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)