The Local Pro-p Grothendieck Conjecture ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

Lecture Notes _Ჴ

The Local Pro-p Grothendieck Conjecture ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

I _{ᲨɼܭྸƷኰʼᲴ}

Theorem : Let p be a prime number. Let K be a finite extension of Qp. Let X_K → Spec(K) and X_K → Spec(K) be smooth, proper, geometrically connected curves over K of genus ≥ 2. Let Δ⁽_X^p) (respectively, Δ⁽_X^p)) be the pro-p completion of the geometric fundamental group of X_K (respectively, X_K ). Then the natural map

Isom_K(X_K, X_K )→Out_ρ(Δ⁽_X^p),Δ⁽_X^p))

defined by “looking at the induced morphism on fundamental groups” is bijective.

ᚡӭᲴ ߼ᡀƷIsom_K(X_K, X_K ) ƸŴX_K ƱX_K ƷKɥƷӷ׹ƢǂƯƔǒƳǔᨼӳŵ ɟ૾Ŵ ӫᡀƷOut(Δ⁽_X^p),Δ⁽_X^p)) ƸŴδ : Δ⁽_X^p) ∼= Δ⁽_X^p) ƳǔΔ⁽_X^p)ƱΔ⁽_X^p) ƷˮႻ፭ƱƠƯƷӷ׹ƨƪƷ ᲢδƱϋᢿᐯࠁӷ׹ƱƷӳ঺ǛŴδƱӷɟᙻƢǔƜƱƴǑƬƯࢽǒǕǔᲣӷ͌᫏ƷᨼӳưŴɦ

˄ƖƷρƸŴžK ƷዌݣGalois፭Ʒٳᢿ˺ဇƱɲᇌƢǔǑƏƳNjƷſǛॖԛƢǔŵ

Remarks:

(1.) ᲢݦᧉٳƷ૾ƷƨNJƴᲣƜǜƳǿǤȗƷኽௐǛᡫࠝžGrothendieckʖेſƱƍƏŵƲǜ Ƴज़ơƷኽௐƳƷƔƱᚕƏƱŴӑ୺׹Ʒˊૠ୺ዴƕɨƑǒǕƨǒŴƦƷ࠹˴ႎƳᘮᙴǛᎋƑǔ ƜƱƕưƖŴƦƷᘮᙴᐯ˳NjˊૠႎᲢƭLJǓŴٶ᪮ࡸưܭ፯ưƖǔNjƷᲣƳƷưŴܭ፯૾ᆉࡸ

Ʒ̞ૠǁƷKƷዌݣGalois፭ᲢˌɦŴΓ_K Ʊ୿ƘᲣƷ˺ဇƔǒŴᘮᙴǁƷ Γ_K Ʒ˺ဇƕࡽƖ ឪƜƞǕŴࢼƬƯŴɨƑǒǕƨˊૠ୺ዴƷ࠹˴ႎؕஜ፭ƷиஊᨂᲢLJƨƸŴи pᲣܦͳ҄ǁƷ Γ_K Ʒ˺ဇƕưƖǔŵ ƦƜưŴG.C.ᲢᲷGrothendieck ConjectureᲣƕɼࢌƠƯƍǔƷƸŴ ƦƏƍƏኵӳƤᛯႎƳŴ፭ᛯႎƳऴإᲢұƪŴ࠹˴ႎؕஜ፭Ʒ˴ǒƔƷܦͳ҄ᲥƦƜǁƷGa- loisƷ˺ဇᲣƔǒŴɨƑǒǕƨˊૠ୺ዴᐯ˳ƕࣄΨưƖǔŴƱƍƏƜƱưƢŵ

(2.) X_K ǍX_K ƕproperưƸƳƘŴǢȕǣȳᲢƨƩƠŴ̔໱Ŵӑ୺׹ưƳƍƱƍƚƳƍᲣư ƋƬƯNjƍƍversionǍ᧙ૠ˳versionNjஊǔŵ

(3.) ྚ߷ƞǜƕᚰଢƠƨૠ˳ɥƷиஊᨂǢȕǣȳ G.C.ᲢᲷGrothendieck ConjectureᲣƱൔ

᠋ƠƨئӳŴǢȕǣȳƔǒproperƴᘍƚƨƱƔŴƋǔƍƸŴиஊᨂƔǒиpƴᘍƚƨƱƍƬƨ

ໜƕਫƛǒǕǔƜƱƸٶƍưƢƕŴܱƸŴljƠǖŴžૠ˳Ɣǒޅ৑˳ƴᘍƚƨſƱƍƏƜƱƷ

૾ƕǑƬdžƲ᣻ᙲưஜឋႎưƔƭٻƖƳᡶഩưƋǔƱ࣬ǘǕǔŵ ƭLJǓŴݲƳƘƱNj೅ૠᲪư ƸŴƜƏƍƏG.C.׹ƷኽௐƸŴ ஜឋႎƴƸ ૠ˳ɥٻ؏ႎƳჇƴžૠᛯႎſƳྵᝋưƸƳƘŴ ljƠǖŴኝƴޅ৑ႎƳpᡶᚐௌႎƳྵᝋƩƱ࣬Əŵ ưŴƦǕǛᇌᚰƢǔƔƷǑƏƴŴˌɦኰʼ ƢǔpᡶႎƳǢȗȭȸȁưᚰଢƢǔƱŴproperǍи pƱƍƬƨᲢྚ߷ƞǜƷኽௐƱൔǂƯᲣ ǑǓࢍƍࣱឋƸƝƘᐯ໱ƴЈƯƘǔŵ

(4.) G.C.ƕLJƩܦμƴʖेƩƬƨ᪭ƴƸŴǢȸșȫٶಮ˳ƷTateʖेƷžᢒǢȸșȫ༿ſƱ

ǑƘƍǘǕƨƕŴTateʖेƸޅ৑˳ƷɥưƸμ໱঺ǓᇌƨƳƍஜឋႎƴૠ˳ɥٻ؏ႎƳኽௐư ƋǔƷƴݣƠƯŴᲢᲭᲣưNjਦઇƠƨǑƏƴŴG.C.ƸpᡶᚐௌႎƳʙܱƱƠƯਵƑƨ૾ƕǑ Ǔᐯ໱ưƋǔŵ

(5.) ૠ˳ɥƷG.C.Ʊޅ৑˳ɥƷG.C.ƷŴኽௐƱƠƯƷࢍƞƷ఍ࠀǛᇢႎƴᅆƢNjƷƱƠƯŴ ഏƷǑƏƳᎋݑƕưƖǔŵૠ˳ɥƷˊૠ୺ዴƕɨƑǒǕƨǒŴ୺ዴƷӷ׹᫏ƴƸŴNjƱNjƱӧ ም̾ƷӧᏡࣱƠƔƳƍƠŴƠƔNjŴNjƠƦƷ୺ዴƷTateь፭ǛჷƬƯƍǕƹŴᲢFaltingsƷஊ ӸƳܭྸǛᢘဇƢǕƹƢƙЎƔǔǑƏƴᲣஊᨂ̾ƷӧᏡࣱƠƔƳƍƷưƢŵƦǕƴݣƠƯŴޅ

৑˳ɥƷˊૠ୺ዴƕɨƑǒǕƨǒŴNjƱNjƱ᩼ӧም̾ƷӧᏡࣱƕƋǔƠŴƠƔNjŴ̊ƑŴƦƷ Tateь፭ǛჷƬƯƍǔƱƠƯNjŴɟᑍƴƸŴLJƩ᩼ӧም̾ƷӧᏡࣱƕƋǔƷưƢŵ

II _Შ p _ᡶ Hodge ྸᛯư୺ዴǛࣄΨƢǔᲴ

LƸ(KǛԃlj)pᡶ˳ᲢᲷฆ೅ૠưpᡶܦͳƳᩉ૝ႎ˄͌࿢OLƷՠ˳ᲣƱƠŴƦƷй˷

˳ kLƕŴK Ʒй˷˳ kɥƷɟഏΨ᧙ૠ˳ƩƱˎܭƢǔŵXK Ʒૠᛯႎؕஜ፭ƓǑƼ࠹˴ႎ

ؕஜ፭ƸƦǕƧǕ Π_XŴΔ_X ƱƠŴΔ_X Ʒ pro-pܦͳ҄Ǜ Δ⁽_X^p)Ʊ୿ƘŵƞǒƴŴΠ_X ǛŴ Δ_X → Δ⁽_X^p)ƷkernelưлƬƨNjƷǛ Π⁽_X^p)

K Ʊ୿ƘŵƦƏƢǔƱŴӲݧφ: Spec(L) → X_K ƴ ݣƠƯŴƦǕƧǕƷૠᛯႎؕஜ፭Ʒ᧓ƴᛔݰƞǕǔᡲዓƳ፭แӷ׹αφ : ΓL → Π⁽_X^p)

K Ǜݣࣖƞ

ƤǔƜƱƕưƖǔŵ ƦƏƍƏφƔǒƖƨแӷ׹Γ_L →Π⁽_X^p)

K ǛŴˌɦž࠹˴ႎſƱԠƿŵ

ƜƜưŴFaltingsƷpᡶHodgeྸᛯǛݰλƠƨƍƷưƢƕŴ࣏ᙲƳNjƷƸμᢿࣄ፼ƠLJ

ƢƷưŴКƴƝ܍ჷƳƘƯNj८ƯǔƜƱƸƋǓLJƤǜŵLJƣŴƦƷྸᛯƴǑǔƱŴഏƷǑƏƳ ᐯ໱Ƴӷ׹ƕƋǔᲴ

H¹(Γ_L/K, L^∧(1))∼= ΩL⊗^K K^∧

ᲢƨƩƠŴΓ_L/K = Ker(Γ_L →Γ_K)Ŵ “(1)”Ƹ Tate twistưŴ “∼=”ƱƍƏƷƸŴܱƸ Falt- ingsƷƍƏ“almost isomorphism”ƳƷưƢŵᲣ ƦƠƯŴ୺ዴXKǛŴƦƷȤdzȓǢȳJX Ʒ ɶƴ؈NJᡂLjŴSpec(L)→X_K →J_X Ƴǔӳ঺ǛƱǔƜƱƕưƖǔŵƱƜǖƕŴ pᡶHodge

ྸᛯƷ࠙ኽƷɟƭƱƠƯŴJX ƕ ஜ࢘ƸK ɥƷǢȸșȫٶಮ˳ưƋǔƴNjƔƔǘǒƣŴᲢݲ ƳƘƱNjŴ঻ŷƕჷǓƨƍƜƱƴ᧙ƠƯƸᲣƋƨƔNjG^gmƴӷ׹ưƋǔƔƷǑƏƴਰᑈƏƷư ƢŵƠƨƕƬƯŴƦƷG^gmɥƷ୍ᡫƷࡈ೅tiᲢƜƜưŴi= 1, . . . , gᲣǛƱƬƯŴSpec(L)→ J_X ƴǑǔˌɦƷᜂŷƷݣᝋƷࡽƖ৏ƠǛƱǔƱŴഏƷǑƏƳ׋ࡸƕЈஹǔᲴ

?∈H¹(Γ_L/K, L^∧(1)) {φ^∗dt_tⁱ}

←−

t_i^p1∞ƴǑǔZp(1)−torsorƨƪ

dt_i

t ƳǔࣇЎƨƪ

ƜƜưŴɥƷᘍƸžπ₁ ႎƳऴإſưŴɦƷᘍƸŴƦƷπ₁ႎƳऴإƴݣࣖƢǔŴžᡲ੗ޖႎƳ ऴإſƳƷưƢŵ ƱƜǖƕŴpᡶ HodgeྸᛯƷɼƳɼࢌƱƍƏƷƸŴɥƷᘍƷžπ₁ႎƳऴ إſƱɦƷᘍƷžᡲ੗ޖႎƳऴإſƕᲢCp ƱȆȳǵȸƠƯŴɧ٭ᢿЎǛƱƬƨǓƢǔƜƱƴ ǑƬƯᲣᐯ໱ƴݣࣖƠƯƍǔŴƱƍƏNjƷưƢŵࢼƬƯŴཎƴŴɥƷᘍǛჷƬƯƍǕƹŴɦƷ ᘍNjჷƬƯƍǔƜƱƴƳǔƷưŴƭLJǓŴφƴᛔݰƞǕǔݧ

H⁰(X_K,Ω_XK/K)→Ω_L⊗K K^∧

ǛჷƬƯƍǔƜƱƴƳǔŵƠƔƠŴƦƏƢǔƱŴݲƳƘƱNj NjƠX_KƕnonhyperellipticƳǒ ᲢƱƜǖƕŴXK Ʒᢘ࢘ƳᘮᙴǁƍƘƜƱƴǑƬƯŴৢƬƯƍǔ୺ዴƕnonhyperellipticưƋ ǔƱŴƍƭưNjˎܭƠƯNjǑƍᲣŴφ∈X_K(L)ƱƍƏໜƷ

X_K →P ^def= P(H⁰(X_K,Ω_XK/K))

Ƴǔ೅แ؈NJᡂLjƴǑǔP ƴƓƚǔ΂ǛჷƬƯƍǔƜƱƴƳǔŵƱƜǖƕŴφ : Spec(L) → XK ƕNjƠ᩼ᡚ҄ƳǒƹŴdominantƴƳǔƷưŴƦǕưXKǛŴP ƷɶƷᢿЎٶಮ˳ƱƠƯ

ࣄΨưƖƨƜƱƴƳǔŵ

ƭLJǓŴƜǕLJư˴ƕưƖƨƔƱƍƏƱŴαφƱƍƏؕஜ፭Ʒ᧓Ʒแӷ׹ƠƔ̅ǘƣƴŴ X_K ǛࣄΨƢǔƜƱƕưƖƨƷưƢŵƠƨƕƬƯŴɥƷɼܭྸƷᚰଢǛ

(∗)^geo ž࠹˴ႎƳแӷ׹Γ_L →Π⁽_X^p)

K Ǜኝƴ፭ᛯႎƴཎࣉ˄ƚǒǕǔƔſ ƱƍƏբ᫆ƴ࠙ბƢǔƜƱƕưƖƨƷưƢŵ

III ᲨஊྸዴளƔǒஊྸໜLJưᲴ

ƞƯŴʻࡇƸਁᝋႎƳᲢΓK ɥƷᲣᡲዓƳ፭แӷ׹α : ΓL → Π⁽_X^p)

K ǛᎋƑLJƠǐƏŵα ƸμݧΠ⁽_X^p)

L → Γ_LᲢƜƜưŴ X_L ^def= X_K ⊗K LᲣƷsection α_L : Γ_L → Π⁽_X^p)

LǛܭ፯ƠŴ Im(αL) ⊆ Π⁽_X^p)

L Ƴǔ᧍ƳᢿЎ፭ƸXLƷ໯ᨂഏ´etale ᘮᙴY_L^∞ → XLǛܭ፯ƢǔŵƦƠƯŴ ƦƷ໯ᨂഏᘮᙴǛŴƋǔஊᨂഏᘮᙴY_Lⁿ → X_L (ƨƩƠŴnƸ᩼᝟ƷૢૠǛƸƠǔᲣƷኒƷᡞ ಊᨂƱƠƯ୿ƘƜƱƕưƖǔŵƱƜǖƕŴИሁႎGaloisྸᛯƔǒƢƙЎƔǔǑƏƴŴαƕ࠹˴

ႎưƋǔƷƱŴY_L^∞(L)ƕᆰưƳƍƷƱƕӷ͌ƳƷưŴᙲƸŴžY_L^∞(L) = ∅ſƷ፭ᛯႎΪЎவ ˑǛᙸƭƚǔƜƱưƋǔŵƠƔƠŴƜǕƸLjǔƔǒƴƸƪǐƬƱᩊƠᢅƗǔƷưŴˌɦᲢIV.Უ ưƸŴ

(∗)^rat ž∃ pƱእƳૢૠ m s.t. P ic^m_Yn

L(L) = ∅ſ ᲢƨƩƠŴP ic^m_Yn L(L)Ƹ Y_LⁿɥƷഏૠmƷline bundleƨƪƔǒƳǔᨼӳưƋǔŵᲣ

Ʒ፭ᛯႎ࣏ᙲΪЎவˑǛ੩ᅆƢǔƕŴƜƜ(III.)ưƸŴƳƥ(∗)^ratƔǒžY_L^∞(L) = ∅ſƕࢼƏ ƔƴƭƍƯᛟଢƢǔŵ

ƦǕưƸŴᲢƢǂƯƷn ≥ 0ƴݣƠƯᲣ (∗)^ratƕ঺ǓᇌƭƱˎܭƠǑƏŵƦƏƢǔƱŴ Y_LⁿɥƷŴഏૠƕpƱእƳvery ampleƳline bundle LǛƱǔƜƱƕưƖǔŵƠƔƠŴLƕ very ampleƳƷưŴL = OY_Lⁿ(D) ᲢƨƩƠŴD ⊆ Y_LⁿƸLɥ´etaleƳ׆܇ᲣƱ୿ƘƜƱƕ ưƖǔŵƱƜǖƕŴDƸLɥ´etaleƳƷưŴ࠹ƭƔƷSpec(L_i)ᲢƨƩƠŴL_iƸLƷஊᨂഏਘ

ٻᲣƷႺԧƱƠƯ୿ƘƜƱƕưƖǔŵƠƔNjŴLƷഏૠƕpƱእƳƨNJŴL_i ƨƪƷŴݲƳƘ ƱNjƲǕƔƻƱƭŴƨƱƑƹŴL₁ƷŴLɥƷഏૠƕpƱእưƋǔƜƱƴƳǔŵƠƔƠŴƦƏ ƢǔƱŴL₁ƸLƷtameƳਘٻƴƳǔƷưŴұƪ

(∗)^tm žY_Lⁿ(L^tm)=∅ſ

ᲢƨƩƠŴL^tmƸLƷஇٻtameਘٻᲣƕЈǔŵ

ƦǕưƸŴy_n ∈ Y_Lⁿ(L^tm)ƨƪǛŴtameƳஊྸໜƷᲢnƴ᧙ƠƯƷᲣЗƱƠǑƏŵ ƞƬ ƖŴpᡶHodgeྸᛯǛࣄ፼ƠƨƱƖŴƦƷQ^p ༿ƠƔৢǘƳƔƬƨƕŴܱƸŴmod p^N ༿Nj ƋǔƷưŴƦǕǛᢘဇƢǕƹŴy_nƨƪƷP ƴƓƚǔ΂ƕᲢαƩƚưൿLJǔᲣ(L^tm)^∧-ஊྸໜ λ_∞ ∈ P((L^tm)^∧)ƴpᡶႎƴӓளƢǔƜƱƕЎƔǔŵƦƠƯŴʻƷᜭᛯƸŴP ƱƍƏŴX_K ɥƷٻ؏ႎࣇЎƷᆰ᧓ƴݣࣖƢǔݧࢨᆰ᧓ƩƚưƸƳƘŴ˓ॖƷY_LⁿɥƷٻ؏ႎࣇЎƷᆰ᧓ƴ ݣࣖƢǔݧࢨᆰ᧓ƴᢘဇƢǔƜƱƕưƖǔƠŴƠƔNj ᲢݲƳƘƱNjn≥2ƳǒƹᲣY_LⁿƸnon- hyperellipticƳƷưŴƦƷݧࢨᆰ᧓ǁƷݧƕ؈NJᡂLjƴƳǓŴࢼƬƯŴ˓ॖƷn₀ ≥ 2ƴݣƠ ƯŴy_nƨƪᲢƨƩƠŴ n ≥ n₀ᲣƷY_Lⁿ⁰ ƴƓƚǔ΂ƕŴᲢαưɟॖႎƴൿLJǔᲣ(L^tm)^∧-ஊ

ྸໜ ∈Y_Lⁿ((L^tm)^∧)ƴpᡶႎƴӓளƢǔƜƱƕЎƔǔŵ

ƭLJǓŴynƨƪƸᲢƍƬƯLjǕƹᲣαưɟॖႎƴܭLJǔy_∞ ∈ Y_L^∞((L^tm)^∧)ƴӓளƢǔ ƷƩŵ ƦƠƯŴGal(L^tm/L) ƕY_L^∞((L^tm)^∧)ƴᐯ໱ƴ˺ဇƠƯƍǔƠŴᲢƞƬƖƷᜭᛯƔǒ ǘƔǔǑƏƴᲣy_∞ƕY_L^∞ƷɟॖႎƳ(L^tm)^∧-ஊྸໜƳƷưŴy_∞ ƸŴܱƸLɥܭ፯ƞǕƯƍ ǔŴƱƍƏኽᛯƴƳǔŵ ұƪŴভకƩƬƨ

(∗)^rat =⇒ žY_L^∞(L)=∅ſᲢ⇐⇒ žαƕ࠹˴ႎưƋǔſᲣ ƕᅆƞǕƨƜƱƴƳǔŵ

IV ᲨஊྸዴளƷ܍נǯȩǤȆȪǪȳᲴ

ƋƱƸŴ(∗)^ratǛ፭ᛯႎƳᚕᓶƴᎇᚪƞƑưƖǕƹŴᚰଢƸܦ঺Ƣǔŵ LJƣƸŴ࠹ƭƔ

᩼ஜឋႎư ƭLJǒƳƍ২ᘐႎƳբ᫆ǛׅᢤƢǔƨNJŴL = KŴY_Lⁿ = X_K ƱˎܭƠǑƏŵ ᲢƭLJǓŴƍƍƔƑǕƹŴቇҥƷƨNJƴŴY_LⁿưƸƳƘŴXK ƴƭƍƯᎋƑǑƏŵᲣ ƢǔƱŴ (∗)^ratƸഏƷǑƏƴƳǔᲴ

(∗)^rat ž∃ pƱእƳૢૠm such that P ic^m_X(K)=∅ſ ƍƣǕƸChern᫏ƴƭƍƯᎋƑƨƍƷưŴLJƣƸHX def

= H²(X_K,Zp(1)) Ტ“H²” Ƹ ´etale cohomologyᲣƷನᡯǛᄩᛐƠƯƓƜƏŵ Leray-Serre ǹȚǯȈȫኒЗǛᢘဇƢǔƱŴH^X ƴ F^∗(−)Ƴǔᐯ໱ƳfiltrationƕλǓŴƳƓŴJ^{X def}= H¹(X_K,Z^p(1)) ᲢƭLJǓŴȤdzȓǢȳƷ pᡶTateь፭ᲣƱፗƘƱŴfiltrationƷᢿЎՠƸഏƷǑƏƴƳǔᲴ

F²/F¹ ⊆H²(X_K,Zp(1))∼=Zp; F¹/F⁰ =H¹(K,JX); F⁰ =H²(K,Zp(1))

ഏƴŴKummer exact sequenceƔǒc^arith₁ :P ic(X_K) =H¹(X_K,Gm)→ HXƳǔᡲኽแӷ

׹ᲢᲷૠᛯႎChern᫏ϙ΂ᲣƕưƖƯŴˌɦưƸŴഏૠƕ2g−2ưлǓЏǕǔline bundleƨ ƪᲢᲷˌɦŴP ic⁽²_X^g−2)Z(K)Ʊ୿ƘᲣƷc^arith₁ ƴǑǔ΂Ǜ፭ᛯႎƴࣄΨƠƨƍŵ

LJƣƸŴ೅แ᫏ᲢƭLJǓŴω_XK/K Ʒc^arith₁ )Ǜ፭ᛯႎƴࣄΨƠƨƍŵ(ஜ࢘ƸŴദᄩƴƍ ƏƱŴZp·c^arith₁ (ω_XK/K)ƠƔࣄΨưƖƳƍƕŴƦǕưNjΪЎưƋǔŵᲣƠƔƠŴ

H⁴(XK ×^KXK,Z^p(2))→H⁴(XK,Z^p(2))∼=Z^p

ƱƍƏdiagonalƴǑǔࡽƖ৏Ơϙ΂ƷdualᲢƜƜưŴX_K×K X_K ƷžૠᛯႎPoincar´e Du- alityſǛ̅ƬƯƍǔƕᲣǛƱǔƱŴZ^p →H²(XK×^KXK,Z^p(1))Ƴǔϙ΂ƕưƖƯŴƦǕǛ

ƞǒƴdiagonalưࡽƖ৏ƢƱŴZp → HX ƕưƖǔƕŴᲢǑƘჷǒǕƯƍǔǑƏƴᲣƦƷϙ

΂Ʒ΂ƸZ^p·c^arith₁ (ω_XK/K)ƳƷưƋǔŵ

ഏƴŴഏૠᲪƷline bundleƴƭƍƯᎋƑƨƍƷƩƕŴLJƣŴmod F⁰ưƸŴJ_X(K)ᲢƜ ƜưŴJXƱƸXƷȤdzȓǢȳᲣƷΨηƴݣƠƯŴ ƦƷΨƕƲƷƘǒƍpƷࠉưлǓЏǕǔ ƔǛᎋƑǔƜƱƴǑƬƯŴ

c^arith₁ (η)∈ HX/F⁰(HX)

Ƴǔžmod F⁰ƷChern᫏ſǛݣࣖƞƤǔƜƱƕưƖǔŵƠƔNjŴ[1]ƴЈƯƍǔžǑƘჷǒ ǕƯƍǔʙܱſƱƠƯŴ

c^arith₁ (JX(K)) =Ker(H¹(K,JX)→H¹(K,JX ⊗Zp BDR))

ƕஊǔŵ ƱƜǖƕŴᲢJX ƕPicard᧙৖ƦƷNjƷưƸƳƘŴƦƷޖ҄ƠƔᘙྵƠƯƍƳƍƨNJ ƴᲣJ_X(K)ƷΨηƸ࣏ƣƠNjline bundleƔǒƖƯƍǔƱƸᨂǒƳƍƕŴηƷᢘ࢘Ƴ̿ΨM · ηǛƱǕƹŴ ^def= M ·ηƕline bundleƔǒƖƯƍǔƱˎܭƠƯNjǑƍƠŴ =c^arith₁ (L) ᲢƜ ƜưŴLƸline bundleᲣƱƳǔƱƖŴƋǔtrickƴǑƬƯŴc^arith₁ (L)NjᲢ፭ᛯႎƴᲣࣄΨư Ɩǔŵ ұƪŴ

H^{P ic def}= Q·c^arith₁ (P ic⁽²_X^g−2)Z(K))^∧⊆ H^X ⊗ZQ

ᲢƜƜưŴ“∧”ƸƍƭNjƷǑƏƴŴpᡶܦͳ҄ǛॖԛƢǔᲣǛ፭ᛯႎƴࣄΨưƖƯƍǔƜƱƴ Ƴǔŵ ࢼƬƯŴഏƷŴኝƴ፭ᛯႎƳவˑǛᎋƑǔƜƱƕưƖǔᲴ

(∗)^pic ž∃η ∈ HX such that the image of η in H²(X_K,Zp(1)) generates H²(X_K,Zp(1)), and, moreover, the image of η in HX ⊗Q is contained in H^{P ic}.ſ

ƦƏƢǔƱŴNjƠவˑ(∗)^picƕ঺ᇌƢǕƹŴ

M ·η =p^b(a·η) =c^arith₁ (L)

ƱƳǔǑƏƳᲢᩐưƳƍᲣpᡶૢૠM = a·p^b (ƜƜưŴa ∈ Zp×ᲣƱŴX ɥƷline bundle Lƕ܍נƢǔŵ ƱƜǖƕŴƦƏƠƯƓƘƱŴKummer sequenceƷܭ፯ƔǒƢƙЈǔǑƏƴŴ L=M^⊗pbƳǔline bundle Mƕ܍נƠƯŴMƷഏૠƸZp×ƴλǔƷưŴpƱእƴƳǔŵ

ƭLJǓŴ(∗)^picưNjƬƯŴ(∗)^ratǛ፭ᛯႎƳᚕᓶƴᎇᚪƢǔƜƱƕưƖƨƷưŴɼܭྸƷ ᚰଢƸƜǕưܦ঺Ƣǔŵ

૨ྂ

[1] Bloch, S. and Kato, K., L-Functions and Tamagawa Numbers in The Grothendieck Festschrift, Volume I, Birkh¨auser (1990), pp. 333-400.

[2] Faltings, G., p-adic Hodge Theory, Journal of the Amer. Math. Soc. 1, No. 1, pp.

255-299 (1988).

[3] Mochizuki, S., The Local Pro-p Grothendieck Conjecture for Hyperbolic Curves, RIMS Preprint 1045.

[4] Tamagawa, A., The Grothendieck Conjecture for Affine Curves, preprint.