ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ Grothendieck ʖे 中村博昭, 玉川安騎男, 望月新

ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ Grothendieck ʖे
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
ᘙ᫆Ʒ Grothendieck ʖेƱƸŴɟᚕưƍƏƱƢǕƹŴӑ୺ႎˊૠ୺ዴƷૠᛯႎؕஜ፭Ƹ୺ዴ
ƷˊૠನᡯLJư‫ܦ‬μƴൿNJƯƠLJƏŴƱƍƏʖेưƋǔŵƜƷբ᫆ƷᄂᆮƸŴᓸᎍƷɟʴ (ɶ஭)
ƴǑǓᲲᲪ࠰ˊƷ஛ƴႆᇢƕ᧏ƔǕŴNjƏɟʴ (ྚ߷) ƴǑǓᲳᲪ࠰ˊЭҞƔǒ (ദ೅ૠƷ‫ئ‬ӳǛ
ԃlj) ஜឋႎƳૼ‫᧏ޒ‬ƕNjƨǒƞǕŴƭƮƍƯஇࢸƷɟʴ (ஓஉ) ƴǑǓŴૼƠƍ (p ᡶႎƳ) ᚐ᣷
ǛЈႆໜƱƢǔஇኳႎƳᚐൿƕɨƑǒǕƨŵ
ƜƷᛯᛟưƸŴբ᫆ƷᏑ୎ǍഭӪƴƭƍƯቇҥƴࣄ፼ƠƨƋƱŴʖेƕɤʴƴǑƬƯഏᇹƴᚐ
ଢƞǕƯƍƬƨಮ‫܇‬Ǜ‫إ‬ԓƢǔŵ
§1. ૠᛯႎؕஜ፭ — ˊૠ࠹˴Ʊ፭ᛯƷ௚ƚ೛ —
§1.1. Ǩǿȸȫؕஜ፭
ᡫࠝƷžˮႻ࠹˴ႎƳؕஜ፭ſƸŴǑƘჷǒǕƯƍǔǑƏƴŴ‫࢟׋‬Ʒᡲዓ‫࢟٭‬ưɧ‫٭‬ƳŴƍǘ
ǏǔțȢȈȔȸɧ‫٭‬᣽ưƋǓŴ̊ƑƹdzȳȑǯȈƳᙐእˊૠ୺ዴưƸؕஜ፭ưൿLJǔƷƸƨƔƩ
ƔᆔૠƷLjưƋǔŵࢼƬƯƦƷLJLJưƸ̾ŷƷˊૠ୺ዴƷˊૠನᡯLJưൿNJǔDŽƲƷጞኬƳɧ‫٭‬
᣽ƴƸƳǓࢽƳƍŵܱᨥŴɥƷʖेưᎋƑƯƍǔૠᛯႎؕஜ፭ƸŴA. Grothendieck ƴǑǓ‫ݰ‬λ
ƞǕƨžǨǿȸȫؕஜ፭ſƷಒࣞǛNjƪƍƯžǬȭǢ፭ƷࡨᧈſƱƠƯᐯ໱ƴ‫ܭ‬፯ƞǕǔNjƷư
Ƌǔŵ
ƜƷಒࣞƸᲫᲳᲰᲪ࠰ˊƴ [SGA1] ƴƓƍƯˊૠ࠹˴ƴƓƚǔžǹǭȸȠƷǬȭǢྸᛯſǛ
ወСƢǔNjƷƱƠƯ‫ݰ‬λƞǕƨNjƷưƋǓŴƦǕƴǑǕƹŴᡲኽƳǹǭȸȠ X ƱƦƷɥƷˊૠ
᧍ໜ x̄ ƕɨƑǒǕƨƱƖŴǨǿȸȫؕஜ፭ π1 (X, x̄) ƸഏƷǑƏƳžᚐᨼӳſƷኒЗƷፗ੭፭Ʊ
ƠƯ‫ܭ‬፯ƞǕǔŵұƪ X ƷஊᨂഏǨǿȸȫᘮᙴ (ˌࢸƠƹƠƹஊᨂഏᘮᙴƱဦƢ) Y ǛμƯឥ
ǒƤǔƱƖŴx̄ ɥƷ fiber set Y (x̄) ƨƪƸஊᨂᨼӳƷ‫ࢨݧ‬ኒǛƳƢƕŴƜƷኒЗƷᐯࠁፗ੭μ
˳ƷƳƢ፭ƕ π1 (X, x̄) ưƋǔŵƜƷ፭ƴƸ̾ŷƷஊᨂᨼӳ Y (x̄) ɥƷፗ੭፭Ʒ‫ࢨݧ‬ಊᨂƱƠƯ
profinite ˮႻ፭1) Ʒನᡯƕᐯ໱ƴλǔƜƱƴදॖƠƯƓƘŵǨǿȸȫؕஜ፭ƷˮႻ፭ƱƠƯƷ
ӷ‫׹‬᫏Ƹ‫ܭ‬፯ƴྵǕǔؕໜ x̄ ƷӕǓ૾ƴ̔‫܍‬ƠƳƍƜƱƕǘƔǔƷưŴ፭ᛯႎƳನᡯƷLjƕբ
᫆ƱƳǔƱƖƠƹƠƹؕໜǛႾဦƠŴҥƴ π1 (X) ƷǑƏƴNj୿ƘƜƱƴƢǔŵ
˓ॖƷஊᨂഏᘮᙴ Y → X ƴ‫ݣ‬ƠƯŴY (x̄) Ƹ π1 (X, x̄) ƷᡲዓƳஊᨂፗ੭ᘙྵǛɨƑǔƕŴƜ
Ʒ‫ࣖݣ‬ƴǑǓŴX ƷஊᨂഏᘮᙴƷμ˳Ʊ π1 (X, x̄) Ʒᡲዓஊᨂፗ੭ᘙྵƷμ˳ƕ‫ח‬ӷ͌ƴƳǔŵཎ
ƴ (ᡲኽƳ Y ƴ‫ݣ‬ƠƯ) Y (x̄) ƷӲໜƴƓƚǔ‫҄ܭܤ‬፭Ƹ π1 (X, x̄) Ʒ᧏ᢿЎ፭2) (Ʒσࢫ᫏) Ǜ‫ܭ‬
NJŴᡞƴ π1 (X, x̄) Ʒ᧏ᢿЎ፭ H Ƹ߼й˷᫏ᨼӳǁƷፗ੭ᘙྵƴ‫ࣖݣ‬Ƣǔஊᨂഏᡲኽᘮᙴ Y → X
(Ʒӷ͌᫏) Ǜ‫ܭ‬NJǔŵƜƷ‫ ̞᧙ࣖݣ‬H ↔ Y ƸࢸƷᜭᛯưNjƠƹƠƹྵǕǔƷưŴY = Y H ,
H = HY ƱƍƏᚡӭư‫ࣖݣ‬ƢǔNjƷǛᘙƢƜƱƴƢǔŵཎƴ HY ƕ Y ᐯ៲Ʒؕஜ፭ π1 (Y ) Ʊ
ӷ‫׹‬ƴƳǔƱƍƏᚇ‫ݑ‬ƸؕஜႎưƋǔŵ
Typeset by AMS-TEX
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ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
ᆰ᧓ X ƕɟໜŴཎƴ˳ K ƷǹȚǯȈȩȠ Spec (K) ƱƠƯɨƑǒǕƨƳǒƹŴᡲኽƳஊ
ᨂഏᘮᙴ Y ɥƷ fiber set Y (x̄) Ƹ Y Ǜ‫ܭ‬፯Ƣǔˊૠ૾ᆉࡸƷᚐƷᨼӳƴDŽƔƳǒƣŴؕஜ፭
def
π1 (Spec (K)) ƸƋǒǏǔ૾ᆉࡸƷžᚐƷፗ੭፭ſƷዮ˳ưƋǔዌ‫ݣ‬ǬȭǢ፭ Gal(K) = Gal(K/K)
ƱӷɟᙻƞǕǔŵ(ƜƜư K Ƹ K ƷЎᩉ᧍ѼǛᘙƢŵ)
ɟᑍƴǹǭȸȠƷ‫ ݧ‬f : X1 → X2 ƓǑƼ X1 ɥƷˊૠ᧍ໜ x̄1 ƕɨƑǒǕǔƱŴx̄1 Ʒ΂
Ǜ x̄2 ƱƠƯᐯ໱Ƴแӷ‫׹‬π1 (X1 , x̄1 ) → π1 (X2 , x̄2 ) ƕࡽƖឪƜƞǕǔŵܱᨥŴஊᨂഏǨǿȸȫ
ᘮᙴ Y → X2 ƷࡽƖ৏Ơ (ȕǡǤȐȸᆢ) Y → X1 ƸǍƸǓஊᨂഏǨǿȸȫᘮᙴưƋǓŴࠝƴ
Y (x̄2 ) ∼
= Y (x̄1 ) ƕ঺ǓᇌƭŵƜƷƜƱƔǒŴ‫ݣ‬ᆅ፭ኒЗǛСᨂƢǔƜƱƴǑƬƯɥƷแӷ‫׹‬ƕ
ဃơǔƷưƋǔŵؕໜ x̄1 ƷӕǓ૾Ǜ‫٭‬ƑƯNjƜƷแӷ‫׹‬Ƹᢘ࢘Ƴӧ੭‫ࡸ׋‬ƷNjƱưӷ͌ƴƳƬ
ƯƠLJƏƷưŴˌɦƠƹƠƹؕໜǁƷᚕӏƸႾဦƠƯŴҥƴ π1 (X1 ) → π1 (X2 ) ƷǑƏƳ୿Ɩ૾
ǛƢǔƜƱƴƢǔŵ
X ƕ˳ K ɥ‫ܭ‬፯ƞǕƨˊૠ‫ٶ‬ಮ˳Ʒ‫ئ‬ӳƴƸŴᐯ໱ƴɨƑǒǕƯƍǔ‫ ݧ‬X → Spec (K) Ɠ
def
ǑƼ Spec K → Spec K ƱƔǒ‫ ݧ‬XK → X (XK = X ×K K) ƕᛔ‫ݰ‬ƞǕŴƜǕǒƔǒؕஜ፭Ʒ
‫ܦ‬μЗ
(1.1)
pr
−→ Gal(K) −→ 1
1 −→ π1 (XK ) −−−−→ π1 (X) −−−X
ƕဃơǔŵ‫ ࢨݧ‬prX ƷఋǛɨƑƯƍǔ፭ π1 (XK ) Ƹ X Ʒž࠹˴ႎſؕஜ፭ƱԠƹǕŴK ƕ೅ૠ
ᲪƷ‫ئ‬ӳƴƸŴ‫ࣖݣ‬Ƣǔᙐእ‫ٶ‬ಮ˳ƷᡫࠝƷˮႻ࠹˴ႎƳؕஜ፭Ʒ profinite ‫( ҄ͳܦ‬μƯƷஊᨂ
ՠ፭Ʒ‫ࢨݧ‬ಊᨂ) Ʊӷ‫׹‬ƴƳǔƷưŴཎƴ‫࢟٭‬ɧ‫٭‬᣽ưƋǔ3) ŵƦƜưžૠᛯႎſؕஜ፭ π1 (X)
Ʒ Gal(K)-ਘ‫ٻ‬፭ƱƠƯƷನᡯ (1.1) ƕᆰ᧓Ʒ‫࢟٭‬ƴ᧙ƠƯƲƏ‫҄٭‬ƢǔƔƕբ᫆ƱƳǔŵ
ɥƷ‫ܦ‬μЗ (1.1) ƴƓƍƯ π1 (XK ) Ƹ π1 (X) ƷദᙹᢿЎ፭ưƋǔƔǒŴσࢫǛƱǔƜƱƴ
ǑƬƯแӷ‫׹‬π1 (X) → Aut(π1 (XK )) ƕ‫ܭ‬LJǔŵଢǒƔƴƜƷแӷ‫׹‬Ƹ π1 (XK ) ǛƦƷϋᢿᐯࠁ
ӷ‫׹‬፭ƴϙƢƔǒŴՠǛƱǕƹŴGal(K) Ɣǒ࠹˴ႎؕஜ፭Ʒ‫ٳ‬ᢿᐯࠁӷ‫׹‬፭ Out(π1 (XK )) ǁ
Ʒแӷ‫ٳ( ׹‬ǬȭǢᘙྵƱƍƏ)
(1.2)
ρX : Gal(K) → Out(π1 (XK ))
ƕဃơǔŵʻƸ prX : π1 (X) → Gal(K) ƔǒЈႆƠƯρX : Gal(K) → Out(π1 (XK )) Ǜ፭ᛯႎƳ
દ˺ƴǑƬƯ‫ܭ‬፯ƠƨƕŴπ1 (XK ) Ʒɶ࣎ƕᐯଢƳ‫ئ‬ӳƴƸŴᡞƴ ρX Ɣǒ፭ᛯႎƳદ˺ƴǑƬ
Ư prX ǛࣄΨƢǔƜƱƕưƖǔŵ̊ƑƹŴ೅ૠᲪƷ˳ɥ‫ܭ‬፯ƞǕƨӑ୺ႎˊૠ୺ዴ (ұƪŴ๖ǒ
ƔƳˊૠ୺ዴưŴᆔૠǛ g Ŵ໯ᨂᢒໜƷૠǛ n ƱƢǕƹ (g, n) = (0, 0), (0, 1), (0, 2), (1, 0) ưƋ
ǔǑƏƳNjƷ) ưƸŴ࠹˴ႎؕஜ፭Ƹ᩼ǢȸșȫƳᐯဌ፭ƳƍƠ୺᩿፭Ʒ profinite ‫҄ͳܦ‬Ʊӷ‫׹‬
ƴƳǓŴƦƷɶ࣎ƸᐯଢưƋǔƜƱƕჷǒǕƯƍǔƷưƜƷ‫ئ‬ӳƴƋƯƸLJǔŵƜƷǑƏƳƱƖ
ƸŴ
žπ1 (XK ) ǁƷ‫ٳ‬ǬȭǢ˺ဇ ρX ſǛᎋƑǔƜƱƱžGal(K) Ʒਘ‫ٻ‬፭ π1 (X)ſǛᎋƑǔƜƱ
ƸӷሁƱƳǔƷưƋǔŵ
§1.2. Grothendieck ƷᢒǢȸșȫʖे
Grothendieck([G1-3]) ƕɼࢌƠƨƷƸŴ ˊૠ‫ٶ‬ಮ˳ X ƕ ‘ᢒǢȸșȫ (anabelian) ˊૠ‫ٶ‬ಮ
˳’ ƱԠƹǕǔŴӑ୺ႎˊૠ୺ዴǛԃljƋǔᆔƷˊૠ‫ٶ‬ಮ˳Ʒǯȩǹƴ‫ޓ‬ƠŴؕᄽ˳ K ƕእ˳ɥ
ஊᨂဃ঺ƳǒƹŴGal(K) Ʒਘ‫ٻ‬፭ƱƠƯƷ π1 (X) Ʒನᡯ (1.1) Ƹ X Ʒ࠹˴‫ܖ‬ǛdzȳȈȭȸȫ
ƢǔưƋǖƏŴƱƍƏႺज़ǛЈႆໜƱƢǔʖे፭ưƋƬƨŵ
LJƣᇹɟƴදॖǛƻƘƷƸŴGrothendieck ƕžᢒǢȸșȫˊૠ࠹˴ƷؕஜʖेſƱԠǜưƍ
ǔഏƷዮᛯႎɼࢌưƋǖƏŵ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
3
(GC1) ؕஜžʖेſ. እ˳ɥஊᨂဃ঺Ƴ˳ K ɥƷᢒǢȸșȫˊૠ‫ٶ‬ಮ˳ X Ƹૠᛯႎؕஜ፭
π1 (X) ƷˮႻ፭ನᡯƓǑƼ˄᨟Ƣǔμ‫ ݧ‬prX : π1 (X) → Gal(K) ƷನᡯƔǒ ‘ࣄΨ’ ưƖǔŵ
ƜƜưžᢒǢȸșȫˊૠ‫ٶ‬ಮ˳ſƸžǢȸșȫ፭ƔǒDŽƲᢒƍؕஜ፭ƴǑƬƯƦƷ࠹˴ƕወ
СƞǕǔſˊૠ‫ٶ‬ಮ˳ƱƍƏǑƏƳॖԛưŴGrothendieck ƴǑǓоЈƞǕƨᚕᓶưƋǔƕŴɟ
ᑍഏΨƴƓƚǔദᄩƳ‫ܭ‬፯Ǜ Grothendieck ƕ̬သƠƯƠLJƬƨƷƱŴɥƷ ‘ࣄΨ (reconstitute)
Ƣǔ’ ƷॖԛNjୱଭƳLJLJസƞǕƨƨNJŴྵ‫נ‬ưNj᭗ഏΨưƷʖेƷ঺ᇌር‫׊‬ǛɧᄩƔƳNjƷƱ
ƠƯƍǔ4) ŵƠƔƠ೅ૠᲪƷˊૠ୺ዴƴƭƍƯƸ Grothendieck ᐯ៲ƴǑǓഏƷǑƏƳଢᅆႎƳ
ʖेƕᇌƯǒǕƨŵ
(GC2) Hom ʖे. ஊྸૠ˳ɥஊᨂဃ঺Ƴ˳ K ɥƷӑ୺ႎˊૠ୺ዴ X, Y ƴ‫ݣ‬ƠƯ
HomK (X, Y ) → HomGal(K) (π1 (X), π1 (Y ))/ ∼
ƴǑƬƯૅᣐႎƳ K-‫ݧ‬ƨƪƱ Gal(K)-ɲᇌႎƳ᧏แӷ‫׹‬ϙ΂Ʒ (ӫƔǒƷ π1 (YK ) Ʒσࢫ˺ဇ
ƴǑǔ) ӷ͌᫏ƨƪƱƷ᧓Ʒɟ‫ݣ‬ɟ‫ࣖݣ‬ƕɨƑǒǕǔŵ(ƭLJǓŴؕஜ፭Ʒ᧏แӷ‫׹‬Ƹˊૠ࠹˴ႎ
Ƴ‫ݧ‬ƴဌஹƢǔNjƷƴᨂǔŵ)
Grothendieck ᐯ៲NjදॖƠƯƍǔǑƏƴŴɥƷʖेƸ G.Faltings ([F1]) ƴǑƬƯᚰଢƞǕƨ
Ǣȸșȫ‫ٶ‬ಮ˳ƷᲫഏΨǨǿȸȫȷțȢȭǸȸ፭ƴ᧙Ƣǔ Tate ʖे:
HomK (A, B) ⊗ Ẑ ∼
= HomGal(K) (H1 (AK , Ẑ), H1 (BK , Ẑ))
(ƜƜư A, B Ƹ‫ ˳؏ٻ‬K ɥ‫ܭ‬፯ƞǕƨǢȸșȫ‫ٶ‬ಮ˳ưŴẐ Ƹ Z Ʒ profinite ‫ )҄ͳܦ‬Ʊ᫏˩
ƠƯƍǔŵLJƨŴƜǕƱӷ଺ƴ Faltings ƴǑǓᚐൿƞǕƨ isogeny ‫( ྸܭ‬Ǎ Shafarevich ʖेሁ)
ǛဇƍǕƹŴȤdzȓ‫ٶ‬ಮ˳ǛᎋƑǔƜƱƴǑǓŴɨƑǒǕƨᆔૠ 2 ˌɥƷ proper ˊૠ୺ዴƱțȢ
ȭǸȸ፭ H1 ƕǬȭǢӷ‫׹‬ƱƳǔ୺ዴƸ᭗ŷஊᨂ̾ưƋǔƜƱƕႺƪƴࢼƏŵH1 ƕ π1 ƷǢȸș
ȫ҄ưƋǔƜƱƴදॖƢǕƹŴؕஜʖे (GC1) ƸŴțȢȭǸȸ፭Ɣǒؕஜ፭ƴऴ‫إ‬ǛࢍNJǔƜ
ƱƴǑƬƯŴӷơɧ‫٭‬᣽Ǜਤƭ୺ዴƷӧᏡࣱǛஊᨂ̾ƔǒᲫ̾ƴ጑NJǔƜƱǛᙲᛪƠƯƍǔƱᙸ
ǔƜƱƕưƖǔŵƨƩƠŴƜƷᆔƷஊᨂࣱ‫ྸܭ‬Ʒ effectivity Ƹ (̊‫ٳ‬ႎƳ‫ئ‬ӳ5) ǛᨊƍƯ) order
ƕᡦ૾NjƳƘ‫ٻ‬ƖƍƷƕ୍ᡫưƋǓŴɟᑍƴƸӑ୺ႎ୺ዴƷ Grothendieck ʖे (GC1)(GC2) Ʊ
ƦƷȤdzȓ‫ٶ‬ಮ˳Ʒ Tate ʖेሁƱƸƔƳǓƷᨠƨǓƕƋǔŵGrothendieck ƸࢬƷʖेƷఌਗ
ƱƠƯૠᛯႎؕஜ፭ π1 (X) ƕž‫ࠝݮ‬ƳǒƟǔбࣱſǛஊƢǔƜƱŴᚕƍƔƑǕƹƦƷૠᛯႎƳ
‘ՠ’ Gal(K) Ʒ ࠹˴ႎ ‘ᢿЎ’ π1 (XK ) ǁƷ‫˺ٳ‬ဇ (1.2) ƕž‫ࠝݮ‬ưƳƍDŽƲࢍƍſƸƣưƋǔƜ
ƱǛ (dzțȢȭǸȸྸᛯƴƓƍƯ A.Weil Ǎ P.Deligne ǒƴǑǓᚐଢƞǕƯƖƨǬȭǢᘙྵƷ᩼
ᐯଢࣱƱൔ᠋ƠƯ) ਫƛƯƍǔ ([G3])ŵ
இࢸƴNjƏɟƭŴƖƪǜƱ‫҄ࡸܭ‬ưƖǔ (சᚐൿ) ʖेƱƠƯᐻԛขƍNjƷƴഏƷ Section ʖ
ेƕƋǔŵ˳ K ɥƷˊૠ‫ٶ‬ಮ˳ X Ʒ K-ஊྸໜ x ∈ X(K) Ƹನᡯ‫ ݧ‬X → Spec K ƴ‫ݣ‬Ƣǔ
section ‫ ݧ‬x : Spec K → X ƷƜƱưƋǔƔǒŴK-ஊྸໜ x ƕɨƑǒǕǔƝƱƴɥƷؕஜ‫ܦ‬μЗ
(1.1) ǛЎᘷƞƤǔ section แӷ‫׹‬αx : Gal(K) → π1 (X) (Ʒπ1 (XK )-σࢫ᫏) ƕᛔ‫ݰ‬ƞǕǔŵ
(GC3) Section ʖे. (GC2) Ʊӷơ X/K ƴ‫ݣ‬ƠƯŴprX : π1 (X) → Gal(K) Ʒ section แӷ
‫ ׹‬α : Gal(K) → π1 (X) ƸŴX Ʒ K-ஊྸໜƴဌஹƢǔNjƷƱŴ‘໯ᨂᢒ K-ஊྸໜ’ ƴဌஹƢǔ
NjƷƴᨂǔ6) ŵ
4
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
໯ᨂᢒໜƴဌஹƢǔ (tangential) section ƴƭƍƯƸഏƷራưᚑǕǔŵLJƨ Hom ʖेƴƭƍ
ƯƸŴGrothendieck ([G2]) Ƹ୼ƴ X ƕɟᑍƷ๖ǒƔƳˊૠ‫ٶ‬ಮ˳ư Y ƕИሁႎᢒǢȸșȫˊ
ૠ‫ٶ‬ಮ˳ (ӑ୺ႎ୺ዴଈƷӒࣄႎ smooth fibration ƱƠƯࢽǒǕǔ‫ٶ‬ಮ˳) Ʒ‫ئ‬ӳƴਘࢌƢǔӧ
ᏡࣱƴƭƍƯᚕӏƠƯƍǔ7) ŵSection ʖेƸŴHom ʖेƴƓƍƯ X Ǜؕᄽ˳ƷǹȚǯȈȩȠ
ưፗƖ੭Ƒƨ‫ئ‬ӳƷ variant ưƋǔ8) ŵ୼ƴ X Ǎ Y ƕ᧙ૠ˳ƷǹȚǯȈȩȠƴƳƬƨ‫ئ‬ӳƴƭ
ƍƯNjᎋॾƕƋǓŴɲ૾ƱNjƦƏưƋǔƱƖǛžᢒǢȸșȫʖेƷӑஊྸ༿9) ſƱƠƯŴƜǕNj
ʖेƱƠƯƍǔŵ
§1.3. ૠᛯႎؕஜ፭ǛNJƙƬƯ
Grothendieck Ʒ૨୿ ([G1-3]) ƴƸŴ§1.2 ưኰʼƠƨǑƏƳʖेƷDŽƔƴNjŴˮႻ୺᩿ɥƷǰ
ȩȕ (dessin d’enfant) ƴǑǔૠ˳ɥƷˊૠ୺ዴƷወਙƷӧᏡࣱǍŴ୺ዴƷȢǸȥȩǤᆰ᧓Ʒૠᛯ
ႎؕஜ፭ƨƪƷ᧓Ʒ݅੗ƳႻʝ᧙̞ƷᚡᡓŴ‫ח‬ᛯႎƳૼƠƍᙻໜƴǑǔᆰ᧓ಒࣞƷ‫٭‬᪃ŴƳƲƷ
‫ٶ‬ƘƷ‫ٹ‬ƕᛖǒǕƯƍǔŵLJƨŴK = Q, X = P1 − {0, 1, ∞} ƴ‫ݣ‬Ƣǔ‫ٳ‬ǬȭǢᘙྵ (1.2) ƕҥ
‫ݧ‬ưƋǔƱƍƏᲫᲳᲱᲪ࠰ˊ஛Ʒ G.V.Belyi ([B]) ƷኽௐƸŴǬȭǢ፭Ʊؕஜ፭Ʒ᧙ǘǓƕ᭗ࡇ
ƴ᩼ᐯଢưƋǔχ‫̊׹‬ƱƠƯ‫ٶ‬ƘƷૠ‫ܖ‬ᎍƔǒදႸƞǕƨŵƦƷЭࢸƔǒŴ[G1-3] Ǎ˂Ʒ཯ᇌƳ
ឪเǛਤƭ࠹ƭNjƷૠᛯႎؕஜ፭ƴ᧙Ƣǔᄂᆮᛢ᫆Ǎசᚐൿբ᫆ƕ‫ݲ‬Ơƣƭ᧙ᡲƮƚǒǕƯᛐᜤ
ƞǕǔǑƏƴƳǓŴྵ‫נ‬NjLJƢLJƢ෇ႆƳᄂᆮƕᘍǘǕƯƍǔ (ǬȭǢƷᡞբ᫆ŴฆӳȢȁȸȕŴ
Ǣȇȸȫႎཎഷ᧙ૠŴGrothendieck-Teichmüller ፭ƳƲ)ŵƜǕǒ‫ٶ‬ૠƷ᣻ᙲƳɼ᫆ǍஇᡈƷᡶ
‫ޒ‬ƳƲƴƭƍƯŴஜᛯᛟưƸኡૠƷᣃӳɥŴ࠹ƭƔƷӋᎋ‫[( ୿׋‬1–6]) ƓǑƼɼᙲ‫إ‬૨ ([I2], [H])
ǛਫƛǔƴƱƲNJŴφ˳ႎƳϋܾƴᇌƪλǔƜƱƷưƖƳƍƜƱǛƓᛀƼဎƠɥƛƨƍŵɥƷǑ
ƏƳૠᛯႎؕஜ፭ǛNJƙǔૠ‫ٶ‬ƘƷᄂᆮᛢ᫆Ʒ‫ٻ‬೔ƷɶƴƋƬƯŴஜᛯᛟưৢƏžGrothendieck
ʖेſƸŴǍǍᚇࣞᛯႎƳᑥӳƍƷຜƍ௑ƷɟƭƱƍƏƜƱƕЈஹǔưƋǖƏŵ
§2. ஊᨂࣱ‫ྸܭ‬Ɣǒбࣱ‫ྸܭ‬ǁ (ɼƴ X :ᆔૠᲪ, K :ˊૠ˳Ʒ‫ئ‬ӳ)
§2.1. Anderson-˙ҾƷ‫ྸܭ‬
Grothendieck ʖेǁƷǢȗȭȸȁƸŴૠᛯႎؕஜ፭Ʒਘ‫ٻ‬ನᡯ (1.1) ǍƦƜƔǒဃơǔ‫ٳ‬Ǭ
ȭǢᘙྵ (1.2) ƷƲƜǛᙸǕƹΨƷᆰ᧓ƷˊૠನᡯƕᑣƘӒପƞǕƯƍǔƔŴƱƍƏբ᫆ǛᎋƑ
ǔƜƱƔǒ‫ڼ‬LJǔŵƱƜǖưእૠ l Ǜ‫ܭ׍‬ƠƨƱƖŴ‫ٳ‬ǬȭǢᘙྵ (1.2) Ƹ π1 (XK ) Ʒஇ‫ ٻ‬pro-l
(l)
ՠ፭10) π1 (XK ) ƷɥǁƷ‫˺ٳ‬ဇ
(l)
(l)
ρX : Gal(K) → Out(π1 (XK ))
ǛNjᐯ໱ƴࡽƖឪƜƠƯƍǔŵᲫᲳᲲᲪ࠰ˊƴ˙Ҿࡍᨙ ([I1]) ƸŴƦǕˌЭƷᄂᆮኺዾƔǒ
Grothendieck, Deligne ሁƱƸ཯ᇌƴ X = P1 − {0, 1, ∞} ƴ‫ݣ‬Ƣǔ pro-l ‫ٳ‬ǬȭǢᘙྵƷᄂᆮǛ
о‫ڼ‬ƠŴƦƷขƍૠᛯࣱ (ȤdzȓԧŴόҥૠƱƷ᧙ǘǓ) ǛଢǒƔƴƠƨ11) ŵƦƠƯƜƷ˙ҾƷ
঺ыǛ‫ڎ‬ೞƴŴƦƷࣖဇǍ˂ƷƍǖƍǖƳ୺ዴǁƷɟᑍ҄ƷแͳƳƲƕɼƴǘƕ‫׎‬ƷᄂᆮᎍǛɶ
࣎ƴᡶNJǒǕƨ12) ŵ
ᲲᲪ࠰ˊࢸҞƴƸŴଏƴഏƷǑƏƳʙܱƕ G.Anderson-˙Ҿ ([AI]) Ʒɟ‫ྸܭ‬ƱƠƯჷǒǕƯ
ƍƨŵஊᨂᨼӳ Λ ⊂ P1 (K) ƕ 0, 1, ∞ ǛԃljƱƖŴ
(l)
‫( ྸܭ‬Anderson-˙Ҿ). ᆔૠᲪƷ୺ዴ X = P1K − Λ ƴ‫ݣ‬Ƣǔ pro-l ‫ٳ‬ǬȭǢᘙྵρX ƷఋƷ‫ܭ׍‬
(l)
˳ KX Ƹᨼӳ Λ ƔǒᙐൔƱ l ʈఌǛӕǔદ˺ǛጮǓᡉƠƯဃơǔˊૠႎૠǛμƯؕᄽ˳ K ƴ
ชьƠƯࢽǒǕǔਘ‫˳ٻ‬ưƋǔŵ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
(l)
5
(l)
ƜƷ‫ྸܭ‬ƸŴpro-l ‫ٳ‬ǬȭǢᘙྵ ρX Ɣǒᐯ໱ƴဃƣǔ K ƷᢿЎ˳ KX ǛŴЎ‫ޟ‬ໜᨼӳ
Λ ⊂ P1 (K) Ʒࡈ೅Ɣǒɟ‫ܭ‬Ʒ৖᪯ưဃ঺ƞǕǔžૠſƷኒЗƴǑƬƯᚡᡓƢǔNjƷưƋǔƕŴ
(l)
ᙸ૾Ǜ‫٭‬ƑǔƱŴૠᛯႎؕஜ፭ π1 (X) Ʒ Gal(K)-ਘ‫ٻ‬፭ƱƠƯƷನᡯƔǒ KX ƱƍƏ K Ʒᢿ
Ў˳ƴ͌Ǜਤƭ፭ᛯႎɧ‫٭‬᣽ǛŴእૠ l ƝƱƴᚡᡓႎƴನ঺ƠƯƍǔƱNjLjƳƤǔŵɶ஭ƸŴƜ
ƷǑƏƳ K ƷᢿЎ˳ƱƠƯ‫ܭ‬፯ƞǕǔૠᛯႎؕஜ፭Ʒɧ‫٭‬᣽ǛǑǓኒወႎƴನ঺ưƖǕƹŴƦ
ǕǒƴǑƬƯᆔૠᲪƷˊૠ୺ዴǛኬƔƘғКƢǔƱƍƏ࢟ư Grothendieck ʖेǁƷǢȗȭȸȁ
ƕЈஹǔƔNjƠǕƳƍŴƱᎋƑƨŵ
§2.2. ǬȭǢፗ੭Ʒž፭ᛯႎſᚡᡓ
Anderson-˙ҾƷ૾ඥƸŴ π1 (P1 − Λ) ƴƓƚǔ pro-l ‫ٳ‬ǬȭǢᘙྵǛŴ P1 − Λ Ʒ “pro-l ୍
ᢄᘮᙴƷጂ” ƴЎࠋƢǔ Λ ɥƷ “pro-cusp” ໜƨƪƷǬȭǢፗ੭ƷᚕᓶƴᎇᚪƠŴƦǕǛᆔૠᲪ
(Ʒ୺ዴƨƪƷƳƢ) ᘮᙴ‫ذ‬Ʒऴ‫إ‬ƴ࠙ბƢǔƱƜǖƕȝǤȳȈƴƳƬƯƍǔŵƦƜưLJƣŴžஊ
ᨂഏᘮᙴƷ cusp ໜƨƪƕǬȭǢፗ੭ƞǕǔſƱƍƏ࠹˴ႎƳྵᝋǛžૠᛯႎؕஜ፭Ʒ (ਘ‫ )ٻ‬፭
ನᡯſƷLjǛဇƍǔ፭ᛯႎƳᚕᓶƴᎇᚪƢǔƜƱǛᎋƑǔŵ
ɟᑍƴ X Ǜ K ɥ‫ܭ‬፯ƞǕƨ (˓ॖƷᆔૠƷ) affine ӑ୺ႎ୺ዴƱƠŴY ǛƦƷஊᨂഏᘮᙴŴ
def
Y ∗ ǛƦƷ᩼ཎီdzȳȑǯȈ҄ƱƢǕƹŴY Ʒ cusp ໜᨼӳƱƸΣY = Y ∗ − Y ƷƜƱưƋǔŵLJ
ƣŴᘮᙴ Y Ʒᐯ໱Ƴ‫ܭ‬፯˳ƸŴ‫ࣖݣ‬Ƣǔ ᧏ᢿЎ፭ HY = π1 (Y ) Ʒ prX : π1 (X) → Gal(K) ƴǑ
ǔ΂Ʒ‫ ˳ܭ׍‬KY ƱƠƯࢽǒǕǔŵLJƨŴY Ʒ࠹˴ႎؕஜ፭ƸσᡫᢿЎ HY ∩ π1 (XK ) ƱƠƯࣄ
ΨƞǕǔŵƜǕƸŴY ∗ Ʒᆔૠƕ gY ŴΣY (K) Ʒຜࡇƕ nY ưƋǔƱƢǕƹŴ᨞ૠ 2gY + nY − 1
Ʒ᩼Ǣȸșȫᐯဌ profinite ፭ưƋǔŵƦƷஇ‫ ٻ‬pro-l ǢȸșȫՠƱƠƯ lᡶǨǿȸȫȷțȢȭ
(l)
Ǹȸ፭ H1 (YK , Zl ) (= π1 (YK )ab ) ƕࢽǒǕŴσࢫǛӕǔƜƱƴǑǓŴƦƜƴƸ Gal(KY )-ь፭
ƱƠƯƷನᡯNjλƬƯƍǔŵƜƜƴŴcusp ໜᨼӳƴƓƚǔόЎፗ੭ᘙྵ13) ƕ (᨞ૠ nY − 1 Ʒ
ᢿЎь፭ƱƠƯ) DŽDžλƬƯƍǔƷưŴƜǕǛ፭ᛯႎƴӕǓЈƤǕƹǑƍŵƦǕǛ̬ᚰƢǔƷƕ
Riemann-Weil ʖेưƋǔŵܱᨥŴբ᫆Ʒ ‘cusp ᢿЎ’ ƴǑǔ H1 (YK , Zl ) Ʒՠь፭Ƹ᨞ૠ 2gY
ƱƳǓŴY ∗ ƷȤdzȓ‫ٶ‬ಮ˳Ʒ l-ࠉЎໜƨƪƷƳƢ l ᡶ Tate ь፭Ʊӷ‫׹‬ưƋǔŵRiemann-Weil
ʖेƸŴƦƜƴƓƚǔ Gal(KY ) Ɣǒဃơǔ Frobenius ˺ဇƷ‫׍‬ஊ͌Ҟࢲ (weight) ƕŴόЎ˺ဇ
Ʒ‫ئ‬ӳƱǹDZȸȫƕီƳǔƜƱǛƍƬƯƍǔƷưŴH1 (YK , Zl ) Ʒ cusp ᢿЎƕ፭ᛯႎƴᜤКƞǕ
ǔƜƱƴƳǔŵ
§2.3. ஊᨂࣱ‫[( ྸܭ‬N1])
√
ƦƜư̊Ƒƹ X = P1K − Λ (Λ ⊃ {0, 1, ∞}) Ʒᘮᙴ Y ƷƏƪŴ˳ K( N 1) Ǜ‫ܭ‬፯˳ƱƠŴ
XK( N√1) ɥ (Z/N Z)|Λ|−1 ǛǬȭǢ፭ƱƢǔǬȭǢᘮᙴƱƳƬƯƍǔNjƷǛᎋƑǔŵƦƠƯӲ
H1 (YK , Zl ) Ʒ cusp ᢿЎƴƓƚǔǬȭǢᘙྵƷఋƷ‫˳ܭ׍‬ƷŴɥƷǑƏƳ Y Ʊእૠ l ƴǘƨǔ
σᡫᢿЎǛᚘምƠƯLjǔƱŴ˳ K((λ − λ )1/N | λ, λ ∈ Λ − {∞}) ƕЈƯƘǔŵƜǕƸ፭Ʒμ‫ݧ‬
prX : π1 (X) → Gal(K) Ɣǒᐯ໱ૠ N ǛɨƑǔƝƱƴ፭ᛯႎƴਁЈƞǕǔɧ‫٭‬᣽ưƋǔŵ୼ƴ
N ǛѣƔƤƹŴቇҥƳǯȳȞȸᜭᛯ (Ʊҥૠ፭Ʒஊᨂဃ঺ࣱ) ƴǑǓŴʈඥ፭ K × ƷƳƔưஊᨂ
ᨼӳ {λ − λ | λ, λ ∈ Λ − {∞}, λ = λ } ưဃ঺ƞǕǔᢿЎ፭ƕɧ‫٭‬᣽ƴƳǔƜƱƕǘƔǔŵƜ
Ʒɧ‫٭‬᣽ƴǑƬƯŴӷơૠᛯႎؕஜ፭ǛɨƑǔΛ ⊂ P1 (K) ƷӕǓ૾ƕ (ɟഏЎૠ‫٭‬੭ǛඥƱƠ
Ư) ஊᨂᡫǓƠƔƳƍƜƱŴLJƨ K ƕཎКƳˊૠ˳Ʒ‫ئ‬ӳƴƸؕஜ፭ (ƷȡǿȷǢȸșȫ҄) ƕ
P1 − { Ხໜ } ǛൿNJƯƠLJƏƜƱƳƲƕᅆƞǕǔŵ
§2.4. бࣱ‫ྸܭ‬
ǑǓјྙǑƘ cusp ໜƨƪƷǬȭǢፗ੭Ʒऴ‫إ‬ǛМဇƢǔƨNJƴŴʻࡇƸπ1 (X) ƷμƯƷ᧏ᢿ
6
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
Ў፭ H ǛȑȩȡȸǿƱƠƯឥǒƤƯŴ‫ࣖݣ‬Ƣǔᘮᙴ Y H ƴ˄᨟Ƣǔ cusp ᢿЎ ⊂ H1 ((Y H )K , Zl )
Ʒऴ‫إ‬ǛμᢿᨼNJƯƘǔƜƱǛᎋƑǔŵƜƷ cusp ᢿЎƸ (§2.2 ưᡓǂƨǑƏƴ) weight filtration
ƴǑƬƯཎࣉƮƚǒǕƯƍǔƔǒŴμƯƷ᧏ᢿЎ፭ H ƓǑƼእૠ l ƴǘƨƬƯ I ∩ H ƕࠝƴƦ
Ʒ cusp ᢿЎƴᓳƪǔǑƏƳπ1 (XK ) Ʒ߹‫ׅ‬ᢿЎ፭ I Ʒӳ́Ǜ˺ǕƹŴGal(K)-ਘ‫ٻ‬፭ π1 (X) Ɣ
ǒ፭ᛯႎƴӕǓЈƤǔ π1 (XK ) ƷᢿЎᨼӳƕࢽǒǕǔŵ
ƜƷᢿЎᨼӳƕ ‘cuspidal isotropy ᢿЎ፭’ Ʒ (σࢫ᫏) ԧᨼӳƴႻ࢘ƢǔNjƷŴƍƍƔƑǕ
ƹ X Ʒ໯ᨂᢒໜ (ұƪ X ᐯ៲ƴ‫ݣ‬Ƣǔ cusp ໜ) ƴƓƚǔ π1 (XK ) ϋƷॄࣱ፭ (∼
= Ẑ) μ˳Ʒ
ӳ́ƴƳǔƜƱƕᚰଢƞǕǔ (ᢒǢȸșȫ weight filtration [N2,4])ŵӲॄࣱ፭Ʒദᙹ҄፭ƱƠƯ
π1 (X) ϋƷЎᚐ፭ƕࣄΨƞǕǔƕŴƜƷЎᚐ፭Ʒɶƴ΂ǛਤƭǑƏƳ prX : π1 (X) → Gal(K)
Ʒ section แӷ‫ ׹‬α : Gal(K) → π1 (X) ƨƪǛ X Ʒ໯ᨂᢒ K-ஊྸໜƔǒႆƢǔ tangential
section ƱԠƿŵGrothendieck Ʒ Section ʖे (GC3) ưբ᫆ƱƞǕǔ section แӷ‫׹‬ƷƏƪ໯
ᨂᢒ K-ஊྸໜƴဌஹƢǔNjƷƴ᧙ƠƯƸŴƜƷǑƏƴƠƯ፭ᛯႎƳཎࣉƮƚƕɨƑǒǕǔŵ
ƞƯŴॄࣱ፭ (Ǎ‫ࣖݣ‬ƢǔЎᚐ፭) Ʒᨼӳƕ፭ᛯႎƴཎࣉƮƚǒǕƨƨNJŴ̊ƑƹʚƭƷ୺ዴ
X1 , X2 /K Ʒૠᛯႎؕஜ፭Ʒ᧓ƴ Gal(K) ɥƷӷ‫ ׹‬π1 (X1 ) ∼
= π1 (X2 ) ƕɨƑǒǕƨƳǒƹŴƦ
Ʒ፭ӷ‫׹‬Ƹᐯѣႎƴॄࣱ፭ƷᨼӳǛ̬ƬƯƍƳƘƯƸƳǒƣŴƔƭӲŷƴ‫ࣖݣ‬Ƣǔ໯ᨂᢒໜƷй
˷˳Nj (Ўᚐ፭Ʒ prX ƴǑǔ΂Ʒ‫˳ܭ׍‬ƱƠƯ) ̬ƨǕǔƜƱƴƳǔŵࢼƬƯ X1 ƕ X1 ƴ᧏؈
λƞǕǔƳǒŴX2 Njπ1 (X1 ) ∼
= π1 (X2 ) ƱƳǔ X2 ƴ᧏؈λƞǕƳƘƯƸƳǒƳƍŵཎƴŴᆔૠ
1
ᲪƷ୺ዴ P − {n ໜ } ƷࣄΨƷբ᫆Ƹ P1 − { Ხໜ } Ʒ‫ئ‬ӳƴ࠙ბƞǕǔƜƱƴƳǔŵ୼ƴ࠹˴
ႎƳ߹‫ׅ‬ᘮᙴưᲬໜưƷLjЎ‫ޟ‬ƢǔǑƏƳNjƷƕؕஜ፭Ʊॄࣱ፭Ʒᚕᓶưཎ‫ܭ‬ưƖǔƜƱǛМဇ
ƢǔƱŴ[N1] Ʒ৖ඥǛੲဇƠƯπ1 (P1 − {0, 1, ∞, λ}) Ʒɧ‫٭‬᣽ƱƠƯ K × ƷʈඥᢿЎ፭Ʒɤƭ
λ
ኵ λ, 1 − λ, λ−1
ǛӕǓЈƤǔƜƱƕǘƔǔŵƜǕƕܱᨥ P1 − {0, 1, ∞, λ} Ʒӷ‫׹‬᫏Ǜཎ
ࣉƮƚǔŵƜƷǑƏƴƠƯ Q ɥஊᨂဃ঺Ƴ˳ɥƷᆔૠᲪƷӑ୺ႎˊૠ୺ዴƸؕஜ፭Ɣǒ ‘ࣄΨ’
ƞǕǔƜƱǛᅆƢƜƱƕưƖǔ ([N2])ŵ
LJƨŴॄࣱ፭ƷཎࣉƮƚƕɟᑍƷᆔૠƷ affine ୺ዴƷ pro-l ؕஜ፭ưNjӧᏡƳƜƱƔǒŴ
Grothendieck ʖेǛ pro-l ؕஜ፭ƴ‫ݣ‬ƠƯ‫҄ࡸܭ‬ƠƳƓƠƨNjƷƴŴ፭Ʒᨀɶ࣎ЗƳƲǛᡫƠƯ
‘‫ܭ‬᣽ႎƴ’ ǢȗȭȸȁƢǔӧᏡࣱƕဃLJǕƨŵཎƴ pro-l ؕஜ፭Ʒᐯࠁӷ‫׹‬፭Ʒᚸ̖Ʒբ᫆ƴཎഷ
҄ƢǔƱŴ᭗ᆔૠƷ୺ዴǍƦƷᣐፗᆰ᧓Ʒ pro-l ‫ٳ‬ǬȭǢᘙྵƷჷᜤǛᆔŷƷ filtration Ʒ૾ඥư
ኵLjӳǘƤƯŴᏉ‫ܭ‬ႎኽௐǛ࠹ǒƔኵጢႎƴࢽǔƜƱƕưƖǔ (ᛯᛟ [N7] Ǎ [NTs],[NTa],[MT] Ƴ
ƲǛӋༀ)ŵƜƷǑƏƳŴؕஜ፭Ʒᐯࠁӷ‫׹‬፭Ɣǒ୺ዴƷƦǕǛࣄΨƢǔբ᫆ƸŴ(GC2) Ʒ Isom
༿ǁƷǹȆȃȗƷɟᢿЎƱLjƳƢƜƱNjưƖǔŵƠƔƠŴǑǓஜឋႎƳ৙᧏ǛɨƑǔƴƸŴLJƣ
ྚ߷Ʒˁʙ (§3) ǛࢳƨƳƚǕƹƳǒƳƔƬƨŵ
ƱƜǖưɥƴƓƍƯŴ໯ᨂᢒໜƔǒဃƣǔ tangential section ƴ‫ݣ‬ƠƯᘍƳƬƨǑƏƳŴ
ˊૠ୺ዴƷૠᛯႎؕஜ፭Ʒਘ‫ٻ‬፭ನᡯ prX : π1 (X) → Gal(K) ƴ‫ݣ‬Ơ
˴ǒƔƷ࠹˴ႎƳ঺‫׆‬ǛਤƭǑƏƳ section แӷ‫ ׹‬α : Gal(K) → π1 (X)
Ǜ (˂ƷɟᑍƷ section แӷ‫׹‬Ɣǒ) ፭ᛯႎƴᜤКƢǔŴ
ƱƍƏǹȆȃȗƸŴˌࢸƷྚ߷ǍஓஉƷˁʙƴƓƍƯNjσᡫƠƯᘍǘǕǔᢿЎƱƳƬƯƍǔŵƦ
ƷᨥŴπ1 (X) Ʒ᧏ᢿЎ፭ H ǛȑȩȡȸǿƷǑƏƴឥǒƤƯŴ‫ࣖݣ‬Ƣǔᘮᙴ୺ዴ Y H ƨƪƷǨ
ǿȸȫȷdzțȢȭǸȸƴԃLJǕǔૠᛯ࠹˴ႎऴ‫إ‬ǛžᢒǢȸșȫႎƴſᨼኖƞƤǔŴƱƍƏ৖ඥ
ƸŴ೅แႎƴဇƍǒǕǔǑƏƴƳƬƨŵƨƩƠŴƞLJƟLJƳૠᛯႎᚨ‫ܭ‬ƷNjƱưŴY H ƷƲƷǑ
ƏƳ࠹˴ႎऴ‫إ‬ǛǨǿȸȫȷdzțȢȭǸȸƷɶƴᙸЈƠŴƲƷǑƏƴƦǕǒǛᨼኖƞƤƯኽௐƴ
ኽƼƭƚǔƔƱƍƏໜƸŴǑǓች߽Ƴ২ᘐƱૼᮗƳႆेǛᙲƢǔ‫ٻ‬ƖƳբ᫆ưƋǔŵˌɦƷӲራ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
7
ƴƓƍƯŴƜƷǑƏƳբ᫆ƴ‫ݣ‬Ƣǔྚ߷ȷஓஉƷŴƦǕƧǕƷૠᛯႎཞඞƴƓƚǔǢǤȇǣǢƷ
‫᧏ޒ‬ǛᚐᛟƢǔƕŴ‫ٳᧉݦ‬ƷᛠᎍƷ૾ƴNjŴƦƷࡁƴ්ǕǔσᡫƷբ᫆ॖᜤƷ࠼ƕǓ૾Ǜ (‫ݲٶ‬
ƳǓƱNj) ԛǘƬƯ᪬ƚǔǑƏƴŴɟഩɟഩᇿǛᡶNJǔƜƱƴƠǑƏŵ
§3. ദ೅ૠˊૠ୺ዴƷؕஜ፭Ʊ Grothendieck ʖे
§3.1. ஊᨂ˳ɥƷ Grothendieck ʖे
ƜƷራưƸŴk Ǜஊᨂ˳ƱƠŴX ǛƦƷɥƷ (᩼ཎီ) affine ୺ዴƱƢǔŵྚ߷ ([T1]) Ʒɼኽ
ௐƷɟƭƸŴǹǭȸȠ X ƕ π1 (X) ƔǒࣄΨƞǕǔƜƱ (ǑǓദᄩƴƸ (GC2) Ʒ Isom ༿Ʒ᫏
˩) ǛᅆƠƨƜƱưƋǔŵƦƷᚰଢƷ૾ᤆƸŴ᧙ૠ˳ k(X) ƕƦƷዌ‫ݣ‬ǬȭǢ፭ Gal(k(X)) Ɣǒ
ࣄΨƞǕǔƜƱǛᅆƠƨϋဋᐻʚƷˁʙ ([U]) ǛȢȇȫƴƠƯƓǓŴ‫ٻ‬ƖƘɤƭƷǹȆȃȗƴЎ
ƔǕǔᲴ
(i) X ∗ ƷӲ᧍ໜƴ‫ݣ‬ƢǔЎᚐ፭Ʒ፭ᛯႎཎࣉƮƚᲵ
(ii) ʈඥ፭ k(X)× ƷࣄΨᲵ
(iii) k(X) = k(X)× ∪ {0} ƷɥƷьඥನᡯƷࣄΨŵ
ƜƜưŴ§2 ƷᡫǓŴX Ʒ᩼ཎီdzȳȑǯȈ҄Ǜ X ∗ ƱᚡƢŵ
ǹȆȃȗ (i) ưƸŴϋဋƸ Brauer ፭Ǜဇƍƨ Neukirch ƷǢǤȇǣǢǛ̅ƬƨƕŴǘǕǘǕ
Ʒ‫ئ‬ӳŴX ɥƷ᧍ໜƴ‫ݣ‬ƠƯƸŴॄࣱ፭ƕෞƑƯЎᚐ፭ƕй˷˳ (ࢼƬƯஊᨂ˳) Ʒዌ‫ݣ‬ǬȭǢ
፭Ʊӷ‫׹‬ƴƳǓŴH 2 ƕෞƑƯƠLJƏƷưŴˊǘǓƴŴˌɦᛟଢƢǔǑƏƳǢǤȇǣǢǛဇƍǔŵ
LJƣŴX ƷɥƷӲ᧍ໜ x ƸŴᡲዓ፭แӷ‫׹‬
αx : Gal(k(x)) = π1 (Spec (k(x))) → π1 (X)
ưƋƬƯ prX ◦ αx ƕᐯ໱Ƴҥ‫ ݧ‬Gal(k(x)) → Gal(k) ƴɟᐲƢǔǑƏƳNjƷǛɨƑŴαx Ʒ΂
ƕ x ƷЎᚐ፭ƱƳǔƜƱƴදॖƢǔŵཎƴŴx ƕ k-ஊྸໜƷ‫ئ‬ӳƴƸŴแӷ‫ ׹‬αx Ƹ prX Ʒ
section ǛɨƑǔŵˌɦŴቇҥƷƨNJŴƜƷ‫ئ‬ӳǛᎋƑǔŵբ᫆ƸŴѨ৖ƴɨƑǒǕƨ prX Ʒ
section แӷ‫ ׹‬α ƕŴƋǔ x ∈ X(k) ƴ‫ݣ‬Ƣǔ αx ƱƳǔƨNJƷவˑǛ፭ᛯႎƴᚡᡓƢǔƜƱư
Ƌǔŵ(Gal(k) ∼
= Ẑ ƕᐯဌ profinite ፭ƳƷưŴSection ʖे (GC3) Ʒ᫏˩ƦƷNjƷƸ঺ᇌƠƑ
ƳƍƜƱƴදॖŵ) LJƣஇИƴǘƔǔƷƸŴբ᫆ƷவˑƕŴ
α Ʒ΂ Im(α) ǛԃljǑƏƳ π1 (X) Ʒ˓ॖƷ᧏ᢿЎ፭ H ƴ‫ݣ‬ƠƯŴ
(∗)
‫ࣖݣ‬Ƣǔ X Ʒᘮᙴ Y H Ʒ k-ஊྸໜᨼӳ Y H (k) ƸᆰƴƳǒƳƍ
Ʊӷ͌ưƋǔƱƍƏƜƱưƋǔŵܱᨥŴ࣏ᙲࣱƸŴЎᚐ፭ƷؕஜႎࣱឋƔǒႺƪƴࢼƍŴҗЎ
ࣱƸŴᆰưƳƍஊᨂᨼӳƷ‫ࢨݧ‬ಊᨂƸᆰƴƳǒƳƍƱƍƏʙܱƴǑǓŴIm(α) ƴ‫ࣖݣ‬Ƣǔ X Ʒ
(pro-) ᘮᙴNjLJƨ k-ஊྸໜǛਤƭƜƱƕࢼƏƷưŴƦƷ k-ஊྸໜǛɟƭƱƬƯ X ƴƓƚǔ΂Ǜ
x ƱƢǕƹŴα = αx ƱƳǔƜƱƔǒǘƔǔŵƋƱƸŴ(∗) ƕ঺ᇌƢǔƔƲƏƔŴƋǔƍƸɟᑍ
ƴஊᨂ˳ k ɥƷ୺ዴƕஊྸໜǛਤƭƔƲƏƔǛŴૠᛯႎؕஜ፭Ɣǒ፭ᛯႎƴЙ‫ܭ‬ƢǔƱƍƏբ
᫆ƕസǔƕŴƜǕƸŴஊྸໜƷ̾ૠ (≥ 0) Ǜ l ᡶǨǿȸȫȷdzțȢȭǸȸ፭ (l Ƹ k Ʒ೅ૠƱီ
Ƴǔእૠ) ǁƷ Frobenius ΨƷ˺ဇưᚡᡓƢǔ Lefschetz ួπࡸƴǑƬƯᚐൿƞǕǔŵ໯ᨂᢒໜ
def
x ∈ Σ = X ∗ − X ƴ‫ݣ‬ƠƯƸŴॄࣱ፭ƕෞƑƳƍƷưŴЎᚐ፭ƷɶǁƷ section แӷ‫ ׹‬αx Ƹɟ
ॖƴƸൿLJǒƣŴLJƨŴαx Ʒ΂ƸЎᚐ፭ƷჇƷᢿЎ፭ƴƳƬƯƠLJƏŵƠƔƠŴƜƷ‫ئ‬ӳƴNjŴ
8
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
ɥƷᜭᛯǛᒉ࠮̲ദƠƯᢘဇƢǔƱŴӲ x ∈ Σ ƴ‫ݣ‬ƠƯ໯ᨂ̾Ʒ (ƓƜǓƏǔμƯƷ) αx ƕ፭
ᛯႎƴࣄΨƞǕǔƷưŴƦǕǒƷ΂ưဃ঺ƞǕǔ π1 (X) ƷᢿЎ፭ƱƠƯŴ x ƷЎᚐ፭ƕࣄΨƞ
Ǖǔŵ
ǹȆȃȗ (ii) Ƹ [U] ƱDŽDžӷಮưŴ᧙ૠ˳ k(X) Ʒ᫏˳ᛯƷႻʝࢷǛ̅Əŵ᫏˳ᛯƸŴNjƱNj
Ʊʈඥ፭ƴǑƬƯǬȭǢ፭ƷǢȸșȫ҄ǛᚡᡓƢǔNjƷƩƕŴᡞƴƜǕǛǬȭǢ፭Ʒɶƴʈඥ፭
Ʒऴ‫إ‬ƕλƬƯƍǔƱᙸǔƷưƋǔŵǹȆȃȗ (i) ư X ∗ ƷӲ᧍ໜ x ƴ‫ݣ‬ƢǔЎᚐ፭ƕࣄΨưƖ
x× /O
x×
ƨƷưŴ‫ޅ‬৑᫏˳ᛯƴǑǓŴƦƷǢȸșȫ҄ƷžWeil ፭ᢿЎſƱƠƯŴx ∈ X Ƴǒƹ K
× ƕŴᐯ໱Ƴ‫ ݧ‬K
× /O
× → π1 (X)ab ƱƱNjƴ
× → π1 (X)ab ƳƍƠ K
(∼
= Z)Ŵx ∈ Σ Ƴǒƹ K
x
x
x
x
x Ƹ‫ޅ‬৑࿢ OX ∗ ,x Ʒ‫҄ͳܦ‬ŴK
x ƸƦƷՠ˳ǛᘙƢŵ) ƦƜ
ࣄΨƞǕƨƜƱƴƳǔŵ(ƜƜưŴO
ưŴArtin ϙ΂
× ×
× /O
× −→ π1 (X)ab
K
K
x
x∈X
x
x
x∈Σ
NjLJƨ π1 (X) ƷLjƔǒ፭ᛯႎƴࣄΨƞǕŴƜƷఋƱƠƯʈඥ፭ k(X)× ƕࣄΨƞǕǔŵ(ƜƜưŴ
X ƕ affine ưƋǔƜƱǛИNJƯဇƍǔŵ X = X ∗ ƩƱɼ‫܇׆‬፭ k(X)× /k × ƠƔࣄΨưƖƳƍŵ)
ǹȆȃȗ (iii) ƸŴ২ᘐႎƴƸNjƬƱNjᩊƠƍǹȆȃȗưƋǔŵLJƣŴǹȆȃȗ (ii) ưŴʈඥ
፭ƩƚưƸƳƘŴX ∗ ƷӲ᧍ໜ x ƴ˄᨟Ƣǔᩉ૝˄͌ ordx : k(X)× → Z ǍŴx ∈ Σ Ƴǒƹ୼
×
× Ʒఋ ƕࣄΨƞǕǔƜƱƴදॖƢǔŵƞƯŴ
ƴ reduction ϙ΂ Ker(ordx ) = OX
(
)
[U]
∗ ,x → k(x)
∗
ƴƓƍƯƸŴؕᄽ˳ k (ƋǔƍƸk̄) ƷьඥನᡯǛLJƣࣄΨƠŴഏƴŴ(“Σ = X ” ƳƷư) ໯ᨂ
̾Ʒໜƴ‫ݣ‬Ƣǔ reduction ϙ΂ǛဇƍƯŴ᧙ૠ˳ƷьඥǛй˷˳ƷьඥƔǒࣄΨƢǔƜƱƕưƖ
ƨŵǘǕǘǕƷ‫ئ‬ӳƴƸŴ(Σ ƕஊᨂᨼӳƷƨNJ) இࢸƷᢿЎƕƏLJƘƍƔƳƍƷưŴˊǘǓƴŴ
(X = P1k − {0, 1, ∞} = Spec (k[t, t−1 , (t − 1)−1 ]) ƴƓƚǔ t ƷǑƏƳ)žǑƍſ᧙ૠ f ∈ k(X)
ǛžƨƘƞǜſӕǓŴஊྸ᧙ૠ˳ƷཎഷࣱǛဇƍƯ k(f ) ⊂ k(X) ƷьඥನᡯǛࣄΨƠŴƦǕǒ
ǛžƸǓӳǘƤƯſk(X) ƷьඥನᡯǛࣄΨƢǔŵ
§3.2. ஊᨂ˳Ɣǒஊᨂဃ঺˳ǁ
§3.1 ưᛟଢƠƨኽௐƓǑƼᚰଢƸŴX ƕ୼ƴӑ୺ႎƳ‫ئ‬ӳƴƸŴ(ૠᛯႎ) ؕஜ፭ π1 (X) Ǜ
ƦƷՠưƋǔ tame ؕஜ፭ π1tame (X) ƴ‫٭‬ƑƯNj঺ǓᇌƭŵƜƷራưƸŴஊྸૠ˳ɥஊᨂဃ঺Ƴ
˳ƷɥƷ affine ӑ୺ႎ୺ዴƴ‫ݣ‬Ƣǔ Grothendieck ʖे (GC2) Ʒ Isom ༿ƕŴஊᨂ˳ɥƷ affine
ӑ୺ႎ୺ዴƷ tame ؕஜ፭ƴ᧙ƢǔƜƷኽௐƔǒ‫ݰ‬ƚǔƜƱǛᛟଢƢǔŵ§1.2 ư Tate ʖेƱ
Grothendieck ʖेƷ᫏˩ǛਦઇƠƨƕŴTate ʖेƷஊᨂ˳༿ưƋǔ Tate Ʒ‫ྸܭ‬ƔǒŴჇƴ‫ٻ‬
؏ႎƳᎋ‫ݑ‬ǛԃLJƳƍҥኝƳᜭᛯƴǑƬƯ Faltings Ʒ‫ྸܭ‬ƕ‫ݰ‬ƚǔƱƸᎋƑƴƘƍŵTate ʖे
Ʊ Grothendieck ʖेƷૠᛯႎࣱឋƷɟƭƷᢌƍƕƜƜƴƋǔŵ(§4.1 NjӋༀŵ)
ૠ˳Ʒ‫ئ‬ӳƔǒஊྸૠ˳ɥஊᨂဃ঺Ƴ˳Ʒ‫ئ‬ӳǛ‫ݰ‬ƘƷƸܾତƳƷưŴˌɦưƸŴK Ǜૠ˳
ƱƠŴK ɥƷ affine ӑ୺ႎ୺ዴ X ǛৢƏŵբ᫆ƸŴK ƷӲஊᨂእໜƴƓƚǔ X Ʒ reduction
Ʒ tame ؕஜ፭ǛŴX Ʒૠᛯႎؕஜ፭Ɣǒ፭ᛯႎƴࣄΨƢǔƜƱưƋǔŵƜƜưƸŴ‫ޅ‬৑˳ɥƷ
ӑ୺ႎ୺ዴƕ good reduction ǛਤƭƔƲƏƔǛ pro-l ؕஜ፭ (l Ƹй˷˳Ʒ೅ૠƱီƳǔእૠ)
ǁƷॄࣱ፭Ʒ‫˺ٳ‬ဇƕᐯଢƔƲƏƔưЙ‫ܭ‬ưƖǔƱƍƏʙܱ (Ǣȸșȫ‫ٶ‬ಮ˳Ʒ good reduction
Ʒ Serre-Tate Й‫ܭ‬ඥƷ᫏˩) ƕᦆƱƳǔŵ(ƜƷŴӑ୺ႎ୺ዴƷ good reduction Ʒ፭ᛯႎЙ‫ܭ‬ඥ
ƸŴproper Ƴ‫ئ‬ӳƸጢဋܑ࠳ ([O1,2]) ƴǑǔŵ)
ʻŴv Ǜ K ƷஊᨂእໜƱƠŴKv Ǜ K Ʒ v ᡶ‫҄ͳܦ‬ŴOv ǛƦƷૢૠ࿢Ŵkv ǛƦƷй˷˳
ƱƠǑƏŵKv Ʒዌ‫ݣ‬ǬȭǢ፭Ƹᐯ໱ƴ K Ʒዌ‫ݣ‬ǬȭǢ፭ƷᢿЎ፭ƱLjƳƞǕŴLJƨŴX Ʒ࠹
˴ႎؕஜ፭Ʊ XKv Ʒ࠹˴ႎؕஜ፭ƸɟᐲƢǔƷưŴXKv Ʒૠᛯႎؕஜ፭ π1 (XKv ) ƸŴ(1.1)
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
9
Ʒᚡӭư pr−1
X (Gal(Kv )) ƱƠƯႺƪƴࣄΨƞǕǔŵƜǕƴɥᚡƷЙ‫ܭ‬ඥǛᢘဇƢǕƹŴXKv ƕ
good reduction ǛਤƭƔƲƏƔƕ፭ᛯႎƴЙ‫ܭ‬ƞǕǔƷưŴˌɦưƸŴXKv ƕ good reduction
ǛਤƭƱˎ‫ܭ‬ƠŴXOv Ǜ XKv Ʒ Ov ɥƷžǑƍſȢȇȫŴXkv ǛƦƷ reduction ƱƢǔŵƜ
ƷƱƖŴπ1tame (Xkv ) Ʊ π1 (XOv ) Ʒ᧓ƴᐯ໱Ƴӷ‫׹‬ƕƋǔƜƱƕჷǒǕƯƍǔƷưŴπ1tame (Xkv )
ǛŴ π1 (XKv ) Ʒՠ፭ π1 (XOv ) ƱӷɟᙻƢǔƜƱƕưƖǔŵƜƷՠǛ፭ᛯႎƴࣄΨƢǔƨNJƴ
ƸŴXKv ƷӲஊᨂഏǨǿȸȫ (ǬȭǢ) ᘮᙴƴ‫ݣ‬ƠŴƦƷᘮᙴƕ XOv ƷǨǿȸȫᘮᙴƴLJưࡨ
ᧈưƖǔƔƲƏƔǛ፭ᛯႎƴЙ‫ܭ‬ưƖǕƹǑƍƕŴƜǕƸŴӑ୺ࣱƴǑǓŴܱƸŴᘮᙴƷ୺ዴNj
LJƨ good reduction ǛਤƭƱƍƏ (ɟᙸࢊƍ) வˑƱӷ͌ƴƳǔƷưŴέƷЙ‫ܭ‬ඥƴǑǓ፭ᛯႎ
ƴЙ‫ܭ‬ưƖǔŵˌɥƴǑǓŴૠ˳ K ɥƷӑ୺ႎ୺ዴ X ƕእໜ v ư good reduction Ǜਤƭ‫ئ‬ӳ
ƴŴƦƷ reduction Xkv Ʒ tame ؕஜ፭ƕŴ(X Ʒૠᛯႎؕஜ፭ƷᢿЎՠƱƠƯ) ፭ᛯႎƴࣄΨ
ƞǕƨƜƱƴƳǔŵ
ƞƯŴૠ˳ K ɥƷʚƭƷ affine ӑ୺ႎ୺ዴ X1 , X2 Ʒૠᛯႎؕஜ፭Ʒ᧓ƴŴGal(K) ɥƷӷ‫׹‬
π1 (X1 ) ∼
= π1 (X2 ) ƕɨƑǒǕǔƱŴɥƷᜭᛯƴǑǓŴ(X1 , X2 ƕƦƜư good reduction ǛਤƭǑ
ƏƳ) DŽƱǜƲμƯƷ K Ʒእໜ v ƴƓƍƯπ1tame ((X1 )kv ) ∼
= π1tame ((X2 )kv ) ƕᛔ‫ݰ‬ƞǕŴࢼƬƯ
ஊᨂ˳ɥƷ affine ӑ୺ႎ୺ዴƷ tame ؕஜ፭ƴ᧙ƢǔɥᚡƷኽௐƴǑǓŴӷ‫( ׹‬X1 )kv ∼
= (X2 )kv
ƕ‫ݰ‬ƔǕǔŵƱƜǖƕʻŴ୺ዴƷӑ୺ࣱ (ƴǑǔ Isom ǹǭȸȠƷஊᨂࣱ) Ɣǒ
Isom(X1 , X2 ) ∼
= Isom((X1 )kv , (X2 )kv )
ƕDŽƱǜƲμƯƷ v ƴ‫ݣ‬ƠƯ঺ǓᇌƭƷưŴX1 ∼
= X2 ƕࢼƏŵƜƏƠƯŴஊᨂ˳ɥƷ affine ӑ
୺ႎ୺ዴƷ tame ؕஜ፭ƴ᧙ƢǔɥᚡƷኽௐƔǒૠ˳ɥƷ affine ӑ୺ႎ୺ዴƴ‫ݣ‬Ƣǔ (GC2) Ʒ
Isom ༿Ǜ‫ݰ‬Ƙᚰଢƕ‫ܦ‬ኽƢǔŵ
ƳƓŴ[M1] ưƸŴஊᨂ˳ɥƷ affine ӑ୺ႎ୺ዴƷ tame ؕஜ፭ƴ᧙ƢǔɥᚡƷኽௐƔǒŴૠ
˳ɥƷ proper Ƴӑ୺ႎ୺ዴƴ‫ݣ‬Ƣǔ (GC2) Ʒ Isom ༿Ǜ‫ݰ‬ƍƯƍǔŵƜƜưŴʚƭƷኽௐǛኽ
ƿᦆƸŴஊᨂ˳ɥƷ (๖ǒƔưƳƍ) ‫୺ܭܤ‬ዴƷžlog ؕஜ፭ſ ƴ‫ݣ‬Ƣǔ (GC2) Ʒ Isom ༿Ʒ᫏
˩ưƋǔŵ
§3.3. ദ೅ૠˊૠ୺ዴƷ࠹˴ႎؕஜ፭
೅ૠ 0 Ʒ‫ئ‬ӳŴ୺ዴƷ࠹˴ႎؕஜ፭Ʒӷ‫׹‬᫏ƸŴᆔૠ g Ʊ໯ᨂᢒໜƷૠ n ƷLjưൿLJǔƕŴ
ദ೅ૠƷ‫ئ‬ӳƸƦƏưƸƳƍŵܱᨥŴ[T2] ưƸŴFp ɥƷᆔૠ 0 Ʒ୺ዴƷǹǭȸȠƱƠƯƷӷ‫׹‬
᫏ƕŴƦƷؕஜ፭ƴǑƬƯ‫ܦ‬μƴൿ‫ܭ‬ƞǕǔƜƱƕᚰଢƞǕƯƍǔŵƜƷǑƏƳኽௐƕŴദ೅ૠ
ˊૠ᧍˳ɥƷ (ӑ୺ႎ) ˊૠ୺ዴƴ‫ݣ‬ƠƯɟᑍƴ঺ǓᇌƭƷƔƲƏƔŴƋǔƍƸ tame ؕஜ፭ƴ
ƭƍƯNj᫏˩Ʒኽௐƕ঺ǓᇌƭƷƔƲƏƔŴᐻԛƕਤƨǕǔŵ
§4. ‫ޅ‬৑˳༿ƴᐱǔኺዾ
§4.1. ‫˳؏ٻ‬Ʊ‫ޅ‬৑˳
§1 ưƸ Grothendieck ʖेǛŴஊྸૠ˳ɥஊᨂဃ঺Ƴ˳ŴұƪእໜǛᝅ݈ƴਤƭǑƏƳž‫؏ٻ‬
ႎƳ˳ſɥƷ‫ݣ‬ᝋƴ᧙ƢǔNjƷƱƠƯኰʼƠƨƕŴஓஉƷɟᡲƷᛯ૨ ([M1-4]) ƸŴƜƷʖेǛ
‫˳؏ٻ‬ǑǓNjljƠǖஜឋႎƴ‫ޅ‬৑˳ɥƷ p ᡶᚐௌႎྵᝋƱƠƯਵƑǔŴƱƍƏૼƠƍᙻໜǛ‫ݰ‬λ
ƠƨŵƜƷᙻໜǛؕƴࢽǒǕƨ‫ޅ‬৑˳༿Ʒ Grothendieck ʖे (§5.1) ƸŴஊྸૠ˳ɥஊᨂဃ঺Ƴ
˳ɥƴ‫҄ࡸܭ‬ƞǕƯƍƨΨƷʖे (GC2) ǛԃljŴǑǓɟᑍႎƳኽௐƱƳƬƯƍǔƕŴƜƷૼƠ
ƍᙻໜǍኽௐƷᛟଢƴλǔЭƴŴLJƣஓஉˌЭƷെ᨞ưŴؕᄽ˳ƴ‫؏ٻ‬ႎƳ˳ǛƱǔƜƱƕᐯ໱
Ʊ࣬ǘǕƯƍƨᏑ୎Ǜ౨ᚰƠƯLjǑƏŵ
10
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
(A) Ǣȸșȫ‫ٶ‬ಮ˳Ʒ Tate ʖेᲴ§1 ưNjᚑǕƨǑƏƴŴGrothendieck ƕᢒǢȸșȫʖेǛ‫ܭ‬
ࡸ҄ƢǔƴƋƨƬƯਦઇƠƨƜƱƷɟƭƱƠƯŴFaltings ([F1]) ƴǑƬƯᚰଢƞǕƨ Tate ʖेƱ
Ʒ᫏˩ƕஊƬƨƕŴFaltings ƷƦƷᚰଢưƸŴૠ˳ɥƷ height ƷྸᛯǍŴǢȸșȫ‫ٶ‬ಮ˳Ʒ᧓Ʒ
ӷᆔƷᨥƴဃơǔŴஊᨂእໜƴǑǔ݃ɨƱ໯ᨂእໜƴǑǔ݃ɨƕƪǐƏƲႻࣱᑣƘෞӊƠƋƬƯ
ƍǔƱƍƏʙܱƳƲŴஜឋႎƴ‫؏ٻ‬ႎƳཞඞƕஊјƴ̅ǘǕƯƍǔŵɟ૾ưૠ˳ (Ǎஊᨂ˳) Ʊ
ᢌƬƯŴQp ƷǑƏƳ‫ޅ‬৑˳ƷɥưƸŴTate ʖेƸƨƩ঺ᇌƠƳƍƷLjƳǒƣŴӷ‫׹‬ƕʖेƞǕ
ǔʚƭƷь፭Ʒࠀƕ‫ئ‬ӳƴǑƬƯƸ᩼ࠝƴ‫ٻ‬ƖƘƳƬƯƠLJƏŵࢼƬƯŴ
žGrothendieck ʖे=Tate ʖेƷӑ୺ႎ୺ዴ༿ſ
ƱƍƏᙻໜƴᇌƯƹŴ(GC2) ƷǑƏƳʖेƷ঺ᇌǛLJƣ‫˳؏ٻ‬ɥưƷLjᎋ‫ݑ‬ƠǑƏƱƢǔ૾Ӽƕ
ᐯ໱ƱƍƑǑƏŵ
(B) Diophantus ࠹˴ǁƷࣖဇᲴᢒǢȸșȫՋ‫ܖ‬Ʒੵቒ஖ƴᇌƪ˟Ƭƨૠ‫ܖ‬ᎍƷ᧓ưƸŴGrothendieck ʖेƸŴDiophantus ࠹˴Ŵұƪ‫؏ٻ‬ႎƳ˳ɥƷ‫ٶ‬ಮ˳ƷஊྸໜƷᄂᆮǁƷૼƠƍǢȗȭȸ
ȁƱƠƯᛐᜤƞǕƯƍƨǑƏưƋǔŵƦƷǢȗȭȸȁǛˊᘙƢǔNjƷƱƠƯŴഏƷǑƏƳᜭᛯƕ
ƋǔŵƋǔˊૠ‫ٶ‬ಮ˳ƕஊᨂ̾ƷஊྸໜƠƔਤƨƳƍƜƱǛᚰଢƢǔ૾ඥƱƠƯŴNjƠˎƴ໯ᨂ
̾ஊƬƨƱƠƨǒŴƦƷžಊᨂſƱƠƯႆဃƢǔஊྸໜƕಮŷƳžᑣᢅƗǔſࣱឋǛਤƨƟǔǛ
ࢽƳƍƜƱƔǒჳႽǛ‫ݰ‬ƖƨƍƱƠǑƏŵƨƩƠŴƦƷᜭᛯǛܱᘍƢǔƨNJƴƸŴžಊᨂſƕ‫܍‬
‫נ‬ƠƳƍƱƍƚƳƍŵƱƜǖƕŴૠ˳ƷǑƏƳ˳Ƹ‫˴ڦ‬Ƴǔ (᩼ᐯଢƳ) ˮႻƴ᧙ƠƯNj‫ͳܦ‬ƴ
ƸƳǒƳƍƷưŴƦƷǑƏƳಊᨂƷ‫נ܍‬ƸൿƠƯႺƪƴƸЎƔǒƳƍŵɟ૾ưŴ(ࡸ (1.2) ƷǑƏ
Ƴ) ǬȭǢᘙྵƸƋǔॖԛưƸžᚐௌႎſƳNjƷƳƷưŴƦƷǑƏƳǬȭǢᘙྵƷЗƴ‫ݣ‬ƠƯŴ
ಊᨂǛਤƭᢿЎЗƕ࣏ƣ‫נ܍‬ƢǔƜƱƸൔ᠋ႎܾତƴᚰଢưƖǔŵƦƜưŴSection ʖे (GC3)
ƕɼࢌƠƯƍǔǑƏƴŴ ஊྸໜƱ (ƋǔவˑǛ฼ƨƢ) ǬȭǢᘙྵƕܱƸӷ͌Ƴ‫ݣ‬ᝋưƋǔŴƱ
ƍƏƜƱƕЎƔǕƹŴǬȭǢᘙྵƷЗƷಊᨂƷ‫נ܍‬ƔǒƦǕƴ‫ࣖݣ‬ƢǔஊྸໜƷЗƷಊᨂƷ‫נ܍‬
ƕ‫ݰ‬ƔǕǔƜƱƴƳǔŵɥᡓƷǑƏƳሂᢊƷᜭᛯǛǑǓች݅҄ƢǕƹŴ̊ƑƹŴ᭗ᆔૠƷˊૠ୺
ዴƴ‫ݣ‬ƢǔžMordell ʖे14) ſƷКᚰଢƕ Section ʖे (GC3) Ɣǒ‫ݰ‬ƚǔӧᏡࣱƕƋǔŵ ƜƷ
ǑƏƳஜឋႎƴ‫؏ٻ‬ႎƳࣖဇǛ᪽ࣞƴፗƍƯLjǔƱŴGrothendieck ʖेǛ‫˳؏ٻ‬ɥưƷLjᎋ‫ݑ‬
ƠǑƏƱƢǔƷƕǍƸǓᐯ໱Ƴ්ǕƱƍƑǑƏŵ
ƜƷǑƏƳཞඞƷɶưŴž‫˳؏ٻ‬Ɣǒ p ᡶ˳ǁſƱƍƏႆेƷ᠃੭ƕஓஉƷˁʙƴǑƬƯNjƨ
ǒƞǕƨŵƦƷƖƬƔƚƱƳƬƨᚇໜƴƭƍƯŴˌɦưᚐᛟƠƯLjǔƜƱƴƢǔŵ
§4.2. ӑ୺‫׹‬ȪȸȞȳ᩿Ʒɟॖ҄ྸᛯƱƷ᫏˩
Grothendieck ʖेƷɼଓƸŴɟᚕưᚕƑƹŴӑ୺ႎ୺ዴƴ˄᨟Ƣǔ‫ٳ‬ǬȭǢᘙྵ (1.2)Ŵұƪ
୺ዴƷ࠹˴ႎؕஜ፭ᲥƦǕƴλǔᐯ໱Ƴžૠᛯႎ˄ьನᡯſ
Ɣǒ୺ዴᐯ៲ǛࣄΨưƖǔưƋǖƏƱƍƏNjƷưƋǔŵƱƜǖƕŴ˳Ʒ‫ࣱ؏ٻ‬ǛབྷཌƴƠŴžૠ
ᛯႎ˄ьನᡯſƱƍƏᘙྵǛ࠼፯ƴᚐ᣷ƢǕƹŴƜǕƴ᫏˩ƢǔྵᝋƕଏƴҗʋɭኔɶƴჷǒǕ
ƯƍƨƜƱƴൢ˄ƘŵƦƷྵᝋƱƸŴӑ୺‫׹‬ȪȸȞȳ᩿Ʒɟॖ҄ྸᛯƷƜƱưƋǔŵ
ᙐእૠ˳ C ɥƷӑ୺ႎ୺ዴ X ƕɨƑǒǕƨǒŴX Ɣǒӑ୺‫׹‬ȪȸȞȳ᩿ X ƕ‫ܭ‬LJǓŴƦƷ
X Ʒ୍ᢄᘮᙴX˜ → X ƴNjȪȸȞȳ᩿Ʒನᡯƕᐯ໱ƴλǔŵƱƜǖƕŴȪȸȞȳ᩿Ʒɟॖ҄‫ܭ‬
def
ྸƔǒŴX̃ ƕŴɥҞ࠯᩿ H = {z ∈ C | Im(z) > 0} ƴദЩƴӷ‫׹‬ƴƳǔƜƱƕЎƔǔŵࢼƬƯŴ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
11
Aut(X˜ ) ∼
= Aut(H) = SL2 (R)/{±1} Ǜ̅ƏƱŴX Ʒ (୍ᡫƷˮႻ࠹˴‫ܖ‬ႎƳ) ؕஜ፭π1 (X ) Ʒ
X˜ ǁƷ˺ဇƔǒ
(4.1)
ρX : π1 (X ) → SL2 (R)/{±1}
ƱƍƏ (SL2 (R)/{±1} ƴǑǔσࢫǛᨊƍƯ) ೅แႎƳᘙྵƕൿLJǔŵᡞƴŴρX ƕɨƑǒǕǕƹŴ
π1 (X ) ƷɥҞ࠯᩿ H ǁƷ˺ဇƕ‫ܭ‬LJǓŴH ǛƦƷ˺ဇưлǔƜƱƴǑƬƯŴȪȸȞȳ᩿ X Ŵƻ
ƍƯƸΨƷˊૠ୺ዴƷ X LJưŴƍƱNjᡯ˺ƳƘࣄΨƞǕǔŵஓஉƸᙐእᚐௌƷɭမƴƓƚǔˌɥ
ƷཞඞƴბႸƠŴ୼ƴ (1.2) ƷρX Ʊ (4.1) ƷρX ƕŴƍƣǕNjž୺ዴƷ࠹˴ႎؕஜ፭ᲥƦǕƴλǔ
ᐯ໱Ƴૠᛯႎ˄ьನᡯſƱƍƏȑǿȸȳƴᛆ࢘ƢǔƱƷᚇ‫ݑ‬ƴᚑႆƞǕŴ
ᘙྵ ρX Ɣǒ (ƔƘNjႺ੗Ɣƭᐯ໱ƴ) X Ǜ (ϐ) ನ঺ưƖǔ
ƱƍƏྵᝋƷ p ᡶ༿ƸƳƍƔᲹ
ƱᚨբƢǔƴᐱƬƨŵܱᨥŴˌɦ (§5) ưኰʼƢǔ Grothendieck ʖेƷ p ᡶ༿ (‫ ྸܭ‬5.1) ƸŴƜ
Ʒբƍƴ‫ݣ‬ƠƯɟᆔƷᏉ‫ܭ‬ႎƳᚐሉǛ੩ᅆƠƯƍǔNjƷƱᙸǔƜƱƕЈஹǔ15) ŵ
࠹˴ႎؕஜ፭π1 (X ) Ʒ H ǁƷ˺ဇƔǒˊૠ୺ዴ X Ǜφ˳ႎƴನ঺ƢǔƨNJƴƸŴӞχႎƴǑ
ƘဇƍǒǕǔ৖ඥƱƠƯŴπ1 (X ) Ʒ˺ဇƷɦưɧ‫٭‬Ƴ H ɥƷࣇЎ࢟ࡸǛ˺ǔƱƍƏNjƷƕƋǔŵ
ƦƷǑƏƳࣇЎ࢟ࡸǛΪЎƴඑ‫˺ޛ‬ǕǕƹŴɥҞ࠯᩿ H ƔǒŴ˴ǒƔƷ‫ࢨݧ‬ᆰ᧓ PX ǁƷ‫ݧ‬
φ : H → PX
ƕ‫ܭ‬፯ƞǕŴƦƷ‫ݧ‬Ʒ΂ƕŴɟᑍᛯƔǒˊૠ‫ٶ‬ಮ˳ƴƳǔƜƱƸƢƙЎƔǔƷƩƕŴƜƷ‫ئ‬ӳŴ
(X ƴ‫ݣ‬ƠƯŴƋǔࢊƍ২ᘐႎƳவˑǛᛢƤƹ) X ᐯ៲ƴƳǔƜƱƕࢼƏŵƜƷᛯඥƸ࣓஭‫ٶ‬ಮ˳
Ǜ‫ݣ‬ᆅᆰ᧓ƷՠƱƠƯನ঺ƢǔᨥƴŴƦƷՠƕˊૠ‫ٶ‬ಮ˳ƴƳǔƜƱǛᚰଢƢǔƷƴ̅ǘǕǔƷ
ƱӷಮƳNjƷưƋǔŵ‫ݲ‬ƠૢྸƢǕƹŴƜƷ৖ƷᜭᛯƷᙲໜƸŴφǛ‫ܭ‬፯ƢǔƷƴ̅ƏࣇЎ࢟ࡸ
ƸѸᛯஇኳႎƴƸ X ɥˊૠႎƳNjƷƴƳǔƷƩƕŴನ঺Ʒെ᨞ƴƓƍƯƸŴH ɥᚐௌႎƳNjƷ
ưƠƔƳƍƱƍƏƜƱưƋǔŵƜƷ žˊૠႎƳࣇЎ࢟ࡸƷᚐௌႎƳᘙᅆǛৢƏſƱƍƏᚇໜƸ
ˌɦưኰʼƢǔ‫ ྸܭ‬5.1 ƷᚰଢƴƓƍƯNj᣻ᙲƳࢫлǛௐƨƠƯƍǔŵ
§4.3. p ᡶ Hodge ྸᛯƱƷ᧙̞
ƞƯŴ‫ ྸܭ‬5.1 ǛᚰଢƢǔƴƋƨƬƯŴЭራưኰʼƠƨᙐእૠ˳ɥƷྸᛯƱƷ᫏˩ƸɟƭƷ৖
ੑƔǓƴƸƳǔƕŴƦƷ᫏˩Ǜ p ᡶƷɭမƴƓƍƯܱᨥƴܱྵƢǔƨNJƴƸ᭗ࡇƳ২ᘐƕ࣏ᙲ
ƴƳǔŵƦƷ২ᘐǛɨƑƯƘǕƯƍǔNjƷƸ Faltings Ʒ p ᡶ Hodge ྸᛯ ([F2]) ưƋǔŵžp ᡶ
Hodge ྸᛯſƱԠƹǕǔNjƷƴƸŴᲫᲳᲰᲪ࠰ˊҞƹƷ Tate ƷέᬝႎƳˁʙƴᢓǔᧈƍഭӪƕ
ஊǔƕŴƜƜư᣻ᙲƳƜƱƸŴƦƷྸᛯƱ Grothendieck ʖेƱƷขƍ᫏˩ࣱưƋǔŵp ᡶ Hodge
ྸᛯƷɼƳȆȸȞƸŴp ᡶ˳ (̊ƑƹŴQp Ʒஊᨂഏਘ‫ )˳ٻ‬ɥƷ‫ٶ‬ಮ˳Ʒ (ǬȭǢ˺ဇ˄ƖƷ) Ǩ
ǿȸȫȷdzțȢȭǸȸƱŴȉȷȩȸȠȷdzțȢȭǸȸƱƷ᧓ƷƍǘǏǔൔ᠋‫ྸܭ‬ưƋǔŵұƪŴ
ƜƷʚᆔ᫏ƷdzțȢȭǸȸɧ‫٭‬᣽ǛŴʝƍƴ‫٭‬੭ƠƋƏǑƏƳžᜋƷ᧙৖ſ(mysterious functor)
ƕ‫נ܍‬ƢǔưƋǖƏŴƱƍƏज़ᙾƳƍƠʖेƕ p ᡶ Hodge ྸᛯƷҾໜưƋǔŵƜƜưŴȉȷȩȸ
ȠȷdzțȢȭǸȸƸ X ɥƷ‫ٶ‬᪮ࡸ᧙ૠǍƦƷ᧙ૠƷࣇЎƷࣱឋǛଓƘɟƭƷᙐӳ˳ƴLJƱNJǔ
ƜƱƴǑƬƯࢽǒǕǔɧ‫٭‬᣽ưƋǔƕŴ(GC2) Ʒ߼ᡀƴЈƯƘǔˊૠ‫ٶ‬ಮ˳Ʒ᧓Ʒ‫ݧ‬ƱƍƬƨ
NjƷNjӷơˊૠ࠹˴ (Ჷ‫ٶ‬᪮ࡸ) ƷɭမƷ˰ʴưƋǔŵɟ૾ư (GC2) ƷӫᡀƴЈƯƘǔ‫ٳ‬ǬȭǢ
12
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
˺ဇ˄ƖƷؕஜ፭ƸŴǢȸșȫȷ᩼ǢȸșȫƷࠀƜƦƋǕŴǨǿȸȫȷdzțȢȭǸȸƱӷơƘŴ
բ᫆Ʒ‫ٶ‬ಮ˳ X ƷǨǿȸȫȷǵǤȈƷ᣻ᙲƔƭᐯ໱Ƴɧ‫٭‬᣽ưƋǔŵƜƷǑƏƴᎋƑƯLjǔƱŴ
mysterious functor ʖेƴƠƯNjŴ(GC2) ƴˊᘙƞǕǔᢒǢȸșȫʖे16) ƴƠƯNjŴƍƣǕNj
žˊૠ࠹˴ႎನᡯ ⇐⇒ ǨǿȸȫˮႻᲥǬȭǢ˺ဇſ
ƱƍƏ೅ᛖƴЩƬƨžൔ᠋‫ྸܭ‬ſƷ঺ᇌǛƏƨƬƯƍǔNjƷƱᙸǔƜƱƕЈஹǑƏŵƨƩƠŴp
ᡶ Hodge ྸᛯƱ Grothendieck ʖेƷ᧓ƴƜƷǑƏƳž‫ח‬ᛯႎ᫏˩ࣱſƸஊƬƯNjŴǢȸșȫƱ
᩼ǢȸșȫƷࠀƴƸ᩼ᐯଢƳNjƷƕஊǓŴƦƷࠀǛ؈NJǔƜƱƕŴ‫ ྸܭ‬5.1 ǛᚰଢƢǔɥưƸ‫ٻ‬
ƖƳ২ᘐႎᛢ᫆ƱƳƬƨŵ
§5. ‫ޅ‬৑˳ɥƷ Grothendieck ʖे
§5.1. ɼ‫ྸܭ‬Ʒኰʼ
ˌɦưƸŴእૠ p Ǜ‫ܭ׍‬ƠŴQp Ʒஊᨂဃ঺ਘ‫˳ٻ‬ƷᢿЎ˳ƱƠƯܱྵӧᏡƳ˳Ǜžэ p ᡶ˳ſ
(sub-p-adic field) ƱԠƿƜƱƴƢǔŵэ p ᡶ˳ƷˊᘙႎƳ̊ƱƠƯŴQ ӍƸ Qp Ʒஊᨂဃ঺Ƴਘ
‫˳ٻ‬Ʒ˂ƴŴദૢૠ N ƴ‫ݣ‬ƠƯஊྸૠ˳ Q ƷμƯƷ N ഏˊૠਘ‫˳ٻ‬Ǜӳ঺ƢǔƜƱƴǑƬƯࢽ
ǒǕǔ (Q Ʒ໯ᨂഏˊૠਘ‫˳ )ٻ‬ሁƕਫƛǒǕǔŵஓஉ ([M3]) ƷɼƳኽௐƸഏƷ‫ྸܭ‬ưƋǔŵ
‫ ྸܭ‬5.1. э p ᡶ˳ K ɥƷ˓ॖƷ๖ǒƔƳˊૠ‫ٶ‬ಮ˳ S Ʊӑ୺ႎ୺ዴ X ƴ‫ݣ‬ƠƯŴ
open
Homdom
K (S, X) → HomGal(K) (π1 (S), π1 (X))
(p)
(p)
→ Homopen
Gal(K) (π1 (S), π1 (X))
ƱƍƏᐯ໱Ƴϙ΂ƨƪƸμҥ‫ݧ‬ƴƳǔŵƨƩƠŴƜƜưŴHomdom
K ƸžૅᣐႎƳ K ɥƷ‫ݧ‬ƨƪ
open
μ˳ſŴHomGal(K) ƸžGal(K) ǁƷ‫ࢨݧ‬ƱɲᇌƢǔ᧏แӷ‫׹‬Ʒ (ӫƔǒƷπ1 (XK ) Ʒσࢫ˺ဇƴ
(p)
᧙Ƣǔ) ӷ͌᫏ƨƪμ˳ſǛॖԛƠŴLJƨŴπ1 (V ) Ƹπ1 (V ) Ʒᐯ໱Ƴ pro-p ‫ࣖݣ‬ཋ (π1 (V ) Ʒ
(p)
Ker(π1 (VK ) → π1 (VK )) ƴǑǔՠ፭) ǛᘙƢƱƢǔŵ
ƜƷ‫ྸܭ‬Ƹʖे (GC2) Ǜࢍƍ࢟ưᚐൿƠƯƍǔNjƷưƋǔŵ§4.2 ưኰʼƠƨɟॖ҄ྸᛯƱƷ
᫏˩ưᚕƑƹŴ߼ᡀƕ X Ʒ S ͌ஊྸໜŴұƪˊૠ୺ዴ X ƷžཋྸႎƳܱ˳ſưƋǔƷƴ‫ݣ‬ƠƯŴ
ӫᡀƕૠᛯႎ˄ьನᡯ˄ƖƷ࠹˴ႎؕஜ፭ƔǒႺ੗ဃơǔžᚐௌႎƳໜſưƋǔŵƭLJǓŴȪȸ
Ȟȳ᩿Ʒɟॖ҄ྸᛯƱӷಮƴŴ‫ ྸܭ‬5.1 Ƹӑ୺ႎˊૠ୺ዴƷཋྸႎƳܱ˳ƱŴƦƷૠᛯႎ˄ьನ
ᡯ˄ƖƷ࠹˴ႎؕஜ፭ƔǒႺ੗ဃơǔᚐௌႎƳ࠹˴ႎ‫ݣ‬ᝋƷӷࣱ͌ǛᚫƑƯƍǔǘƚưƋǔŵ
‫ݵ‬Ŵ‫ ྸܭ‬5.1 Ʒᒉ࠮Ʒɟᑍ҄ (Ჷ [M3] Ʒ‫ ྸܭ‬A) ƷኒƱƠƯഏƷ Grothendieck ʖेƷӑஊྸ
༿ƕஊǔŵ
ኒ 5.2. э p ᡶ˳ K Ǜ‫ܭ‬ૠ˳ƱƢǔ˓ॖഏΨƷദЩƳ᧙ૠ˳ L Ʊ M ƴ‫ݣ‬ƠƯŴ
HomK (M, L) → Homopen
Gal(K) (Gal(L), Gal(M ))
ƱƍƏᐯ໱Ƴϙ΂Ƹμҥ‫ݧ‬ƴƳǔŵƨƩƠŴƜƜưŴHomK ƸžK ɥƷ࿢แӷ‫׹‬ƨƪμ˳ſŴ
ƸžGal(K) ǁƷ‫ࢨݧ‬ƱɲᇌƢǔ᧏แӷ‫׹‬Ʒ (ӫƔǒƷ Gal(M ⊗K K) Ʒσࢫ˺ဇƴ
Homopen
Gal(K)
᧙Ƣǔ) ӷ͌᫏ƨƪμ˳ſǛॖԛƢǔƱƢǔŵ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
13
ؕᄽ˳ K ƕஊྸૠ˳ɥஊᨂဃ঺Ƴ˳ƷƱƖŴƜƷኒƷ Isom ༿Ǜ F.Pop ƕ [M3] ˌЭƴμƘᢌƏ
૾ඥƴǑƬƯᚰଢƠƯƍǔ ([P2])ŵ
දŵ(i) ‫ ྸܭ‬5.1 Ƹ˳ K ɥƷ‫ٶ‬ಮ˳Ǎӑ୺ႎ୺ዴƴ᧙ƢǔNjƷƩƕŴbase ƱƠƯэ p ᡶ˳Ʒˊǘ
Ǔƴэ p ᡶ˳ɥ๖ǒƔƳˊૠ‫ٶ‬ಮ˳ B ǛӕǓŴS Ʊ X ƕƦǕƧǕ B ɥƷ‫ٶ‬ಮ˳Ŵӑ୺ႎ୺ዴƷ๖
ǒƔƳଈƱƳǔƜƱǛˎ‫ܭ‬ƢǕƹŴ‫ ྸܭ‬5.1 ƱӷಮƳ‫ྸܭ‬ƕ঺ǓᇌƭŵܱᨥŴƦƷǑƏƳኽௐƸ
(B Ʒ᧙ૠ˳ƕϐƼэ p ᡶ˳ƴƳǔƜƱƴදॖƢǕƹ) ‫ ྸܭ‬5.1 ƔǒႺƪƴࢼƏƷưƋǔŵ
(ii) ‫ ྸܭ‬5.1 ƷNjƏɟƭƷ࠙ኽƱƠƯŴӑ୺ႎ୺ዴɥƷ smooth Ƴӑ୺ႎ୺ዴଈƷμᆰ᧓ƱƠ
ƯࢽǒǕǔǑƏƳˊૠ୺᩿ƴ᧙Ƣǔ (GC2) Ʒ Isom ༿ƕ‫ݰ‬ƚǔŵᛇƠƘƸ [M4] ǛӋༀƷƜƱŵ
§5.2. ‫ ྸܭ‬5.1 ƷᚰଢƷ૾ᤆ
˳ K ƕ Qp Ʒஊᨂഏਘ‫ٻ‬ƴƳǔƱƍƏNjƬƱNjஜឋႎƳ‫ئ‬ӳƴСᨂƠƯᛅǛᡶNJǔƜƱƴƢ
ǔŵ ୼ƴŴቇҥƷƨNJƴŴX(ƓǑƼ S) ƕ proper Ɣƭ non-hyperelliptic Ƴӑ୺ႎ୺ዴưƋǔ
ƱƢǔŵܱᨥŴƜǕǒƷᜂŷƷவˑƸᚰଢƷஜឋƴƸɟӼƴᚑǕƳƍƷưŴɟᑍƷ‫ئ‬ӳƸƜǕǒ
ƷவˑǛˎ‫ܭ‬Ơƨ‫ئ‬ӳƴႺƪƴ࠙ბƞǕǔƷưƋǔŵஇࢸƴŴ‫ ྸܭ‬5.1 ưƸŴɤƭƷžHomſƕ
ЈƯƘǔƕŴ(NjƬƱNjஜឋႎƳNjƷưƋǔ) ɟ̾ႸƱɤ̾ႸƷ Hom Ʒ᧓Ʒϙ΂ƴᨼɶƢǔƜƱ
(p)
ƴƢǔŵբ᫆Ƹ‫˴ڦ‬ƴƠƯπ1 (X) → Gal(K) Ɣǒ X ǛࣄΨƢǔƔƱƍƏƜƱưƋǔŵ
def
(p)
LJƣŴT = π1 (XK )ab ƱƢǔŵࢼƬƯŴX ƕᆔૠ g Ʒ୺ዴƩƱƢǔƱŴT Ƹ Zp ɥƷ᨞ૠ
2g Ʒᐯဌь፭ƱƳǓŴGal(K)-ь፭ƱƠƯƷᐯ໱ƳನᡯNjλǔŵƱƜǖƕŴžp ᡶ Hodge ྸᛯſ
ƷŴTate ƴLJưᢓǔNjƬƱNjӞƍᢿЎƷ࠙ኽƱƠƯŴK Ʒ p ᡶ‫҄ͳܦ‬Ǜ Cp ưᘙƢƱƢǔƱŴ
def
(T ⊗Zp Cp )Gal(K) ∼
= DX = H 0 (X, ωX/K )
ƱƍƏᐯ໱Ƴӷ‫׹‬ƕ঺ǓᇌƭŵƜƜưŴ߼ᡀƸŴਙࢆϋƷь፭Ʒ Gal(K)-ɧ‫٭‬ᢿЎưŴӫᡀƸŴ
X ɥƍƨǔƱƜǖദЩƳࣇЎμ˳ƔǒƳǔ K ɥƷ g ഏΨșǯȈȫᆰ᧓ưƋǔŵഏƴŴDX ƴ‫ࣖݣ‬
Ƣǔ‫ࢨݧ‬ᆰ᧓Ǜ PX ưᘙƢƱƢǔƱŴX ƕ non-hyperelliptic ưƋǔƱƍƏˎ‫ܭ‬ƔǒŴX ƕ PX Ʒ
ɶƴ೅แႎƴ؈NJᡂLJǕǔƜƱƕŴИሁႎˊૠ࠹˴ƴǑǓƢƙЎƔǔŵƭLJǓŴX Ʒ೅แႎƳ
žλǕNjƷſƱƳǔ PX Ǜ ρX Ɣǒ‫ܦ‬μƴž፭ᛯႎƴſࣄΨƢǔƜƱƴ঺ыƠƯƍǔŵࢼƬƯբ
᫆Ƹ‫˴ڦ‬ƴƠƯ PX ƷিǔཎКƳᢿЎ‫ٶ‬ಮ˳ (ƭLJǓŴX) Ǜ፭ᛯႎƴࣄΨƢǔƔŴƱƍƏƜƱƴ
Ƴǔŵ
ƜƜưŴ§4.2 ưႇ‫ئ‬ƠƨᚐௌႎƳ‫ݧ‬φ : H → PX Ǜ࣬ƍЈƠƯ᪬ƖƨƍŵƦƜưᛇᡓƠƨಮƴŴ
ƜƷ‫ݧ‬ƸˊૠႎƳࣇЎ࢟ࡸǛᚐௌႎƳ‫ݣ‬ᝋƱƠƯನ঺ƢǔƜƱƴǑƬƯ‫ܭ‬፯ƞǕǔNjƷưƋǔŵ
ƜǕǛȒȳȈƴŴ࢘Ʒ p ᡶႎƳᚨ‫ܭ‬ư (িǔॖԛư) ᫏˩ႎƳನ঺ǛᘍƍƨƍƷƩƕŴƦƜưȝǤ
ȳȈƱƳǔƷƸŴH ƴӕƬƯˊǘǔNjƷƸ˴ƔŴƱƍƏƜƱưƋǔŵ[M2] Ǎ [M3] ƷᚰଢưƸŴ
ƦƷࢫႸǛ๫ơƯƘǕǔNjƷƸŴX Ʒ᧙ૠ˳ǛিǔžǑƍſࣱឋǛਤƬƨ p ᡶ˄͌ư‫҄ͳܦ‬ƠŴ
ƦƷ‫҄ͳܦ‬Ʒஇ‫ ٻ‬tame ਘ‫ٻ‬ǛӕǓŴ୼ƴƦƷஇ‫ ٻ‬tame ਘ‫ٻ‬Ǜ p ᡶ‫҄ͳܦ‬ƢǔƜƱƴǑƬƯࢽ
ǒǕǔ˳ưƋǔŵƜƷ˳ǛˌɦưƸ L Ʊ୿ƘƜƱƴƢǔ17) ŵƜƷ˳ L ƸŴQp Ʒஊᨂഏਘ‫ٻ‬Ʊӷ
ಮŴ‫ͳܦ‬Ƴ p ᡶ˄͌ƷλƬƨ˄͌˳ưƸƋǔƕŴQp Ʒஊᨂഏਘ‫ٻ‬ƱᢌƬƯŴž࠹˴ႎƳഏΨſǛ
ɟ̾ԃஊƠƯƍǔNjƷưƋǔŵ̊ƑƹŴƦƷƜƱƷྵǕƷɟƭƱƠƯŴL Ʒй˷˳Ƹஊᨂ˳ɥƷ
ɟ‫٭‬ૠ᧙ૠ˳Ʒஇ‫ٻ‬Ўᩉਘ‫ٻ‬ƱƳǔŵƦƠƯŴL ƷNjƏɟƭƷ᣻ᙲƳࣱឋƱƠƯŴ‫ܭ‬፯ƔǒႺƪ
ƴЎƔǔǑƏƴŴᐯ໱Ƴ‫ݧ‬
ξ : Spec(L) → X
14
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
ƕ‫נ܍‬Ƣǔŵ୼ƴNjƏɟƭƷදႸƢǂƖࣱឋƱƠƯŴL ƱƍƏ˳Ʒӷ‫׹‬᫏Ƹ X ƷȢǸȥȩǤƴƸ
̔‫܍‬ƠƳƍŵƜƷࣱឋƴƸŴH ƷȪȸȞȳ᩿ƱƠƯƷӷ‫׹‬᫏ƕ X ƷȢǸȥȩǤƴ̔‫܍‬ƠƳƍƱƍ
ƏʙܱǛᡲेƞƤǔNjƷƕƋǔŵ
L Ʒ࠹˴ႎƳഏΨƷ‫נ܍‬ƕ̬ᚰƠƯƘǕǔƜƱƷɟƭƱƠƯŴX ɥƷᩐưƳƍࣇЎǛ (ξ ưNjƬ
Ư) L ƴࡽƖ৏ƢƜƱƴǑƬƯࢽǒǕǔ Spec(L) ɥƷࣇЎƸ࣏ƣᩐưƳƍNjƷƴƳǔŵࢼƬƯŴ
X ɥƷࣇЎǛ Spec(L) ƴࡽƖ৏ƢƱƍƏદ˺ƸܱࣙƳદ˺ưƋǓŴƦƷࡽƖ৏ƠƨࣇЎǛŴΨ
ƷࣇЎƷžᚐௌႎƳᘙᅆſƱᙸǔƜƱƕЈஹǔŵᙐእᚐௌႎƳ‫ئ‬ӳƱƷ᫏˩ưᚕƑƹŴƜƷદ˺
ƸdzȳȑǯȈƳȪȸȞȳ᩿ X ɥƷࣇЎǛɥҞ࠯᩿ H ɥƷࣇЎƴࡽƖ৏ƢƱƍƏદ˺ƴ‫ࣖݣ‬ƠƯ
ƍǔŵ
ƞƯŴPX ƷᢿЎ‫ٶ‬ಮ˳ƱƠƯƷ X Ʒ፭ᛯႎࣄΨƷբ᫆ƴ৏ǓƨƍƕŴFaltings Ʒ p ᡶ Hodge
ྸᛯƷ࠙ኽƷɟƭƱƠƯŴ(িǔࢊƍ፭ᛯႎƳவˑǛ฼ƨƢ) ˓ॖƷᡲዓƳแӷ‫׹‬α : Gal(L) →
(p)
π1 (X) ƴ‫ݣ‬ƠƯŴK ɥƷ‫ݧ‬
φα : Spec(L) → PX
ƕ‫ܭ‬፯ƞǕǔŵƭLJǓŴ
žᚐௌႎƳ L-ஊྸໜſƴ‫ݣ‬ƠƯŴ§4.2 ƷᙐእᚐௌƷᛅƴЈƯƖƨφ : H →
PX ƱƍƏ‫ݧ‬ƴ᫏˩ႎƳŴp ᡶᚐௌႎƳ‫ݧ‬φα ƕ‫ܭ‬LJǔŵբ᫆ƸŴφα Ʒ΂ƕƲƏƳǔƔƱƍƏƜƱ
ưƋǔŵ̊ƑƹŴαƕξ ƷǑƏƳž࠹˴ႎƳſL-ஊྸໜ (ƭLJǓŴX(L) ƷΨ) Ɣǒဃơǔ18) ‫ئ‬ӳ
ƴƸŴφα ƷǹǭȸȠᛯႎƳ΂Ʒ᧍Ѽƕ X ƱƽƬƨǓɟᐲƢǔŵࢼƬƯŴ
žα ƕ࠹˴ႎƴဃơǔſƱƍƏவˑǛŴX ƴ˄᨟ƢǔಮŷƳ‫ݣ‬ᝋƷƏƪ
(p)
π1 (X) → Gal(K) ƩƚƠƔᘙᑈӨƴЈƯƜƳƍǑƏƳᘙྵƴ୿ƖႺƢ
ƜƱƞƑЈஹǕƹŴᚰଢƸ‫঺ܦ‬Ƣǔŵ
ƦƜưŴྚ߷Ʒᚰଢ19) ƴȒȳȈǛࢽƨഏƷᛯඥƕஊјƴƳǔŵแӷ‫׹‬αƸŴX Ʒؕᄽ˳Ǜ K
(p)
Ɣǒ L ƴ base-change ƢǔƜƱƴǑƬƯࢽǒǕǔ୺ዴ XL Ʒૠᛯႎؕஜ፭π1 (XL ) → Gal(L)
(p)
(p)
Ʒ section αL : Gal(L) → π1 (XL ) Ǜ‫ܭ‬፯ƠƯƍǔŵƦƷ section แӷ‫׹‬αL Ʒπ1 (XL ) Ʒɶư
(p)
Ʒ΂ Im(αL ) ƸŴ Gal(L) ƴӷ‫׹‬ƳŴπ1 (XL ) Ʒ᧍ᢿЎ፭ƴƳǔƕŴƦƷᢿЎ፭ǛԃljӲ᧏ᢿ
(p)
Ў፭ H ⊆ π1 (XL ) ƴ‫ݣ‬ƠƯŴY H → XL ƱƍƏஊᨂഏǨǿȸȫᘮᙴƕ‫ܭ‬LJǔŵƜƜưŴY H Ƹ
L ɥ࠹˴ႎƴᡲኽƳӑ୺ႎˊૠ୺ዴƴƳǔƕŴදႸƢǂƖƜƱƸŴƜƷǑƏƴဃơǔᘮᙴƨƪ
{Y H → XL } μ˳ƔǒƳǔžᘮᙴƷଈſƕαƴ̔‫܍‬ƠƯƍǔƱƍƏƜƱưƋǔŵࢼƬƯŴแӷ‫׹‬
α ƔǒࡽƖឪƜƞǕǔ section แӷ‫׹‬αL ƴ‫ݣ‬ƢǔவˑƱƠƯŴഏƷNjƷƕ‫҄ࡸܭ‬ưƖǔᲴ
(p)
(∗)
Im(αL ) Ǜԃlj π1 (XL ) ƷӲ᧏ᢿЎ፭ H ƴ‫ݣ‬ƠƯŴ
Y H Ʒ L-ஊྸໜƷᨼӳ Y H (L) ƸᆰƴƳǒƳƍŵ
ˎƴƜƷவˑƕ঺ᇌƢǔƜƱƕᚕƑƨƱƠǑƏŵƦƏƢǔƱŴ᧏ᢿЎ፭ H Ǜᢘ࢘ƴឥǒƤƨƱ
ƖŴY H (L) = ∅ ƳƷưŴY H (L) Ʒ΂ƱƠƯŴXL (L) ƴಮŷƳໜƨƪƕᓳƪƯƘǔŵƱƜǖƕŴ
[F2] Ʒ mod pN ༿ǛᢘဇƢǔƜƱƴǑƬƯŴƜƷǑƏƴဃơǔ XL (L) ƷӲໜƴ‫ݣ‬ƠƯŴέᆉƷ
ϙ΂φα Ʒ mod pN ༿Ǜನ঺ưƖŴƦƷϙ΂ƨƪǛဇƍǔƜƱƴǑƬƯƜƷ XL (L) ƷໜƨƪƕŴ
িǔཎ‫ܭ‬Ʒໜ x∞ ∈ XL (L) ƴӓளƢǔƜƱƕᚰଢưƖǔŵƠƔNjŴƜƷໜ x∞ Ɣǒဃơǔแӷ‫׹‬
(p)
Gal(L) → π1 (X) ƕΨƷแӷ‫ ׹‬α Ʊ࣏ƣɟᐲƢǔƜƱƸŴƜƷನ঺ƔǒႺƪƴࢼƏŵƭLJǓŴ
αƷ࠹˴ࣱƕᚕƑƨƜƱƴƳǔŵࢼƬƯŴ‫ݲ‬ƠૢྸƢǔƱŴαƴ‫ݣ‬Ƣǔவˑ (∗) ƕኝƴž፭ᛯႎſ
ưƋǔƜƱƞƑᚕƑǕƹŴᚰଢƸኳǘǔŵ
ˊૠ୺ዴƷؕஜ፭ƴ᧙Ƣǔ GROTHENDIECK ʖे
15
բ᫆ƸᙲƢǔƴ Y H Ʒ L-ஊྸໜƷ‫נ܍‬Ǜž፭ᛯႎſƴЙ‫ܭ‬ưƖǔƔƲƏƔƩƕŴƜƷբ᫆ƸŴ
ྚ߷ƕৢƬƨஊᨂ˳Ʒ‫ئ‬ӳƷǑƏƴŴஊྸໜƷ̾ૠƷѥ‫ܭ‬ሁƴǑǔႺ੗ႎƳǢȗȭȸȁƴƸᬘ௨
LJƣŴᚐൿƴƸ‫᧓ݲٶ‬੗ႎƳᜭᛯǛ࣏ᙲƱƢǔNjƷưƋǔŵұƪŴL ɥஊྸƳໜƷˊǘǓƴŴL
ɥஊྸƳ (p ƱእƳഏૠƷ) line bundle Ʒ‫נ܍‬ǛᎋƑǔƷưƋǔŵƦƷྸဌƱƠƯŴline bundle
Ʒ૾ƸŴƦƷ Chern ᫏ǛƱǔƜƱƴǑƬƯŴ୺ዴ Y H ƷǨǿȸȫȷdzțȢȭǸȸ᫏ƱƠƯᙸǔƜ
ƱƕЈஹŴƠƔNjӑ୺ႎ୺ዴƷǨǿȸȫȷdzțȢȭǸȸƸƦƷૠᛯႎؕஜ፭Ʒ፭dzțȢȭǸȸƱ
ᐯ໱ƴӷ‫׹‬ƴƳǔƷư‫ܦ‬μƴž፭ᛯႎſƳ‫ݣ‬ᝋƱƳǔŵƢǔƱŴdzțȢȭǸȸ፭ƷɶưŴp ƱእƳ
ഏૠƷ line bundle Ʒ Chern ᫏ƱƠƯဃơǔ᫏ƨƪƷ፭ᛯႎཎࣉƮƚƕբ᫆ƴƳǔƕŴƜƷբ᫆
Ƹ [BK] Ʒ p ᡶ exponential map ƷྸᛯǛᢘဇƢǔƜƱƴǑƬƯΪЎƴ‫ݣ‬ϼӧᏡƳբ᫆ƴ࠙ბƞ
Ǖǔ20) ŵƭLJǓŴY H ɥƷŴ L ɥஊྸƳໜƱᢌƬƯŴL ɥஊྸƳ (p ƱእƳഏૠƷ) line bundle
Ʒ‫נ܍‬ƸŴൔ᠋ႎႺ੗ႎƳ፭ᛯႎЙ‫ܭ‬ඥǛᚩƢǘƚưƋǔŵƱƜǖƕŴИሁႎˊૠ࠹˴ƔǒƢƙ
ЎƔǔǑƏƴŴY H ɥư p ƱእƳഏૠƷ L ɥஊྸƳ line bundle Ʒ‫נ܍‬ƕᚕƑǕƹŴp ƱእƳഏૠ
Ʒ L-ஊྸƳ ample Ƴ line bundle Ʒ‫נ܍‬ƕᚕƑŴƦƷ line bundle Ǜ effective Ƴ‫܇׆‬ƱƠƯ୿
ƘƜƱƴǑƬƯŴY H ƕŴp ƱእƳഏૠƷ L Ʒਘ‫˳ٻ‬ƷɥưஊྸໜǛਤƭƜƱƕЎƔǔŵƠƔƠŴ
ƦƷǑƏƳਘ‫˳ٻ‬ƕ࣏ƣ L Ʒ tame Ƴਘ‫ٻ‬ƴƳǔƜƱƱŴL ƕƦƷ‫ܭ‬፯ǑǓ᩼ᐯଢƳ tame ਘ‫ٻ‬
ǛਤƨƳƍƜƱƔǒŴY H ƕଏƴ L ɥƷஊྸໜǛਤƬƯƍǔƜƱƕ࠙ኽƞǕǔŵƭLJǓŴˌɥƷ
வˑǛ฼ƨƢ line bundle Ʒ‫נ܍‬Й‫ܭ‬ඥƸᐯѣႎƴ L-ஊྸໜƷ‫נ܍‬Й‫ܭ‬ඥƴNjƳǔƱƍƏƜƱư
ƋǔŵࢼƬƯŴƜƷЙ‫ܭ‬ඥƷᄩᇌƴǑƬƯŴ‫ ྸܭ‬5.1 ƷᚰଢƸᬍ‫ރ‬ᑣƘ‫ܦ‬ኽǛᙸǔƜƱƱƳǔŵ
ද
1) ஊᨂ፭Ʒ‫ࢨݧ‬ಊᨂƱƠƯᘙƤǔˮႻ፭Ǜ profinite ፭ (иஊᨂ፭) ƱƍƏŵdzȳȑǯȈƳ‫ܦ‬
μɧᡲኽȏǦǹȉȫȕˮႻ፭ƱƍƬƯNjӷ͌ŵ
2) profinite ፭ƴƓƍƯƸ᧏ᢿЎ፭ưƋǔƜƱƱਦૠஊᨂƳ᧍ᢿЎ፭ưƋǔƜƱƸӷ͌ŵ
3) இᡈƷྚ߷ƷᄂᆮưƸŴദ೅ૠƷ‫ئ‬ӳƴƸŴ‫࢟٭‬ƴஜឋႎƴ̔‫܍‬Ƣǔऴ‫إ‬ƕ࠹˴ႎؕஜ፭
ƷɶƴNjԃLJǕƯƍǔƜƱƕᚐଢƞǕƭƭƋǔŵ[H] ƓǑƼ §3.3 ǛӋༀƷƜƱŵ
4) Grothendieck Ƹӑ୺ႎ୺ዴƷDŽƔŴƦǕǒƷӒࣄႎ smooth fibration ƱƠƯᘙƤǔᆰ᧓
ǍȢǸȥȩǤᆰ᧓ƳƲǛᢒǢȸșȫ‫ٶ‬ಮ˳ƷͅᙀƱƠƯᅆՐƠƯƍǔŵஇᡈƷᄂᆮưƸŴᢒǢȸ
șȫႎƱƳǔƨNJƷ࣏ᙲவˑƱƠƯŴ࠹˴ႎؕஜ፭ƕ ‘ᘍЗ፭ႎ’ ƱƍƏǑǓ ‘ᐯဌ፭ႎ’ ƳNjƷ
ưƋǔƜƱƳƲƕਫƛǒǕƯƍǔŵ[IN] Ӌༀŵ
5) ̊ƑƹŴᖎૠʈඥǛਤƨƳƍ౹ό୺ዴưƸ Faltings ƷኽௐǛኵLjӳǘƤƯᲫ̾ƴƢǔᜭᛯ
NjӧᏡưƋǔŵ[N6] 5.4 Ӌༀŵ
6) ؕᄽ˳ K ƴᲫƷࠉఌǛμƯชьƠƨ˳Ǜ K∞ Ʊ୿ƘƱƖŴX Ʒ K-ஊྸໜƴဌஹƢǔNj
ƷƸ α(Gal(K∞ )) Ʒ π1 (XK ) ǁƷσࢫ˺ဇƕ᩼ᐯଢƳ‫ܭ׍‬ໜǛਤƨƳƍNjƷƱƠƯཎࣉƮƚǒ
ǕǔƜƱNjʖेƞǕƯƍǔŵ
7) ƜǕǑǓŴ Grothendieck Ƹ X Ǜ‫٭‬ૠႎƴᎋƑƯᢒǢȸșȫ‫ٶ‬ಮ˳ Y Ʒ X-ஊྸໜƷᨼӳ
Y (X) = Hom(X, Y ) Ǜؕஜ፭ƴǑǓࣄΨƢǔƱƍƏ‫ח‬ᛯႎྸेǛ࣬ƍ੨ƍƯƍƨƱᎋƑǒǕǔŵ
ƜƷྸेǛИNJƯ (ᢿЎႎƴ) ܱྵƠࢽƨƷƸஓஉ ([M3]) ưƋǔŵ§5 Ӌༀŵ
8) ႻီƳǔஊྸໜƴ‫ݣ‬ƠƯƸσࢫưƳƍ section แӷ‫׹‬ƕ‫ࣖݣ‬ƢǔŴƱƍƏƜƱƸŴMordellWeil Ʒ‫ྸܭ‬ƷࣖဇƱƠƯ Grothendieck ([G3]) ƴǑǓᅆƞǕƯƍǔŵƜƷƜƱƷࣖဇƱƠƯŴᙐ
እӑ୺࠹˴‫ܖ‬ƴƓƚǔžჿဋʖेſƷˊૠ࠹˴ႎ᫏˩ǛƋǔᆔƷӑ୺‫ٶ‬ಮ˳ƴƭƍƯᅆƢƜƱƕ
žႻီƳǔஊྸໜƴ‫ࣖݣ‬Ƣǔ section
Јஹǔ ([N5], [N7]2.2 Ӌༀ)ŵƳƓŴஓஉƸŴ‫ ྸܭ‬5.1 ƔǒŴ
แӷ‫׹‬ƕσࢫƴƸƳǒƳƍſƱƍƏʙܱƷ pro-p ༿Ǜ‫ݰ‬ƍƯƍǔ ([M3], Theorem C ǛӋༀ)ŵ
16
ɶ஭Ҧଯ, ྚ߷‫ܤ‬ᬱဏ, ஓஉૼɟ
9) ƜǕƴƭƍƯƸ F.Pop ([P1,2])Ŵஓஉ (§5 Ӌༀ) ƴǑǔᝡྂƕƋǔŵƳƓŴPop ƷᄂᆮƸŴ
ƦƷ৖ඥNjԃNJƯŴᲫᲳᲰᲪ࠰ˊࢸҞƷ J.Neukirch ([Ne]) Ʒ཯оƴ‫ڼ‬LJǓŴ൷ဋദ᬴Ŵ‫ޥ‬ຓͤ
ӴǒƷˁʙǛኺƯᲱᲪ࠰ˊࢸҞƴϋဋᐻʚƴǑƬƯ‫঺ܦ‬ƞǕƨžˊૠ˳Ʒዌ‫ݣ‬ǬȭǢ፭ƔǒƷࣄ
ΨſƱƍƏᄂᆮƷ්ǕǛൽljNjƷưƋǔŵ
10) profinite ፭ G ƷˮႻႎƳՠ፭ưஊᨂ l ፭Ʒ‫ࢨݧ‬ಊᨂƱƠƯ୿ƚǔNjƷƷƏƪஇ‫ٻ‬ƷNjƷ
Ǜ G Ʒஇ‫ ٻ‬pro-l ՠ፭ƱƍƏŵ
11) [I1] Ʒࡀ૨ƴᅆՐƞǕƯƍǔǑƏƴŴ˙ҾƸᲫᲳᲰᲪ࠰ˊǑǓ཯ᐯƷᚇໜƔǒஊᨂ˳ɥ
Ʒ modular ᧙ૠ˳ƴ‫ݣ‬Ƣǔ᩼Ǣȸșȫ᫏˳ᛯƷ࡫ᚨǛ‫ڼ‬NJŴᲱᲪ࠰ˊИ᪽LJưƴƸ Fp2 ɥƷ
P1λ − {0, 1, ∞} Ʒ tame ᘮᙴƷƏƪ SL2 (Z[ 1p ]) Ʒ (ӳӷ) ᢿЎ፭ưወСƞǕǔଈƕŴ‘supersingular
Ƴ λ-እໜᨼӳƷ‫ܦ‬μЎᚐ’ ƱƍƏૠᛯႎƳவˑưཎࣉƮƚǒǕǔƜƱƳƲǛᅆƠƯƍǔŵƜƷǑƏ
Ƴ‫ࢨݧ‬ႺዴƷᲭໜЎ‫ؕޟ‬ஜ፭ƴԃLJǕǔขƍૠᛯࣱƷᄂᆮƸŴ§1.1 ưᡓǂƨǑƏƳ Grothendieck
Ʒˊૠ࠹˴ႎƳǬȭǢྸᛯ࡫ᚨƷѣೞƱƸμƘီឋƳឪเǛਤƭŴ˙Ҿ཯ཎƷ (᩼Ǣȸșȫ) ᫏
˳ᛯႎƳᚇໜƔǒဃLJǕƯƍǔŵƳƓŴ[2] ৑ӓƷ Deligne Ʒᛯ૨ƸŴȢȁȸȕՋ‫ܖ‬Ʒ්ǕƷɶ
ưƜƷᲭໜЎ‫ؕޟ‬ஜ፭Ʒžࠉҥˊૠ፭҄ſǛӕǓɥƛƯƍǔƕŴƜǕNjLJƨ཯ᇌƷᚇໜǛឪเƱ
ƠƯƍǔƱƍƑǔŵ
12) ƜƷ଺஖Ʒᡶ‫ޒ‬ƴƭƍƯƸɼƴ [I2] ƓǑƼ [1] ৑ӓƷᜂᛯ૨ǛӋༀŵ
13) ƜƜưƸǬȭǢ፭ƷᲫƷࠉఌǁƷ˺ဇƔǒဃơǔᐯ໱ƳɟഏΨ l ᡶᘙྵ Zl (1) Ǜቇҥƴ
όЎ˺ဇƱԠƼŴƜǕƱ cusp ໜƷᨼӳƴƓƚǔፗ੭ᘙྵƱƷȆȳǽȫᘙྵǛ ‘όЎፗ੭ᘙྵ’ Ʊ
Ԡǜưƍǔŵ
14) ૠ˳ɥƷᆔૠ 2 ˌɥƷ୺ዴƷஊྸໜƕஊᨂ̾ƠƔƳƍƱƍƏʖेŵFaltings ƴǑƬƯ Tate
ʖेƱӷơᛯ૨ ([F1]) ưᚰଢƞǕƨŵ
15) ܱƸŴƜƷբƍƴ‫ݣ‬ƠƯƸŴNjƏɟƭƷŴ‫ݲٶ‬ᢌƏࣱឋƷᏉ‫ܭ‬ႎᚐሉNjࢽǒǕƯƍǔ (ᛇ
ƠƘƸ [M5-8] ǛӋༀ)ŵ
16) ܱƸŴmysterious functor Ʒ‫נ܍‬ǛʖᚕƠƨƷNj˂ưNjƳƍ Grothendieck ƩƕŴʚƭƷ
ʖेƷ᧙̞ƴƭƍƯƸŴ‫ٻ‬ᩃ৭Ƴ࢟Ʒ᫏˩ࣱƸƱNjƔƘŴ§5.2 ưኰʼƢǔǑƏƳᚰଢƴƭƳƕǔ
ᆉ݅੗ƳNjƷƩƱƍƏᛐᜤǛஜʴƕਤƬƯƍƨ࢟ួƸƳƍŵƦƏƍƬƨཞඞƷᏑ୎ƴƭƍƯƸŴ
§4.1 ǛӋༀŵ
17) ܱƸŴL Ʒ‫ܭ‬፯ƱƠƯƜǕǛ੔ဇƢǔƱŴˌɦƷᜭᛯƕ‫ݲ‬ƠƸɧദᄩƴƳǔƕŴ˷Ǔஜឋ
ႎưƳƍ২ᘐႎƳᛅǛஇ‫ݱ‬ᨂƴƱƲNJǔƨNJƴŴƜƷǑƏƳƜƱƸࣂܾឡ᪬ƘƜƱƴƢǔŵ
18)ž࠹˴ႎƳஊྸໜ Spec(L) → X ƔǒဃơǔſƱƸƭLJǓŴ‫ ݧ‬Spec(L) → X ƴ‫ݣ‬ƠƯŴπ1
(p)
ƱƍƏ᧙৖Ǜ଀ƢƜƱƴǑƬƯࢽǒǕǔ‫ ݧ‬Gal(L) = π1 (Spec(L)) → π1 (X) → π1 (X) ƱƠƯ
ဃơǔƱƍƏॖԛưƋǔŵ
19) ᛇƠƘƸŴ§3.1 (i) ǛӋༀŵ
20) ᛇƠƘƸŴ[M3] ǛӋༀŵ
૨
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[B]
[BK]
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