An Introduction to p-adic Teichm¨uller Theory

An Introduction to p-adic Teichm¨uller Theory

ஓஉ ૼɟ

1 The Stack of Nilcurves

1.1 ᙐእ࠹˴ƔǒƷ motivation

ஜራƴƭƍƯƸ[Ord], IntroǛӋༀŵ

XǛCɥƷhyperbolic curve (smooth, proper, connected, genusgminus rpoints 2g−2 +r≥0)ŴX Ǜ˄᨟ƢǔȪȸȞȳ᩿ƱƢǔŵؕஜ፭π1(X)ƕ

୍ᢄᘮᙴX˜ƴ˺ဇƢǔŵ

KoebeƷɟॖ҄ܭྸ X ∼˜ =H:={z∈C|Im(z)>0} Remark 1.1. Mumford-Schottkyɟॖ҄ƱƸᢌƏŵ

բ᫆ᲴƜǕƷpᡶ᫏˩ ƕDŽƠƍŵ

pᡶႎƴৢƏƨNJƴɥƷܭྸƷˊૠႎƳᢿЎǛӕǓЈƢŵ π1(X)→PSL2(R)→PGL2(C) ƔǒᲢPSL2(R)ƸHƷദЩᐯࠁӷ׹፭μ˳ƱLjƳƤǔᲣ

( ˜X ×P¹_C)/π1(X)→X˜/π1(X) ǛࢽǔŵƜƷˊૠ҄Ǜ

P →X ƱƢǔŵ

H∼= ˜X →X ט P¹_C ǑǓsectionσ:X→PƱ੗ዓ∇PƕܭLJǓŴ

ݱ࠯-Spencer morphism ∇P(σ) :τX→∼ σ^∗τP /X

Ƹӷ׹ƱƳǔŵƜƷǑƏƳ(P,∇P)Ǜ׍ஊளᲢindigenous bundleᲦ᳃᲼ᲣƱ ƍƏŵ᳃᲼μ˳ƸH⁰(X, ω_X^⊗2)ɥƷtorsorƴƳƬƯƍǔᲢr= 0ƷƱƖᲣᲴ

¯S^g,r→M¯^g,r

ƸΩ^log_M¯

g,r-torsorŵƜƜưŴ¯S^g,r, ¯M^g,rƸƦǕƧǕstable curve+IB, type(g, r) stable curveƷmoduliᆰ᧓ŵKoebeƷɟॖ҄Ɣǒ೅แႎᲢcanonicalᲣƳܱ

ᚐௌႎƳЏૺƕܭLJǔᲴ

(¯Sg,r)_C //( ¯Mg,r)_C

sH

vv

ƜǕƷpᡶ᫏˩ƕDŽƠƍ

1.2 ׍ஊளƷؕஜႎࣱឋ

ᛇƠƘƸ[Ord], Ch.I,§1,2ǛӋༀŵ

1.2.1 ܭ፯

f : X → S Ǜproper, smooth, genus g curveƷଈƱƢǔᲢቇҥƷƨNJ r= 0)ŵ

Definition 1.2. (P →^π X,∇P)ƕ׍ஊளᲢ᳃᲼ᲣưƋǔƱƸŴ੗ዓ˄ƖP¹- ȐȳȉȫưƋƬƯŴƋǔЏૺσ:X →P ƕ܍נƠƯ∇P(σ) :τX →σ^∗τP /X

ƕӷ׹ưƋǔƜƱǛƍƏŵ

Remark 1.3. ƭLJǓlocalƴƸɟॖ҄ƴƳƬƯƍǔƱƍƏƜƱŵ

1.2.2 Filtration & de Rham cohomology

A= Ad(P) =π_∗τP /XƱƓƘŵƜǕƸXɥƷȩȳǯᲭƷșǯȈȫளưŴޅ

৑ႎƴsl2Ʊӷ׹ƳLie࿢ƷನᡯǛNjƭŵഏƷ׋ࡸ

π_∗(τP /X) ⁱ ////σ^∗τP /X ∼=τX

Ker?OO i ^j ////

(Iσ/I²σ)⊗τP /X∼=OX

Ker?OO j ^∼= //

(I²σ/I³σ)⊗τP /X ∼=ωX

ƔǒAƷfiltrationƕᛔݰƞǕǔᲴ

(F⁻¹/F⁰)(A) =τX,(F⁰/F¹)(A) =OX,(F¹/F²)(A) =ωX. ƠƨƕƬƯŴR

.

Proposition 1.4. i= 1 ƷƱƖŴ

RⁱfDR,∗(A,∇A) = 0.

i= 1 ƷƱƖഏƷܦμኒЗƕƋǔ:

0→f_∗ω^⊗_X²→R¹fDR,∗(A,∇A)→R¹f_∗τX →0.

ᇹɟ᪮Ტᇹʚ᪮Ŵᇹɤ᪮ᲣƸ᳃᲼ƱƠƯƷ٭࢟Ტ(P,∇P)Ʒ٭࢟ŴX ƷКƷ

٭࢟Ʒ᳃᲼ƱƳǔ٭࢟ᲣǛᚘǔŵᇹʚ᪮Ɣǒᇹɤ᪮ǁƷϙ΂ƸcurveƷ٭࢟

ưƋƬƯ(P,∇P)ƕƦƷ٭࢟ɥ᳃᲼ƱƳǔNjƷǛݣࣖƞƤǔŵ

1.2.3 Formal uniformization

ഏƷ׋ࡸǛᙸǔŵӷ׹ϙ΂Ƹ∇P Ʒ࢟ࡸᆢЎƴǑǔᲵσƸ᳃᲼Ǜܭ፯Ƣǔ ƱƖƴྵǕǔsectionŵ

π^∗₁P ^∼= //

##H

HH HH HH

HH π^∗₂P

{{vvvvvvvvv

@

@@

@@

@@

@@ X×^PDS X

π^∗₁σ

]]

π₁

zzvvvvvvvvv ^π2

$$H

HH HH HH

HH P

~~}}}}}}}}}

X X

DǛπ2ƴǑǔOƷpush-forwardƱƢǔŵӳ঺ǛƱǔƱɦƷໜዴƷݧƕư ƖǔᲴ

X

σ

((Q

QQ QQ QQ QQ QQ QQ QQ QQ

d

SpecD_ _ _ _ _ _ _//

##G

GG GG GG

GG P



X

ƜƜưdƸdiagonalX →X ×^PDS XƔǒƘǔNjƷŵƜǕƔǒŴ࢟ࡸɟॖ҄ƕ ࢽǒǕǔᲴ

Oˆ^PD@σP

∼ξ

→D.

ӷ׹ưƋǔƜƱƸŴ

Iσ/I²σ //I_D/I²_D

σ^∗ωP /X

∼

= //ωX

ƔǒǘƔǔŵƜƜưɦƷӷ׹Ƹݱ࠯-SpencerƔǒŵ Corollary 1.5.

P ∼=P(ID/I^[3]_D) ƭLJǓŴ᳃᲼Ƹ∇PưൿLJǔŵ

Proof.

P∼=P(Iσ/I^[3]σ ) (P¹-bundleƷtautology)

∼=P(I_D/I^[3]_D) (formal uniformizationǑǓ)

Corollary 1.6.

{᳃᲼μ˳}= (f_∗ω_X^⊗2)-torsor Proof.

0 //I^[2]_D/I^[3]_D //ID/I^[3]_D //ID/I^[2]_D //0

ω^⊗_X²

∼=

OO

ωX

∼=

OO

∇P,∇PǛʚƭƷ੗ዓƱƢǔƱŴ∇P−∇P ∈F⁰(Ad(I_D/I^[3]_D))⊗ωXŵ∇P,∇P

Ǜ˄᨟ƢǔξƕidƱƳǔǑƏƴ੔ǔƜƱƴǑǓŴܱƸ∈F¹(Ad(I_D/I^[3]_D))⊗ ωX=ω^⊗_X²ŵ

1.3 ദ೅ૠưƷྸᛯ

ᛇƠƘƸ[Ord], Ch.II,§2ǛӋༀŵ

1.3.1 Motivation

Mell Ǜ౹ό୺ዴƷZp ɥƷstackŴE →^f Mell Ǜtautological౹ό୺ዴŴ E=R¹fDR,∗O^EƱƢǔŵEƸM^ellɥƷȩȳǯᲬƷșǯȈȫளư∇EǛGauss- Manin੗ዓƱƢǔŵƜƜưP =P,∇P =P(∇E)ƱƓƘƱ(P,∇P)Ƹ᳃᲼ƴ ƳǔᲢCɥưNj ೅แႎ᳃᲼ ƴƳǔᲣŵɟ૾ŴpᡶႎƴƸŴFrobeniusƷ˺ဇƕ ƋǔŵƜƜưƸƠƔƠnaiveƳFrobeniusưƸҗЎưƳƍƨNJrenormalized FrobeniusF^∗ ǛဇƍǔᲢᛇƠƘƸMain Theorem IIǛӋༀᲣŵ

Frobenius invarianceF^∗(E,∇E)∼= (E,∇E) F^∗(P,∇P)∼= (P,∇P)

ƕ঺ǓᇌƪŴƜƷࣱឋƴǑǓž೅แႎſƳNjƷǛ੕ƢɥưƷٻƖƳȒȳȈƕ ࢽǒǕǔᲴ

(E,∇E)Ʒp-curvatureƸsquare nilpotent

Remark 1.7. p-curvatureƱƸAd(E)⊗Φ^∗_Mellω_Mell ƷΨưŴδƕderivation ƳǒδǛ∇δ^p−(∇δ)^pƴƏƭƢŵ

1.3.2 NilcurvesƷstack

NilcurveƷstackN^g,rƱƸhyperbolic curveƱ᳃᲼ƷኵưƋƬƯp-curvature ƕsquare nilpotentƴƳǔᢿЎƱƢǔᲴ

Ng,r HHHHHHHH//H(##S^g,r)_Fp

(Mg,r)_Fp Theorem 1.8 (Main Theorem I).

N¯g,r→( ¯Mg,r)_Fp

Ƹfinite, flat, local complete intersection, degree=p^3g−3+rŵ

ƭLJǓŴup to isogenyưƸƜǕƕ೅แႎƳsectionƱƳƬƯƍǔŵ

1.3.3 ܭྸƷᚰଢ

(a)

(¯Sg,r)_Fp V //

$$J

JJ JJ JJ JJ

J Q¯g,r=V(Φ^∗Ω^log_M¯

g,r)

wwnnnnnnnnnnn

( ¯Mg,r)_Fp

૲NJƷϙ΂ƸƲƪǒNj(3g−3 +r)-dim relative affine spaceŵVƸ᳃᲼ɥƷ Verschiebung(ɦƷɟᘍႸưܭ፯ᲣŵC¯→^f M¯g,rǛuniversal curveƱƢǔŵ

(P,∇P) → −det(p-curvature∈Ad(P)⊗Φ^∗_XωX/S)

= 1

2Tr(p-curvature²)∈f_∗Φ^∗_Xω^⊗_X/S² ƔƭX/S-horizontal

⇒∈Φ^∗_S(f_∗ω^⊗_X/S² )

⇒N^g,r =V⁻¹(0)

ƠƨƕƬƯܭྸǛᅆƢƴƸŴXiƨƪǛ( ¯M^g,r)_FpɥƷ relative affine ࡈ೅Ʊ ƢǔƱƖŴ

V:Xi →X_i^p+ (deg< p) i= 1, . . . ,3g−3 +r

ƭLJǓŴdeg≤pƔƭɼᙲ᪮ƕᐯ໱ƳݧΦ^∗Ω→S^pΩưƋǔƜƱǛƍƑƹǑ ƍŵƦǕƴƸ

1

2Tr(p-curvature²)ǛᚘምƢǕƹǑƍ (b)

(P,∇P)Ǜ᳃᲼ƱƢǔƱPƸƍƭNjӷơƩƔǒ

∇P =∇+θ

∇Ƹ׍ܭƞǕƨNjƷưŴθ∈F¹(Ad(P))ƕѣƘᢿЎƱưƖǔŵƭLJǓŴ¹₂Tr({(∇+ θ)^p}²)ǛθƷ᧙ૠƱƠƯᚘምƠŴdeg ≤pŴɼᙲ᪮ƕɥƷǑƏưƋǔŴƜƱ ǛᅆƤƹǑƍŵ

(c)

LJƣƸŴ(∇+θ)^pǛᎋƑǔŵθ²= 0Ǜ̅ƏƱŴθƕɟဪٶƍ᪮Ƹ θ∇θ . . . θ∇θƱ∇θ∇. . .∇θ∇

ЭᎍƸʚʈƠƨǒŴ0ƴƳǔƷư 1

2Tr{(∇+θ)^p}²=1

2Tr(θ∇θ . . . θ∇θ∇θ . . . θ∇) + (deg_θ< p) ǑǓdegθ≤p.

(d)

θ²= 0ƴදॖƢǔƱ

θ∇θ∇θ . . . θ∇θ∇= [θ,∇]^p−1θ

ƱƜǖƕŴ[θ,∇]ƸOX-linearƔƭŴ∈F⁰(Ad(P))ŵƠƨƕƬƯŴ[θ,∇]^p−1θ NjOX-linearƔƭ∈F¹(Ad(P))ŵƠƔNjŴ∇ƷF⁻¹/F⁰ᢿЎƸOX-linearŵ ݱ࠯-Spencerϙ΂ƕidưƋǔƜƱǛ̅ƏƱŴ

1

2Tr(. . .) = (abs.const)θ^p+ (deg_θ≤p)

ƱᘙƤǔŵࢸƸabs.constƷᚘምŵƜǕƸnodeǍmarked pointƷȢȎȉȭ ȟȸǛᚘምƢǕƹǑƍᲴ(M =ȢȎȉȭȟȸ˺ဇእ)

(td

dtǛ˺ဇƞƤǔƱ)p-curvature =M^p−M =M((detM)^p−12 −1)

⇒ 1

2T r(p-curvature²) =−det(M)((detM)^p−12 −1)²

⇒deg_θ= degdetM =p

2 Canonical p -adic Liftings

2.1 Ordinary nilcurves Ʒ canonical liftings

ᛇƠƘƸ[Ord], Ch.II,§3; Ch.IIIǛӋༀŵ

Ng,r⊇(N^ordg,r)_Fp={M¯g,rɥ´etale locus} =∅

ǛᎋƑǔŵƜǕǛordinary nilcurves ƱǑƿŵJacobianƕordinaryƱƸᢌ Əŵ“ordinary”ƷNjƏɟƭƷܭ፯ǛɨƑǑƏŵX →S, (P,∇P)ǛFp ɥƷ nilcurveƱƠŴΦưFrobeniusǛᘙƢŵ

p-curvature∈Ad(P)⊗Φ^∗_XωX/S

⇒Φ^∗_XτX/S →Ad(P)

⇒Φ^∗_XτX/S →Ad(P)→(F⁻¹/F⁰)(Ad(P)) =τX/S

ǑǓഏǛࢽǔᲴ Φ^∗_SR¹f_∗τX/S

Φ^∗_X

→ R¹f_∗Φ^∗_XτX/S →R¹f_∗τX/S(= Θ_Mg,r|S).

Theorem 2.1.

(X →S,(P,∇^P))ordinary⇔Φ^∗_SR¹f_∗τX/S R¹f_∗τX/S

Proof. ဦŵƩƚƲ

´

etale⇔Ng,rƷtangent spaceMg,rƷtangent space ƸƝኛࢽƍƨƩƚǔƸƣŵ

2.1.1 ordinary nilcurveƷ canonical lifting

X_Fp → S_Fp ⊆S = Spec(W(R)), R :perfect/Fp,P:= (P,∇P)_FpǛ೅ૠp Ʒordinary nilpotent᳃᲼ƱƢǔŵƜǕǛ(X_Zp,P_Zp)ƴcanonicalƴਤƪɥ ƛƨƍŵ

Theorem 2.2 (Main Theorem II). ɦᚡƷவˑǛ฼ƨƢuniqueƳ (X_Zp,P_Zp)ƕƋǔŵ

வˑ: F^∗PZpPZp

Remark 2.3. PZpǛCrys(X_Fp/S)ɥƷP¹-bundle Ʒ crystal ƱƢǔŵP = P(E)ƷƱƖŴF¹(E)⊆EƱƍƏrank 1ƷșǯȈȫளƕ᳃᲼Ʒܭ፯ƴЈƯƘ ǔσǑǓൿLJǔŵΦX ǛFrobeniusƱƢǔƱƖŴ

Φ^∗_X(E,∇E)⊗F^p⊇Φ^∗_XF¹(E)⊗F^p.

renormalized FrobeniusƴǑǔpull-backF^∗(E,∇E)ǛΦ^∗_X(E,∇E)Ʒsection ưƋƬƯŴmodpƠƨƱƖƴΦ^∗_XF¹(E)⊗Fp ƴλǔNjƷƱܭ፯ƢǔŵƜǕƸ rank 2șǯȈȫளƷcrystalƴƳǔŵF^∗(P,∇_P)ƸƦƷݧࢨ҄ƱƢǔŵ

LJƣŴ٭࢟Ʒಮ܇Ǜ࣬ƍЈƢŵᇹɟ᪮ƸF²Ŵᇹɤ᪮ƸF⁻¹/F⁰ưƋƬƨŵ 0→f_∗ω_X/S^⊗2 →R¹fDR,∗Ad(P)→R¹f_∗τX/S→0

Proof. ׋ࡸưᎋƑǔŵɦƷӲᘙƸPZ^pǛᘙƢŵƢǂƯƕOKƴƳǕƹᚰଢƕ

ኳǘǔŵ

(1) modpƴƭƍƯƸǘƔƬƯƍǔŵ

F² F⁻¹/F⁰

OK OK modp

? ? modp²

? ? modp³

Lemma 2.4. F^∗ƢǔƱ

F⁻¹/F⁰ →^∼ p⁻¹· F⁻¹/F⁰ F² →p²· F²

Remark 2.5. ᚰଢƴƭƍƯƸ[Ord], Ch.III, Lemmas 2.5, 2.6ǛӋༀŵ (2) LemmaǑǓp· F⁻¹/F⁰ modp²ƕൿܭƞǕǔƷưŴ

F² F⁻¹/F⁰

OK OK modp

? OK modp²

? ? modp³

ƭLJǓŴX_Z/p2 ƕൿLJƬƨŵ

(3)F⁻¹/F⁰modp³ƴƭƍƯƸŴܭྸƷவˑƷӷ׹ƔǒൿܭƞǕǔᲴ F² F⁻¹/F⁰

OK OK modp

? OK modp²

? OK modp³

? ? modp⁴

ƭLJǓŴჇɥƷOKƔǒOKƱƳǔŵ

(4) LemmaǑǓ(F^∗P)_Z/p2 ƸF²modp² ƴǑǒƳƍƷưவˑǑǓƜǕNj

ൿLJǔŵ

F² F⁻¹/F⁰

OK OK modp

OK OK modp²

? OK modp³

? ? modp⁴

ƜǕƸ(2)ƱӷơǑƏƳཞඞƴƳƬƯƍǔƷư৖᪯ǛጮǓᡉƤƹǑƍŵ

2.2 moduli Ʒئӳ

ᛇƠƘƸ[Ord], Ch.IIIǛӋༀŵ (N^ordg,r)_Fp

´

→et (Mg,r)_Fp Ƹ ´etaleƳƷưuniqueƳp-adicƳਤƪɥƛN :=

(N^ordg,r)_Zp →^´et (Mg,r)_Zp ƕƋǔŵέDŽƲƷ§1ƷᜭᛯưŴS_Fp :=N_F^{P F}_p ƱƠƯƓ ƘƱŴ

X_Zp →S=W(N_F^{P F}p )

⇒S →(Mg,r)_Zp s.t. NǛኺဌƢǔ

⇒S →N

Ǜࢽǔŵ

Lemma 2.6. ∃! Frobenius liftingΦ_N:N→Nsuch that S −−−−→ N

⏐

⏐^ΦS ⏐⏐^ΦN S −−−−→ N

Corollary 2.7. ∃!(Φ_N,(P,∇P))such that F^∗(P,P)∼= (P,∇P).

Remark 2.8. ᇹᲫᇘ§1ƷsH Ʒ᫏˩ưƋǔp-adic formal stack᧓Ʒϙ΂ƕ ࢽǒǕǔᲴ

(S^g,rˆ)_Zp

N

==z

zz

zz //(Mg,rˆ)_Zp

Remark 2.9. CƷئӳᲢBersɟॖ҄ᲣƱӷಮŴΦ_NƸNɥƷ೅แႎƳࡈ೅Ǜ ɨƑǔᲛᲢSerre-TateƱ᫏˩ႎƳƕǒTorelliƴ᧙ƠƯcompatibleưƸƳƍᲣŵ Remark 2.10. Φ_N ƷǑƏƳ Ordinary Frobenius liftingƸܱᚐௌႎDZȸȩȸ ᚘ᣽Ʒpᡶ᫏˩ưƋǔŵ

p-adic C

hyperbolic curves p-adic Teichm¨uller Weil-Petersson metric

abelian varieties Serre-Tate Siegel upper half-plane metric

2.3 Galois ᘙྵ

ᛇƠƘƸ[Ord], Ch. IV, VǛӋༀŵ

2.3.1 k perfect, char pƷƱƖ

X → S = Spec(W(k)); (P,∇P)Ǜ canonical liftingƱƢǔŵP ɥƴƸ Hodge filtrationŴconnection ∇PŴFrobenius actionF^∗(P,∇P) ∼= (P,∇P) ƕλǔƷưŴFaltingsƷMF^∇(X)ƷobjectƴƳǔŵFaltingsƷྸᛯǛᢘဇ ƢǔƱcanonicalƳcrystalline Galois representationƕưƖǔᲴ

ρX:π1(X_Qp)→PGL2(Zp).

Remark 2.11. ᇹᲫᇘ§1 CƷئӳƷπ1(X_C)→PSL2(R)ǛӋༀŵ

2.3.2 moduliƷƱƖ

S =W(N^{P F}_Fp )ƱƠƨƱƖŴӷಮƴGalois representationƕưƖǔƕŴܱ

ƸNjƬƱ ݱƞƍbaseƷɥưܭ፯ƞǕƯƍǔŵN^∞:= lim←−(· · · →N^Φ→^N N^Φ→^N

· · ·^Φ→^NN)ƱƢǔƱŴFaltingsƷྸᛯǑǓŴ

ρX_N∞ :π1(X_N∞⊗Qp)→PGL2(Zp) Remark 2.12.

(N^∞ˆ) =W(N_Fp)

N^∞ƸformalƳNjƷŵƜǕǛᐯ໱ƴƋǔNjƷưᚐ᣷Ơƨƍŵ

M= (Mg,r)_Qp,X→Muniversal curveƱƢǔŵOuter homǛᎋƑǔᲴ 1→π1(X^η¯)→π1(X)→π1(M)→1

⇒π1(M) acts on Rep(π1(Xη¯),PGL2(Zp))

π1ƷɟᑍᛯǑǓ໯ᨂഏᘮᙴ R→M ƕưƖǔŵʻŴMZ^p Ǜp-adic formal stackƱᙸƯRϋƷnormalizationǛƱǔƱŴR_Zp →M_ZpƕưƖŴρ_XN∞ Ǒ ǓഏƷኒƕưƖǔŵ

Corollary 2.13. p-adic open immersion N^∞→R_Zp

ƕưƖǔŵ

Remark 2.14. ౹ό୺ዴƷئӳƸKatz-MazurǛӋༀŵ

2.3.3 Crystalline induction

N^∞ → NƸ Zp(1)^3g−3+r-coveringƳƷưρX_N∞ : π1(X_N∞ ⊗Qp) → PGL2(Zp)ǛᲢᘙྵᛯƷॖԛưᲣinductionƢǔƱ

ρX_N :π1(X_N⊗Qp)→PGL2(ٻƖƳ࿢) ƕưƖǔŵܱƸŴƜǕNj“crystalline”ŵƭLJǓŴ

MF^∇ //Galois representation

Crystalline induction //

::

representationƷinduction

::

Crystalline inductionǛƢǔƱŴρX_N ǛႆဃƢǔᲢٻƖƳᲣMF^∇-objectƕ

˺Ǖǔŵ

3 Irreducibility of Moduli

3.1 Admissible locus Ʒ affine ࣱ

ᛇƠƘƸ[Ord], Ch.II,§2; [Fnd], Ch.III,§2ǛӋༀŵ

X →^f S/FpgenusgcurveƷଈᲵ(P,∇P) nilpotent᳃᲼ƱƢǔŵp-curvature ƸAd(P)→Φ^∗ωX/SưɨƑǒǕǔŵ

Definition 3.1. Nilcurveƕ admissibleƱƸŴp-curvatureƕμݧưƋǔƜ ƱŵAdmissible locusƸ open substackƴƳǔᲴN^admg,r ⊆Ng,r.

ʻŴ

R¹fDR,∗(Ad(P)) −−−−→ R¹fDR,∗(Φ^∗ωX/S)

⏐⏐ Φ^∗_Sf_∗(ω^⊗_X/S² ) ᲢጏƷݧƸCartier operatorᲣŵ

Lemma 3.2. ӳ঺ƕμݧ⇔Ad(P)→Φ^∗ωX/S ƕμݧ

Proof. Riemann-Roch ƷܭྸƔǒࢼƏŵ̊ƑƹŴ⇒ ƸᩐໜƕஊƬƨǒŴӳ

঺ƸμݧưƳƍŵ

Corollary 3.3. Ng,rν ƴݣƠƯŴ

N^admg,r ν ⇔Ng,r smooth over Fp atν Proof. ܭ፯ǑǓŴ

ᲢCorƷ߼ᡀᲣ⇔ᲢLemƷӫᡀᲣ

⇔ᲢLemƷ߼ᡀᲣ⇔ᲢCorƷӫᡀᲣ

Remark 3.4. ࢼƬƯŴordinary(=´etale locus)ƸadmissibleŵᲢᩊƠƍܭྸƩ ƚƲᲣܱƸŴgeneric admissibleƸordinaryŵ

ഏƷܭྸǛᅆƢŵ

Theorem 3.5. Ng,r⊇N^admg,r Ƹ(quasi-)affineŵ

Proof. (P,∇P)Ǜnilpotent admissible᳃᲼ƱƢǔƱƖŴHasse invariantƷ᫏

˩hXƕƋǔᲴωX/S=F¹(Ad(P)→Ad(P)Φ^∗_XωX/Sŵ{hX = 0}^open⊆ X ǛcurveƷordinary locusƱƍƍŴΣ := {hX¹² = 0} ^closed⊆ X ǛcurveƷ supersingular locusƱƍƏŵƜƷƱƖŴΣ→SƸ(finite) ´etaleưƋǔŵᚰଢ ƸclassicalƳئӳƷIgusa’s thmƱӷಮŵ

Ad(P)Φ^∗_XωX/SƸsquare nilpotentưƋǔƜƱǑǓŴ“conjugate filtra- tion”ƕưƖǔᲴ

Ad(P) ⁱ ////Φ^∗_XωX/S

0 ∗ 0 0

Keri ^j ////

OO

OX

∗ 0 0 −∗

Kerj ^∼= //

OO

Φ^∗_XτX/S

0 0

∗ 0

ΣɥưƸŴωX/S ⊆ Ad(P)ƕnilpotentǑǓωX/S|^Σ → O^X|^ΣƸǼȭƴ ƳǔŵǑƬƯωX/S|Σ→Φ^∗_XτX/S|Σ=τ_X/S^⊗p |ΣǛࢽǔƕŴƜǕƸclassicalƳ supersingular elliptic curveƷƱƖƱӷಮŴᐱǔ৑nonzeroƳΓ(Σ, τ_X/S^⊗(p+1)|Σ) ƷΨǛɨƑǔŵƱƜǖƕŴΣ→SƸfinite ´etaleƔƭωX/S|ΣƸampleŴǑƬ ƯܭྸƕᅆƞǕƨŵ

3.2 Dormant locus Ʒ smooth ࣱ

ᛇƠƘƸ[Fnd], Ch.II, Ch.III,§1ǛӋༀŵ

X →^f S/FpgenusgcurveƷଈᲵ(P,∇P) nilpotent᳃᲼ƱƢǔŵp-curvature ƸAd(P)→Φ^∗ωX/SưɨƑǒǕǔ(horizontal)ŵ

Definition 3.6. NilcurveƕdormantƱƸŴp-curvature≡0ŵDormant locus ǛNg[∞]ưᘙƢŵNg[∞]^closed⊆ NgưƋǔŵ

DormantƱˎܭƢǔƱŴp-curvatureƷɟᑍᛯǑǓŴഏƷǑƏƳP¹-bundle ƕ܍נƢǔᲴ

P //

Q

X _Φ

X/S

//X^F

Ɣƭ {P Ʒ horizontal sections}={QƷ sections}. ࢼƬƯŴS = V(I) ⊆ T, I² = 0ƱƢǔƱŴX →SƷਤƪɥƛXT →T ƴݣƠƯŴX_T^F Ƹਤƪɥ ƛƴǑǒƳƍᲢƳƥƳǒŴI^p = 0ǑǓOT

()^p

→ OT ƸOS ǛኺဌƢǔƔǒᲣŵ Q→X^F ƷѨ৖ƳਤƪɥƛQT →X_T^F ƴݣƠƯŴPT := (Φ^∗_XT/TQT)→XT

Ƹ੗ዓ∇^PT ƭƖP¹-bundleƱƳǓŴƠƔNjCrys(X/T)ɥƷcrystalƱƠƯ XTƴǑǒƳƍ ŵɟ૾ŴP → X Ʒ ٭࢟ ưƋǔƔǒŴR¹fDR,∗(Ad(P)) → R¹f_∗(τX/S)ƕμݧưƋǔƜƱǑǓŴ(PT → XT,∇^PT)ƕXT ɥƷ᳃᲼ƱƳ

ǔǑƏƳXT →T ƕuniqueƴ܍נƢǔƜƱƕǘƔǔŵ׆LjƴŴѨ৖Ƴ٭࢟

QT →X_T^FƔǒЈႆƠƨǘƚƩƠŴRiemann-RochǑǓŴ

H⁰(XF,Ad(Q)) = 0 bundleƷautom.

H¹(XF,Ad(Q)) = (rank 3g−3) ٭࢟Ʒmoduli H²(XF,Ad(Q)) = 0 ٭࢟Ʒobstruction ǑƬƯŴ

Theorem 3.7. Ng[∞]ƸsmoothưdimNg[∞] = dimNg= 3g−3ŵ Remark 3.8. Admissible locus(cf. §1, Cor)ǛN^g[0]ưᘙƢŵ0Ʊ∞Ʒ᧓ƴ ɶ᧓ႎƳNg[d](spiked locus)NjஊǓŴNg=

∞ d=0

Ng[d]ƱƳǔŵᚰଢƸNjƏݲ ƠᩊƠƘƳǔƕŴN^g[d]Ƹμᢿsmoothŵ

3.3 p -adic Teichm¨ uller theory ƴǑǔ moduli Ʒଏኖࣱ

Theorem 3.9. pgƳǒ(Mg)_FpƸଏኖŵ

Remark 3.10. CɥƷTeichm¨uller theoryƷɟဪؕஜႎƳࣖဇƷɟƭƸLJƞ ƴƜƷଏኖࣱưƋǔŵ

Proof. g ƴ᧙Ƣǔ࠙ኛඥŵž˓ॖƷconnected component I ⊆ (M^g)_Fp Ƹ properƴƳǒƳƍŵſǛᅆƤƹҗЎᲢboundaryƸ˯ƍgƷMgư୿ƚǔᲣŵ I properƱƢǔŵNg|I →IǛᎋƑǔŵNg|I Ʒgeneric pointη ƴݣƠdη:=

(η ∈N^g[dη]ƱƳǔNjƷᲣƱƓƘŵdηƕஇٻƱƳǔǑƏƳηǛƱƬƯƖƯƦƷ closureJ :={η} ⊆Ng|IǛᎋƑǔŵ

Claim 1JƸproper, smooth,⊆N[dη].

Proof. ProperƸܭ፯ǑǓƨƩƪƴࢼƏŵƋǔdƴݣƠŴν∈J∩Ng[d]ƱƢǔ ƱŴνƕη ƷspecializationưƋǔƜƱǑǓŴd≥dηŵNg[d]ƕsmoothưƋ ǔƜƱƱŴdηƕஇٻưƋǔƜƱǑǓd=dηŴǑƬƯν∈J∩Ng[dη]⊆Ng[dη] ŵࢼƬƯŴJ ^closed⊆ N[dη]ŵӫᡀƸsmoothưഏΨƸ3g−3ŴLJƨŴJƷഏΨ Nj3g−3ƳƷưJNjᐱǔƱƜǖsmoothŵ

Claim 2dη = 0

Proof. dη = 0ƱƢǔƱŴTheorem 3.5ǑǓNg[0]Ƹquasi-affineƩƬƨƔǒŴ ƦƷclosed subset JNjquasi-affineŵɟ૾ŴJ ƸproperƩƬƨƔǒŴquasi- affineưNjƋǔƱƢǔƱŴFpɥfiniteƴƳǓŴdimJ = 3g−3= 0ƴჳႽŵ ࢼƬƯŴCorollary 3.3ǑǓŴNgƸηưƸreducedưƳƍ(reduced ƱƢǔ ƱŴJƷgeneralƳໜưƸŴN^gƸF^pɥsmoothŵƦƷǑƏƳໜƸN^g[0]ƴޓ Ƣǔ)ŵǑƬƯɦƷ׋ƷjƸfinite, flat, deg< p^3g−3ŵ

Ng|I

fin, flat, deg=p^3g−3 //I

J?OO

j

55k

kk kk kk kk kk kk kk kk kk k

ɟ૾Ŵ(Sg)_ZP →(Mg)_ZPƸMgɥƷƋǔample line bundleLƴλǔconnec- tionƷtorsorŵǑƬƯŴc^crys₁ (L)Ƹϙ΂

H_crys² ((Mg)_ZP)→H_crys² ((Sg)_ZP)

ƴǑǓF²⊆H_crys² ((Sg)_ZP)ƴᡛǒǕǔŵƍLJŴpg,LampleǑǓŴc1(L)^3g−3|I ∈ Z^∗pƱƠƯNjǑƍŵ

c^crys₁ (L)^3g−3|^J= deg(J/I)(c^crys₁ (L)^3g−3|^I)

߼ᡀƸF^6g−6H_crys^6g−6(J/Zp) = p^3g−3H_crys^6g−6(J/Zp) = p^3g−3ZpƷΨưƋǔƔ ǒŴp^3g−3|deg(J/I)ƱƳǔƕŴƜǕƸdeg(J/I)< p^3g−3ƴჳႽƢǔŵ

Ӌᎋ૨ྂ

[Ord] S. Mochizuki. A theory of ordinary p-adic curves. Publ. Res. Inst.

Math. Sci., Vol. 32, No. 6, pp. 957-1152, 1996.

[Fnd] S. Mochizuki. Foundations of p-adic Teichm¨uller Theory. AMS/IP Studies in Advanced Mathematics 11, 1999.

TEX ^{ᡈᕲ ୓}