ᇹɟᛅᲴ Faltings Ʒ p ᡶțȃǸྸᛯƷኰʼᲴ Grothendieck ƷžᜋƷ᧙৖ſʖेƷ

ȬǯȁȣȸȷȎȸȈᲴ ஓஉ ૼɟ ᲢʮٻૠྸᄂᲣ

ᇹɟᛅᲴ Faltings _Ʒ p _{ᡶțȃǸྸᛯƷኰʼᲴ} Grothendieck ƷžᜋƷ᧙৖ſʖेƷ

ᚇໜƔǒ

ႸഏᲴ

I. țȃǸྸᛯƷؕஜႸ೅

II. Galois cohomologyƷޅ৑ႎᚘምƱžവǜƲૠܖſ III. p-divisible groupƷئӳ

I ᲨțȃǸྸᛯƷؕஜႸ೅Ჴ

(A.)ᙐእૠ˳CɥƷțȃǸྸᛯᲴ LJƣŴCɥƷئӳǛ࣬ƍЈƠƯƓƜƏŵXǛŴCɥproper

ưsmoothƳٶಮ˳ᲢưܭLJǔᙐእٶಮ˳ᲣƱƢǔŵƦƏƢǔƱŴɟ૾ưƸŴ

H_singⁱ (X,C)

Ƴǔsingular cohomologyƷdzțȢȭǸȸь፭ƕƋǓŴ˂૾ưƸŴ

p+q=i

H^p(X,Ω^q_X)

ƳǔHodge cohomologyƷdzțȢȭǸȸь፭ƕƋǔŵ ЭᎍƷཎࣉƱƠƯŴXƷӨˮႻᆰ᧓Ʃ

ƚưൿLJǔƱƍƏࣱឋƕƋǓŴƦǕƴݣƠƯŴࢸᎍƸŴݲƳƘƱNjܭ፯ƷɥưƸŴXƷᙐእನ ᡯƴࢍƘ̔܍ƠƯƍǔŵ ƦƜưŴHodgeྸᛯƕɼࢌƠƯƍǔƜƱƸŴƜƷʚᎍƕܱƸ೅แႎ ƴӷ׹ưƋǔᲴ

H_singⁱ (X,C)∼=

p+q=i

H^p(X,Ω^q_X)

ƱƍƏƜƱưƋǔŵƭLJǓŴᚕƍ੭ƑǕƹŴƜƷɟᙸμ໱ီឋƦƏƳʚƭƷɭမᲢᲷ Ȉȝȭ ǸȸƷɭမƱŴദЩLJƨƸˊૠႎƳ᧙ૠƷɭမᲣƕ೅แႎƴƭƳƕƬƯƍǔƷưƋǔŵ

(B.) pᡶႎƳئӳƷؕஜᚨܭᲴ KƸ QpƷஊᨂഏਘٻƱƠŴA ⊆ KƸƦƷૢૠ࿢ƱƢǔŵ X → Spec(A)ƸAɥsmoothưproperƳǹǭȸȠƱƢǔŵ ƳƓŴX ^def= X ⊗^A KƱƢǔŵ Hodge cohomology፭ƸஜឋႎƴƸˊૠႎƳNjƷƳƷưŴ(A.)Ʒܭ፯ƸʻࡇƷpᡶႎƳཞඞƷ NjƱưNjƦƷLJLJᡫဇƢǔƚƲŴsingular cohomologyƷ૾ƸᲢ࢘໱ƷƜƱƳƕǒᲣ´etale co-

homologyƴፗƖ੭ƑƳƍƱƍƚƳƍŵ ƦƏƢǔƱŴഏƷǑƏƳʚᆔ᫏Ʒᐯ໱ƳdzțȢȭǸȸ

፭ƕưƖǔᲴ

(1) H_etⁱ (X⊗K K,Zp) ᲢƜƜưŴK ƸK Ʒˊૠ᧍ѼᲣ (2) ⊕p+q=i H^p(X,Ω^q_X/A)

ࢼƬƯŴᲢᲫᲣƱᲢᲬᲣǛൔ᠋ƠǑƏƱƢǔƷƸᐯ໱ưƔƭಒƶദᚐƩƕŴƨƩƠŴƪǐƬƱ Ơƨ২ᘐႎƳբ᫆ƱƠƯŴᲢᲫᲣưƸZp ɥƷrankƕBettiૠƴƳǔƷƴݣƠƯŴᲢᲬᲣƷ૾

ưƸŴAɥƷrankƕBettiૠƴƳǔŵƦƠƯŴNjƏƪǐƬƱขЦƳբ᫆ƱƠƯŴᲢᲫᲣƴƸŴ

Γ_K ^def= Gal(K/K)ƴǑǔᲢɟᑍƴƸᲣ᩼ࠝƴ᩼ᐯଢƳ˺ဇƕᐯ໱ƴͳǘƬƯƍǔƷƴݣƠƯŴ ᲢᲬᲣƷ૾ƴƸŴΓK Ʒ˺ဇǒƠƖNjƷƸƲƜƴNjᙸ࢘ƨǒƳƍŵ

ƜƷʚƭƷբ᫆ƸɟᙸᲢᲫᲣƱᲢᲬᲣƷ᧓ƴƸᐯ໱Ƴӷ׹ƳƲஊǓࢽǔNjƷưƸƳƍƜƱ ǛཋᛖƬƯƍǔǑƏƴ࣬ƑǔƔNjƠǕƳƍƕŴƦǕưNjŴƳǜƱƔᢘ࢘ƳᲢऀǒƘܭ፯ƢǔƷ ƕ᩼ࠝƴ᩼ᐯଢƳᲣ᧙৖Ǜ଀ƢƜƱƴǑƬƯᲢᲫᲣƱᲢᲬᲣǛƨƕƍƴᐯ໱ƴ٭੭ƢǔƜƱƕ ưƖǔưƋǖƏƱƍƏƷƕŴGrothendieckƷžmysterious functorſʖेưƋǔŵ ƪƳLjƴŴ ʻƷʖेƸᲢᢒǢȸșȫπ₁ƴ᧙ƢǔᲣƍǘǏǔžGrothendieckʖेſƱƸᢌƏƷƩƕŴžmys- teriousſƱƍƏᚕᓶƕॖԛƠƯƍǔƱ࣬ǘǕǔŴȈȝȭǸȸƱˊૠႎ᧙ૠᲢᲷٶ᪮ࡸᲣƱƍƏ ʚƭƷμƘီឋƦƏƳɭမǛኽƼ˄ƚǔɧ࣬ᜭƳž˴ƔſƷ܍נǛʖेƠƯƍǔƱƍƏໜưƸŴ ƜƷʚƭƷžGrothendieckʖेſƸݲƳƘƱNjՋܖႎƴƸ࣏ƣƠNj໯ጂưƸƳƍŵƠƔNjŴܱ

ᨥŴ[3]ưƸŴƪǐƏƲžmysterious functorſʖेƷFaltingsƴǑǔᚐൿǛဇƍƯŴᲢ୺ዴƷ ئӳƷᲣᢒǢȸșȫʖेǛᚰଢƠƯƍǔƜƱƔǒNjᇄƑǔǑƏƴŴƜƷƾƨƭƷʖेƷƭƳƕ ǓƴƸӈ݅ƳૠܖႎƳ᩿ͨNjஊǔŵ

(C.) όЎਦ೅ƱFaltings [2]ƷɼܭྸᲴ ƦǕưƸŴɼܭྸǛᡓǂǔƷƴ࣏ᙲƳᚡӭǛኰʼƠ ƯƓƜƏŵχ : ΓK → Z^×p ƸόЎਦ೅ƱƠŴƦǕƴݣࣖƢǔΓK-ь፭ǛZ^p(1)Ʊ୿ƘŵƳƓŴ Γ_K-ь፭M Ʊn∈ZƴݣƠƯŴM(n)^def= M⊗ZpZp(1)^⊗nƱƢǔŵ ƦƏƢǔƱŴFaltings[2]

ƷɼܭྸƸഏƷǑƏƴƳǔᲴ

Theorem 1: Let X → Spec(A) be proper and smooth. Then there exists a natural Γ_K-equivariant isomorphism (for alli ∈Z)

H_etⁱ (X⊗K K,Zp)⊗Zp K^∧ ∼=

p+q=i

H^p(X,Ω^q_X/K)⊗K K^∧(−q) Moreover, there also exist integral and mod pⁿ versions of this isomorphism.

II _Შ Galois cohomology Ʒޅ৑ႎᚘምƱžവǜƲૠܖſᲴ

(A.) CɥƷئӳᲴ StokesƷܭྸǁƷ࠙ბᲴ FaltingsƷܭྸƷ᭽щƷƻƱƭƸŴƦƷᚰଢƕ ƋǔॖԛưƸŴCɥƷئӳƷൔ᠋ܭྸᲢᲷ(I.)Ʒ(A.)ᲣƷᚰଢƱǑƘ˩ƯƍǔƱƍƏƜƱưƋ ǔŵƲƜƕ˩ƯƍǔƔƱƍƏƱŴproperƳٶಮ˳ƴ᧙Ƣǔ᩼ᐯଢƳٻ؏ႎƳɼࢌǛŴޅ৑ႎ Ƴᚘምƴ࠙ბƠƯƍǔƱƍƏƱƜǖƩŵƦƠƯŴgeneral nonsenseƠƔ̅ǘƣƴŴƦƷޅ৑ႎ ƳᚘምƔǒЈǔNjƷƨƪǛࢌǓӳǘƤǔƜƱƴǑƬƯٻ؏ႎƳኽௐǛЈƢŵ̊ƑƹŴCɥƷئ ӳƕƲƏƩƬƨƔǛ࣬ƍЈƠƯLjǔƱŴX ɥƷᲢ୍ᡫƷᚐௌႎƳˮႻƴƓƚǔᲣܭૠޖCƷ resolutionǛƾƨƭᎋƑǔŵƻƱƭƸŴC^∞ ኢƷࣇЎ࢟ࡸƴǑǔde Rham complexưŴNjƏ ƻƱƭƸŴžޅ৑ႎƳҥ˳ſɥƷ᧙ૠƴǑǔcomplexưƋǔŵഏƴŴࣇЎ࢟ࡸǛҥ˳ƴƦƬƯ ᆢЎƢǔƜƱƴǑƬƯЭᎍƔǒࢸᎍǁƷᐯ໱Ƴϙ΂ƕࢽǒǕǔŵƨƩƠŴƦƷᐯ໱Ƴϙ΂ƕdz țȢȭǸȸƷɥưNjݧǛࡽƖឪƜƢƜƱǛƍƏƨNJƴƸŴϙ΂ƕҥƳǔь፭Ʒ᧓Ʒϙ΂ưƋǔ ƷLjƳǒƣƴŴcomplexƷ᧓Ʒϙ΂ưƋǔƜƱǛᚕǘƳƍƱƍƚƳƍŵƠƔƠŴƦǕƸƪǐƏ ƲŴnഏΨᇌ૾˳ɥƷStokesƷܭྸƷϋܾưƋǔŵ ƢƳǘƪŴኽޅŴҥˮғ᧓ɥƷࣇᆢЎඥ ƷؕஜܭྸᲢƱƍƏޅ৑ႎƳኽௐᲣƴ࠙ბƠƯƍǔƱƍƏǘƚƩŵpᡶႎƳئӳƴƸŴƜƷStokes ƷܭྸƴݣࣖƢǔNjƷƸʻƔǒኰʼƢǔGalois cohomologyƷޅ৑ႎƳᚘምưƋǔŵ

(B.)ЎޟǛžവǜƲſൈƢƨNJƷZp-ਘٻᲴAƸ(I.)Ʒ(B.)ƷǑƏƳ࿢ƱƠŴRƸAɥsmooth ưႻݣႎഏΨdƷӧ੭࿢ƱƢǔŵቇҥƷƨNJƴŴRƷɶƴŴdlog(u_i)ƨƪƕΩ_R/A ƷؕࡁƱƳ ǔǑƏƳҥΨƨƪu₁, . . . udƕƋǔƱˎܭƢǔŵ ƞǒƴŴRǛŴ೅ૠᲪưƸǨǿȸȫƳRƷஇ ٻƷਘٻƱƠŴΓ_R ^def= Gal(R_K/R_K)ƱƢǔŵᙲƸŴΓ_RƷGalois cohomologyǛᚘምƠƨƍ ƷƩƕŴႺ੗ᚘምƠǑƏƱƢǕƹᩊƠᢅƗǔƷưŴഏƷǑƏƳನ঺ǛƢǔŵLJƣŴ

R_∞ ^def= R[ζ_p∞, u¹_i^/p∞]⊆R

ᲢƜƜưŴζ_p∞ ƸɟƷpࠉʈఌμ˳ǛॖԛƢǔᲣƱƍƏRƷᢿЎ࿢ǛݰλƢǔŵ ƦƏƢǔƱŴ ഏƷǑƏƳ᣻ᙲƳᚇݑƕƋǔᲴ

(∗^alm) RƸR_∞ƷਘٻƱƠƯŴžവǜƲǨǿȸȫſưƋǔŵ

ƋǔB-࿢C ƕBɥžവǜƲǨǿȸȫſưƋǔƱƸƲƏƍƏƜƱƔƱƍƏƱŴᲢƭLJǒƳƍ২ ᘐႎƳவˑǛႾဦƢǔƱᲣ೅ૠᲪưƸǨǿȸȫưŴƔƭ e ∈ C⊗B C[1/p]ƱƍƏŴݣᚌ؈NJᡂ LjƴݣࣖƢǔݧࢨ܇ƕഏƷவˑǛ฼ƨƠƯƍǔᲴ˓ॖƷ > 0ƴݣƠƯŴp·eƕC ⊗^B C → C ⊗B C[1/p]Ʒ΂ƴλƬƯƍǔŵ ƭLJǓŴNjƠCƕ୍ᡫƷॖԛưDŽǜƱƏƴBɥǨǿȸȫ ƩƬƨǒŴdiagonalƕSpec(C⊗^B C)ƷɶƷᲢ࠹ƭƔƷᲣᡲኽ঺ЎƴƳǔƷưŴe∈C⊗^BC ƱƍƏƜƱƴƳǔƚƲŴžവǜƲǨǿȸȫſƳƱƖƴƸŴC ⊗B CƴƸžവǜƲλƬƯƍǔſ ƚƲŴ࣏ƣƠNjƽƬƨǓλƬƯƍǔƱƸᨂǒƳƍŵƋǔƍƸŴЙКࡸƷᚕᓶưƍƏƱŴЙКࡸ

Ʒ˄͌ƕ˓ॖƴݱƞƘƳǔǑƏƳCƷɶƷB-latticeƕƱǕǔŴƱƍƏƜƱƩŵ̊ƑƹŴஇNj

ؕஜႎƳ̊ƸŴA[ζ_p∞]-࿢ƱƠƯƷAᲢ^def= KƷૢૠ࿢ᲣưƋǔŵ

ƪƳLjƴŴ(∗^alm)ƷᚰଢƩƕŴ২ᘐႎƴᡂLjλƬƯƍǔƷưŴƜƜưƸƋLJǓขλǓƠƨ ƘƳƍƷƩƕŴቇҥƴƍƏƱŴRǛheight 1Ʒprimeưޅ৑҄ƢǔƜƱƴǑƬƯDVRƴ᧙Ƣ ǔբ᫆ƴ࠙ბƠƯƓƍƯŴƦǕư DVRƷӞχႎƳЎޟྸᛯƔǒኽௐǛݰƘƱƍƏƷƕؕஜ૾

ᤆƩŵ

(C.) žവǜƲǨǿȸȫſਘٻƴ᧙ƢǔࣇЎǍdzțȢȭǸȸƷ᧙৖ႎਰᑈƍᲴ žവǜƲǨǿȸ ȫſਘٻƷƍƍƱƜǖƸɟᚕưƍƏƱŴവǜƲŴǨǿȸȫਘٻƷǑƏƴਰᑈƏƱƍƏƜƱƩŵ

̊ƑƹŴႻݣႎࣇЎь፭ƸžവǜƲǼȭƴƳǔſᲢᲷƭLJǓŴpƴᩐ҄ƞǕǔᲣŵ ƋǔƍƸŴ Galois cohomologyƴƠƯNjŴNjƠCƕBɥGaloisưŴGalois፭ƕGƳǒƹŴC[G]-ь፭ ƷᲢ᭗ഏᲣdzțȢȭǸȸNjവǜƲǼȭƴƳǔŵཎƴŴƜǕǒƷʙܱǛ̅ƏƜƱƴǑƬƯŴഏƷ ǑƏƳᚘምƕưƖǔᲴ LJƣŴࣇЎь፭ƷؕஜܦμኒЗƷƻƱƭưƋǔ

Ω_A/A⊗_AR→Ω_R/R →Ω_R/R⊗

AA →0

ǛਤƪЈƢŵƦƏƢǔƱŴƞƬƖƷᜂŷƷᚇݑƱΩ_A/A ∼= K/ρ⁻¹A(1) ᲢƜƜưŴρ ∈ A − {0}ᲣƱƍƏǑƘჷǒǕƯƍǔ೅แႎƳӷ׹ǛᢘဇƢǔƱŴɥƷܦμኒЗǛ

0→(R[1/p]/ρ⁻¹R)(1)→ ?→ Ω_R/A ⊗R(R[1/p]/R)→0

Ʊ٭࢟ƢǔƜƱƕưƖǔŵ୼ƴŴHom(Q^p/Z^p(1),·)Ǜ଀ƢƱŴGaloisь፭Ʒ೅แႎƳਘٻ 0→ρ⁻¹R^∧ →Eρ →Ω_R/V ⊗RR^∧(−1)→0

ƕࢽǒǕǔŵഏƴŴΔR def

= Ker(ΓR →ΓK)ƱƠƯƓƘƱŴΓK-ь፭Ʒᐯ໱ݧ Ω_R/V ⊗RR^∧_K(−1)→H¹(Δ_R, R^∧_K)

ƕࢽǒǕǔŵ ƱƜǖƕŴHⁱ(Δ_R, R^∧_K)ƱƍƏGalois cohomology፭ƸŴR_∞/RưNjƬƯᚘም ƢǔƜƱƕưƖǔᲢදᲴGal(R_∞/R) ∼=Z^dp⁺¹ ƳƷưŴƦƷGalois cohomologyƸ᩼ࠝƴᚘም ƠǍƢƍᲣƷưŴƦƏƢǔƱŴƜƷᐯ໱ݧƕܱƸŴ೅แႎƳᲢΓK-ӷ٭ƳᲣӷ׹

φⁱ : Ωⁱ_R/V ⊗^RR^∧_K(−1)→Hⁱ(ΔR, R^∧_K)

ǛࡽƖឪƜƠƯƍǔƜƱƕЎƔǔŵɼܭྸƸƪǐƏƲX ƴSpec(R)Ʊ୿ƚǔǢȕǣȳƨƪƴ ǑǔᘮᙴǛƱƬƯƖƯŴƦǕƧǕƷSpec(R)ƴݣࣖƢǔφⁱƨƪǛࢌǓӳǘƤǔƜƱƴǑƬƯ ᅆƢƷưƋǔŵ

Remark: Faltings [2]ƷɼܭྸƷᚰଢƷᙲưƋǔӷ׹φⁱƱDŽƱǜƲӷ͌ƳNjƷƸŴFaltingsƱ DŽDžӷ଺஖ƴτ᪽඙൞ƴǑƬƯNjᲢ̲ٟᛯ૨ưᲛᲣ཯ᇌƴႆᙸƞǕƯƍǔƦƏưƋǔŵ

(D.) ɼܭྸƷᚰଢᲴ LJƣŴǨǿȸȫƳSpec(R) → X ƴݣƠƯŴR_K ƱƍƏΓ_R-ь፭Ǜݣࣖ

ƞƤǔƜƱƴǑƬƯŴᲢٻᩃ৭ƴƍƏƱᲣXetɥƷޖR^K ƕࢽǒǕǔŵཎƴŴᐯ໱ݧ H_etⁱ (X_K,Qp)→H_etⁱ (X_K,RK)

ƕࢽǒǕǔŵƱƜǖƕŴφⁱƨƪǛࢌǓӳǘƤƯŴdzțȢȭǸȸǛƱǔƱŴ H_etⁱ (X_K,R^K)∼=

p+q=i

H^p(X,Ω^q_X/K(−q))⊗^KK^∧

Ƴǔӷ׹ƕࡽƖឪƜƞǕǔŵ ƦƠƯŴƜƷƾƨƭƷϙ΂Ʒӳ঺ǛӕǔƱŴ Γ_K-ӷ٭Ƴᐯ໱ݧ

H_etⁱ (X_K,Q^p)⊗Q_pK^∧→

p+q=i

H^p(X,Ω^q_X/K(−q))⊗^K K^∧

ƕưƖǔŵ ƜƷᐯ໱ݧƕӷ׹ƴƳǔƜƱƕƍƑǕƹŴᚰଢƸܦ঺ƢǔƷƩƕŴƦǕǛƍƏƨNJ

ƴƸŴPoincar´e dualityǛ̅ƬƯŴLJƣᡞϙ΂ᲢƷͅᙀᲣǛ˺ƬƯƓƍƯŴƦƠƯŴɥƷᐯ໱

ݧƱƜƷᡞϙ΂Ʒͅᙀƕɲ૾ƱNjChern᫏ƱɲᇌƢǔƱƍƏᐯଢƳʙܱǛଓƘ᧙৖ႎƴƍơǔ ƜƱƴǑƬƯŴɥƷᐯ໱ݧƕܱᨥӷ׹ƴƳǔƜƱǛኽᛯƢǔŵ

III _Შ p-divisible group _ƷئӳᲴ

(A.) TateƷྸᛯᲴ G → Spec(A)Ǜp-divisible groupƱƢǔŵᲢܭ፯ƴ᧙ƠƯƸŴ[5]ǛӋ ༀŵᲣज़ᙾႎƴƸŴp-divisible groupƸᲢ͞ಊƕ࣏ƣƠNjλǔƱƸᨂǒƳƍᲣǢȸșȫᙐእٶ ಮ˳Ʒpᡶ༿ƷǑƏƳNjƷƩŵཎƴŴദჇദ᥄ƷǢȸșȫٶಮ˳ƕɨƑǒǕƨ଺ŴƦƷKer(pⁿ) ƨƪǛƱǔƜƱƴǑƬƯŴp-divisible groupǛƭƘǔƜƱƕưƖǔŵNjƬƱɟᑍƴŴ ˓ॖƷ p-divisible group GƸ࣏ƣഏƷǑƏƴဃơǔᲴG → Spec(A)ƱƍƏᲢǨǿȸȫƳᢿЎǛԃlj ƔNjƠǕƳƍᲣ࢟ࡸ፭ƕƋƬƯŴƦƷGƷKer(pⁿ)ƨƪƷunionƕGƴƳǔŵƱƜǖƕŴƜ ƷGƸᲢ೅แႎƳӷ׹ǛᨊƍƯᲣuniqueƳƷưŴGƷ੗ᆰ᧓ Θ_G ǛŴGƷǼȭƴƓƚǔ੗ᆰ ᧓Ʊܭ፯ƢǔƜƱƕưƖǔŵɟ૾ŴGǛSpec(K)ƴࡽƖ৏ƠŴHom(Q^p/Z^p,·)ǛƱǔƜƱƴ ǑƬƯŴƍǘǏǔTateь፭ T(G)ǛƭƘǔƜƱƕưƖǔŵǑƘჷǒǕƯƍǔǑƏƴŴT(G)ƴ Ƹᐯ໱ƳΓK-˺ဇƕλǔŵ ƦƏƢǔƱŴഏƷǑƏƳܭྸƕ঺ǓᇌƭᲢ[5]ᲣᲴ

Theorem 2: Let G →Spec(A) be a p-divisible group. Then there is a natural isomor- phism of Γ_K-modules

T(G)⊗QpK^∧∼= ((ΩG^∗)⊕Θ_G(1))⊗AK^∧

(Here, Ω_G∗ is the A-dual of theA-module Θ_G∗, andG^∗ is the dualp-divisible group toG.) Moreover, whenG arises from an abelian variety, this isomorphism commutes with that of Theorem 1 (due to Faltings).

(B.)ዌݣɧЎޟƳئӳᲴAƕZp ɥɧЎޟƳƱƖƴƸŴNjƬƱች݅ƳྸᛯᲢ[6]ᲣƕƋǔŵ ƜƜ ưƸŴኬƔƍƜƱƸႾဦƢǔƕŴଔƍᛅƠŴp-divisible group GƴݣƠƯŴfiltration˄ƖƷ ஊᨂᐯဌA-ь፭ᲢƍǘǏǔ Dieudonn´e-ь፭ᲣF¹(M) ⊆ M ƱŴƦƷM ǁƷFrobeniusƷ

˺ဇᲢᲷAƷFrobeniusƴ᧙ƠƯҞዴ࢟ႎƳᐯࠁแӷ׹Φ_M : M → MᲣǛݣࣖƞƤǔƜƱƕ ưƖǔŵƠƔNjŴ

F¹(M)∼= ΩG; M/F¹(M)∼= ΘG^∗

ƱƍƬƨ೅แႎƳӷ׹ƕƋǔƔǒŴM ƸGƷde Rham cohomologyƷǑƏƳNjƷƩŵƦƠ ƯŴ(F¹(M) ⊆ M,ΦM)ƱƍƏȇȸǿƔǒŴNjƱƷGǛܦμƴࣄΨƢǔƜƱƕưƖǔŵ̊Ƒ ƹŴTateь፭T(G)ƸŴB_crysƱƍƏ᩼ࠝƴٻƖƘƯᙐᩃƳ࿢ƷɥưƷM ƷFrobeniusᲢᲷ ΦMᲣɧ٭᣽ǛƱǔƜƱƴǑƬƯࣄΨƢǔƜƱƕưƖǔŵ

ᇹʚᛅᲴ Grothendieck _ʖेƱ

೅แႎᡫࠝ୺ዴƷ፭ᛯႎཎࣉ˄ƚ

ႸഏᲴ

I. ೅แႎᡫࠝ୺ዴƷܭ፯Ʊؕஜࣱឋ

II. Local pro-p Grothendieck ConjectureƱƷ᧙̞

I Შ ೅แႎᡫࠝ୺ዴƷܭ፯Ʊؕஜࣱឋ

(A.) Serre-TateྸᛯƷhyperbolic༿ᲴpᡶႎƳbaseɥƷǢȸșȫٶಮ˳ƷྸᛯưƸŴSerre- TateƴǑǔ೅แႎƳਤƪɥƛƷྸᛯƕƋǔƜƱƸԗჷƷᡫǓƩƕŴܱƸŴӑ୺ႎƳ୺ዴƷئӳ ƴNjŴƦǕƱƪǐƏƲ᫏˩ႎƳ೅แႎਤƪɥƛྸᛯƕƋǔŵLJƣŴӋᎋLJưƴŴǢȸșȫٶಮ

˳ƷئӳǛݲƠࣄ፼ƠƯLjǔƱŴƍǖǜƳܭࡸ҄Ʒˁ૾ƸƋǔƕŴhyperbolicƳئӳǁƷɟᑍ

҄ƴஇNjᢘƠƯƍǔƷƸഏƷܭࡸ҄ưƋǔŵ kƸ೅ૠpƷܦμ˳ƱƠŴA ^def= W(k)ƸƦƷ Witt࿢ƱƢǔŵ ƦƏƢǔƱŴA^g ᲢᲷZ^p ɥƷɼ͞ಊǢȸșȫٶಮ˳ƷȢǸȥȩǤȷǹǿȃ ǯᲣƷɥƴŴpᡶႎƳ´etale formal stack

A^ordg → Ag

ƕஊƬƯŴA^ordg Ʒɥƴᐯ໱ƳFrobeniusਤƪɥƛΦ_A : A^ordg → A^ordg ƕ˺ဇƠƯƍƯŴƦƷ Φ_Aƴ׍ܭƞǕǔA-ஊྸໜƕŴSerre-TateƷॖԛưƷž೅แਤƪɥƛſưƋǔŵ ɟ૾Ŵӑ୺

ႎ୺ዴƷئӳŴMg,r ᲢᲷZpɥƷ(g, r)׹ܤܭ୺ዴƷȢǸȥȩǤȷǹǿȃǯᲣƷɥưƸŴᲢܭ

፯ƸႾဦƢǔƕᲣ᩼ࠝƴᐯ໱ƳpᡶႎƳ´etale formal stack N^ordg,r → M^g,r

ƕஊƬƯŴN^ordg,r Ʒɥƴᐯ໱ƳFrobeniusਤƪɥƛΦ_N :N^ordg,r → N^ordg,r ƕ˺ဇƠƯƍƯŴƦƷ Φ_N ƴ׍ܭƞǕǔA-ஊྸໜǛŴN^ordg,r Ʒž೅แႎƳໜſƱLjǔƜƱƕưƖǔŵǢȸșȫٶಮ˳

ƷئӳƱᢌƬƯŴN^ordg,r → Mg,rƸopen immersionƴƸƳǒƳƍƕŴN^ordg,r Ʒž೅แႎƳໜſ ǛMg,r ƴᓳƢƜƱƴǑƬƯŴMg,rƷž೅แႎƳA-ஊྸໜſŴұƪŴAɥƷž೅แႎƳ୺ዴ X →Spec(A)ſƕࢽǒǕǔŵʻଐƷᛅƠƸƦƏƍƏXƴ᧙ƢǔNjƷưƋǔŵ

(B.) Frobeniusɧ٭Ƴ׍ஊளƴǑǔཎࣉ˄ƚ:ž೅แ୺ዴſǛ(A.)ƷǑƏƴܭ፯ƠƯƠLJƏƱŴ N^ordg,r ǍΦ_N ƱƍƬƨφ˳ႎƴƸ᩼ࠝƴᚘምƠƴƘƍݣᝋƨƪƕЈƯƖƯŴ୺ዴƴ᧙ƠƯƸ˷

ǓϋנႎƳܭ፯ƴƸƳǒƳƍƷưŴ(A.)Ʒܭ፯ǛNjƏݲƠϋנႎƳ࢟ƴႺƠƨƍŵƦǕǛᢋ঺

ƢǔƨNJƴƸŴLJƣž׍ஊளſƱƍƏ˷ǓǑƘჷǒǕƯƍƳƍಒࣞǛݰλƠƳƍƱƍƚƳƍƷ ƩƕŴX →Spec(A)ƱƍƏᆔૠg ≥2Ʒ୺ዴƕɨƑǒǕƨƱƖŴഏƷǑƏƴܭ፯Ƣǔŵ Definition 1: (P →X,∇P)ƸXɥƷŴ੗ዓ˄ƖƷP¹-bundleƱƢǔŵsection σ :X → P ƕɨƑǒǕƨǒŴσǛ∇^P ưࣇЎƢǔƜƱƴǑƬƯŴσƷKodaira-Spencer mapτX →σ^∗τ_P/X ᲢƜƜưŴžτſƸtangent bundleƷॖԛᲣƸƍƭưNj˺ǕǔƕŴKodaira-Spencer mapƕ ӷ׹ƱƳǔǑƏƳσ ǛᚩܾƢǔ(P,∇^P)Ǜž׍ஊளſᲢindigenous bundleᲣƱƍƏŵ

XɥƷ׍ஊளP ^def= (P,∇P)ƕɨƑǒǕƨǒŴP ǛCrys(X⊗Ak/A)ɥƷcrystalƱLjǔƜƱ ƕưƖǔŵƦƏƢǔƱŴXk def

= X ⊗AkɥƷFrobeniusưࡽƖ৏ƢƜƱƴǑƬƯŴΦ^∗_XP Ʊƍ ƏૼƨƳCrys(X ⊗Ak/A)ɥƷcrystalƕࢽǒǕǔŵ ঻ŷƸŴܱƸŴஇኳႎƴƸŴFrobenius ưࡽƖ৏ƠƯNjᐯЎᐯ៲ƴ৏ǔǑƏƳžFrobenius ɧ٭ſƳP ƴƭƍƯᎋƑƨƍƷƩƕŴƋǔ ২ᘐႎƳྸဌƴǑƬƯŴžFrobenius ɧ٭ſǛΦ^∗_XP ^def= P ưܭ፯ƢǔƱଓƘᘍƔƳƍŵ ࢼƬ ƯŴƜƜưƸ˷ǓขλǓƠƨƘƳƍƷƩƕŴΦ^∗_XP Ʒintegral structureǛݲƠᛦૢƢǔᲢᲷƭ LJǓŴQpɥƷನᡯƸ٭ǘǒƳƍᲣƜƱƴǑƬƯŴF^∗P ƱƍƏŴFrobeniusࡽƖ৏ƠƷžǑǓ Ǒƍſܭ፯ᲢᲷrenormalizedᲢϐദᙹ҄ᲣFrobeniusᲣƕࢽǒǕƯŴƦƬƪƷ૾Ʒܭ፯Ǜ੔ဇ ƢǔƜƱƴǑƬƯŴǑǓଓƘᘍƘǑƏƳžFrobenius ɧ٭ࣱſƷܭ፯ƕࢽǒǕǔŵ

Definition 2: ׍ஊளP ƕF^∗P ∼=P Ǜ฼ƨƠƯƍǔƱƖŴP ǛžFrobeniusɧ٭ſƳ׍ஊ ளƱƍƏŵ

ƦƏƢǔƱŴഏƷܭྸ([4])ƕ঺Ǔᇌƭŵ

Theorem 1: X →Spec(A)ƕŴ(A.)Ʒॖԛưž೅แ୺ዴſưƋǔƷƱŴžordinaryſᲢƱ ƍƏŴিǔ২ᘐႎƳவˑᲣǛ฼ƨƠƯƍǔžFrobenius ɧ٭ſƳ׍ஊளP ǛᚩܾƢǔƷƱƕŴ ӷ͌ưƋǔŵ

(C.) Galoisᘙྵ: X →Spec(A)Ƹ೅แ୺ዴƱƢǔŵƢǔƱŴTheorem 1ƔǒЎƔǔǑƏƴŴ Frobenius ɧ٭Ƴ׍ஊளP ƕXƴλǔŵƱƜǖƕŴDieudonn´eь፭ƷྸᛯƷ࠙ኽƱƠƯŴƦ ƏƍƏP Ƹ࣏ƣX ɥƷp-divisible group G → XƷDieudonn´eь፭ᲢƷݧࢨ҄P(−)ᲣƱƠ ƯဃơǔƷƩŵƭLJǓŴᚕƍ੭ƑǕƹŴƜƷDieudonn´eྸᛯƷ࠙ኽƱƍƏƷƸŴᇹɟᛅƷᚕᓶ ưƍƏƱŴ(III.)Ʒ(B.)Ʒparametrized versionƳƷưƋǔŵƠƨƕƬƯŴ୼ƴŴGǛX_K ^def= X⊗AK ᲢƜƜưŴK ƸAƷՠ˳ᲣƴСᨂƢǔƱŴΠ_X ^def= π₁(X_K)ᲢؕໜƷƜƱƸቇҥƷƨ NJŴൢƴƠƳƍᲣƷ೅แႎƳᘙྵ

κ_X : Π_X →GL₂(Zp)

ƕࢽǒǕǔŵƭLJǓŴX ƕ೅แ୺ዴưƋǔƱˎܭƢǔƩƚưŴκ_X ƷǑƏƳcrystallineᘙྵ ᲢᲷ ƜƷئӳƴƸŴҥƴŴp-divisible groupƔǒဃơƨƱƍƏॖԛᲣƕ܍נƢǔƱƍƏ᩼ࠝƴ᩼ᐯ ଢƳ࠙ኽǛݰƘƜƱƕưƖƨŵΠ_X ƕƜƷǑƏƳcrystallineᘙྵǛᚩܾƢǔƱƍƏƜƱƸŴ୺

ዴXƷࣱឋƱƠƯƸŴ᩼ࠝƴྦྷƠƍࣱឋưƋǔŵƱƍƏƜƱƸŴᲢDŽƔƴNj࠹ƭƔ˷Ǔ᣻ᙲư Ƴƍ২ᘐႎƳவˑNjஊǔƕᲣŴܱƸŴٻᩃ৭ƴƍƏƱŴκX ƷǑƏƳcrystallineᘙྵǛਤƭX Ƹ࣏ƣ೅แႎƴƳǔŵ

(D.) բ᫆੩ឪ: V → Spec(A)ƕǢȸșȫٶಮ˳ƳǒƹŴᲢǑƘჷǒǕƯƍǔǑƏƴᲣƦƷp ᡶ Tateь፭ Tp(V)ǛᙸǔᲢƭLJǓŴTp(V)ƕǨǿȸȫƳᢿЎƱʈඥႎƳᢿЎƷႺԧƴЎᚐƢ ǔƱƍƏவˑƩƕᲣƜƱƴǑƬƯŴV ƕ೅แႎƳƷƔƲƏƔƕቇҥƴЙܭưƖǔŵƱƍƏƜƱ ƸŴᆔૠg ≥ 2Ʒ୺ዴX → Spec(A)ƕɨƑǒǕƨƱƖŴΠXᲢিƍƸŴNjƬƱദᄩƴƍƏ ƱŴΓK def

= Gal(K)Ʒ࠹˴ႎπ₁ŴΔX ⊆ ΠXŴǁƷٳᢿ˺ဇᲣƠƔ̅ǘƣƴŴXƕ೅แႎ ƳƷƔƕЙܭưƖǔሀưƋǔŵ ƱƜǖƕŴƦǕƕܱᨥЈஹǔƷƩŴƱƍƏƷƕʻଐƷᛅƠƷɼ ȆȸȞưƋǔŵ

II _Შ Local pro-p Grothendieck Conjecture _ƱƷ᧙̞

(A.) [3]ƷɼܭྸƷࣄ፼: LJƣŴ[3]ƷɼܭྸᲢƷʻƷᛅƠƴ᧙ᡲƢǔཎКƳئӳᲣǛ࣬ƍЈƠ

ƯƓƜƏŵ

Theorem 2: Let K be as above. Let X_K →Spec(K) and X_K → Spec(K) be smooth, proper, geometrically connected curves over K of genus ≥ 2. Let ΔX (respectively, ΔX) 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,Δ_X)

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

“Outρ” denotes outer isomorphisms between the two groups in parentheses that are com- patible with the natural outer actions of Γ_K.

ƭLJǓŴቇҥƴƍƏƱŴ୺ዴXK Ʒӷ׹᫏ƸΔX ǁƷΓK Ʒٳᢿ˺ဇƩƚư᧙৖ႎƴൿLJǔŴ ƱƍƏϋܾƷܭྸưƋǔŵƱƍƏƜƱƸŴিǔॖԛưƸŴƜƷܭྸƸ(I.)Ʒ(D.)ư੩ឪƞǕƨ բ᫆ǁƷɟᆔƷሉƑǛଏƴ੩ᅆƠƯƍǔƷưƋǔŵƳƥƳǒŴTheorem 2ƴǑǔƱŴXK Ʒ ӷ׹᫏LJưƕΠ_XưଏƴൿLJƬƯƠLJƏƠŴƠƔNjž೅แႎſƳƷƔƲƏƔƸଢǒƔƴNjX_K Ʒӷ׹᫏ƴƠƔ̔܍ƠƳƍƷưŴƦǕưŴX_K ƕ೅แႎƳƷƔƲƏƔƕΠ_XǛᙸǔƩƚưЙܭ ưƖƨƜƱƴƳǔŵ

ƠƔƠŴƜƷǑƏƳሉƑƩƚưƸƪǐƬƱ฼ឱưƖƳƍŵƳƥƳǒŴLJƣŴЙܭඥƱƠƯ ƸŴƪǐƬƱ᧓੗ႎᢅƗǔƱƍƏƷNjɧ฼ƳໜƷƻƱƭƩƕŴNjƬƱٻƖƳբ᫆ໜƱƠƯŴƜ ƷǑƏƳሉƑƸ(I.)Ʒ(D.)ƷችᅕƔǒƸƣǕƯƍǔƷưƋǔŵ ƱƍƏƜƱƸŴিǔᘙྵΠ_X → GL₂(Zp)ƕɨƑǒǕƨƱƖŴƦǕƕƲǜƳ଺ƴκ_XƴƳǔƷƔŴƭLJǓŴƲǜƳ଺ƴcrystalline ƴƳǔƷƔŴǛ፭ᛯႎƴЙܭƞƤƯƘǕǔǑƏƳྸᛯƕഒƠƍƷưƋǔŵᲢܱƸŴXƕ೅แႎ ƴƳǔƨNJƴƸŴƦƷᘙྵƕcrystallineưƋǔƱƍƏவˑˌٳƴNj࠹ƭƔƷኬƔƍ২ᘐႎƳவ ˑǛ฼ƨƞƳƍƱƍƚƳƍƕŴƦǕǒƷவˑƸ˷ǓஜឋႎưƸƳƍƠŴƠƔNj፭ᛯႎƳᚕᓶƴ ƳƓƢƷƸൔ᠋ႎᐯଢưƋǔƷưŴƦƷᛅƠƸƜƜưƸႾဦƞƤƯNjǒƏŵᲣ

ƱƴƔƘŴΠ_XƷᘙྵƕƲǜƳƱƖƴcrystallineƳƷƔᲢᲷXɥƷp-divisible groupƔ ǒဃơǔƔᲣƷ፭ᛯႎƳЙܭඥƕഒƠƍƷƩƕŴƦƏƍƏЙܭඥƸܱƸŴTheorem 2ᐯ៲Ɣǒ ƸЈƳƍNjƷƷŴƦƷᚰଢƔǒƸЈǔƷưƋǔŵ

(B.) [3]Ʒࣄ፼: LƸ(K Ǜԃlj)pᡶ˳ᲢᲷฆ೅ૠưpᡶܦͳƳᩉ૝ႎ˄͌࿢OLƷՠ˳ᲣƱ ƠŴƦƷй˷˳ kLƕŴK Ʒй˷˳ kɥƷɟഏΨ᧙ૠ˳ƩƱˎܭƢǔŵƦƏƢǔƱŴӲᲢ᩼ᡚ

҄ƳᲣݧ

φ:Spec(L)→X_K

ƴݣƠƯŴƦǕƧǕƷૠᛯႎؕஜ፭Ʒ᧓ƴᛔݰƞǕǔᡲዓƳ፭แӷ׹αφ : ΓL → ΠX Ǜݣࣖƞ ƤǔƜƱƕưƖǔŵਁᝋႎƳᡲዓ፭แӷ׹α : Γ_L → Π_X ƴݣƠƯŴƦƏƍƏφƔǒဃơƨแ ӷ׹Γ_L → Π_X ǛŴˌɦž࠹˴ႎſƱԠƿŵ ƦƏƢǔƱŴিǔॖԛưƸŴ[3]ƷჇƷɼܭྸƸ ഏƷܭྸưƋǔᲴ

Theorem 3: αƕ࠹˴ႎƳƷƔƲƏƔƸŴܦμƴ፭ᛯႎƴЙܭưƖǔŵ

ƱƜǖƕŴž፭ᛯႎƴЙܭưƖǔſƱƋǔƕŴφ˳ႎƴŴƲƏƍƏ፭ᛯႎƳவˑưЙܭưƖǔ Ɣƴ᧙ƠƯƸŴ଺᧓Ʒ᧙̞ưƜƜưƸႾဦƢǔƕŴᛇƠƘƸ[3]ƷSection 7ƱSection 10Ǜ ƝӋༀɦƞƍŵ

ƍƬƨǜTheorem 3ǛᛐNJǔƱŴƦƜƔǒTheorem 2ǛЈƢƷƸŴᇹɟᛅƷFaltingsƷ

ྸᛯǛᢘဇƢǕƹܾତƴЈஹǔƜƱưƋǔŵƭLJǓŴƲƏƍƏαƕ࠹˴ႎƳƷƔƕЎƔǕƹŴ

࠹˴ႎƳα =αφǛƱƬƯƖƯŴഏƷನ঺ƕưƖǔŵ LJƣŴᇹɟᛅưƸRƱԠǜưƍƨNjƷƷ ƔǘǓǛƭƱNJǔƷƸŴƜƜưƸŴLƳǜƩƚƲŴRƱLƷ᧓ưƸž࿢ſƱž˳ſƷᢌƍƸஊƬ ƯNjŴЭƱμƘӷơǑƏƴŴLƷGalois cohomologyǛŴവǜƲǨǿȸȫਘٻǛ̅ƬƯᚘምƢ ǔƜƱƕưƖƯŴƦƏƢǔƱŴഏƷǑƏƳᐯ໱Ƴӷ׹ƕࢽǒǕǔᲴ

H¹(Δ_L, L^∧(1))∼= ΩL⊗LL^∧

ᲢƜƜưŴΔ_L ^def= Ker(Γ_L → Γ_K)ŵᲣ ɟ૾ŴX_K ƕžK(π,1)ſƴƳƬƯƍǔƱƍƏʙܱ

ƱŴᇹɟᛅƷTheorem 1ǛᢘဇƢǔƱŴ

H¹(ΔX, K^∧(1))∼=H_et¹(X_K, K^∧(1))∼= (H⁰(XK,Ω_XK/K)⊗KK^∧)⊕(H¹(XK,OX_K)⊗KK^∧(1)) Ƴǔᐯ໱Ƴӷ׹ƕ঺ǓᇌƭƜƱƕЎƔǔŵƱƜǖƕŴžࡽƖ৏ƠſƱƍƏદ˺ƴ᧙ƠƯƸŴ̊

Ƒ፭dzțȢȭǸȸưƋǖƏƕŴᡲ੗ޖƷZariskidzțȢȭǸȸưƋǖƏƕŴ˴NjƔNjƕᐯ໱Ƴƨ NJŴα =αφƴࡽƖឪƜƞǕǔݧ

H¹(Δ_X, K^∧(1))^ΓK →H¹(Δ_L, L^∧(1)) ᲢᏅƴƷƬƯƍǔžΓKſƸΓK-ɧ٭ᢿЎƷॖᲣǛLjǔƷƱŴ

H⁰(XK,Ω_XK/K)→ΩL⊗^LL^∧

ǛLjǔƷƱƕŴӷ͌ƩƱƍƏƜƱƴƳǔŵƱƜǖƕŴࢸᎍƷ૾ƸŴǑƘᎋƑƯLjǔƱŴ(ݲƳ ƘƱNjŴX_K ƕnonhyperellipticƳئӳƴƸᲣX_K → P ^def= P(H⁰(X_K,Ω_XK/K))ƱƍƏ೅

แႎ؈NJᡂLjƴƓƚǔφ ∈ XK(L) ⊆ XK(L^∧)ƷᘍƘέƷݧࢨࡈ೅ƴ˂ƳǒƳƍŵƠƔNjŴ φƕ᩼ᡚ҄ƳݧƱˎܭƠƯƍǔƨNJŴཎƴdominantƴƳǔƷưŴX_KǛŴφƷP ƴƓƚǔ΂

Ʒ᧍ѼƱƠƯࣄΨƢǔƜƱƕưƖǔŵƜǕưŴTheorem 2ǛᲢFaltingsƷྸᛯǛဇƍƯᲣThe-

orem 3ƔǒݰƚƨƜƱƴƳǔŵܱƸŴTheorem 3ᐯ˳NjƪǐƏƲƜƷǑƏƳᜭᛯưᚰଢƢǔ

ƷưƋǔŵ

(C.) ೅แࣱƷ፭ᛯႎЙܭඥᲴƠƔƠŴЭƴNjᚑǕƨǑƏƴŴʻଐƷᛅƠƷɼƳႸႎƸŴTheo-

rem 2ˌٳƴNjTheorem 3ƔǒЈǔ᩿ႉƍ࠙ኽƕƋǔƜƱǛਦઇƢǔƜƱƴƋǔŵƦǕƴƸŴ

FaltingsƷLJƨКƷܭྸǛݰλƢǔ࣏ᙲƕƋǔŵƦƷܭྸƷϋܾƱƸŴቇҥƴƍƏƱŴᘙྵλ :

ΠX → P GL₂(Z^p)ƴݣƠƯŴλƕXɥƷp-divisible group GƔǒဃơǔƨNJƴƸŴ (B.) ƴЈƯƖƨǑƏƳӲα_φ : Γ_L → Π_X ƴݣƠƯŴλƷΓ_LǁƷᲢαφƴǑǔᲣСᨂƕSpec(OL) ɥƷp-divisible groupƔǒဃơǕƹǑƍƜƱǛᚕƬƯƍǔŵƱƍƏƜƱƸŴƜƷܭྸƱŴ(B.) ƷTheorem 3ŴƦƠƯ୼ƴ(I.)Ʒ(C.)ǛኵLjӳǘƤǔƱŴഏƷǑƏƳ࠙ኽᲢ[3], Theorem 10.6, and [4], Chapter IV, Theorem 1.3ᲣƕЈǔᲴ

Theorem 4: In order that a curve X → Spec(A) = Spec(W(k)) (where k is a perfect field) be canonical, it is necessary and sufficient that there exist a representation κ_X : ΠX → P GL₂(Z^p) (whose corresponding ΠX-module we denote by V) such that: (i) the Γ_K-modules Hⁱ(Δ_X, Ad(V)) satisfy certain (not so important) properties (which we omit here for the sake of brevity); (ii) det(V)is the cyclotomic character; (iii) the restriction of κ_X to Γ_L with respect to every geometric α : Γ_L →Π_X arises from a p-divisible group of dimension 1 on Spec(O^L).

ұƪŴƪǐƏƲஓLjᡫǓƷŴ೅แࣱƷ፭ᛯႎЙܭඥƕưƖƨǘƚưƋǔŵ

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] Mochizuki, S.,A Theory of Ordinary p-adic Curves, RIMS Preprint 1033 (1995).

[5] Tate, J., p-divisible Groups, in Driebergen Conference on Local Fields, 1966 (T. A.

Springer, ed.), Springer-Verlag, Berlin, pp. 158-183 (1967).

[6] Fontaine, J. M., and Laffaille, G.,Construction de repr´esentations p-adiques,Ann. Sci.

Ec. Norm. Super. 15, pp. 547-608 (1982).