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English

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 Résumé. — 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)iTr ( Fn, Hic(X) ) , where F denotes the Frobenius endomorphism of Xk¯ and Hic(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 onHic(X) is an algebraic number, and all the complex conjugates of ? have absolute value q w2 for some non-negative integer w ≤ i, with equality if X is smooth and complete.

  • then gx

  • ?? ax

  • a?g?x ?

  • quotient group

  • finite fields

  • g2 ?

  • algebraic variety

  • any connected algebraic

  • linear algebraic

  • varieties over


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COUNTINGPOINTSOFHOMOGENEOUSVARIETIESOVERFINITEFIELDSMichelBrionandEmmanuelPeyreRésumé.—LetXbeanalgebraicvarietyoveraniteeldFq,homogeneousunderalinearalgebraicgroup.WeshowthatthereexistsanintegerNsuchthatforanypositiveintegerninaxedresidueclassmodN,thenumberofrationalpointsofXoverFqnisapolynomialfunctionofqnwithintegercoefcients.Moreover,theshiftedpolynomials,whereqnisformallyreplacedwithqn+1,havenon-negativecoefcients.1.IntroductionandstatementoftheresultsGivenanalgebraicvarietyXoveraniteeldk=Fq,onemayconsiderthepointsofXwhicharerationaloveranarbitraryniteeldextensionFqn.ThenumberofthesepointsisgivenbyGrothendieck'straceformula,(1.1)|X(Fqn)|=(1)iTrFn,Hci(X),X0iwhereFdenotestheFrobeniusendomorphismofXk¯andHci(X)standsfortheith-adiccohomologygroupofXk¯withpropersupports,beingaprimenotdividingq(seee.g.[De77,Thm.3.2,p.86]).Moreover,bycelebratedresultsofDeligne(see[De74,De80]),eacheigenvalueαofFactingwonHci(X)isanalgebraicnumber,andallthecomplexconjugatesofαhaveabsolutevalueq2forsomenon-negativeintegerwi,withequalityifXissmoothandcomplete.Thisimpliesthegeneralpropertiesofthecountingfunctionn7→|X(Fqn)|predictedbytheWeilconjectures.WeshallobtainmorespecicpropertiesofthatfunctionundertheassumptionthatXishomogeneous,i.e.,admitsanactionofanalgebraicgroupGoverksuchthatX(k¯)isauniqueorbitofG(k¯);thenXisofcoursesmooth,butpossiblynon-complete.Webeginwithastructureresultforthesevarieties:Theorem1.1LetXbeahomogeneousvarietyoveraniteeldk.Then(1.2)X=(A×Y)/Γ,whereAisanabeliank-variety,Yisahomogeneousk-varietyunderaconnectedlinearalgebraick-groupH,andΓisanitecommutativek-groupschemewhichactsfaithfullyon