ȢǸȥȩȸ࢟ࡸ 1 (2)(A.) ӑ୺׹ȪȸȞȳ᩿Ʒɟॖ҄ X: ӑ୺ႎ୺ዴ/C: ๖ǒƔưŴ properŴ ᡲኽƳᆔૠ g Ʒ ˊૠ୺ዴ − r ̾ƷໜŴ s.t

ˊૠ୺ዴƴ᧙Ƣǔ Grothendieck ʖे Ü p _{ᡶ࠹˴ƷᙻໜƔǒ}

ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

I. _λᧉ

A. ӑ୺׹ȪȸȞȳ᩿Ʒɟॖ҄

B. _{ૠᛯႎؕஜ፭} C. _ኽௐ

D. _{ȢǸȥȩȸ࢟ࡸ}

1

(A.) ӑ୺׹ȪȸȞȳ᩿Ʒɟॖ҄

X: _{ӑ୺ႎ୺ዴ}/C: _๖ǒƔưŴ

proper_{Ŵ ᡲኽƳᆔૠ} g _{Ʒ ˊૠ୺ዴ}

− r _̾ƷໜŴ s.t. 2g − 2 + r > 0 ^{ӑ୺ႎȪȸȞȳ᩿} X

K¨obe _{Ʒɟॖ҄ܭྸ}:

X ^{Ʒ୍ᢄᘮᙴ} X ∼ = ɥҞ࠯᩿ H

2

ᚕƍ੭ƑǕƹᲴ

žX ƱƍƏᲢኝᲣˊૠႎƳNjƷƕŴ

H/π₁(X ) ƱƍƏ࠹˴ႎƳƍƠᚐௌႎƳ ᘙᅆǛNjƭſ

ƭLJǓŴ

ˊૠ୺ዴ X ⇐⇒ π₁(X ) H ƜƜưŴ

ӫᡀᲷ π₁(X ) + ƋǔžૠᛯႎſƳȇȸǿ

↑ ↑

[_{ȢǸȥȩǤƴǑǒƣ}] + [_{ȢǸȥȩǤƴǑǔ}] ᲢƨƩƠŴƜƜưŴ

žૠᛯႎſᲷž໯ᨂእໜƴƓƚǔૠᛯſᲣ 3

(B.) _{ૠᛯႎؕஜ፭}

K: _{Ტ೅ૠᲪƷᲣ˳Ჵ} X_K: K _{ɥƷˊૠٶಮ˳}

ƢǔƱŴX_K → π₁(X_K)

...(compact _{ƳᲣиஊᨂ} (profinite) _፭Ŵ X_K ƷஊᨂഏǨǿȸȫᘮᙴǛወСƢǔŵ (_̊ƑƹᲴ K = C _ƷƱƖŴ

π₁^top(_{ݣࣖƢǔᙐእٶಮ˳} X ) Ʒиஊᨂܦͳ҄ŵᲣ

4

K _{ƕ᧍˳ưƳƍ଺Ŵ} 1 → π₁(X

K) → π₁(X_K) → Gal(K/K) → 1 ဇᛖᲴ

࠹˴ႎؕஜ፭ π₁(X

K)

ǬȭǢ፭ Γ_K ^def= Gal(K/K) ૠᛯႎؕஜ፭ π₁(X_K)

K=_೅ૠᲪ =⇒ ^{࠹˴ႎؕஜ፭Ƹ} X

K Ʒ ȢǸȥȩǤƴǑǒƣ

Ტ୺ዴƷƱƖŴ(g, r) _{ƩƚưൿLJǔᲣ} K=_೅ૠ p =⇒ ^{࠹˴ႎؕஜ፭Ƹ} X

K Ʒ ȢǸȥȩǤǛൿܭƠƕƪ

Ტྚ߷ᲴᆔૠᲷᲪƱƖᲣ

5

ˌɦưƸŴɥƷኒЗƷՠǛᎋƑǔᲴ LJƣŴእૠ p _Ǜ׍ܭƢǔŵ

Δ_X ^def= π₁(X_K) _Ʒஇٻ pro-p _ՠ Π_XK ^def= Δ_X/Ker(π₁(X

K) → Δ_X) ƢǔƱŴ

1 → Δ_X → Π_XK → Γ_K → 1

=⇒ Γ_K → Out(Δ_X) ^def= Aut(Δ_X)/Inn(Δ_X) žΓ_{K ƕ} Δ_{X ƴٳ˺ဇƢǔŵſ}

ᲢऴإƱƠƯŴٳ˺ဇ ⇐⇒ ^{ɥƷܦμኒЗŴ} ȢǸȥȩǤƴǑǔᲛᲣ

6

ƭLJǓŴ

X_K → {π₁(X

K) + Γ_K π₁(X

K) } ӍƸ {Δ_X + Γ_K Δ_X }

Ტ࢟Ƹ C ɥƷɟॖ҄ܭྸǛᡲेƞƤǔᲛᲣ ^{ᐯ໱ƳբƍੑƚᲴž}→^{ſƸᲢ˴ǒƔƷ}

ॖԛưᲣž↔^{ſưፗƖ੭ƑǔƜƱƸ} ưƖƳƍƔᲹ

ƜǕƕƍǘǏǔžGrothendieck _ʖेſ

ᲢX_K Ჷӑ୺ႎ୺ዴŴӍƸžᢒǢȸșȫſ (anabelian) _Ƴٶಮ˳Ŵ

K ᲷžҗЎƴૠᛯႎſƳ˳Უ 7

(C.) _ኽௐ

K_Ჴžэ p _ᡶ˳ſ⊆ Q_{p ɥஊᨂဃ঺Ƴ˳}

̊ K_ᲴQ _{ɥஊᨂဃ঺}

K_ᲴQ_{p ɥஊᨂဃ঺}

K = ∪

[K:Q]≤NK ܭྸᲫ

X_{KᲴӑ୺ႎˊૠ୺ዴ} /K

S_KᲴsmooth _ˊૠٶಮ˳ /K ƜƷƱƖŴ

X_K(S_K)^dom→^∼ Hom^open_Γ

K (π₁(S_K), π₁(X_K))

→∼ Hom^open_Γ

K (Π_SK, Π_XK ) 8

ܭྸᲬ

L, M_{Ჴ ᧙ૠ˳} /K _{Ტ˓ॖഏΨᲣ} ƜƷƱƖŴ

Hom_K(M, L)→^∼ Hom^open_Γ

K (Γ_L, Γ_M ) දᲴ

(i) ƋǔவˑǛ฼ƨƢžӑ୺ႎƳ୺᩿ſ ƴ᧙Ƣǔž୺᩿༿ſNjƋǔŵ

(ii) _{߼ᡀǛžˊૠ୺ዴ} X_{K ƷˊૠႎƳໜſŴ} ӫᡀǛžΠ_XK ƱƍƏᚐௌႎƳݣᝋƷໜſ ƱLjǕƹŴK¨obe _{ƷܭྸƱ࢟ƕ᫏˩ƠƯ} ƍǔŵܱᨥŴƪǐƏƲƜƷƜƱƴȒȳȈ

ǛࢽƯŴžp _{ᡶ࠹˴ſưᚰଢƢǔŵ}

9

(iii) ΨŷƸŴǢȸșȫٶಮ˳Ʒ Tate _ʖे

Ʒ᫏˩ =⇒ Tate _{ʖेƱӷಮŴٻ؏ႎƳ} ૠ˳ƷɥưƠƔ঺ᇌƠƳƍNjƷƱƠƯ ᎋƑǒǕƯƍƨŵƭLJǓŴ(ii) _ƷǑƏƳ

ᙻໜƸʖे࢘଺ƳƔƬƨǒƠƍŵƦƷ Ҿ׆ƷɟᲢᲹᲣᲴǢȸșȫٶಮ˳Ʒ

Tate _ʖेƸ p ᡶ˳ɥưƸ঺ᇌƠƳƍŵ (iv) ܭྸᲫˌЭƴŴ೅ૠᲪƷዌݣஊᨂဃ঺˳

ƷɥưƸŴ

· ɶ஭Ҧଯ൞ ᲢᆔૠᲪŴIsom _༿Უ

· ྚ߷ܤᬱဏ൞ ᲢǢȕǣȳŴIsom _༿Უ ƷኽௐƕƋǓŴཎƴŴɶ஭൞ƕஊјƴ

̅ƍŴྚ߷൞ƕஜឋႎƴોᑣƠƨžᘮᙴ ƷذǛኵጢႎƴМဇƢǔſ৖ඥƸܭྸᲫ

ƷᚰଢưNjဃƖƯƍǔŵ

(v) K _ƕ Q ɥஊᨂဃ঺Ʒ଺ŴܭྸᲬƷ Isom

༿Ƹ Pop _Ʒܭྸŵ 10

(D.) _{ȢǸȥȩȸ࢟ࡸ}

բ᫆: _{ɟ˳ƲƏǍƬƯ} {Δ_X + Γ_K Δ_X} Ɣǒˊૠ୺ዴ X_{K ǛࣄΨƢǔƔᲹ}

ȒȳȈᲴ C _{ɥƷئӳŴɥҞ࠯᩿} H ^ɥư

ȢǸȥȩȸ࢟ࡸǛᚐௌႎƴᙌᡯƠƯ ݧࢨᆰ᧓ǁƷ؈NJᡂLjǛನ঺Ƣǔŵ

Ტ̊Ჴ࣓஭ٶಮ˳Ʒನ঺, e.g.,

H/SL₂(Z), Poincar´e _ኢૠሁᲣ ɥҞ࠯᩿ H → ^ݧࢨᆰ᧓

↓

ˊૠ୺ዴ → ^ݧࢨᆰ᧓

11

. . . . . . . .

SL₂(Z) _Ʒئӳ

ȝǤȳȈᲴȢǸȥȩȸ࢟ࡸƷᚐௌႎƳᘙᅆ ǛᎋƑǔŵ

=⇒ ƜǕƱ᫏˩ႎƳƜƱǛžp _ᡶƷɭမſ ưǍǓƨƍŵ

=⇒ ^žp _ᡶ Hodge _{ྸᛯſ Ǜဇƍǔŵ} 12

Grothendieck _ʖे

ٶಮ˳Ʒݧ ൔ᠋

←→ ^{ؕஜ፭ᲢƷݧᲣ} žˊૠ࠹˴ſ

žٶ᪮ࡸ

=_᧙ૠſ

ž´etale _ˮႻ+ Galois _˺ဇſ

p _ᡶ Hodge _ྸᛯ

p ᡶ˳ɥƷˊૠٶಮ˳ƴݣƠƯ de Rham

(crystalline) cohomology

ൔ᠋←→ p _ᡶ ´etale cohomology žٶ᪮ࡸ

=_᧙ૠŴ ƦƷࣇЎſ

ž´etale _ˮႻ+ Galois _˺ဇſ

13

ቇҥƷƨNJŴ [K : Q_p] < ∞,

X_K = X_K, non-hyperelliptic ƱˎܭƢǔŵ

=⇒ Δ^ab_X T_p(J_X

K ) ᲢJ_X

KᲴX

K Ʒ Jacobi _ٶಮ˳Უ Hodge-Tate _ЎᚐᲢC_p = K _ᲣᲴ

Δ^ab_X ⊗

Zp

C_p ∼= {D_X⊗

K

C_p} ⊕ {D_X^∨ ⊗

K

C_p(1)} ƨƩƠŴD_X ^def= H⁰(X_K, ω_XK/K)

ƳƓŴnon-hyperelliptic =⇒

X_K → P_X ^def= P(D_X) 14

ࢼƬƯŴX_{K ƱӷơˎܭǛ฼ƨƢ} Y_{K ƱŴ} Γ_{K Ʒٳ˺ဇƱɲᇌƢǔ}

Δ_X ∼= Δ_Y ƕɨƑǒǕƨǒŴ

=⇒ Δ^ab_X ∼= Δ^ab_Y

=⇒ D_X = (Δ^ab_X ⊗ K )^ΓK

∼= (Δ^ab_Y ⊗ K )^ΓK = D_Y

⇓

P(D_X) = P_X ∼= P_Y = P(D_Y ) X_K ^? Y_K

15

P_X ∼= P_Y X_K ^? Y_K

բ᫆Ჴ X_K, Y_K Ǜܭ፯ƢǔᲢٶ᪮ࡸ׹Უ ᧙̞ࡸƕ̬ƨǕǔƔƲƏƔŵ

ž᧙̞ࡸƷ̬܍ſ

D_X = H⁰(X_K, ω_XK/K) _{ƷΨƨƪŴƭLJǓŴ} ȢǸȥȩȸ࢟ࡸƷᲢp _{ᡶᲣᚐௌႎᘙᅆ}

Ǜ̅ƬƯᚰଢƢǔŵ

16

ˊૠ୺ዴƴ᧙Ƣǔ Grothendieck ʖे Ü p _{ᡶ࠹˴ƷᙻໜƔǒ}

ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

II. p _ᡶ Hodge _ྸᛯ

A. _{ݧࢨᆰ᧓Ʒ᧓Ʒݧ}

B. ɥҞ࠯᩿ƷˊஆཋƷႇئ C. Faltings _Ʒྸᛯ

D. J-_࠹˴ࣱ

E. Malˇcev _ܦͳ҄Ʒ p _ᡶ Hodge _ྸᛯ

1

(A.) _{ݧࢨᆰ᧓Ʒ᧓Ʒݧ} ЭׅᲴ [K : Q_p] < ∞;

ӑ୺ႎ୺ዴ X_K, Y_KᲵ

Γ_K Δ_X ∼= Δ_Y ƔǒЈႆƠƯ

⇓

D_X = (Δ^ab_X ⊗ K )^ΓK ∼= (Δ^ab_Y ⊗ K )^ΓK = D_Y ƨƩƠŴD_X ^def= H⁰(X_K, ω_XK )

D_Y ^def= H⁰(Y_K, ω_YK )

⇓

P(D_X) = P_X ∼= P_Y = P(D_Y ) X_K ^? Y_K

2

ž᧙̞ࡸƷ̬܍ſ

⇐⇒ ∀i ≥ 1, _ӳ঺

R_i ⊆ ⊗ⁱ D_X →^∼ ⊗ⁱ D_Y → D_Yⁱ ƸŴᲪƔᲹ

ƨƩƠŴ

D_X^{i def}= H⁰(X_K, ω_X^⊗i

K)) D_Y^{i def}= H⁰(Y_K, ω_Y^⊗i

K)) R_i ^def= Ker(⊗ⁱ D_X → D_Xⁱ )

↑

ž᧙̞ࡸſ 3

(B.) ɥҞ࠯᩿ƷˊஆཋƷႇئ

∃ stable Y → Spec(O_K) s.t.

Y ⊗_OK K = Y_{K ƱˎܭƠǑƏŵ} ƳƓŴ℘ ∈ Y ⊗_OK k _ƕŴ

special fiber _ɶ generic _{ƩƱˎܭƢǔŵ}

⇓

O_L ^def= (O_Y^unram_,℘ )^∧; Ω_L ^def= {L _{ƷᡲዓƳࣇЎ} }

=⇒ dim_L(Ω_L) = 1;

∃ ^{ӷ፯ӒࣄႎƳ} ξ_Y : Spec(L) → Y; ž᧙̞ࡸƷ̬܍ſ ⇐⇒

R_i → D_Y^{i ξ}→^Y Ω^⊗i_{L ƕᲪƴƳǔ} 4

ƭLJǓŴ

⊗i D_X → Ω^⊗i_L

ǛᚘምƠƯŴƦƷ R_i ⊆ ⊗ⁱ D_{X ǁƷСᨂƕ} ᲪƴƳǔƜƱǛᚕƍƨƍŵ

දᲴ

(i) D_Y → Ω_L Ƹ žȢǸȥȩȸ࢟ࡸƷ ᚐௌႎޒ᧏ſƷǑƏƳNjƷŵ

⇓

Spf(O_L) ƸɥҞ࠯᩿ƷࢫǛ๫ơƯƍǔŵ (ii) ഏራưƸŴƜƷޒ᧏Ǜžπ_{1ſƷᚕᓶƴ}

ᎇᚪƠƯŴD_X → Ω_{L ǛᚘምƢǔŵ}

5

Spf(O_L) → Y (iii) _ܱᨥŴ

O_L ∼= {(Z_p[t])^∧₍ ^{, unram}

p) }^∧

ཎƴŴSpf(O_L) _ƸŴY_{KᲢƷȢǸȥȩǤᲣ} ƴǑǒƳƍŴ࠹˴ႎഏΨᲷᲫƷ࠹˴ႎ

ݣᝋŵƭLJǓŴSpf(O_L) _Ƹஜ࢘ƴ Y_{K Ǜ} žɟॖ҄ſƠƯƍǔƷưƋǔŵ

ᚕƍ੭ƑǕƹŴžSpf(O_L)_{ſƷನ঺ƸŴ} Y_{K ƔǒЈႆƠǑƏƱŴ}X_{K ƔǒЈႆ}

ƠǑƏƱŴ٭ǘǒƳƍƷưŴX_{K Ʊ} Y_K Ǜൔ᠋ƢǔƷƴžᢘ˓ſưƋǔŵ

6

(C.) Faltings _Ʒྸᛯ

Γ_L/K ^def= Ker(Γ_L → Γ_K) = Γ_L·K

=⇒ Faltings, _{τ᪽ ඙൞Ჴ}

∃ ^ᐯ໱ưŴΓ_K-_ӷ٭Ƴӷ࢟

H¹(Γ_L/K, L(1)) ∼= Ω_L⊗_KK t^1/p∞ ↔ (dt/t) ɲᡀᲴ “log”_ƬdžƍNjƷ · L ⊗_K K žޅ৑ႎƳ Hodge-Tate _ЎᚐſᲴ

ࢌǓӳǘƤƨǒ =⇒

୍ᡫƷ Hodge-Tate _Ўᚐ

7

ƜǕưŴࣇЎ࢟ࡸƷޒ᧏ϙ΂ D_Y → Ω_L Ǜܦμƴ π_{1 ƷᚕᓶƴᎇᚪưƖǔ}:

H¹(Δ_Y , K (1))^ΓK → H¹(Γ_L/K, L(1))

↑

D_Y → Ω_L⊗_KK

ƜƷ׋ࡸǛഏƷ׋ࡸƷӫƴƘƬƭƚǔƱŴ

H¹(Δ_X, K (1))^ΓK →^∼ H¹(Δ_Y , K (1))^ΓK

D_X →^∼ D_Y

(_{ɦƷᘍƷӳ঺})^⊗i _{ᲷᚘምƠƨƍNjƷ} ɥƷᘍƷӳ঺Ჷ π_{1 ƷɭမƷNjƷ}

8

ƞƯŴϙ΂

α_Y ^def= π₁(ξ_Y ) : Γ_L → Π_YK ƱŴɨƑǒǕƨӷ࢟

Π_YX ∼= Π_XK

Ʒӳ঺Ǜ α_X : Γ_L → Π_{XK Ʊ୿ƘƱŴ}

(∗^geom) α_{X ƕŴ}

∃ ξ_X : Spec(L) → X_{K Ɣǒဃơǔŵ}

ƭLJǓŴα_{X ƕž࠹˴ႎſƴƳǔƔƲƏƔǛ} բƏƜƱƕưƖǔŵ

9

NjƠ (∗^geom) ƕ঺ᇌƢǕƹŴέDŽƲƷ žӳ঺ӧ੭׋ࡸſ

H¹(Δ_X, K (1))^ΓK → H¹(Γ_L/K, L(1))

↑

D_X → Ω_L⊗_KK

ƷɦƷᘍƕŴᲢFaltings _{ƷྸᛯƷ᧙৖ࣱǑǓᲣ} ξ_X ƔǒဃơǔƜƱƴƳǔŵ

=⇒ ξ_X^∗ (R_i) = 0 ұƪŴ

(∗^geom) _঺ᇌ =⇒ ^{ž᧙̞ࡸƷ̬܍ſ঺ᇌ} ࢼƬƯŴ(∗^geom) _{ǛᅆƤƹŴҗЎŵ}

10

(D.) J-_࠹˴ࣱ

(∗^geom) ⇐⇒ {α_{X Ʒ࠹˴ࣱ} } ƸᩊƠᢅƗǔ

⇓

ӳ঺ α^J_X : Γ_L → Π_XK → Π

J_XK⁽¹⁾

ǛᎋƑǔŵ ƨƩƠŴJ_X⁽¹⁾

K Ƹ X_{K Ʒ} Albanese_Ჴ 1 → Δ^ab_X → Π

J_XK⁽¹⁾ → Γ_K → 1 α^J_{X Ʒ࠹˴ࣱᲢᲷ} ? ∈ J_X⁽¹⁾

K(L) _{ƔǒဃơǔᲣ} ǛŴα_{X Ʒ} J-࠹˴ࣱƱԠƿŵஜራƱഏራư ƸŴα_{X Ʒ} J-࠹˴ࣱƴƭƍƯᛟଢƢǔŵ

11

แӷ࢟ α^J_{X Ʒ࠹˴ࣱ}:

(i) _{࠹˴ႎƳᲢᲷ} ? ∈ Y_K(K) _{ƔǒဃơǔᲣ} β_Y : Γ_K → Π_YK

ƴݣƠƯŴ

β_X^J : Γ_K → Π_YK ∼= Π_XK → Π

J_XK⁽¹⁾

Ʒ࠹˴ࣱǛᅆƢŵ ᲢഏራưᛇᡓŵᲣ (ii) _ࠀ “α^J_X − β_X^J ”

∈ Im(H¹(Γ_L, Δ^ab_Y )) ⊆ H¹(Γ_L, Δ^ab_X ) Ʒ࠹˴ࣱǛᅆƢŵᲢஜራưᛟଢŵᲣ

12

(ii) _ƷᚰଢᲴ

Tate _Ʒܭྸ =⇒ Δ^ab_X ∼= Δ^ab_Y Ƹ

Formal gp.(J_XK) ∼= Formal gp.(J_YK ) ƔǒဃơǔŵࢼƬƯŴ

ࠀ “α^J_Y − β_Y^J ” ∈ H¹(Γ_L, Δ^ab_Y )

Ƹܭ፯ǑǓ࠹˴ႎ =⇒ ^{ƜƷࠀǛŴɥƷӷ࢟}

ưᡛƬƯNjŴЈƯƘǔΨ

Ჷ ࠀ “α^J_X − β_X^J ” Ƹ̔໱ƱƠƯ࠹˴ႎŵᚰଢኳŵ

13

(E.) Malˇcev _ܦͳ҄Ʒ p _ᡶ Hodge _ྸᛯ แӷ࢟ β_Y : Γ_K → Π_{YK Ƹ}

1 → Δ_X → Π_XK → Γ_K → 1 ƷЎᘷǛܭ፯ =⇒ Γ_K ^βY Δ_Y

Ტٳ˺ဇưƸƳƘŴஜཋƷ˺ဇŵᲣ

⇓

Δ_{Y ƷᲢʚഏƷᲣ}K _ɥƷ Malˇcev _ܦͳ҄Ʒ ž᣻ƞᲪſƷՠᲢ⇐⇒ (1), (2) _ሁŴnonzero

Tate twist ƕƳƍஇٻƷNjƷᲣƕܭ፯ƞǕ ǔᲴ

0 → ∧² D_Y ⊗_K K → Z_Y →

D_Y ⊗_K K → 0 14

Z_Y ^Ჴ K _ɥƷ Lie _࿢ŴΓ_K-_ь፭ŵ

ƱƜǖƕŴBloch-_{ьᕲƷྸᛯᲥƋǔᚘም}

⇓

{β_Y^J _Ʒ J-_࠹˴ࣱ } ⇐⇒ { Z_Y ^{ƕ ЎᘷƢǔ} } ᲢƨƩƠŴӫᡀᲷ

žɥƷ Γ_K-ь፭ƷܦμኒЗƕЎᘷƢǔſᲣ ɟ૾Ŵܭ፯ǑǓŴ

{^βY Δ_Y } ∼= {^βX Δ_X}

=⇒ Z_Y ^Γ∼=^K Z_X

15

ࢼƬƯŴɥƱኵLjӳǘƤǔƱŴ

{β_{Y Ʒ} J-_࠹˴ࣱ } ⇐⇒ {β_{X Ʒ} J-_࠹˴ࣱ } ǑƬƯŴ(i)_ᲢᲷ β_X^J _{Ʒ࠹˴ࣱᲣƕ঺ᇌŵ}

16

ˊૠ୺ዴƴ᧙Ƣǔ Grothendieck ʖे Ü p _{ᡶ࠹˴ƷᙻໜƔǒ}

ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

III. _{ஊྸໜƷನ঺}

A. J-_࠹˴ࣱƱ Chern _᫏ B. _ӓளƷᚨܭ

C. _{ӓளƷᚰଢ} D. _{ɼܭྸƷᚰଢ}

1

(A.) J-_࠹˴ࣱƱ Chern _᫏ ЭׅᲴ ࠹˴ႎƳ

α_Y ^def= π₁(ξ_Y ) : Γ_L → Π_YK

ƱŴɨƑǒǕƨӷ࢟ Π_YX ∼= Π_XK Ɣǒ ӳ঺α_X : Γ_L → Π_{XKǛ˺ǓŴƞǒƴ}

α^J_X : Γ_L → Π_XK → Π

J_XK⁽¹⁾

Ʒ࠹˴ࣱǛᅆƠƨŵࢼƬƯŴӳ঺

Π_XL ^(id−→^,αX) Π_XL×LXL −→ Π

XL×LJ_XL⁽¹⁾

Nj࠹˴ႎưƋǔŵ ᲢදᲴ “[π₁, _Ⴚᆢ] = 0”_Უ 2

c₁(diagonal) ∈ H²(Π_XL×LXL, Z_p(1)) ǛᎋƑǑƏŵИሁႎˊૠ࠹˴ƔǒŴƜƷ᫏

∈ Q · Im{c₁(M) ∈ H²(Π

XL×LJ_XL⁽¹⁾ , Z_p(1))}

=⇒ ᲢɥƷӳ঺Ʒ࠹˴ࣱ Ქ

Kummer exact sequence _ǑǓᲣ η_X = c₁(^∃L) + torsion

ƨƩƠŴ

η_X ^def= (id, α_X)^∗c₁(diagonal) Ƹžʴ߻ႎƳ c₁(O_XL(ξ_X))_ſŵ

දᲴ ஜ࢘Ʒξ_XƷ܍נƸLJƩЎƔǒƳƍŵ

3

=⇒ ^Ტždeg(L) _Ƹᐯѣႎƴ 1 _{ƴƳǔſǑǓᲣ} X_Lɥƴ deg=1 _Ʒ line bundle _{ƕ܍נƢǔᲛ}

=⇒ L^⊗m = O_XL(D) ƨƩƠŴ

m=_ٻŴp _Ʊእ D =

Spec(L_i); [L_i : L] < ∞

=⇒ ([^∃L_i : L], p) = 1

=⇒ L_i/L tamely ramified!

ኽᛯᲴY_K(L)^᩼ᡚ҄ = ∅ =⇒ X_K(L^tm) = ∅

4

(B.) _ӓளƷᚨܭ

แӷ࢟α_YL : Γ_L → Π_{YLƔǒЈႆƠƯŴ}

∀ i ≥ 0 ƴݣƠƯŴᢿЎ፭Ǜܭ፯Ჴ

Δ_Y ^{i def}= (Image(α_YL)) · Δ^<i>_Y ⊆ Π_YL ƨƩƠŴ

Δ^{<0> def}= Δ

Δ^{<i+1> def}= (Δ^<i>)^p · [Δ^<i>, Δ^<i>]

⇓

࠹˴ႎƴᡲኽᲢƳƥƳǒŴΔ_Y ⁱ Γ_LᲣƳ ஊᨂഏ´etale _{ᘮᙴƷذƕႆဃᲴ}

. . . → Y_Lⁱ⁺¹ → Y_Lⁱ → . . . → Y_L

5

ӷಮƴŴα_XL : Γ_L → Π_{XLƴݣƠƯNjŴ} ᘮᙴƷذƕܭ፯ƞǕǔᲴ

. . . → X_Lⁱ⁺¹ → X_Lⁱ → . . . → X_L ƱƜǖƕŴα_Y = π₁(ξ_Y ) _Ƹ࠹˴ႎŴ

Image(α_YL : Γ_L → Π_YL) ⊆ Δ_Y ⁱ

⇓

ξ_Y ∈ Y_L(L) _Ƹᐯ໱Ƴξ_Yⁱ ∈ Y_Lⁱ(L) _ƴਤƪ ɥƕǓŴ(A.) ƷžኽᛯſǛᢘဇƢǔƱŴ

⇓

∀ i ≥ 0, X_Lⁱ (L^tm) = ∅

6

ӓளܭྸᲴƜƷǑƏƳཞඞƷɦŴ∀i ≥ 0, ξ_Xⁱ _i ∈ X_Lⁱ (L^tm); ξ_X^{i def}= Image_X(ξ_Xⁱ _i)

⇓

ξ_X^{i p}−→^{ᡶ ∃!}ξ_X ∈ X_L(L) ⊆ X_L((L^tm)^∧) s.t. α_X = π₁(ξ_X).

ද:

(i) ƜƷܭྸƷᚰଢƸഏራưኰʼƢǔŵ (ii) _ƜǕưŴ(∗^geom) _{ƷᚰଢƸܦኽƢǔŵ}

ࢼƬƯŴܭྸᲫƷᚰଢNjܦኽƢǔƕŴ ᧈƘƯᙐᩃƳƷưŴ(D.) _{ưࣄ፼Ƣǔŵ}

7

(iii) ƜƷǑƏƴᘮᙴƷذǛኵጢႎƴМဇƢǔ ƱƍƏ৖ඥƸŴAnderson-_˙ҾǍɶ஭

ƷˁʙƴLJưᢓǓŴƜƜưƸŴྚ߷Ʒ ᚰଢƴǍǍᡈƍ࢟ưੲဇƠƯƍǔŵ ƨƩƠŴྚ߷ƷئӳŴؕᄽ˳Ƹஊᨂ˳

ƩƬƨƨNJŴӓளƸᐯѣႎŵƭLJǓŴ p _ᡶ Hodge ྸᛯƷǑƏƳᩊƠƍྸᛯƷ щǛ͈ǓƯɥƷžӓளܭྸſƷǑƏƳ NjƷǛᚰଢƢǔ࣏ᙲƸƳƔƬƨŵ

8

(C.) _{ӓளƷᚰଢ}

ഏƷӧ੭׋ࡸƴදॖƠǑƏᲴ

“Δ_Xⁱ”

Γ_L ^π1(ξ

Xii )

−→ Π_Xⁱ

L −→ Γ_L

“α_X” ↓ Γ_L ^π1(ξ

Xi )

−→ Π_XL −→ Π_XL/Δ^<i>_X ᲢƨƩƠŴɥƷᘍƷӳ঺Ƹ id_{ΓL ưƋǔŵᲣ}

ƭLJǓŴmodulo Δ^<i>_X ,

“α_X” ≡ π₁(ξ_Xⁱ ) : Γ_L → Π_XL 9

NjƠŴˎƴ i = ∞ ^{ƩƬƨƱƢǔƱŴ}Fal- tings _{ƷྸᛯƷ᧙৖ࣱŴұƪ} (II.) (C.) _Ʒ ӧ੭׋ࡸ

H¹(Δ_X, K (1)) ^π1(ξ

Xi )^∗

−→ H¹(Γ_L/K, L(1))

D_X ⊗_K K ^dξ

Xi

−→ Ω_L⊗_KK ƷɥƷᘍƕܦμƴൿLJƬƯƠLJƍ

⇓

ξ_X^i=∞_Ʒ P_XƴƓƚǔݧࢨࡈ೅ᲢᲷɦƷᘍᲣ NjŴܦμƴൿLJǔƜƱƴƳǔŵ

10

ƭLJǓŴξ_X^i=∞ ∈ X_L(L) ⊆ P_X (L) _ᐯ៲ NjܦμƴൿLJǔŵ

ѸᛯŴܱᨥƴƸŴi = ∞ ^ƩƕŴFaltings _Ʒ

ྸᛯƷžp _{ᡶႎᡲዓࣱſƔǒŴ}∀i ≥ 0,

“α_X” ≡ π₁(ξ_Xⁱ ) : Γ_L → Π_XL modulo Δ^<i>_X

⇓

ξ_Xⁱ _{Ʒݧࢨࡈ೅NjŴ}mod p^i−∃c_ưൿLJǔŵ ƱƜǖƕŴX_K ⊆ P_X =⇒ ^ƜǕưໜЗ

{ξ_Xⁱ } ^ƕ p ᡶႎƴӓளƢǔƜƱƕЎƔǔŵ

ಊᨂƷɟॖࣱǍࣱឋNjŴӷಮƳᜭᛯǑǓ ࢼƏŵ

11

(D.) _{ɼܭྸƷᚰଢ} ܭྸᲫ

K:_žэ p _ᡶ˳ſ⊆ Q_{pɥஊᨂဃ঺Ƴ˳}

X_{KᲴӑ୺ႎˊૠ୺ዴ}/K

S_KᲴsmooth _ˊૠٶಮ˳/K ƜƷƱƖŴ

X_K(S_K)^dom→^∼ Hom^open_Γ

K (π₁(S_K), π₁(X_K))

→∼ Hom^open_Γ

K (Π_SK, Π_XK )

ᚰଢᲴ [K : Q_p] < ∞ ^{ƷئӳƴƢƙ࠙ბŵ}

=⇒ p _ᡶ Hodge _{ྸᛯ̅ဇӧŵ} ƳƓŴቇҥƷƨNJŴX_K, S_K = Y_Kƕ proper, non-hyperelliptic _{Ƴӑ୺ႎ୺ዴ} ưƋǔƱˎܭƠǑƏŵ

12

ቇҥƷƨNJŴ˓ॖƷ open _{Ƴแӷ࢟ưƸƳƘ} ӷ࢟ Γ_K Δ_X ∼= Δ_Y ƔǒЈႆƠƯŴ

D_X = (Δ^ab_X ⊗ K )^ΓK ∼= (Δ^ab_Y ⊗ K )^ΓK = D_Y ƨƩƠŴD_X ^def= H⁰(X_K, ω_XK )

D_Y ^def= H⁰(Y_K, ω_YK )

⇓

P(D_X) = P_X ∼= P_Y = P(D_Y ) X_K ^? Y_K

ž᧙̞ࡸƷ̬܍ſ

13

୺ዴ Y_KƷ stable model Y → Spec(O_K) ƴݣƠƯŴӷ፯ӒࣄႎƳžໜſ

Spec(L) → Y_K Ǜ˺ǔŵƨƩƠŴ

L ^def= (O_Y^unram_,℘ )^∧_Ʒՠ˳

℘ = Y^Ʒ special fiber _Ʒ^∃ generic point ƢǔƱŴžπ₁Ʒ᧙৖ࣱſǑǓŴแӷ࢟

α_Y : Γ_L → Π_YK

ǍŴα_Y ƱɨƑǒǕƨӷ࢟ƱƷӳ঺ƕܭ፯ ƞǕǔᲴ

α_X : Γ_L → Π_XK 14

ƠƔNjŴž᧙̞ࡸƷ̬܍ſǛᚕƏƴƸŴ α_XƷ࠹˴ࣱ

ᲢᲷ? ∈ X_K(L) ƔǒဃơǔᲣǛᚕƑǕƹ Ǒƍŵ

ƱƜǖƕŴα_XƷ࠹˴ࣱǛႺ੗ᚰଢƢǔ ƜƱƸᩊƠᢅƗǔŵ ࢼƬƯŴ

α^J_X : Γ_L → Π

J_XK⁽¹⁾ Ʒ࠹˴ࣱǛŴ

Bloch-ьᕲƷྸᛯሁǛ̅ƬƯᚰଢƢǔŵ

⇓

Chern _{᫏ƷᜭᛯǑǓŴ}X_L(L^tm) = ∅

15

இࢸƴŴα_XǛ̅ƬƯŴX_{LƷᘮᙴƷذ}

Ǜ˺ǓŴɥƷ tame ஊྸໜƷ܍נƔǒŴƜƷ ذƷӲᘮᙴƕ tame ஊྸໜǛਤƭƜƱǛ࠙ኽ Ƣǔŵ

⇓

ƦƷໜƨƪƕžξ_Xſƴ p _{ᡶӓளƠŴƠƔNjŴ} α_X = π₁(ξ_X)

⇓

α_{X Ƹ࠹˴ႎŵ}

ᚰଢኳŵ 16