We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.
RésuméNousétudions le comportement asymptotique de l'espace des modules des morphismes de degré anticanonique donné d'une courbe rationelle vers une variété torique déployée, lorsque ce degré tend vers l'infini. Nous obtenons dans ce cas un analogue géométrique de la conjecture de Manin sur le nombre de points de hauteur bornée des variétés définies sur un corps global. L'étude se fait via une série génératriceà coefficients dans un anneau de Grothendieck de motifs, la fonction zêta des hauteurs motivique. Afin d'établir des propriétés de convergence de cette fonction, nous utilisons une notion de produit eulérien motivique, laquelle repose sur la construction de Denef et Loeser permettant d'associer un motif virtuelà une formule logique du premier ordre dans le langage des anneaux.
in french ; largely revised and corrected ; in particular we no longer claim that the known conjectural interpretation of the main term of the height zeta function is not valid for every toric variety defined over a global field of positive characteristic.We investigate the anticanonical height zeta function of a (non necessarily split) toric variety defined over a global field of positive characteristic, drawing our inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. By the way, we give a detailed account of their method
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