RigidgeometryandApplications

K.FujiwaraandF.Kato

（Nagoya/Kyoto）

Applications

•ArithmeticgeometryofShimuravarieties

(p-adicperiodmapandlocalmodels,p-adic

automorphicrepresentations)

•Cohomologytheoryofalgebraicvarieties

(`-adicLefschetztraceformulas,p-adic

cohomologytheories)

Possibleapplications

•Mirrorsymmetry

(ConstructionofMirrorpartner)

•p-adicHodgetheory(viaalmostetale)

•Derivedcategoryequivalence

Rigidspacesvsschemes

Meritofusingnon-schemetheoreticalgeometric

objects:

•Topologicalfeature:Admissibletopologyis

finerthanZariskitopology

•Allow“infiniterepetitionconstruction”:

non-coherentobjectsplayanessentialrole

(e.g.p-adicuniformization)

Constantdeformationtechnique

Howtoapplytherichstructuretoschemes:

Variety

Xy

Speck

Formal

scheme

Xk[[t]]y

Spfk[[t]]

Rigidspace

(Xk[[t]])rig

y

(Spfk[[t]])rig

Applications

Finitenessofcoherentcohomologyforproper

stacks

Theorem(Faltings).S:noetherianalgebraic

space

f:X→S:properfppf-algebraicstack/S,F:

coherentsheafonX⇒Rqf∗F:coherentfor

q∈Z.

“Rigidanalytic”proof

Forsimplicity,assumeX:algebraicvariety/k.

K=k((t)).X=XanK,F=F

rigK.

Grauert-Kiehlmethod:

∃Finiteaffinoidcoverings{Ui}i∈I,{Vi}i∈I

suchthat

•UiisrelativelycompactinVi,i.e.,Viisan

affinoidenlargementofUi.

Existenceofthecoverings⇐Kiehlproperness.

Affinoidcovers:Leraycovering(theoremAfor

affinoids)

Cechcalculation:

Hq(X,F)'H

q(C({Vi}i∈I,F))

'Hq(C({Ui}i∈I,F))

(isom.ofK-Banachspaces)

Especially,everycohomologyclassis

overconvergent.

KeyClaim.Inducedisomorphism

Hq(C({Vi}i∈I,F))'H

q(C({Ui}i∈I,F))

isK-linearlycompact.

Unitballisrelativelycompact⇒finiteness.

RelativecompactnessofUi↪→Vi

=⇒Keyclaim.

Example.Enlargement

D1(0,|a|)↪→D

1(0,1),F=O.

Thecorrespondinghomomorphism

K〈〈X〉〉→K〈〈X

a〉〉

isK-linearlycompact.Forexample,{Xn}n∈N

ismappedto{an(

Xa)

n}n∈N,whichconverges

to0.

Toconclude,needtodeducethefiniteness/k

fromtherigidstatement/k((t)).

Frobenius

•S:Fq-scheme,

•Frq:S→S:Frobenius/Fq=q-thpower

map,

•CS:somecategoryofgeometricobjects/S

TheimportanceofFrq:(usually)inducesa

self-map

Fr∗q:CS→CS.

i.e,inducesself-similarity.

Frq-structureFr∗qA'A

←→anFrq-fixedpoint.

Q:WhathappensnearFrq-fixedpoint?

ConstantdeformationandFrobenius

Observation.

A1C→A

1C,X7→X

qiscontractingnear0for

analytictopology

Rigidgeometrysituation.

D1→D

1,X7→X

q:contractingnear0

(Frq(D1(r))⊂D

1(r

q))

TotreatFq:Useconstantdeformation

Fq→Fq[[t]]

X→XFq((t))→X=(XFq((t)))rig

Y⊂X:Fq-invariantsubspace

Y=(YFq((t)))rig

RegardFrqasadynamicalsystem

XFrq

→XFrq

→XFrq

→···

Claim.“FrobeniusiscontractingnearY”,i.e.,

Frqiscontractingnear〈Y〉in〈X〉.

TraceFormulaincharacteristicp

Thisprincipleappearedinthestudyof

Lefschetztraceformulaincharacteristicp

(SolutionofDeligne’sconjecture,[F],Invent.

Math.)

Situation.

X/k:algebraicspace

a:Y→X×kX:correspondence(a1proper,

a2quasi-finite)

K:Q`-complex(1`∈k)withcohomological

correspondencecompatiblewitha.

Lefschetznumber∈Q`isdefinedby

Lef(a,RΓc(X,K))=Trace(a∗,RΓc(X,K)).

Theorem(Deligne’sconjecture).For

k=Fq,ifthedataadmitFrq-structure,

∃N∈Ns.t.

•dimFix(Frnq◦a)=0forq

n>N.

•Forqn

>N,

Lef(Frnq◦a,RΓc(X,K))=

∑

D∈Fix(Frn

q◦a)

naive.locD(Frnq◦a,K).

Herenaive.locD(a,K)vanishesifK|D=0.

Structureofproof:Establishtraceformulafor

rigidanalyticallycontractingcorrespondence

Note:Notcompletelyscheme-theoretical.

Otherapproaches:

X:smoothalgebraicvar.,K:smoothsheaf:

Shpiz,Pink(arround1990):Under∃good

compactificationandtamenessofK

Recentscheme-theoreticproofbyT.Saito

ApplicationsofDeligne’sconjecture:

•Non-abelianclassfieldtheory(Shtuka

moduli(Lafforgue,needmoregeneraltrace

formula),Shimuravarieties(Harris-Taylor,

...))

•RepresentationtheoryofChevalleygroups

(Digne-Rouquier,...)

•Modeltheory(Hrushovski-Macintyre)

Questions:

•Deligne’sconjectureforoverconvergent

F-crystals.

•RigidanalytictechniqueinMoritheory

Numbertheoreticalexamples

Analysisofarithmeticmoduli

M=themodulispaceofellipticcurves/Z

(Euniv

:Universalcurve)

WeviewMasanexampleofShimuravarieties.

Analysisnearcusps

UniversalTatecurveconstruction

N=themodulispaceof1-motives

L=[Z→Gm]∼SpecZ[q,q−1

],17→q

(mixedShimuravariety)

N∼SpecZ[q](partialcompactificationofN)

S=Ncs∼SpecZ[[q]],D={q=0},

U=S\D.

A=Gm/qZ:TatecurveonS

AU:ellipticcurve,AD=Gm(A:

semi-abelian).

⇒Wehaveclassifyingmap

α:U→M,α∗E

univ=AU.

Arithmeticcompactification

Theorem.∃propersmoothalgebraicstack

M/ZcontainingMasanopensubstacks.t.

thecompletionofMalonginfinityis

isomorphictoS=Ncs∼SpecZ[[q]].

•Theconstructionisgeneralizedtomoregeneral

Shimuravarieties(Hilbert-Blumenthal(Rapoport),

Siegelmodular(Faltings-Chai),PEL-type(F.))

•Logarithmicapproachrepresentingfunctors:K.

Kato,Nakayama,Kajiwara.

Applications:

•Integralityofholomorphicmodularforms,

e.g.,

j(q)=q−1

+∑

n≥0

anqn

an∈Zfromthedefinitionofj.

•Congruencesbetweenholomorphicmodular

forms(Constructionofp-adicL-function

(Deligne-Ribet),Mainconjectureforelliptic

curves(Skinner-Urban))

Construction

PatchMandSalongUviaα.

Firststep:Showthatαisformallyetale.

•Takeazero-dimensionalclosedsubscheme

Z⊂Usupportedataclosedpointx.

•Applyinfinitesimalcriterionforformal

etalenessatx.Reducedto:

AnydeformationofAx=Gm/qZxasan

ellipticcurvecomesfromAU.

•UniformizationtheoryonZ⇒deformation

ofAx=deformationof1-motive

[Z→Gm](17→qx),hencecomesfromU.

Secondstep:Algebraize(A,S)by

approximationtheorem:Wehavea

semi-abelianfamily(A,S)whichareoffinite

typeoverZ.D=thelocuswhereAisnotan

ellipticcurve.Then

•ˆS|D'S|D.

•TheclassifyingmapS\D→Misetale.

Finalstep:PatchMandSalongS\Dby

extendingtheequivalencerelation.

PropernessassuredbyGrothendieck’s

semi-stablereductiontheoremforabelian

varieties.

p-adicoverconvergentforms

p:fixed,X=M/Zp.

U=XordFp⊂M/Zp⊂XFp:theordinary

locus.WeviewUasanopensubformalspace

ofX.U=Urig

,X=(X)rig

:Theassociated

p-adicrigidspaces.

Definition.

•ThespaceSkofp-adicmodularformsof

weightk=Γ(U,ωk|U)

•ThespaceSovkofoverconvergentp-adic

modularformsofweightk=Γ(U,ωk|U)

Sovk⊂Sk.

Analysisnearcusps

x=∞Fp:cuspinXFp.

ThetubeCxofx=sp−1

X(x)int

:admissible

opensubsetofU.

CxdependsonlyonXx.Bythelocal

descriptionatcusp,Cx={|q|<1}.

Cx\∞Qp⊂U.

⇒Wegetrigidanalyticq-expansionof

overconvergentforms.

Bygeometricreasons,Up:Sovk→S

ovkisa

compactoperator⇒Coleman-Mazur

“eigencurve”

Cf.Generalconstructionofeigenvarietydueto

Emerton,Urban