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

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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

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1. Introduction Grothendieck ʖे

KᲴஊྸૠ˳ Q ɥஊᨂဃ঺Ƴ˳

ӑ୺ႎ

ˊૠ୺ዴ /K Ტ࠹˴ႎᲣؕஜ፭+ ٳ Galois ˺ဇ

·||·

ૠᛯႎؕஜ፭ [࠹˴ႎݣᝋ] [ˊૠႎݣᝋ]

←−−−−−−−−

ܦμƴൿܭ ᲢࣄΨᲣ

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1.1. ˊૠٶಮ˳Ʒؕஜ፭ǁƷ ٳ Galois ˺ဇ

ˊૠٶಮ˳ƷᲢˮႻႎᲣؕஜ፭ XᲴˊૠٶಮ˳ /C

Ტᙐእٶಮ˳ƱƠƯƷˮႻƴǑǓˮႻᆰ᧓Უ x XᲴؕໜ

ᲢˮႻႎᲣؕஜ፭ π1top(X, x) π1top(X) X Ʒᘮᙴμ˳ǛወСᲴ

(f : Y X) f−1(x)

(ᘮᙴ/X)

π1top(X, x) - ˺ဇ

˄Ɩᨼӳ

ஊᨂഏ ᘮᙴ /X

π1top(X, x) - ˺ဇ

˄Ɩஊᨂᨼӳ

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proﬁnite ܦͳ҄

π1(X) def= π1top(X) ƜƜưŴ፭ Γ ƴݣƠƯ

Γ def= lim

←−ƷƢǂƯƷஊᨂՠ) ᲢǳȳȑǯȈμɧᡲኽˮႻ፭Უ ᐯ໱Ƴݧ Γ ΓƷ΂Ƹᆓ݅ưƋǓŴ

Γ - ˺ဇ

˄Ɩஊᨂᨼӳ

=

Γ- ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

ཎƴ

ஊᨂഏ ᘮᙴ /X

π1(X) - ᡲዓ˺ဇ

˄Ɩஊᨂᨼӳ

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Galois ˺ဇ

X ƕ K( C) ɥܭ፯ƞǕƯƍǔئӳ

Aut(C/K) X

ƩƕŴ X Ʒᙐእٶಮ˳ƱƠƯƷˮႻƴ᧙ƠƯ ᡲዓưƳƍƷư

Aut(C/K) π1top(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) = (fσ : Y σ Xσ = 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 KᲴ˓ॖƷ˳

XᲴˊૠٶಮ˳ /K ﬁnite ´etale ᘮᙴ

X ƷžஊᨂഏᘮᙴſǛኝˊૠႎƴܭ፯

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̊ X = A1K − {0} = Spec(K[T, T −1])

̊ 1 A1K − {0} → A1K − {0} = X K[S, S−1] K[T, T −1]

K = K

SN T

̊ 2 A1K − {0} → A1K − {0} = X K[S, S−1] K[T, T −1]

K = K

aSN T

(a K×)

̊ 3 A1K − {0} → A1K − {0} = X K[T, T −1] K[T, T −1]

K K

T T

K K ƷஊᨂഏЎᩉਘٻᲣ

̊ 4 ƜǕǒƷฆӳ

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ؕஜ፭

π1(X) 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(XK) π1(X) prX Gal(K) 1 π1(X)Ჴૠᛯႎؕஜ፭

π1(XK)Ჴ࠹˴ႎؕஜ፭

ρX : Gal(K) Out(π1(XK))

ද 1 K C Ʒ଺Ƹ π1(XK) = π1(XC) ư ρX Ƹ 1.1 ƷᘙྵƱɟᐲ

ද 2 π1(XK) Ʒɶ࣎ƕᐯଢƳǒƹ

1(X), prX) (π1(XK), ρX)

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pro-l ؕஜ፭ᲢlᲴእૠᲣ (pro-l ) = lim

←−(ஊᨂ l - ) Π proﬁnite

Πl Π Ʒஇٻ pro-l ՠ

def= lim

←−ƷƢǂƯƷˮૠ l ǂƖஊᨂՠ) π1(l)(XK) def= π1(XK)l

π1(l)(X) def=

π1(X)/ Ker(π1(XK) π1(l)(XK))

1 π1(l)(XK) π1(l)(X) pr

(l)

X Gal(K) 1 ρ(Xl) : Gal(K) Out(π1(l)(XK))

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2. Grothendieck ʖे

ʖे

K Q ɥஊᨂဃ঺Ƴ˳

CᲴӑ୺ႎˊૠ୺ዴ /K

S smooth ˊૠٶಮ˳ /K ƜƷ଺Ŵ

HomdomK (S, C)

Hom extopen

Gal(K)1(S), π1(C)) C C ƷǳȳȑǯȈ҄, g C Ʒᆔૠ Σ = C C, ν = (Σ(K))

CᲴӑ୺ႎ ⇐⇒ χ(C) def= 2 2g ν < 0

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⎜⎝

S C dominant ݧ

Spec(K)

⎟⎠

⎜⎝

π1(S) π1(C) Ჴᡲዓ᧏፭แӷ׹

Gal(K)

⎟⎠ modulo Inn(π1(CK))

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ॖԛ

žK ɥƷӑ୺ႎˊૠ୺ዴƸƦƷૠᛯႎؕஜ፭ ƔǒܦμƴࣄΨƞǕǔſ

Ტᙸ૾Უ

(1) S = C Ტӑ୺ႎˊૠ୺ዴᲣ ӑ୺ႎˊૠ

୺ዴ /K

dom

π1

proﬁnite / Gal(K)

open

ext

ƕܱࣙΪ฼ŵ ཎƴ

žૠᛯႎؕஜ፭ƕӷ׹Ƴ K ɥƷӑ୺ႎˊૠ

୺ዴƸӷ׹ſ

(2) ߼ᡀ = C Ʒ S - ஊྸໜᨼӳ

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(i) Ǣȸșȫٶಮ˳Ʒ Tate ʖेᲢFaltings Ʒ ܭྸᲣƱ᫏˩

HomK(A, A) Z Zˆ

HomGal(K)1(AK), π1(AK))

(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 CᲴݧࢨႎᲢ ⇐⇒ C = C ʖे

HomK(S, C)

Hom extGal(K)1(S), π1(C))

ཎƴ S = Spec(K) ƱƢǕƹ

C(K) → { prC : π1(C) Gal(K)

ƷᡲዓŴ፭ᛯႎ sections}ext

ᲢΣ = Ʒ଺Ƹ̲ദᙲ “tangential sections”

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3. ɶ஭Ʒˁʙ

K Q ɥஊᨂဃ঺Ƴ˳

C g = 0 ν 3 ቇҥƷƨǊ

C = P1K Σ, P1(K) Σ ⊃ {0, 1, ∞}

ܭྸᲢAnderson - ˙ҾᲣ

ρ(Cl) : Gal(K) Out(π1(l)(CK))

ƷఋƴݣࣖƢǔ K ƷᲢ໯ᨂഏ GaloisᲣਘٻ

˳ KC(l) ƸŴ Σ ƔǒžᙐൔſƱžl ʈఌſǛጮ ǓᡉƠƯဃơǔžૠſǛ K ƴชьƠƨ˳

̊ Q(Pl)1

Q−{0,1,2,∞} = Q(Pl)1

Q−{0,1,3,∞}(l = 3)

૾ᤆᲴ Σ Ʒࡈ೅ǛܦμƴࣄΨƢǔǄƲƷऴإ ǛਤƬƨ K Ʒਘٻ˳Ǜ C Ʒૠᛯႎؕஜ፭Ɣ ǒ፭ᛯႎƴЏǓЈƤƳƍƔᲹ

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CᲴӑ୺ႎ୺ዴᲢgᲴɟᑍᲣ, x Σ

π1(C) Dx

ᲢЎᚐ፭Უ Ix

Ტॄࣱ፭Უ Zˆ 1 Ix Dx Gal(κ(x))1

open

1→π1(CK)→π1(C)prC Gal(K) 1 Ǉƨ Dx = Nπ1(C)(Ix)

ᙀ᫆ (ॄࣱ፭Ʒ፭ᛯႎཎࣉ˄ƚ)

߹ׅᢿЎ፭ J π1(CK) ƴݣƠ J = Ix(∃x Σ) ⇐⇒

ž∀H

open π1(CK) ƴݣƠƯ J H ƕ Hab Ʒ ‘όЎႎᢿЎ’ ƴλǔſǋƷƷɶưಊٻ

Riemann-Weil ʖेƴǑǓࢸƷவˑƸ፭ᛯႎ 19

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C = P1K − {0, 1, ∞, λ} λ K − {0, 1}ν > 4 Ʒ଺Ƹᙀ᫆Ǜ̅ƬƯ ν = 4 ƴ࠙ბᲣ Step1 π1(C) Kn, λ n1 ) (n N)

0, ƷǈưᲢܦμᲣЎޟƢǔ P1K(ζn) Ʒ n

߹ׅᘮᙴƴƓƚǔ 1, λ Ʒ ﬁbers Ʒй˷˳Ʒ ӳ঺˳ǛᎋƑŴƦƷǑƏƳᘮᙴμƯǛឥƬƯ ƦƷσᡫᢿЎǛӕǔᲢᙀ᫆ƴǑǓž፭ᛯႎſᲣ Step2 {Kn, λ n1 )}n∈N λ

Kummer ྸᛯŴҥૠ፭Ʒஊᨂဃ঺ࣱ

Step3

P1K − {0, 1, ∞, λ} λ

|

P1K − {0, 1, ∞, 1 λ} 1 λ

|

P1K − {0, 1, ∞, λ−1λ } λ−1λ

⎫⎪

⎪⎪

⎪⎪

⎪⎭

λ

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pro-l ᫏˩

AutK(C) Aut extGal(K)1(l)(C)) ƱƳǔ̊ǛɨƑƨ

৖ඥᲴ pro-l ϙ΂᫏፭Ტ Out(π1(l)(CK)) Ʒ weight ﬁltration ǛဇƍƯ pro-l ٳ Galois ᘙྵƴƓƚǔ Galois ΂Ǜᚡᡓ

᭗ᆔૠƷ୺ዴǍƦƷᣐፗᆰ᧓Ʒئӳƴǋᄂᆮ Ტᚌႏܨȷ᭗ރݵനƱσӷᲣ

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4. ྚ߷Ʒˁʙ

Grothendieck ʖे / ஊᨂ˳

kᲴஊᨂ˳

CᲴӑ୺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 Ix Dx Gal(κ(x))1

open

1→π1(CK)→π1(C)prC Gal(k) 1 x C(k) Ƴǒƹ

αx : Gal(k) π1(C), prC αx = idGal(k)

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ᡞƴ

α : Gal(k) π1(C), prC α = idGal(k) ƴݣƠƯ

α = αx(∃x C(k))

⇐⇒ α(Gal(k)) ƴݣࣖƢǔ C ƷᲢpro - 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

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π1tame/ ஊᨂ˳Ʒ Grothendieck ʖे

π1/ ஊᨂဃ঺˳ /Q Ʒ Grothendieck ʖे

ද ࠹˴ႎؕஜ፭ƴƭƍƯ CᲴ୺ዴ / ೅ૠ 0 Ʒˊૠ᧍˳

= π1(C) Ƹ (g, ν) ƷǈưൿǇǔ ܭྸ

୺ዴ C/Fp, g = 0 ƷᲢǹǭȸȠƱƠƯƷᲣ ӷ׹᫏Ƹ π1(C) ƴǑƬƯܦμƴൿܭƞǕǔ ᲢgᲴɟᑍƸசᚐൿᲣ

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5. ஓஉƷˁʙ

Grothendieck ʖे /p ᡶ˳

ኽௐ

K Qp ɥஊᨂဃ঺Ƴ˳ƷᢿЎ˳

̊ K Q ɥஊᨂဃ঺

K Qp ɥஊᨂဃ঺

K =

[K:Q]≤NK

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ܭྸᲫ

CᲴӑ୺ႎˊૠ୺ዴ /K

S smooth ˊૠٶಮ˳ /K ƜƷ଺Ŵ

HomdomK (S, C)

Hom extopen

Gal(K)1(S), π1(C))

Hom extopen

Gal(K)1(p)(S), π1(p)(C)) ܭྸᲬ

L, MᲴᲢ˓ॖഏΨᲣˊૠٶಮ˳ /K Ʒ᧙ૠ˳

ƜƷ଺Ŵ

HomK(M, L)

Hom extopen

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

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(i) Ǣȸșȫٶಮ˳Ʒ Tate ʖेƸ p ᡶ˳ɥ ưƸ঺ᇌƠƳƍŵ

(ii) K ƕ Q ɥஊᨂဃ঺Ʒ଺ŴܭྸᲬƷ Isom

༿Ƹ Pop Ʒܭྸ

ᚰଢᲴ p Hodge ྸᛯƷࣖဇ Ტ[K : Qp] <

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Grothendieck ʖे

ٶಮ˳Ʒݧ ൔ᠋

←→ ؕஜ፭ᲢƷݧᲣ

ž᧙ૠſႎ

ˮႻႎ

´etale

Galois ᘙྵႎ p Hodge ྸᛯ

p ᡶ˳ɥƷˊૠٶಮ˳ƴݣƠƯ de Rham

(crystalline) cohomology

←→ൔ᠋ p ´etale cohomology

ࣇЎь፭ႎ ž᧙ૠſႎ

ˮႻႎ

´etale

Galois ᘙྵႎ

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ቇҥƷƨǊ C = C, non-hyperelliptic π1(p)(CK)ab Tp(JK)

J C Ʒ Jacobi ٶಮ˳Უ

Hodge-Tate ЎᚐᲢCp = (K)ᲣᲴ π1(p)(CK)ab

ZpCp {Γ(C, Ω1C/K)

KCp} ⊕ {H1(C, OC)

KCp(1)} ǏƑƴ

1(p)(CK)ab

ZpCp)Gal(K)

= Γ(C, Ω1C/K) = 1 ⊕ · · · ⊕ g canonical embedding C → PgK−1

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΂ƷࣄΨƷƨǊƷ Key

L 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 pN ༿ p Hodge ྸᛯǛ C ƷӲᘮᙴ ƴᢘဇŴƳƲᲣ

ܭྸᲬᲴ

M/K Ʒឬឭഏૠƴ᧙Ƣǔ࠙ኛඥ

ܭྸᲫƴ࠙ბ

31

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