Semistable abelian varieties over Q

23 Pages·2004·0.2 MB·English

manuscriptamath.113,507–529(2004) ©Springer-Verlag2004
FrankCalegari
Q
Semistable abelian varieties over
Received:2October2002/RevisedVersion:17November2003
Publishedonline:3March2004
Abstract. WeprovethatforN =6andN =10,theredonotexistanynon-zerosemista-
bleabelianvarietiesoverQwithgoodreductionoutsideprimesdividingN.Ourresultsare
contingentontheGRHdiscriminantboundsofOdlyzko.Combinedwithrecentresultsof
Brumer–KramerandofSchoof,thisresultisbestpossible:ifN issquarefree,thereexists
anon-zerosemistableabelianvarietyoverQwithgoodreductionoutsideprimesdividing
N preciselywhenN ∈/ {1,2,3,5,6,7,10,13}.
1. Introduction
In 1985, Fontaine [3] proved a conjecture of Shafarevich to the effect that there
donotexistanynonzeroabelianvarietiesoverZ(orequivalently,abelianvarieties
A/Qwithgoodreductioneverywhere).Fontaine’sapproachwasviafinitegroup
schemesoverlocalfields.Inparticular,heprovedthefollowingtheorem:
Theorem1.1(Fontaine).LetG beafiniteflatgroupschemeoverZ killedby(cid:1).
(cid:1) (cid:1)
LetL=Q (G ):=Q (G (Q )).Then
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
1
v(DL/Q(cid:1))<1+ (cid:1)−1
where v is the valuation on L such that v((cid:1)) = 1, and DL/Q(cid:1) is the different of
L/Q .
(cid:1)
IfG istherestrictionofsomefiniteflatgroupschemeG/ZthenQ(G)isafortiori
(cid:1)
unramified at primes outside (cid:1). In this context, the result of Fontaine is striking
sinceitimpliesthatthefieldQ(G)hasparticularlysmallrootdiscriminant.IfA/Q
has good reduction everywhere, then it has a smooth proper Ne´ron model A/Z,
andG := A[(cid:1)]/Zisafiniteflatgroupscheme.Usingthediscriminantboundsof
Odlyzko [8], Fontaine showed that for certain small primes (cid:1), for every n, either
A/Zorsomeisogenousabelianvarietyhasarational(cid:1)n-torsionpoint.ReducingA
modulopforsomeprimepofgoodreduction(inthiscase,anyprime),onefinds
abelianvarietiesoverF withatleast(cid:1)nrationalpoints.Oneknows,however,that
p
F. Calegari: Department of Mathematics, Harvard University, Cambridge, MA 02138,
USA.e-mail:[email protected]
MathematicsSubjectClassification(2000):14K15
DOI:10.1007/s00229-004-0445-1
508 F.Calegari
isogenous abelian varieties over F have an equal and thus bounded number of
p
points.ThiscontradictionprovesthatA/Qcannotexist.
IfoneconsidersabelianvarietiesA/QsuchthatAhasgoodreductionoutside
asingleprimep,onecannolongerexpectnonexistenceresults.Indeed,thereexist
abelian varieties with good reduction everywhere except at p. One such class of
examplesaretheJacobiansofmodularcurvesX (pn),whichhavepositivegenus
0
foreveryp andsufficientlylargen.Anaturalsubclassofabelianvarieties,how-
ever,arethesemistableones.ByconsideringthemodularabelianvarietiesJ (N),
0
onefindsnonzerosemistableabelianvarietiesoverQwhichhavegoodreduction
outsideN forallsquarefreeN ∈/ {1,2,3,5,6,7,10,13}.Areasonableconjecture
tomakeisthattherearenosemistableabelianvarietiesoverZ[1/N]forN inthis
set.Fontaine’sTheoremisthecaseN =1.RecentlyBrumerandKramer[1]proved
this statement for N ∈ {2,3,5,7}, and (by quite different methods) Schoof [12]
for N ∈ {2,3,5,7,13}. In this paper, we treat the remaining cases N ∈ {6,10},
andprovethefollowingtheorems:
Theorem1.2.Let A/Q be an abelian variety with everywhere semistable reduc-
tion,andgoodreductionoutside2and3.IftheGRHdiscriminantboundsofOdlyzko
hold,thenAhasdimension0.
Theorem1.3.Let A/Q be an abelian variety with everywhere semistable reduc-
tion,andgoodreductionoutside2and5.IftheGRHdiscriminantboundsofOdlyzko
hold,thenAhasdimension0.
WenotethatinFontaine[3],Brumer–Kramer[1]andSchoof[12],theGRHisnot
assumed.Ourtechniqueforprovingtheseresultsislinkedstronglytotheideasin
Brumer–Kramer[1]andSchoof[12],andthusweconsideritimportanttobriefly
recallthemainideasofthesepapersnow.Schoof’sapproachissimilarinspiritto
Fontaine’s.InsteadofworkingwithfiniteflatgroupschemesoverZ,oneconsiders
finite flat group schemes over Z[1], where p is prime. In order to restrict to the
p
classofgroupschemespossiblyarisingfromnon-semistableabelianvarieties,one
usesthefollowingfactduetoGrothendieck([4],Expose´ IX,Prop.3.5):
Theorem1.4(Grothendieck).LetAbeanabelianvarietyoverQwithsemistable
reductionatp.LetI ⊂ Gal(Q/Q)denoteachoiceofinertiagroupatp.Then
p
theactionofI onthe(cid:1)n-divisionpointsofAfor(cid:1)(cid:3)=pisranktwounipotent;i.e.,
p
asanendomorphism,forσ ∈I ,
p
(σ −1)2A[(cid:1)n]=0.
Inparticular,I actsthroughitsmaximalpro-(cid:1)quotient,whichisprocyclic.
p
Thus one may restrict attention to finite flat group schemes G/Z[1] of (cid:1)-power
p
ordersuchthatinertiaatp actsthroughitsmaximalpro-(cid:1)quotient.Thekeystep
ofSchoof’sapproachistoshowthatanysuchgroupschemeadmitsafiltrationby
thegroupschemesZ/(cid:1)Zandµ .Usingthisfiltration,alongwithvariousextension
(cid:1)
results(inthespiritofMazur[9],inparticularProposition2.1pg.49andPropo-
sition4.1pg.58)forgroupschemesoverZ[1],oneshowsasinFontainethatfor
p
SemistableabelianvarietiesoverQ 509
eachn,someabelianvarietyisogenoustoAhasrationaltorsionpointsoforder(cid:1)n.
TheapproachofBrumerandKramerisquitedifferent.Although,asinSchoofand
Fontaine, they use discriminant bounds to control Q(A[(cid:1)]) for particular (cid:1), they
seekacontradictionnottoanylocalboundsbuttoatheoremofFaltings.Namely,
theyconstructinfinitelymanypairwisenonisomorphicbutisogenousabelianvari-
eties,contradictingthefinitenessofthisset(asfollowsfromFaltings[2],Satz6,
pg.363).Theessentialdifferenceinthetwoapproaches,however,isthatBrumer
andKramerusetheexplicitdescriptionoftheTatemoduleT ofAataprimepof
(cid:1)
semistablereduction.SuchadescriptionisoncemoreduetoGrothendieck[4].
Both of these approaches fail (at least na¨ıvely) to work when N = 6 or 10.
Using Schoof’s approach one runs into a problem (when N = 10) because µ
3
admits non-isomorphic finite flat group scheme extensions by Z/3Z over Z[ 1 ],
10
whereasnonontrivialextensionsexistovereitherZ[1]orZ[1].Onedifficultythat
2 5
arises in Brumer and Kramer’s approach is that the field Q(A[3]) fails to have a
unique prime above the bad primes 2 or 5, as fortuitously happens in the cases
theyconsider.Wecombinebothmethods,aswellassomenewideas,toproveour
results.InthenextsectionwerecallsomedefinitionsandresultsfromBrumerand
Kramer’spaper.
1.1. Notation
Letp beaprimenumber.LetD = Gal(Q /Q )denotethelocalGaloisgroup
p p p
atp.ForaGaloisextensionofglobalfieldsL/Q,wedenoteadecompositiongroup
at p by D (L/Q). This is well defined up to conjugation, or equivalently, up to
p
anembeddingL(cid:3)→Q whichweshallfixwhennecessary.Inthesamespirit,let
p
I = Gal(Q /Qunr),andletI (L/Q)beaninertiagroupatp asasubgroupof
p p p p
D (L/Q)andofGal(L/Q).OnenotesthatI isnormalinD .ForanyD -module
p p p p
M,letMdenoteM/(cid:1)M;itisaD -modulekilledby(cid:1).WeshalluseM(cid:1)todenote
p
aGal(Q/Q)-modulekilledby(cid:1)constructedfunctoriallyfromM.A“finite”group
schemeG/RwillalwaysmeanagroupschemeGfiniteandflatoverSpecR.For
ˆ
anabelianvarietyA,letAdenotethedualabelianvariety.
2. LocalConsiderations
2.1. Preliminaries
In this section we introduce some notation and results from the paper of Brumer
andKramer[1].
LetA/Qbeanabelianvarietyofdimensiond >0withsemistablereductionat
p.Let(cid:1)beaprimedifferentfromp,andconsidertheTatemoduleT (A/Q ).Let
(cid:1) p
Mf(p) = T(cid:1)(A/Qp)Ip,andletMt(p)bethesubmoduleofT(cid:1)(A/Qp)orthog-
onaltoM (p)(Aˆ)undertheWeilparing
f
e:T (A)×T (Aˆ)−→Z (1).
(cid:1) (cid:1) (cid:1)
510 F.Calegari
InBrumerandKramer,thesemoduleswerereferredtoasM andM respectively.
1 2
Followingasuggestion,weuseinsteadthehopefullymoresuggestivenotationM
f
(f forfiniteorfixed)andM (tfortoric).SinceAissemistable,thereareinclusions
t
0⊆M (p)⊆M (p)⊆T (A/Q ).
t f (cid:1) p
SinceI isnormalinD ,thegroupsM (p)andM (p)areD =Gal(Q /Q )-
p p f t p p p
modules.LetA/ZbetheNe´ronmodelforA.LetA0 betheconnectedcomponent
F
p
of the special fibre of A at p. It is an extension of an abelian variety of dimen-
sion a by a torus of dimension t = d −a . One has rank(M (p)) = t and
p p p t p
rank(M (p))=t +2a =d+a .
f p p p
Definition2.1(Brumer–Kramer).LetAbeanabelianvarietywithbadreductionat
p.Leti(A,(cid:1),p)denotetheminimalintegern≥1suchthatQ (A[(cid:1)n])isramified
p
atp.Calli(A,(cid:1),p)the“effectivestageofinertia”.
Wenotethati(A,(cid:1),p)isfinitebythecriterionofNe´ron–Ogg–Shafarevich.
Let (cid:4) (p) = (A/A0)(F ) be the component group of A at p. For a finite
A p
groupG,letord (G)denotethelargestexponentd suchthat(cid:1)d dividestheorder
(cid:1)
ofG.Recallthefollowingresultfrom[1]:
Theorem2.2(Brumer–Kramer). Let A be a semistable abelian variety with bad
reduction at p. Let M (p) and M (p) denote the projections of M (p) and
f t f
M (p)toA[(cid:1)].Supposethatκ isaGal(Q /Q )-submoduleofA[(cid:1)]andletφ :
t p p
A−→A(cid:7)beaQ -isogenywithkernelκ.Then
p
ord(cid:1)((cid:4)Aˆ(cid:7)(p))−ord(cid:1)((cid:4)Aˆ(p))=dim(κ∩Mt(p))+dim(κ∩Mf(p))−dim κ.
Moreover,ifM (p)⊆κ ⊆M (p),theni(A(cid:7),(cid:1),p)=i(A,(cid:1),p)+1.
t f
Bytakingκ tobeaproperGal(Q/Q)submoduleofA[(cid:1)],BrumerandKramer
usethistheoremtoconstructinfinitelymanynon-isomorphicvarietiesisogenous
toAoverQ.ThiscontradictsFaltings’Theorem.AlthoughweshallalsouseFal-
tings’Theorem, our final contradiction will come from showing that A (or some
isogenousabelianvariety)hastoomanypointsoversomefinitefield,contradicting
Weil’sRiemannhypothesis,muchasintheapproachofSchoof[12].Weshallalso
makeuseofthefollowinglemma.
Lemma2.3.Let σ ∈ I . The image of (σ −1) acting on T (A) lies in M (A).
p (cid:1) t
Theimageof(σ −1)onA[(cid:1)]liesinM (p).
t
Proof. Lety∈M (p)(Aˆ),andx∈T (A).Thene((σ−1)x,y)=e(xσ,y)/e(x,y).
f (cid:1)
Sincebothy andZ (1)arefixedbyσ,weconcludethat
(cid:1)
e((σ −1)x,y)=e(xσ,yσ)/e(x,y)=e(x,y)σ/e(x,y)=1.
Thus (σ −1)x ∈ M (p).The second statement of Lemma 2.3 follows from the
t
first. (cid:9)(cid:10)
SemistableabelianvarietiesoverQ 511
2.2. Results
InprovingTheorem1.2(or1.3),wemayassumethatAhasbadreductionatboth
2 and 3 (respectively, both 2 and 5), since otherwise we may apply the previous
resultsofBrumer–Kramer[1],Schoof[12],orFontaine[3].
TheproofofTheorem1.3isverysimilartotheproofofTheorem1.2,although
some additional complications arise. Thus we restrict ourselves first to the case
N = 6,andthenlaterexplainhowourproofcanbeadaptedtoworkforN = 10.
Onemainingredientisthefollowingresult,provedinsection3:
Theorem2.4.LetG/Z[1]beafinitegroupschemeof 5-powerordersuchthat:
6
1.Inertiaat2and3actsthroughacyclic5-group.
2.TheactionofinertiaonthesubquotientsG[5n](Q)/G[5n−1](Q)isthroughan
order5quotientforalln.
AssumetheGRHdiscriminantboundsofOdlyzko.ThenGhasafiltrationbythe
groupsch√emes√Z/5Zandµ5.Moreover,ifGiskilledby5,thenQ(G)⊆K,where
K :=Q(52, 53,ζ ).
5
Inparticular,ifA/Qisasemistableabelianvarietywithgoodreductionoutside
2and3,andA/ZisitsNe´ronmodel,thenbyTheorem1.4theconditionsofTheo-
rem2.4aresatisfiedbythefinitegroupschemeA[5n]/Z[1]foreachn.ThusA[5n]
6
has a filtration by the group schemes Z/5Z and µ , and Q(A[5]) ⊆ K. These
5
resultsandtheirproofsareofthesameflavourasresultsinSchoof[12].Onesuch
resultfromthatpaperweuseexplicitlyisthefollowing(aspecialcaseofTheorem
3.3andtheproofofCorollary3.4inloc.cit.):
Theorem2.5(Schoof).Letp = 2 or3.LetG/Z[1]beafinitegroupschemeof
p
5-power order such that inertia at p acts through a cyclic 5-group. Then G has
a filtration by the group schemes Z/5Z and µ . Moreover, the extension group
5
Ext1(µ ,Z/5Z) of group schemes over Z[1] is trivial, and there exists an exact
5 p
sequenceofgroupschemes
0−→M −→G−→C −→0
whereM isadiagonalizablegroupschemeoverZ[1],andC isaconstantgroup
p
scheme.
Insections2.3,2.4and2.5weshallassumethereexistsasemistableabelian
varietyA/Z[1],andderiveacontradictionusingTheorem2.4.
6
2.3. ConstructionofGaloisSubmodules
TheproofofBrumerandKramerreliesonthefactthatforabelianvarietieswith
badsemistablereductionatoneprimep ∈ {2,3,5,7},thereexistsan(cid:1)suchthat
thereisauniqueprimeabovepinQ(A[(cid:1)]).Inthiscase,theD -modulesM (p)
p f
and M (p) are automatically Gal(Q/Q)-modules, and so one has a source of
t
Gal(Q/Q)-moduleswithwhichtoapplyTheorem2.2.Thisapproachfailsinour
512 F.Calegari
case,(atleastif(cid:1)=5)sinceTheorem2.4allowsthepossibilitythatQ(A[5])could
beasbigasK :=Q(21/5,31/5,ζ ),and2and3splitinto5distinctprimesinO .
5 K
On the other hand, something fortuitous does happen, and that is that the inertia
subgroupsI (K/Q)forp =2,3arenormalsubgroupsofGal(K/Q),whenapri-
p
oritheyareonlynormalsubgroupsofD (K/Q).Usingthisfactwemayconstruct
p
globalGaloismodulesfromthelocalD (K/Q)-modulesM (p)asfollows.
p f
Lemma2.6.Let F = Q(A[(cid:1)]), (cid:8) = Gal(F/Q), and H ⊆ (cid:8) be a normal sub-
group of (cid:8). Let M be a subgroup of A[(cid:1)] fixed pointwise by H. Let M(cid:1)be the
Gal(Q/Q)-submodulegeneratedbythepointsofM.ThenQ(M(cid:1)) ⊆ E,whereE
isthefixedfieldofH.
Proof. By Galois theory, it suffices to show that M(cid:1)is fixed by H.This result is
a special case of the more general fact: If H is any normal subgroup of (cid:8), then
any(cid:8)-modulegeneratedbyH-invariantelementsisitselfH-invariant.Anysumof
elementsfixedbyH isclearlyfixedbyH.Thusitremainstoshowthatanyelement
gP withg ∈(cid:8)isalsofixedbyH.Forthisweobservethat
h(gP)=g(g−1hgP)=gP
sinceg−1hg ∈H. (cid:9)(cid:10)
Definition2.7.LetM(cid:1) (p)betheGal(Q/Q)-modulegeneratedbyM (p),con-
f f
sideredasasubgroupofA[(cid:1)]afterchoosingsomeembeddingQ(cid:3)→Q .
p
SinceallembeddingsQ (cid:3)→ Q differbyanautomorphismofQ,wefindthat
p
M(cid:1) (p) does not depend on the choice of embedding, although M (p) does, in
f f
general.We note that by Faltings theorem, there exist only finitely many abelian
varietiesoverQisogenoustoA.Thusismakessensetochosearepresentativefrom
theisogenyclassofAthatismaximal withrespecttoanywelldefinedproperty.
Lemma2.8.Supposethatord5((cid:4)Aˆ(2))ismaximalamongstallabelianvarieties
isogenoustoA.Then
1.A[5]isunramifiedat2
2.Thereisanexactsequence
0−→µm −→A[5]−→(Z/5Z)n −→0
5
withm+n=2d.Moreover,m=n=d.
Similarly,ifAischosensuchthatord5((cid:4)Aˆ(3))ismaximal,thenA[5]isunramified
at 3 and statement 2 still holds. Finally, A and any variety isogenous to A has
ordinaryreductionat5.
SinceI (K/Q)isanormalsubgroupofGal(K/Q),Lemma2.6impliesthat
p
Q(M(cid:1) (2))⊆Q(ζ ,31/5), Q(M(cid:1) (3))⊆Q(ζ ,21/5).
f 5 f 5
WenowapplyTheorem2.2withκ =M(cid:1) (2).LetA(cid:7) =A/κ.SinceκisaGal(Q/Q)
f
moduleA(cid:7)isanabelianvarietyoverQ.Weseethat
ord5((cid:4)Aˆ(cid:7)(2))−ord5((cid:4)Aˆ(2))=dim κ ∩Mt(2)+dim κ ∩Mf(2)−dim κ.
SemistableabelianvarietiesoverQ 513
SincebyconstructionM (2)⊆M (2)⊆κ,therighthandsideisequalto
t f
2d−dim κ ≥0.
Yet from the maximality of ord5((cid:4)Aˆ(2)), it follows that 2d −dim κ ≤ 0.Thus
dim κ = d, and in particular M(cid:1) (2) = κ = A[5].Thus by Lemma 2.6 A[5] is
f
unramifiedat2.Notethatthissameconstructioncanbeappliedmutatismutandis
when2isreplacedby3.SinceA[5]isunramifiedat2,itfollowsfromastandard
patchingargument([9],1.2(b),p.44)thatA[5]prolongstoafinitegroupscheme
overZ[1].ThuswemaynowapplyTheorem2.5,andconcludethatthereexistsan
3
exactsequenceofgroupschemesoverZ[1]
3
0−→µm −→A[5]−→(Z/5Z)n −→0
5
wherem+n=2d.Itnowremainstoshowthatm=n=d.
Let A(cid:7) = A/µm.The morphism A → A(cid:7) induces a proper map (Z/5Z)n =
5
A[5]/µm →A(cid:7).Byanfppfabeliansheafargument,weseethatthismapisacat-
5
egoricalmonomorphismandhencebyEGAIV 8.11.5([5])aclosedimmersion.
3
SpecializingtothefibreoverF wefindthat
5
(Z/5Z)n (cid:3)→A(cid:7) [5].
F
5
Thep-rankofthep-torsionsubgroupofanabelianvarietyoveranalgebraically
closedfieldofcharacteristicpisatmostthedimensiond,withequalityonlyifA
isordinaryatp.Thusn≤d.ApplyingthesameargumenttoAˆwefindthatm≤d
andthusn=m=d,andAhasordinaryreductionat5.Sinceordinaryreduction
ispreservedunderisogeny,wearedone. (cid:9)(cid:10)
Wenowdivideourproofbycontradictionintotwocases.Inthefirstcasewe
assume that A has mixed reduction at at least one of 2 or 3 (i.e. the connected
componentofthespecialfibreistheextensionofanon-trivial abelianvarietyof
dimensiona (cid:3)= 0byatorusofdimensiont = d −a ).Inthesecondcasewe
p p p
assumethatAhaspurelytoricreductionatboth2and3.
2.4. AhasMixedReductionat2or3
Let ord5((cid:4)Aˆ(2)) be maximal. Then from Lemma 2.8 there is an exact sequence
overZ[1]
3
0−→µd −→A[5]−→(Z/5Z)d −→0.
5
If A has mixed reduction at 2 then a > 0, and M (2) has rank t + 2a =
2 f 2 2
d +a >d.Inparticular,κ :=M (2)∩µd isnontrivialanddefinesadiagonal-
2 f 5
izableGal(Q/Q)-submoduleofA[5](hereweusethefactthateverysubgroupof
µd(Q)isGal(Q/Q)stable).WenowapplyTheorem2.2.LetA(cid:7) = A/κ.Wefind
5
that
ord5((cid:4)Aˆ(cid:7)(2))−ord5((cid:4)Aˆ(2))=dim κ ∩Mt(2)+dim κ ∩Mf(2)−dim κ.
514 F.Calegari
Sinceκ ⊆ Mf(2),thelasttwotermscancel,andord5((cid:4)Aˆ(cid:7)(2))isalsomaximal.
Hence we may repeat this process, thereby constructing morphisms A −→ A(n)
withlargerandlargerkernelsκ ,whereκ hasafiltrationbycopiesofthefinite
n n
groupschemeµ .
5
Lemma2.9.Anyextensionofdiagonalizablegroupschemesof5-powerorderover
Z[1]isdiagonalizable.
6
Proof. By taking Cartier duals, it suffices to prove the analogous statement for
constantgroupschemes:Anyextensionof5-powerorderconstantgroupschemes
overZ[1]isconstant.TheactionofGal(Q/Q)onanysuchextensionisunramified
6
outside2and3,andactsviaa5-group.Sincep-groupsaresolvable,itsufficesto
provethattherearenoGalois5-extensionsofQunramifiedoutside2and3.Easy
classfieldtheory(forexample,theKronecker–Webertheorem)showsthatnosuch
extensionsexist. (cid:9)(cid:10)
Thuswehaveproventhatforalln,thereexistexactsequences
0−→κ −→A[5k(n)]−→M −→0.
n n
whereκ isadiagonalizablegroupscheme,k(n)thesmallestintegersuchthat5k(n)
n
ˆ ∨
killsκ ,andM isthecokernel.HencethevarietyA/M containsthearbitrarily
n n n
∨
largeconstantgroupschemeκ ,andso,afterchoosingsomeauxiliaryprimeq of
n
goodreduction,weseethat(Aˆ/M∨)(F )canbearbitrarilylarge.Thiscontradicts
n q
theuniformboundednessofthenumberofpointsoverF forallvarietiesisogenous
q
ˆ
toA(indeed,thenumberofpointsforallsuchvarietiesisequal).
IfAdoesnothavepurelytoricreductionat3,asimilarargumentapplies.
2.5. AhasPurelyToricReductionat2and3
Underthisassumption,forp ∈{2,3},wehaveM (p)=M (p),andsowewrite
t f
both as M(p).Again we assume that ord5((cid:4)Aˆ(2)) is maximal. In particular, we
mayassumethatM(cid:1)(2)=A[5],thatQ(A[5])iscontainedinQ(ζ ,31/5),andthat
5
wehaveanexactsequenceofgroupschemesoverZ[1]:
3
0−→µd −→A[5]−→(Z/5Z)d −→0.
5
ByabuseofnotationwemayalsothinkofthisasanexactsequenceofGal(Q/Q)-
modules.
Lemma2.10.TheGaloismodulesM(cid:1)(3)andµd coincide.Equivalently,thereis
5
anequalityofGaloismodules:M(cid:1)(3)=µd.
5
Proof. FirstweshowthatM(2)∩µd ={0}.Ifnot,thensincedim M(2)=d,the
5
moduleM(2)wouldnotsurjectonto(Z/5Z)d,andtheelementsofM(2)could
SemistableabelianvarietiesoverQ 515
notpossiblygenerateA[5]asaGal(K/Q)-module1.Thusbydimensionconsider-
ations,asaF -vectorspace,wehavethatA[5]=µd ⊕M(2).
5 5
LetL := Q(ζ ,31/5).Thenaswehavenoted,Q(A[5]) ⊆ L.Letσ generate
5
I (L/Q).FromGrothendieck’sTheorem(Theorem1.4),wehave(σ−1)2 =0as
3
anendomorphismonA[5].ThusM(2)+σM(2)isawelldefinedI -module.On
3
theotherhand,M(2)isaD (L/Q)-module,andI isasetofrepresentativesforthe
2 3
leftcosetsofD (L/Q)inGal(L/Q).ThusM(2)+σM(2)isaGal(L/Q)-module,
2
andso
M(cid:1)(2)=M(2)+σM(2).
SincedimF M(cid:1)(2) = dimF (A[5]) = 2d,bydimensionconsiderationsonemust
5 5
haveσM(2)∩M(2)=0.
ThedecompositiongroupofL := Q(ζ ,31/5)at3istheentireGaloisgroup
5
Gal(L/Q),andtheinertiagroupI isequalto(cid:15)σ(cid:16).Weshowthatµd ⊂M(cid:1)(3)and
3 5
M(cid:1)(3)⊂µd.
5
SinceσM(2)∩M(2)={0},wehaveker(σ −1)∩M(2)={0}.Anelement
killedbyσ−1isexactlyfixedbyI .ThustheonlyelementsofA[5]fixedbyI are
3 3
thoseinµd.Since(byLemma2.6)M(cid:1)(3)isunramifiedat3,wehaveM(cid:1)(3)⊆µd.
5 5
Ontheotherhand,|M(cid:1)(3)|≥|M(3)|=5d =|µd|.Thuswearedone. (cid:9)(cid:10)
5
WenowapplyTheorem2.2againwithκ =M(cid:1)(3)=µd.IfA(cid:7) =A/µd,then
5 5
sinceM(3)=M(cid:1)(3),wehavei(A(cid:7),5,3)=i(A,5,3)+1≥2.Ontheotherhand,
weseefromtheexactsequenceforA[5]that(Z/5Z)d ⊂ A(cid:7)[5].ByTheorem2.4
andtheproofofLemma2.8weinferthatthereexistsanexactsequenceofgroup
schemesoverZ[1]:
6
0−→(Z/5Z)d −→A(cid:7)[5]−→µd −→0.
5
ReplaceAbyA(cid:7).SinceQ(A[5])isunramifiedat3,weknowthatismustbecon-
tainedwithinQ(ζ ,21/5).SinceAisordinaryat5,however,wemayprovemore.
5
Lemma2.11.ThefieldQ(A[5])isQ(ζ ).Thereisonlyoneprimeabove3inthe
5
extensionQ(A[5])/Q.
Proof. ConsidertheactionofI onA[5].ByLemma2.8,Aisordinaryat5.Thus
5
A[5]asanI -moduleisanextensionofaconstantmoduleofrankd byacyclo-
5
tomicmoduleofrankd.The(Z/5Z)d insideA[5]mustintersecttriviallywiththis
cyclotomic module. Thus it provides a splitting of A[5] as an I -module into a
5
productofacyclotomicmoduleandaconstantmodule.ThusQ (A[5])isunrami-
5
fiedoverQ (ζ ).ThemaximalextensionofQ(ζ )insideKunramifiedat1−ζ is
5 5 5 5
Q(ζ ,181/5).SinceQ(A[5])isalsounramifiedover3(asi(A,5,3)≥2),Q(A[5])
5
must be exactly Q(ζ ).The second statement of the lemma clearly follows from
5
thefirst. (cid:9)(cid:10)
1 Another way to reduce to the case where M(2)∩µd = {0} is as follows: if this
5
intersection was nontrivial, we could take quotients repeatedly until the resulting inter-
sectionwastrivial.Ifthisprocessrepeatedindefinitely,wecouldapplytheargumentsof
section2.4toproduceacontradiction.
516 F.Calegari
ThesecondpartofLemma2.11impliesthatM(3)isaGal(Q/Q)-module,as
in [1].ApplyingTheorem 2.2 once more, with κ = M(cid:1)(3) = M(3), and setting
A(cid:7) =A/κ,wefindthat
i(A(cid:7),5,3)=i(A,5,3)+1≥3.
ReplaceAbyA(cid:7).Inparticular,Q(A[52])isunramifiedat3.ThusbyTheorem2.5
thereexistsafiltrationofgroupschemesoverZ[1]:
2
0−→M −→A[52]−→C −→0
whereM isadiagonalizablegroupscheme,andCisaconstantgroupscheme.Let
q ∈Zbeaprimeofgoodreduction.WeobservethatthevarietiesA/MandAˆ/C∨
contain constant subgroup schemes of order #C and #M respectively. It follows
fromWeil’sRiemannHypothesisthatabelianvarietiesofdimensiondoverF have
√ √ √ q
atmost(1+ q)2d points.Thus#C ≤(1+ q)2d and#M ≤(1+ q)2d,and
√
54d =#A[52]=#C#M ≤(1+ q)4d.
√
Choosingq = 7,say,thensince5 > 1+ 7,wehaveacontradictionifd > 0.
ThiscompletestheproofofTheorem1.2exceptforTheorem2.4,whichweprove
now.
3. GroupSchemesoverZ[1/6]
First,somepreliminaryremarksongroupschemes.HerewefollowSchoof[12].
Let((cid:1),N)=1.LetC bethecategoryoffinitegroupschemesGoverZ[1/N]
satisfyingthefollowingproperties:
1. Giskilledby(cid:1):G=G[(cid:1)].
2. Forallp|N,theactionofσ ∈I onG(Q )iseithertrivialorcyclicoforder(cid:1).
p p
Forexample,Z/(cid:1)Zandµ areobjectsofC.Asremarkedin[12],thiscategoryis
(cid:1)
closedunderdirectproducts,flatsubgroupsandflatquotients.Thus,toprovethat
anyobjectofChasafiltrationbyZ/(cid:1)Zandµ itsufficestoshowthattheonlysim-
(cid:1)
pleobjectsofCareZ/(cid:1)Zandµ .IfA/Qisasemistableabelianvarietywithgood
(cid:1)
reduction at primes not dividing N, then fromTheorem 1.4, we have A[(cid:1)] ∈ C.
AnotherclassofexamplesarethegroupschemesG definedbyKatz–Mazur([7]
(cid:9)
Chapter 8, Interlude 8.7, [12]); for any unit (cid:9) ∈ Z[1/N] they construct a group
scheme G ∈ C of order (cid:1)2 killed by (cid:1). G is an extension of Z/(cid:1)Z by µ , and
(cid:9) (cid:9) (cid:1)
G (Q)=Q(ζ ,(cid:9)1/(cid:1)).
(cid:9) (cid:1)
LetN = 6and(cid:1) = 5.ToprovethattheonlysimpleobjectsofC areµ and
5
Z/5Z,itsufficestoshowthattheQpointsofanyobjectofC aredefinedoverthe
fieldK,whereK =Q(ζ ,21/5,31/5),becauseofthefollowingresult:
5
Lemma3.1.LetG/Z[1/N]beasimplegroupschemekilledby(cid:1),where(N,(cid:1))=
1. Let L = Q(G(Q)) and suppose that Gal(L/Q(ζ )) is an (cid:1)-group. Then G is
(cid:1)
eitherZ/(cid:1)Zorµ .
(cid:1)