Stratifications on moduli spaces of abelian varieties in positive characteristic

46 Pages·2012·0.24 MB·English

STRATIFICATIONSONMODULISPACESOF
ABELIANVARIETIESINPOSITIVECHARACTERISTIC
Jeffrey D.Achter
ADissertation inMathematics
Presentedto theFaculties oftheUniversity of PennsylvaniainPartialFul(cid:2)llment
oftheRequirementsfor theDegreeof Doctor of Philosophy
1998
Supervisor of Dissertation
GraduateGroup Chairperson
COPYRIGHT
JEFFREY D.ACHTER
1998
Acknowledgments
This thesis would not have been written without Ching-Li Chai, whom I thank for pa-
tiently introducing me to abelian varieties and awaiting this work. Chia-Fu Yu has been
nolesspatient,andIthankhimforbothgeneraldiscussionsandtechnicalremarks.
TheUniversityof Pennsylvania has beena warm place tolearn and domathematics. Let
methankthefacultyingeneralandSteveShatzinparticularforsupportandguidance.
Istartedworkonthisprojectin autumnof1995 atHarvardUniversity,whosehospitality
Ienjoyed. Iparticularlybene(cid:2)tedfromconversationswithJohandeJong.
IamgratefultoScottPaulsandRachelPriesfortheirfriendshipandsupport,bothmathe-
maticalandotherwise;andEileenAnderson,GeoffPikeandToddSinaifortheirs,mainly
otherwise.
Final thanks are due to my parents, Kathy and Gene Achter, and my brother, Mike, for
theirloveandencouragement.
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ABSTRACT
STRATIFICATIONSONMODULISPACESOF
ABELIANVARIETIESINPOSITIVECHARACTERISTIC
JeffreyD.Achter
Ching-LiChai
Over a (cid:2)eld of positive characteristic p, we consider moduli spaces of polarized abelian
varietiesequippedwithanactionbyaringunrami(cid:2)edat p. Usingdeformationtheory,we
show that ordinary points are dense in each of the following situations: the polarization
is separable; the polarization is mildly inseparable, and the ring of endomorphisms is a
totally real number (cid:2)eld; or the polarization is arbitrary, and the ring is a real quadratic
(cid:2)eldactingonabelianfourfolds. Weintroduceanewinvariantwhichmeasurestheextent
to which a polarized Dieudonne· module admits an isotropic splitting lifting the Hodge
(cid:2)ltration, and use it to explain the singularities arising from mildly inseparable polariza-
tions.
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Contents
1 Modulispaces 3
2 Howtodeformanabelianvariety 6
2.1 Kodaira-Spencertheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Cartier-Dieudonne· theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Crystallinecohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Nicedeformstoordinary 26
4 Mildinseparability 30
5 Arbitraryinseparability 37
Bibliography 39
v
Thericharithmeticofabelian varietiescomesfromplayingthegrouplaw againsttheun-
derlyinggeometry. Whiletosomeextentanyabelianvarietybehaveslikeaclassicalcom-
plexprojectivetorus,intriguingfeaturesariseinpositivecharacteristic. Apparentlysimple
questionsaboutthestructureofthetorsiongrouprevealdecisivelynewphemonena.
ConsideranabelianvarietyXofdimensiongoveranalgebraicallyclosed(cid:2)eldkofpositive
characteristic p. There is a number (cid:26) between zero and g, the p-rank of X, such that the
numberof p-torsionpointsin theabeliangroup X(k)is p(cid:26). When(cid:26)isaslargeaspossible,
theabelianvarietyissaidtobeordinary.
Theclassi(cid:2)cationby p-rankinducesastrati(cid:2)cationonanyfamilyofabelianvarieties. This
dissertationexploresthis strati(cid:2)cation on moduli spaces of abelian varieties with a given
endomorphismring.
The problems considered here arise in two distinct but closely related lines of inquiry.
On one hand, Deuring shows that the generic elliptic curve is ordinary [Deu]. Mumford
announces[Mum],andNormanandOortprove[N-O],theobviousgeneralizationofthis
statementtohigherdimension: ordinarypointsaredenseinthemodulispaceofpolarized
abelianvarieties.
On the other hand, moduli spaces of PEL type (cid:150) those parametrizing abelian varieties
with certain polarization, endomorphismand level-structure data (cid:150) are important spaces
in their own right. Roughly speaking, when the characteristic of the ground(cid:2)eld is rela-
tivelyprimetothemoduliproblem,theresultingspaceissmooth. Whenthecharacteristic
resonateswiththemoduli functor, thingsgetinterestingand thespacesgetsingular. The
singularities of such spaces have attracted considerable attention. Forresults along these
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lines see,e.g.,[dJ] and [Nor] for moduli spaceswith inseparable polarizations; [D-P] and
[R-Z] for endomorphism rings rami(cid:2)ed at p; and [C-N], [D-R] and [KaMa] for p level
structure.
Withthiscontext,Isetouttounderstandthelocusparametrizingordinaryabelianschemes
inmodulispacesofPELtype. Themainresultsinthisdirectionare3.3,4.2and5.1. Along
theway,weareabletosaysomethingaboutthegeometryofthesespaces.
This paper is organized in the following way. The (cid:2)rst section gives the precise de(cid:2)ni-
tion of the moduli stacks in question. The second section collects a number of results
on the deformation theoryof abelian varieties. As we avail ourselvesof techniques from
Kodaira-Spencer, Dieudonne· and crystalline theories, we present a utilitarian review of
their main theorems. Subsequently we extend these techniques to the deformation of an
abelianvarietywithgivenendomorphismstructure.
The(cid:2)nalthreesectionsshowthatordinarypointsaredenseinmodulispacesofPELtype
undervaryinghypotheses. Sectionthreeshowsdirectlythatordinarypointsaredensein
any such smooth space. Section four proves a similar result for spaces with singularities
comingfrommildlyinseparablepolarizations. Wealsogiveacompletedescriptionofthe
singularities which arise. The (cid:2)nal section uses slightly different techniques to examine
a slightly different class of spaces; at the expense of serious restrictions on the type of
endomorphismring,weallowarbitrarilyinseparablepolarizations.
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1 Moduli spaces
Thisthesisinvestigatesmodulispacesforpolarizedabelianvarietiesequippedwithendo-
morphisms. Thesespacesarede(cid:2)nedinthefollowingway.
LetO be an order in a (cid:2)nite-dimensional Q-algebra B with positive involution (cid:3). Let E
B
bethere(cid:3)ex(cid:2)eldof B,essentiallythe(cid:2)eldoftracesofelementsof Bontherepresentation
space Lie(X) below, and let D be the product of all primes of E lying over primes in Q
which ramify in B or E. For natural numbers g and d we denote by AOB the category of
g;d
e
triples(X=S;(cid:19);(cid:21))where
i. X ! S!SpecO [1]isanabelianschemeofrelativedimensiong.
E D
(cid:19)
ii. O ,!End(X)isaringhomomorphismtaking1toidX,sothatLie(X)isafreeO (cid:10)
B B
O -module.
S
iii. X!(cid:21) X_isapolarizationofdegreed2,takingthegiveninvolutiononO totheRosati
B
involutionofEnd(X).
Recall that X ! S is an abelian scheme if it is a smooth proper group scheme with [geo-
metrically]connected(cid:2)bers.
Fixanalgebraicallyclosed(cid:2)eldO [1]!kofcharacteristic p>0. Denotethereductionof
E D
theglobalmodulispacemodulo pby
AOB d=efAOB (cid:2) Speck:
g;d g;d SpecOE[D1]
e
Remark 1.1 The demanded compatibilities in (ii) and (iii) are quite reasonable requests
of our moduli space. The freeness constraint in (ii) expresses one instance of Kottwitz’s
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(cid:147)determinantalcondition(cid:148)[Kott]. Whilemodifyingthisconditionstillyieldsareasonable
modulispace,anyothersuchconditionforbidstheexistenceofordinarypoints.Moreover,
Lie(X)isalwaysfreeoverO (cid:10)O ifdisinvertibleon S.
B S
Itmaybeworthmaking(iii)’smeaningexplicit,too. AnamplelinebundleL onanabelian
variety X over a (cid:2)eld k induces an isogeny (cid:30)L : X ! X_ d=ef Pic0(X), x 7!L (cid:10)Tx(cid:3)L(cid:0)1. An
isogenyarisinginthiswayiscalledapolarizationofX=k. IfXisanabelianschemeoverS,
thenapolarizationof Xisamap(cid:21):X! X_ whichisapolarizationofabelianvarietiesat
everygeometricpointof S. Thedegreeofapolarizationissimplyitsdegreeasanisogeny,
thatis,therankofitskernel. AnypolarizationinducesaRosatiinvolutiononEnd(X)(cid:10)Q,
de(cid:2)nedby(cid:11)y =(cid:21)(cid:0)1(cid:14)(cid:11)_(cid:14)(cid:21). Weinsistthat,foranyb2O ,(cid:19)(b(cid:3))=(cid:19)(b)y.
B
Thefunctor(X=S;(cid:19);(cid:21))7! Sclearlyreveals AOB asa(cid:2)beredcategoryoverSch .
g;d OE[D1]
e
Theorem1.2ThecategoryAOB isanalgebraicstackoverO [1].
g;d E D
e
Proof The sketch in Theore(cid:30)me 1.20 of [Rap], which treats the case where O is a totally
B
real(cid:2)eldofdimensiong,isastandardexegesisofArtin’smethodwhichworksforgeneral
B. A standard class-number argument, which I learned from Chia-Fu Yu, showsthat the
forgetful functor AOB !(cid:30) A is quasi(cid:2)nite. Indeed, one can directly prove that, for any
g;d g;d
e e
pair of orders in Q-algebras with positive involutions, Hom((O ;(cid:3) );(O ;(cid:3) )) is (cid:2)nite.
B1 1 B2 2
Moreover,arigiditystatementonhomomorphismsofabelianvarieties([F-C],I.2.7)shows
(cid:30)isproper,too. SoAOB is(cid:2)niteoverA ,itselfwellknowntobeanalgebraicstack. Thus,
g;d g;d
e e
AOK isanalgebraicstack.
g;d
e
Remark1.3GiventheintroductoryremarksonproblemsofPELtype,thereadermayrea-
sonablywonderattheabsenceoflevelstructureinthesemoduliproblems. Leveldatahas
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beenomittedhere,assuchstructurehasnoeffectonthelocalargumentsusedthroughout.
Indeed, all results proved for AOB are true for moduli spaces of polarized O -abelian va-
g;d B
e
rieties with given prime-to-p level structure. If the level structure is suf(cid:2)ciently (cid:2)ne, the
associated(cid:2)nemodulispaceisactuallyascheme;thismayaffordsomesmallpsychologi-
calcomforttothereader.
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