƨNJƷ Key_Ჴ

L_Ჴ K _Ǜԃlj p ᡶܦͳƳᩉ૝˄͌˳Ŵй˷˳

Ƹ K Ʒй˷˳ƷɥƷɟ٭ૠˊૠ᧙ૠ˳

ƜƷ଺ŴɨƑǒǕƨ

Gal(L) →

Gal(K) π₁(C) ƕ࠹˴ႎƔƲƏƔŴ i.e.

Spec(L) →

Spec(K) C

ƔǒஹǔNjƷƔƲƏƔǛ፭ᛯႎƴЙܭ

Ტmod p^N _༿ p _ᡶ Hodge _ྸᛯǛ C _ƷӲᘮᙴ ƴᢘဇŴƳƲᲣ

ܭྸᲬᲴ

M/K Ʒឬឭഏૠƴ᧙Ƣǔ࠙ኛඥ

→ ^{ܭྸᲫƴ࠙ბ}

31