COUNTING POINTS OF HOMOGENEOUS VARIETIES OVER FINITE FIELDS

Description

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

Sujets

• Quotient group
• Finite field
• Algebraic variety

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

COUNTINGPOINTSOFHOMOGENEOUSVARIETIESOVERFINITE FIELDS MichelBrionandEmmanuelPeyre Résumé.—Let X beanalgebraicvarietyoveraniteeld F q ,homogeneousunderalinear algebraicgroup.Weshowthatthereexistsaninteger N suchthatforanypositiveinteger n ina xedresidueclassmod N ,thenumberofrationalpointsof X over F q n isapolynomialfunction of qn withintegercoefcients.Moreover,theshiftedpolynomials,where qn isformallyreplaced with q n+1 ,havenon-negativecoefcients. 1.Introductionandstatementoftheresults Givenanalgebraicvariety X overaniteeld k= F q ,onemayconsiderthepointsof X whicharerationaloveranarbitraryniteeldextension F q n .Thenumberofthesepointsis givenbyGrothendieck'straceformula, (1.1) |X( F q n )|=(−1) i TrF n ,H c i (X), X
� 0≥i where F denotestheFrobeniusendomorphismof X k ¯ and H c i (X) standsforthe i th ℓ -adic cohomologygroupof X k ¯ withpropersupports, ℓ beingaprimenotdividing q (seee.g. [De77,Thm.3.2,p.86]).Moreover,bycelebratedresultsofDeligne(see[De74,De80]),each eigenvalue α of F acting w on H c i (X) isanalgebraicnumber,andallthecomplexconjugatesof α haveabsolutevalue q 2 forsomenon-negativeinteger w≤i ,withequalityif X issmooth andcomplete.Thisimpliesthegeneralpropertiesofthecountingfunction n7→|X( F qn )| predictedbytheWeilconjectures. Weshallobtainmorespecicpropertiesofthatfunctionundertheassumptionthat X ishomogeneous,i.e.,admitsanactionofanalgebraicgroup G over k suchthat X(k ¯ ) isa uniqueorbitof G(k ¯ ) ;then X isofcoursesmooth,butpossiblynon-complete.Webeginwith astructureresultforthesevarieties: Theorem1.1.—Let X beahomogeneousvarietyoveraniteeld k .Then (1.2) X = ∼ (A×Y)/Γ, where A isanabelian k -variety, Y isahomogeneous k -varietyunderaconnectedlinear algebraic k -group H ,and Γ isanitecommutative k -groupschemewhichactsfaithfullyon