A note on functional equations for zeta functions with values in Chow motives

Heinloth, Franziska

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1927-1945.

Nous considérons les fonctions zêta à valeurs dans l’anneau de Grothendieck des motifs de Chow. L’étude de la $\lambda$-structure de cet anneau, nous permet d’obtenir une équation fonctionnelle pour la fonction zêta des variétés abéliennes. En outre nous montrons que l’existence d’une telle équation fonctionnelle est une propriété stable par produit.

We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the $\lambda$–structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the property of having a rational zeta function satisfying a functional equation is preserved under products.

Classification : 14G10, 14F42
Keywords: zeta functions, Chow motives, functional equation
Mot clés : fonctions zêta, motifs de Chow, équation fonctionnelle

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