(X, ι)Ƹܱᙐእٶಮ˳ƱƠŴμX ƸŴӒദЩƳݣӳιƷ˺ဇƷɦưɧ ٭ưƋǓŴƔƭ(X, μ X def= μX|X)ƕSLSƱƳǔǑƏƳŴX ɥƷȪȸȞȳᚘ᣽ƱƢ ǔŵƢǔƱŴɥưܭ፯ƠƨXι x→αxƱƍƏݣࣖƴǑƬƯൿǇǔϙ΂ π0(Xι)→SectGal(C/R)(π1top(Xι)) ᲢƨƩƠŴӫᡀƸŴᲢπtop1 (X) ƴ᧙ƢǔσࢫǛᨊƍƯƷᲣπ1top(Xι

(1)

ܱᙐእٶಮ˳ƷǻǯǷȧȳʖेƱยעዴƷ࠹˴

ஓஉ ૼɟ Ტʮᣃٻܖૠྸᚐௌᄂᆮ৑Უ ᲬᲪᲪᲪ࠰Ჳஉ

FǛ೅ૠ0Ʒ˳ƱƠŴXF ǛFɥƷ๖ǒƔƳˊૠٶಮ˳ƱƢǔŵƜƷƱƖŴˮႻ࠹

˴ƴЈƯƘǔžؕஜ፭ſƷ᫏˩ƱƠƯŴˊૠٶಮ˳XF Ʒžˊૠႎؕஜ፭ſπ1^alg(XF) ƸŴᲫᲳᲰᲪ࠰ˊƴGrothendieckƴǑƬƯܭ፯ƞǕƨŵ˓ॖƷᘮᙴǛወСƢǔ୍ᡫƷ ˮႻ࠹˴Ʒؕஜ፭ƱᢌƬƯŴˊૠႎؕஜ፭ƸŴܦμƴˊૠႎƳחưਵƑǒǕǔƜƱƕ ӧᏡƳŴஊᨂഏǨǿȸȫᲢᲷɧЎޟᲣᘮᙴſǛወСƠƯƍǔŵ̊ƑƹŴXF def

= Spec(F) ƱƠƨƱƖŴ

π1^alg(Spec(F)) = Gal(F /F)

ᲢƭǇǓŴF ƷዌݣǬȭǢ፭ᲣƱƳǓŴɟ૾ŴF =C Ტᙐእૠ˳ᲣƱƠƨƱƖŴ π^alg1 (XF) = lim←−_N π1^top(XF(C))/N

ᲢƨƩƠŴN ⊆π1^top(XF(C))ƸŴᙐእٶಮ˳XF(C)ƷਦૠஊᨂƳദᙹᢿЎ፭ǛឥǔᲣ ƭǇǓŴXF(C)Ʒ୍ᡫƷؕஜ፭Ʒиஊᨂܦͳ҄ƴƳǔŵƳƓŴˮႻ࠹˴ܖƴЈƯƘ ǔțȢȈȔȸܦμኒЗƷ᫏˩ǋƋǓŴƦǕǛXF ƷನᡯݧXF →Spec(F)ƴᢘဇƢ ǔƱŴ

1→π^alg1 (X_F)→π1^alg(XF)→Γ_F →1

ᲢƨƩƠŴX_F ^def= XF ⊗F FŴΓ_F ^def= Gal(F /F)ᲣƷǑƏƳܦμኒЗƕưƖǔŵ ᚕƍ੭ƑǕƹŴɥƷཞඞǛቇҥƴǇƱǊǔƱŴ

XF → {π^alg1 (XF)→Γ_F}

ƱƍƏݣࣖᲢᲷǋƬƱദᄩƴƸŴž᧙৖ſᲣǛܭ፯ƢǔƜƱƕưƖƨŵƦƜưŴGrothen- dieckƸŴ1983࠰ƷFaltingsܮƷ৖ኡƷɶưŴഏƷǑƏƳᎋƑ૾Ǜ੩కƠƨᲴ

җЎƴ‘ૠᛯႎ’Ƴ˳F ⊆CƱŴҗЎƴӑ୺ႎᲢᲷhyperbolicᲣư ƋǓŴƔƭ(C͌ஊྸໜǛƱƬƨƱƖᲣˮႻႎƴƸ ‘K(π,1)’ƱƳƬ Ưƍǔٶಮ˳XF ƴݣƠƯŴƜƷ᧙৖ƸŴΪ฼ܱࣙƴƳƬƯƍǔ ưƋǖƏŵ

ƜƷࣱឋǛ฼ƨƢٶಮ˳XF ƸŴžᢒǢȸșȫ(Ჷanabelian)ſƱƍƏᲢྵ଺ໜư ƸŴ̔໱ƱƠƯૠܖႎƴӈ݅Ƴܭ፯ǛஊƞƳƍᲣӸᆅǛ˄ƚǒǕƨŵƨƩƠŴXF Ʒ

Typeset byAMS-TEX 1

(2)

2 ஓஉ ૼɟ Ტʮᣃٻܖૠྸᚐௌᄂᆮ৑Უ

ഏΨᲷᲫŴƭǇǓ୺ዴƷƱƖŴžӑ୺ႎƳ୺ዴƸᢒǢȸșȫưƋǖƏſƱƍƏૠܖႎƴ ӈ݅ƳॖԛǛǋƭʖेǛƨƯƯƍǔŵƜƷʖेƸŴ˳F ƕૠ˳ᲢᲷஊྸૠ˳QƷஊ ᨂഏਘٻᲣǍpᡶޅ৑˳ᲢᲷQp ƷஊᨂഏਘٻᲣƷǑƏƳᲢǍƸǓʖेƲƓǓᲣ‘ǍǍ ૠᛯႎƳ˳’ƷƱƖƴƸŴᲫᲳᲳᲯ᳸Ჰ࠰ƴྚ߷ܤᬱဏ൞Ʊᇿᎍ([Tama], [Mzk1,2]) ƴǑƬƯᏉܭႎƴᚐൿƞǕƯƍǔŵ

ܱᨥŴGrothendieckƕஇИƴʖेǛƨƯƨƱƖŴƲƏǋؕᄽ˳ƱƠƯŴؕஜႎ

ƴƸૠ˳ǛेܭƠƯƍƨǒƠƍƕŴ[Mzk2]ưƸŴpᡶޅ৑˳ƷɥưǋGrothendieck Ʒʖेƕ঺ǓᇌƭƜƱƕᅆƞǕƯƍǔŵƜƷǑƏƴŴؕᄽ˳ƕૠ˳ƷǑƏƳžٻ؏

˳ſưƸƳƘŴޅ৑˳ƷƱƖưǋ঺ǓᇌƭưƋǖƏƱᇿᎍƕ̮ơǔƴᐱƬƨɟဪƷఌ ਗƸŴpᡶޅ৑˳ǑǓƦƷನᡯǍૠᛯƕᢕƔƴЎƔǓǍƢƍŴܱૠ˳ᲢƱƍƏӷơޅ

৑˳ᲣƷɥưGrothendieck ʖेƷ᫏˩ƕ঺ǓᇌƭƱƍƏႆᙸưƋǔŵᲢܱૠ˳ɥ ƷࣇЎ࠹˴Ʊpᡶૠ˳ɥƷૠᛯ࠹˴Ʒ᧓Ʒ᫏˩ƴƭƍƯƸŴ[Mzk3], IntroductionƓ ǑƼ [Mzk4], Introduction, §0.10ưᛇƠƘᚐᛟƠƯƋǔŵᲣܱƸŴܱૠ˳ƷɥưƸŴ ૠ˳Ǎpᡶޅ৑˳ƷɥưჷǒǕƯƍǔƜƱǑǓǋƣƬƱࢍƍ࢟ưGrothendieckƷᢒ ǢȸșȫՋܖƕ঺ǓᇌƭƜƱƕЎƔƬƯƍǔŵƦƷǑǓࢍƍ࢟ƱƸŴஜᜒ๫ưኰʼƢ ǔžܱᙐእٶಮ˳ƷǻǯǷȧȳʖेſưƋǔŵ

ǇƣŴܱᙐእٶಮ˳ƱƍƏဇᛖǛܭ፯ƠƳƚǕƹƳǒƳƍŵܱᙐእٶಮ˳(X, ι) ƱƸŴᙐእٶಮ˳X ƱƦƷٶಮ˳ƴ˺ဇƢǔӒദЩƳݣӳ ι : X → X ƔǒƳǔኵ ǈƷƜƱưƋǔŵƜƷಒࣞƸŴܱૠ˳Ʒɥưܭ፯ƞǕƨᲢ๖ǒƔƳᲣˊૠٶಮ˳Ʒ ɟᑍ҄ưƋǓŴƦƷǑƏƳˊૠٶಮ˳ƴݣƠƯɟƭƷܱᙐእٶಮ˳ƕᐯ໱ƴܭǇǔ ƕŴˊૠႎƳǋƷƔǒႆဃƠƳƍŴžჇƴᚐௌႎƳſܱᙐእٶಮ˳ǋƋǔŵܱᙐእٶ ಮ˳(X, ι)ƕɨƑǒǕǔƱŴX ǛιƷ˺ဇưŴreal analytic stackᲢᲷreal analytic orbifoldᲣƷחƴƓƍƯлǔƜƱƴǑƬƯŴreal analyticƳstackXι ƕܭǇǓŴƦƷ Ტ୍ᡫƷˮႻ࠹˴ƷॖԛưƷᲣؕஜ፭ǛᎋƑǔƜƱƴǑƬƯŴɥƷˊૠႎƳᛅƴЈƯ

ƖƨܦμኒЗƷ᫏˩ƕưƖǔᲴ

1→π^top1 (X)→π1^top(Xι)→Gal(C/R)→1

ƳƓŴܱૠ˳ɥƷˊૠٶಮ˳ƷR͌ஊྸໜƷ᫏˩ǛŴ X^ι ^def= {x ∈X | ι(x) =x}

Ʊܭ፯ƢǔƱŴstackǇƨƸorbifoldƷܭ፯ǑǓŴX^ι ƷӲໜx∈X^ι ƸŴɥƷܦμኒ ЗƷᐯ໱Ƴsection

αx : Gal(C/R)→π1^top(Xι)

ǛŴᲢπ1^top(X)ƴ᧙ƢǔσࢫǛᨊƍƯᲣܭǊǔŵƠƔǋŴܾତƴᄩᛐƞǕǔǑƏƴŴ section αx ƸŴxƕ৑ޓƢǔX^ι Ʒᡲኽ঺Ў[x]∈π0(X^ι)ƩƚưൿǇǔŵ

ˌɦƷᜭᛯưƸŴӕǓৢƏܱᙐእٶಮ˳ƴݣƠƯഏƷǑƏƳࣇЎ࠹˴ܖႎƳவ ˑǛᛢƠƨƍŵǇƣŴX Ǜ˓ॖƷࣇЎٶಮ˳ƱƠŴμX ǛŴX ɥƷȪȸȞȳᚘ᣽ƱƢ ǔŵǋƠX ƷᲢ୍ᡫƷˮႻ࠹˴ƷॖԛưƷᲣ୍ᢄᘮᙴᆰ᧓ǛX →XƱ୿ƘƱŴμX

ǛXƴࡽƖ৏ƢƜƱƴǑƬƯX ɥƴᚘ᣽μ_XƕܭǇǓŴཎƴŴࣇЎٶಮ˳XƴᲢɟ

(3)

ܱᙐእٶಮ˳ƷǻǯǷȧȳʖेƱยעዴƷ࠹˴ 3

ƭƷᲣžȪȸȞȳ࠹˴ſƕλǔŵƦƷȪȸȞȳ࠹˴˄Ɩᆰ᧓ (X, μ _X) ƕžstraight line spaceſ ᲢˌɦưƸŴSLSƱဦƢᲣưƋǔƱƸᲴ

X Ʒ˓ॖƷႻီƳǔʚໜ x1, x2 ∈ X ƴݣƠƯŴƦƷʚໜǛኽƿ ยעዴƕŴӢɟƭ܍נƢǔŵ

ƱƍƏƜƱưƋǔŵ̊ƑƹŴ୍ᡫƷᚘ᣽ǛλǕƨƱƖŴᙐእ࠯᩿Ტᚘ᣽ᲷȦȸǯȪȃ ȉᚘ᣽ᲣǋɥҞ࠯᩿Ტᚘ᣽ᲷPoincar´eᚘ᣽ᲣǋSLSƴƳǔƕŴᲢ୍ᡫƷᚘ᣽λǓƷᲣ

ྶ᩿S²ǍȦȸǯȪȃȉᚘ᣽λǓƷɟໜ৷ƖƷᙐእ࠯᩿C\{0}ƸŴᲢƢƙᄩᛐƞǕǔ ǑƏƴᲣSLSƴƸƳǒƳƍŵ

ƞƯŴƜǕư࣏ᙲƳဇᛖƕ੤ƬƨƷưŴஜᜒ๫ƷɼܭྸᲢᛇƠƘƸ[Mzk5], The-

orem 3.6ǛӋༀᲣǛኰʼƠƨƍᲴ

Theorem A. (X, ι)Ƹܱᙐእٶಮ˳ƱƠŴμX ƸŴӒദЩƳݣӳιƷ˺ဇƷɦưɧ

٭ưƋǓŴƔƭ(X, μ _X ^def= μX|_X)ƕSLSƱƳǔǑƏƳŴX ɥƷȪȸȞȳᚘ᣽ƱƢ ǔŵƢǔƱŴɥưܭ፯ƠƨX^ι x→αxƱƍƏݣࣖƴǑƬƯൿǇǔϙ΂

π⁰(X^ι)→Sect_Gal(_C/R₎(π1^top(Xι))

ᲢƨƩƠŴӫᡀƸŴᲢπ^top1 (X) ƴ᧙ƢǔσࢫǛᨊƍƯƷᲣπ1^top(Xι) → Gal(C/R) Ʒ

sectionμ˳ǛᘙƢƱƢǔᲣƸμҥݧƴƳǔŵ

ƜƷܭྸƷˎܭǛ฼ƨƢχ׹ႎƳܱᙐእٶಮ˳ƱƠƯŴഏƷ̊ƕਫƛǒǕǔᲴ (1) ܱૠ˳ɥƷӑ୺ႎƳˊૠ୺ዴŵ

(2) ܱૠ˳ɥƷӑ୺ႎƳˊૠ୺ዴƷȢǸȥȩǤȷǹǿȃǯŵ (3) ܱૠ˳ɥƷǢȸșȫٶಮ˳ŵ

(4) ܱૠ˳ɥƷɼ͞ಊǢȸșȫٶಮ˳ƷȢǸȥȩǤȷǹǿȃǯŵ

ཎƴŴƜƷȪǹȈƷɶƴƸŴGrothendieckƷᢒǢȸșȫՋܖƕ঺ǓᇌƪƦƏƳٶಮ

˳ƸμᢿԃǇǕǔƷưƋǔŵ

ƜƷܭྸƷᚰଢƷƋǒƢơƸƩƍƨƍഏƷƱƓǓưƋǔŵǇƣŴXƷ୍ᢄᘮᙴᆰ᧓ XƕŴιƷ˺ဇưлƬƯ˺ƬƨorbifoldXιƷ୍ᢄᘮᙴᆰ᧓ưǋƋǔƜƱƔǒŴπ1^top(Xι) ƸXƴᐯ໱ƴ˺ဇƢǔŵɟ૾ŴቇҥƴᄩᛐưƖǔǑƏƴŴᨼӳSect_Gal(_C/R₎(π1^top(Xι)) ƸŴπ1^top(Xι)ϋƷˮૠ2ƷΨτ Ʒ̓ࢫ᫏Ʊᐯ໱ƴӷɟᙻƞǕŴᘮᙴƱ٭੭፭Ʒɟᑍ

ᛯǛဇƍǔƱŴܭྸƕɼࢌƢǔμҥݧࣱƸŴഏƷƜƱƱӷ͌ưƋǔƜƱǛᅆƢƜƱ ƕưƖǔᲴ

ƦƷǑƏƳˮૠ2ƷΨτ ƴ᧙ƢǔX Ʒɧѣໜᨼӳ X^τ ^def= {x∈X | τ(x) =x} ⊆X

(4)

4 ஓஉ ૼɟ Ტʮᣃٻܖૠྸᚐௌᄂᆮ৑Უ ƸŴᆰưƳƍŴᡲኽƳᨼӳƴƳǔŵ

ƦǕưƸŴǇƣᆰƴƳǒƳƍƜƱǛᚰଢƠǑƏŵx1 ∈ X Ǜ X ƷѨ৖ƳໜƱƠŴ x² ^def= τ(x¹)ƱƢǔŵǋƠx¹ = x²ƳǒƹŴx¹ ∈X^τ ƱƍƏƜƱƴƳǔƔǒŴX^τ ƕ ᆰƴƳǒƳƍƱƍƏᚰଢƸኳǘǔŵɟ૾Ŵx1 =x2 ƷƱƖŴX ƕSLSƴƳǔƱƍƏ ˎܭǛᢘဇƢǔƱŴx¹ Ʊx²ǛŴƋǔᲢuniqueƳᲣยעዴưኽƿƜƱƕưƖǔŵʻŴ ƦƷยעዴƷɶໜxǛƱǔƱŴᚘ᣽μ_Xƕπ1^top(Xι)Ʒ˺ဇƷɦưɧ٭ưƋǔƜƱƱŴ ݣӳτ ƕʚໜᨼӳ{x1, x2}Ǜ̬ƭƜƱƔǒŴτ ƕɶໜxǛǋ̬ƭƜƱƕŴยעዴƷɟ

ॖࣱǑǓႺƪƴࢼƏŵƭǇǓŴx∈X^τ ƱƍƏƜƱƴƳǔƔǒŴƜǕưX^τ ƕᆰƴƳ ǒƳƍƜƱƷᚰଢƕܦኽƢǔŵܱƸŴᡲኽࣱǋӷơǑƏƳᜭᛯƔǒ࠙ኽƞǕǔŵᲢǋ ƠX^τƕᡲኽưƳƔƬƨǒŴႻီƳǔᡲኽ঺ЎƴޓƢǔໜx1Ʊx2ǛᢠƿƱŴƦƷʚ ໜǛኽƿuniqueƳยעዴμ˳ƕŴᲢτ ƕμ_X Ǜ̬ƭƜƱƱŴτ(x¹) =x¹, τ(x²) =x² ǑǓᲣτ ƴ׍ܭƞǕƯƠǇƏŵƭǇǓŴยעዴƷໜμ˳ƕX^τ ƴλƬƯƍǔƜƱƴƳ ǔƔǒŴx¹ Ʊx² ƕX^τ ƷႻီƳǔᡲኽ঺ЎƴޓƠƯƍǔƱƍƏˎܭƴӒƢǔŵᚰ ଢኳŵᲣ

இࢸƴŴɥƷɶໜᜭᛯƩƕŴƜǕƸൿƠƯᇿᎍƕૼƠƘႆᙸƠƨǋƷưƸƳƘŴ

Teichm¨ullerྸᛯưƸƝƘ೅แႎƳ৖ඥưƋǓŴӷྸᛯƷഭӪƷɶưƸŴദƴƜƷǑ

ƏƳᜭᛯƕưƖǔǑƏƳཞඞǛૢƑƯƓƘƜƱƕŴTeichm¨ullerᆰ᧓ƷยעዴƷ࠹˴

ƷᄂᆮƷٻƖƳѣೞ˄ƚƱƞǕƯƖƨŵ

૨ྂ

[Mzk1] S. Mochizuki, The Proﬁnite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields, J. Math. Sci., Univ. Tokyo3 (1996), pp. 571-627.

[Mzk2] S. Mochizuki,The Local Pro-p Anabelian Geometry of Curves,Inv. Math. 138 (1999), pp. 319-423.

[Mzk3] S. Mochizuki, A Theory of Ordinary p-adic Curves,Publ. of RIMS 32 (1996), pp. 957-1151.

[Mzk4] S. Mochizuki, Foundations of p-adic Teichm¨uller Theory, AMS/IP Studies in Advanced Mathematics 11, American Mathematical Society/International Press (1999).

[Mzk5] S. Mochizuki,Topics Surrounding the Anabelian Geometry of Hyperbolic Curves, RIMS Preprint ??.

[Tama] A. Tamagawa, The Grothendieck Conjecture for Aﬃne Curves, Compositio Math. 109, No. 2 (1997), pp. 135-194.