p-shin teichmuller riron no kaisetsu (hokudai 2001-01)

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1、AnIntroductiontop-adicTeichm¨ullerTheoryஓஉૼɟ1TheStackofNilcurves1.1ᙐእ࠹˴ƔǒƷmotivationஜራƴƭƍƯƸ[Ord],IntroǛӋༀŵXǛCɥƷhyperboliccurve(smooth,proper,connected,genusgminusrpoints2g−2+r≥0)ŴXǛ˄᨟ƢǔȪȸȞȳ᩿ƱƢǔŵؕஜ፭π1(X)ƕ୍ᢄᘮᙴX˜ƴ˺ဇƢǔŵKoebeƷɟॖܭྸX˜∼=H:={z∈C

2、Im(z)>0}Remark1.1.Mumford-SchottkyɟॖƱƸᢌƏŵբ᫆ᲴƜǕƷpᡶ᫏˩ƕDŽƠƍŵpᡶႎƴৢ

3、ƏƨNJƴɥƷܭྸƷˊૠႎƳᢿЎǛӕǓЈƢŵπ1(X)→PSL2(R)→PGL2(C)ƔǒᲢPSL2(R)ƸHƷദЩᐯࠁӷ׹፭μ˳ƱLjƳƤǔᲣ(XטP1)/π(X)→X˜/π(X)C11ǛࢽǔŵƜƷˊૠǛP→XƱƢǔŵH∼=X˜→XטP1CǑǓsectionσ:X→PƱ੗ዓ∇PƕܭLJǓŴ∼∗࠯-Spencermorphism∇P(σ):τX→στP/XƸӷ׹ƱƳǔŵƜƷǑƏƳ(P,∇P)Ǜ׍ஊளᲢindigenousbundleᲦ᳃᲼ᲣƱƍƏŵ᳃᲼μ˳ƸH0(X,ω⊗2)ɥƷtorsorƴƳƬƯƍǔᲢr=0ƷƱƖᲣᲴXS¯g,r→M¯g,r1logƸΩM¯-torsorŵƜƜưŴS¯g

4、,r,M¯g,rƸƦǕƧǕstablecurve+IB,type(g,r)g,rstablecurveƷmoduliᆰ᧓ŵKoebeƷɟॖƔǒ೅แႎᲢcanonicalᲣƳܱᚐௌႎƳЏૺƕܭLJǔᲴsH(S¯g,r)C(M¯g,r)CƜǕƷpᡶ᫏˩ƕDŽƠƍ1.2׍ஊளƷؕஜႎࣱឋᛇƠƘƸ[Ord],Ch.I,§1,2ǛӋༀŵ1.2.1ܭ፯f:X→SǛproper,smooth,genusgcurveƷଈƱƢǔᲢቇҥƷƨNJr=0)ŵπ1Definition1.2.(P→X,∇P)ƕ׍ஊளᲢ᳃᲼ᲣưƋǔƱƸŴ੗ዓ˄ƖP-ȐȳȉȫưƋƬƯŴƋǔЏૺσ:X→Pƕ܍נƠƯ∇P(σ):τX→σ∗τP/

5、Xƕӷ׹ưƋǔƜƱǛƍƏŵRemark1.3.ƭLJǓlocalƴƸɟॖƴƳƬƯƍǔƱƍƏƜƱŵ1.2.2Filtration&deRhamcohomologyA=Ad(P)=π∗τP/XƱƓƘŵƜǕƸXɥƷȩȳǯᲭƷșǯȈȫளưŴޅ৑ႎƴsl2Ʊӷ׹ƳLie࿢ƷನᡯǛNjƭŵഏƷ׋ࡸπ(τ)iσ∗τ∼=τ∗P/XP/XXj2∼=OKeri(Iσ/Iσ)⊗τP/XX∼=Kerj(I2/I3)⊗τ∼=ωσσP/XXƔǒAƷfiltrationƕᛔƞǕǔᲴ(F−1/F0)(A)=τ,(F0/F1)(A)=O,(F1/F2)(A)=ω.XXX.ƠƨƕƬƯŴRfDR,∗(A,∇A)

6、ƴfiltrationƕᛔƞǕǔŵ2Proposition1.4.i=1ƷƱƖŴRif(A,∇)=0.DR,∗Ai=1ƷƱƖഏƷܦμኒЗƕƋǔ:0→fω⊗2→R1f(A,∇)→R1fτ→0.∗XDR,∗A∗Xᇹɟ᪮Ტᇹʚ᪮Ŵᇹɤ᪮ᲣƸ᳃᲼ƱƠƯƷ٭࢟Ტ(P,∇P)Ʒ٭࢟ŴXƷКƷ٭࢟Ʒ᳃᲼ƱƳǔ٭࢟ᲣǛᚘǔŵᇹʚ᪮Ɣǒᇹɤ᪮ǁƷϙ΂ƸcurveƷ٭࢟ưƋƬƯ(P,∇P)ƕƦƷ٭࢟ɥ᳃᲼ƱƳǔNjƷǛݣࣖƞƤǔŵ1.2.3FormaluniformizationഏƷ׋ࡸǛᙸǔŵӷ׹ϙ΂Ƹ∇PƷ࢟ࡸᆢЎƴǑǔᲵσƸ᳃᲼Ǜܭ፯ƢǔƱƖƴྵǕǔsectionŵ∼=π∗Pπ∗P12

7、π∗σ1PDX×SXPπ2π1XXDǛπ2ƴǑǔOƷpush-forwardƱƢǔŵӳ঺ǛƱǔƱɦƷໜዴƷݧƕưƖǔᲴXσdSpecDPXPDƜƜưdƸdiagonalX→X×SXƔǒƘǔNjƷŵƜǕƔǒŴ࢟ࡸɟॖƕࢽǒǕǔᲴξOˆPD@σ→∼D.Pӷ׹ưƋǔƜƱƸŴI/I2I/I2σσDD∼=σ∗ωωP/XX3ƔǒǘƔǔŵƜƜưɦƷӷ׹Ƹ࠯-SpencerƔǒŵCorollary1.5.[3]P∼=P(ID/ID)ƭ

8、LJǓŴ᳃᲼Ƹ∇PưൿLJǔŵProof.P∼=P(I/I[3])(P1-bundleƷtautology)σσ∼=P(I/I[3])(formaluniformizationǑǓ)DDCorollary1.6.⊗2{᳃᲼μ˳}=(f∗ωX)-torsorProof.0I[2]/I[3]I/I[3]I/I[2]0DDDDDD∼=∼=ω⊗2ωXX∇,∇ǛʚƭƷ੗ዓƱƢǔƱŴ∇−∇∈F0(Ad(I/I[3]))⊗