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Vsrc:16 endring.texDt G G cmr17Explicitly7tComputingtheEndomorphismRingsof [mJMosdular7tAbelianVVarieties pXXQ cmr12WilliamStein M DecemrbSer15,2004 5 src:17 endring.texK`y
cmr10Let( bGeasubgroupofSLٓR cmr72sj(&
msbm10Z)thatcontains 1|s(
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cmmi10N)forsomesrc:18 endring.texN,]andletJ= JacW(X qȲ)bGetheJacobianofthecorrespondingmodularcurve.P=Theabelianvqarietysrc:19 endring.texJis*cdenedover*csrc:20 endring.texQandhasdimensiong=*.dimS2|s( ).ItisisogenoustoaproGduct(with6multiplicities)ofsimpleabGeliansubvqarietiessrc:22 endring.texA 0er cmmi7f?Z
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cmsy10:JwoattachedtonewformsoflevelUUdividingN.qWhensrc:23 endring.texfhisanewform(oflevelN),theHeckealgebra5l>T=T 8=Z[:::
UO;Tnq~;:::]End(J 9)preservessrc:27 endring.texAf/ ,=osothereisarestrictionhomomorphismsrc:28 endring.texT!End<(Af).Thishomo- morphismUUisalmostsurjective,inthefollowingsense.html: html:9
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cmbx10PropQositionT1.1. src:31 endring.tex) ':
cmti10ThehomomorphismT!End(Af/ )hasnitec}'okernel.*эPr}'oof.UIsrc:34 endring.texBy5[%html:Shi73 html:],theimageofsrc:35 endring.texT0
QinEnd(Af/ =Q)
Qisaeldofdegreesrc:36 endring.texdimUVAf/ .ButtgAfissimpleby[)html:Rib80 html:,dCor.4.2],so["html:Rib92 html:,Thm.2.1]impliesthatsrc:39 endring.texEnd(Af/ =Q)w
Qqalsohasdimensiondim(Af/ ).Thussrc:40 endring.texTH
QqsurjectsontoEnd(Af/ =Q)H
Q,xwhichprovesUUtheclaim.( ffd ff Y ff ff
src:44 endring.texWhen = 0|s(N),wehavecomputedallcokernelssrc:45 endring.texT!End(Af/ )forallN3332.SeeUUExamplehtml:1.12 html:bGelow.html: html:9
Remark1.2.Vsrc:49 endring.texNoteFthatEnd(Af/ =C fe Q)issometimeslargerthanEnd(Af/ =Q).3Theringsrc:50 endring.texEnd(Af/ =C fe Q)canalsobGecomputedviamethodssimilartotheonesdescribedhere,but]inadditiontoHecke]opGeratorsonehastocomputeendomorphismsonmodularsymbGolsattachedtoShimura'sinnertwistopGerators.IhavenotfoundanecientwayPtocomputetheseinnertwistopGeratorsonmodularsymbols.WMotivqation:JIfonecould_makecomputationofsrc:57 endring.texEnd(Af/ =C fe Q)ecient,thencombinedwitha\characteris-ticwzeromeataxe"(thepresumedsub 8jectofAllanSteele'sPh.D.research),
onewouldhave=ageneralalgorithmforcomputingallsrc:60 endring.texQ-curvesofgivenlevel.NotealsothatRibGetUUhasstudiedtheabstractstructureofsrc:62 endring.texEnd(Af/ =C fe Q)indetail(see[)html:Rib80 html:]).*э src:65 endring.texUsingmoGdularsymbols,onecandescribeH1(J;Z)~Z^2g&explicitlyasamoduleoverjKtheHeckejKalgebrasrc:66 endring.texT=Z[:::
UO;Tnq~;:::].Also,forjKeachnewformsrc:67 endring.texf}ڲthereisaninjectionUUAf8,UX!J 9,whichinducesaninclusion5yf8=H
G1Ì(Af/ ;Z),UX!H
G1(J;Z)with7saturatedimage,=i.e.,torsion7freecokernel.gGiven7src:71 endring.texf,=theimageofsrc:72 endring.texH1t(Af/ ;Z)in H1t(J;Z)!canbGeexplicitlycomputedjustusingHecke!operators.+W*eviewsrc:73 endring.texAf AasacomplexUUtorusgivenbythisimage,whichdenesasubtorusofthecomplextorus5|(J 9(C)T͍src:75 endring.tex+3=
UNHO1Q²(J;R)=H *1
(J;Z):html: html: 1 *ԍ X3html: html:_ X3Algorithm|1.3(Saturate).src:80 endring.tex+m#R
cmss10GivenwasubspaceVR[ofQ^nq~,%qthisalgorithmcomputesthe 3intersectionUUsrc:81 endring.texL=Vqĸ\8Z^nq~.3Zhtml: html:81. src:83 endring.tex[Echelon0Form]Usingthereducedrowechelonformofabasismatrixofsrc:84 endring.texV8,7nda matrix/˵A2Mnq~(Z)whoseintegerkernelisL.eD(Inpractice,7Msrc:85 endring.texVhwillbGepresentedby givingUUthereducedrowechelonformofabasismatrix.)html: html:
d82. src:87 endring.tex[Kernel]hcComputeKer(A),&e.g.,ashcdescribGedin[&html:Coh93 html:=,&x2.7.1],usinghctheLLL latticeUUreductionalgorithm.qSincesrc:89 endring.texLissaturatedinZ^nq~,wehaveL=Ker(A).
xhtml: html: Lemma:1.4. src:95 endring.texL}'etXKtbeanumbereld.pgIfanelementsrc:96 endring.texx2Cisxe}'dbyeveryelementofAut(C=K ),thensrc:97 endring.texx2K.xPr}'oof.UIsrc:100 endring.texIfzx\2} fe 5UK9,HthisisstandardGaloistheory*.Ifx62} fe 5UK9,Hthensrc:101 endring.texxistranscendental.Sincensrc:102 endring.texxI_+1isalsotranscendental,tDtheeldssrc:103 endring.tex} fe 5UK 5U(x)and} fe 5UKi(x+1)areisomorphicviaaCmapsendingsrc:104 endring.texxtoxnԲ+1.dEveryCautomorphismofasubeldofsrc:105 endring.texCextendstosrc:106 endring.texC,soUU.extendstoanautomorphismofCthatdoGesnotxsrc:107 endring.texx.e_ ffd ff Y ff ff html: html:
9PropQositionT1.5. src:111 endring.texIfAisasimpleab}'elianvarietyoveranumbereldK ,thentX^ѲEndjI(A=K )=(End(A=K)8
Q)\End#(A);wher}'eeMsrc:114 endring.texA J=H
G1Ì(A;Z)andweimplicitlyembedEndO(A=K )insrc:115 endring.texEnd(A),nsotheinter- se}'ctiontakesplaceinsrc:116 endring.texEnd(A)8
Q.Pr}'oof.UIsrc:120 endring.texTheNinclusionofEnd(A=K )intherighthandsideisobvious,LsosuppGosesrc:121 endring.tex'[2(End(A=K )(
Q)\EndB(A).xThenthereisapGositiveintegersrc:122 endring.texnsuchthatn'L2End(A=K ).[Thussrc:123 endring.texn'inducesacomplex-linearendomorphismofT*an0(A'qy msbm7C5).Hence<src:124 endring.tex'inducesacomplex-linearendomorphismofT*an.0(AC5),AandbyhypGothesissrc:125 endring.tex'preservesA.Anelementofsrc:126 endring.texEnd(A=C)isacomplexlinearmaponT*an}0((AC5)thatpreservesUUsrc:127 endring.texA,so'2End(A=C). src:129 endring.texThereisanactionofGal(C=K )onEnd (A=C),)whichweextendsrc:130 endring.texQ-linearlytoanaction{onEnd)(A=C).
Q.RSince{src:131 endring.texn'2End(A=K )End(A=C){isdenedover{src:132 endring.texK ,
foranyUU"2Galg(C=K ),wehavesrc:133 endring.tex[ٲ(n')=n'.qButtt
[ٲ(n')=([n])(')=[n](');so Q[n]([ٲ(')8 ')=0;Zwhichimpliessrc:138 endring.tex[ٲ(')=',Nsincethekernelof[n]isniteandimageofsrc:139 endring.tex[ٲ(') 'iseitherUUinniteor0.qByLemmahtml:1.4 html:,src:140 endring.tex'2End(A=K ). w( ffd ff Y ff ff
dhtml: html: Algorithm1.6(Computesrc:143 endring.texEnd(Af/ )).!src:144 endring.texLetfڧ2S2|s( )bGeanewform._ThisalgorithmcomputesUUsrc:145 endring.texEnd(Af/ ).html: html:
81. src:147 endring.tex[Initialize]ELetdO=dimw(Af/ ), letEn=2,andEletV3=OQIU'bGethesubspaceof src:148 endring.texEnd(Af/ =Q)8
QEnd(H1t(Af;Q))UUspannedbytheidentitymatrix.html: html:82. src:150 endring.tex[Compute]HeckeOpGerator]UsingmoGdularsymbGols,computetherestrictionsrc:151 endring.texTnq~jAO
\ cmmi5f ofUUtheHeckeopGeratorsrc:152 endring.texTntoH
V1Qɲ(Af/ ;Q).html: html:83. src:153 endring.tex[Increase4V8]Replacesrc:154 endring.texVmbyV0Z+vWc,;+whereW/isthespanofthepGowerssrc:155 endring.texTnq~j^ibAf=,;+for i=1;2;:::;d. 2 Uԍ X3html: html:_3 Xhtml: html:
84. src:156 endring.tex[Finished?]qIfUUsrc:157 endring.texdim(V8) html:
ig85. src:158 endring.tex[Saturate]UUComputeEnd@(Af/ =Q)=Vqĸ\8End#(Af=)UUusingAlgorithmhtml:1.3 html:.."html: html:wyRemark*n1.7.src:164 endring.texOften}wendAf byndingaHeckeopGeratorTc,whichisaran- dom]Mlinearcombinationofsrc:165 endring.texTn˲withnsmall,_KsuchthatthecharacteristicpGolynomialofsrc:166 endring.texT#3onS2|s( )new nissquarefree.Thentheimageofsrc:167 endring.texTinEndR(Af/ =Q)willgeneratesrc:168 endring.texEnd(Af/ =Q)fW
Qasanalgebra.>aIfweusesrc:169 endring.texTinsteadofTn inStephtml:3 html:,wewillalwaysbGeUUnishedinStephtml:4 html:. bhtml: html:
9Remark1.8.*src:174 endring.texAlgorithmEhtml:1.6 html:canbGeextendedtoarbitraryexplicitfactorssrc:175 endring.texAofJacW(X qȲ)UUbydecompGosingsrc:176 endring.texAuptoisogenyasaproGductofAf/ 's.
html: html: #Remark51.9.2ksrc:180 endring.texEverything]wehavedoneformallymakesensewithfreplacedbyanewformz*ofweightz*src:181 endring.texkU>z2.EWhatistheimpGortanceofEndd(Af/ (C))andthecokernelofUUsrc:182 endring.texT!End(Af/ (C))inthiscase? src:186 endring.texOne`canalsocomputetheimageofTinEndKl(Af/ )asasubring,csincesrc:187 endring.texT1|s;:::;Trm,generateUUTasaZ-moGdule,wheresrc:188 endring.texr5=Kk+BmK&