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COUNTING POINTS OF HOMOGENEOUS VARIETIES OVER FINITE FIELDS

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Niveau: Supérieur, Doctorat, Bac+8
COUNTING POINTS OF HOMOGENEOUS VARIETIES OVER FINITE FIELDS MICHEL BRION AND EMMANUEL PEYRE Abstract. Let X be an algebraic variety over a finite field Fq, homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N , the number of rational points of X over Fqn is a polynomial function of qn with integer coefficients. Moreover, the shifted polynomials, where qn is formally replaced with qn + 1, have non-negative coefficients. 1. Introduction and statement of the results Given an algebraic variety X over a finite field k = Fq, one may consider the points of X which are rational over an arbitrary finite field extension Fqn . The number of these points is given by Grothendieck's trace formula, (1.1) |X(Fqn)| = ∑ i≥0 (?1)i Tr ( F n, H ic(X) ) , where F denotes the Frobenius endomorphism of Xk¯ and H i c(X) stands for the ith -adic cohomology group of Xk¯ with proper supports, being a prime not dividing q (see e.g. [De77, Thm. 3.2,p. 86]). Moreover, by celebrated results of Deligne (see [De74, De80]), each eigenvalue ? of F acting on H ic(X) is an algebraic number, and all the complex conjugates of ? have absolute value q w 2 for some non-negative integer w ≤ i, with equality

  • then gx

  • a?g?x ?

  • quotient group

  • group

  • algebraic variety

  • any connected algebraic

  • varieties over

  • over finite

  • connected linear


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COUNTINGPOINTSOFHOMOGENEOUSVARIETIESOVERFINITEFIELDSMICHELBRIONANDEMMANUELPEYREAbstract.LetXbeanalgebraicvarietyoverafinitefieldFq,homogeneousunderalinearalgebraicgroup.WeshowthatthereexistsanintegerNsuchthatforanypositiveintegerninafixedresidueclassmodN,thenumberofrationalpointsofXoverFqnisapolynomialfunctionofqnwithintegercoefficients.Moreover,theshiftedpolynomials,whereqnisformallyreplacedwithqn+1,havenon-negativecoefficients.1.IntroductionandstatementoftheresultsGivenanalgebraicvarietyXoverafinitefieldk=Fq,onemayconsiderthepointsofXwhicharerationaloveranarbitraryfinitefieldextensionFqn.ThenumberofthesepointsisgivenbyGrothendieck’straceformula,X(1.1)|X(Fqn)|=(1)iTrFn,Hci(X),0iwhereFdenotestheFrobeniusendomorphismofXk¯andHci(X)standsfortheith`-adiccohomologygroupofXk¯withpropersupports,`beingaprimenotdividingq(seee.g.[De77,Thm.3.2,p.86]).Moreover,bycelebratedresultsofDeligne(see[De74,De80]),eacheigenvalueαofFactingonHci(X)isanalgebraicnumber,andallthecomplexwconjugatesofαhaveabsolutevalueq2forsomenon-negativeintegerwi,withequalityifXissmoothandcomplete.Thisimpliesthegeneralpropertiesofthecountingfunctionn7→|X(Fqn)|predictedbytheWeilconjectures.WeshallobtainmorespecificpropertiesofthatfunctionundertheassumptionthatXishomogeneous,i.e.,admitsanactionofanalge-braicgroupGoverksuchthatX(k¯)isauniqueorbitofG(k¯);thenXisofcoursesmooth,butpossiblynon-complete.Webeginwithastructureresultforthesevarieties:Theorem1.1.LetXbeahomogeneousvarietyoverafinitefieldk.nehT(1.2)X=(A×Y)/Γ,whereAisanabeliank-variety,Yisahomogeneousk-varietyunderaconnectedlinearalgebraick-groupH,andΓisafinitecommutative1
2MICHELBRIONANDEMMANUELPEYREk-groupschemewhichactsfaithfullyonAbytranslations,andactsfaithfullyonYbyautomorphismscommutingwiththeactionofH.Moreover,A,YandΓareuniqueuptocompatibleisomorphisms,Y/Γisahomogeneousk-varietyunderH,andthereisacanonicalisomorphism(1.3)Hc(X)=H(A)Hc(Y/Γ).Inparticular,(1.4)|X(Fqn)|=|A(Fqn)||(Y/Γ)(Fqn)|.Theorem1.1isdeducedinSection2fromastructureresultforal-gebraicgroupsoverfinitefields,duetoArima(see[Ar60]).Inviewof(1.4)andtheknownresultsonthecountingfunctionofabelianvarieties,wemayconcentrateonhomogeneousvarietiesunderlinearalgebraicgroups.Forthese,weobtain:Theorem1.2.LetXbeavarietyoverFq,homogeneousunderalinearalgebraicgroup.Then|X(Fqn)|isaperiodicpolynomialfonctionofqnwithintegercoefficients.Bythis,wemeanthatthereexistapositiveintegerNandpolyno-mialsP0(t),...,PN1(t)inZ[t]suchthat(1.5)|X(Fqn)|=Pr(qn)whenevernr(modN).WethensaythatNisaperiodofthefunctionqn7→|X(Fqn)|.Noticethat|X(Fqn)|isgenerallynotapolynomialfunctionofqn.Forexample,ifchar(k)6=2,thentheaffineconicXAk2withequationx2ay2=bishomogeneousunderthecorrespondingorthogonalgroupandsatisfies|X(Fqn)|=qnε,whereε=1ifaisasquareinFqn,andε=1otherwise.Theorem1.2isprovedinSection3,byshowingthateacheigenvalueofFactingonHc(X)istheproductofanon-negativeintegerpowerofqwitharootofunity(Proposition3.1).Asaconsequence,thereexistsauniquepolynomialPX(t)Z[t]suchthat(1.6)PX(qn)=|X(Fqn)|foranysufficientlydivisible,positiveintegern.Ourthirdresultyieldsafactorizationofthatpolynomial:Theorem1.3.LetXbeavarietyoverFq,homogeneousunderalinearalgebraicgroup,andletPX(t)bethepolynomialsatisfying(1.6).Thenthereexistsanon-negativeintegerrsuchthat(1.7)PX(t)=(t1)rQX(t),whereQX(t)isapolynomialwithnon-negativeintegercoefficients.Thisresultfollowsfrom[BP02,Thm.1]whenXisobtainedfromacomplexhomogeneousvarietybyreductionmoduloalargeprime.However,certainhomogeneousvarietiesoverfinitefieldsdonotadmit