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.

[1] André, Y. Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], 17, Société Mathématique de France, Paris, 2004 | MR | Zbl

[2] André, Y. Motifs de dimension finie d’après S.-I. Kimura, P. O’Sullivan,$\cdots$, Astérisque, Volume 299 (2005), pp. 115-145 (Séminaire Bourbaki, Vol. 2003/2004) | Numdam | MR | Zbl

[3] Atiyah, M. F.; Tall, D. O. Group representations, $\lambda$-rings and the $J$-homomorphism, Topology, Volume 8 (1969), pp. 253-297 | DOI | MR | Zbl

[4] del Baño, S. On motives and moduli spaces of stable bundles over a curve, PhD thesis, Universitat Politècnica de Catalunya, Barcelona, 1998 (Available from http://www-mal.upc.es/recerca/1998.html)

[5] del Baño, S.; Aznar, V. Navarro On the motive of a quotient variety, Collect. Math., Volume 49 (1998) no. 2-3, pp. 203-226 (Dedicated to the memory of Fernando Serrano) | EuDML | MR | Zbl

[6] Beauville, A. Sur l’anneau de Chow d’une variété abélienne, Math. Ann., Volume 273 (1986) no. 4, pp. 647-651 | DOI | EuDML | MR | Zbl

[7] Deligne, P. Catégories tensorielles, Mosc. Math. J., Volume 2 (2002) no. 2, pp. 227-248 (Dedicated to Yuri I. Manin on the occasion of his 65th birthday) | MR | Zbl

[8] Deninger, C.; Murre, J. Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., Volume 422 (1991), pp. 201-219 | EuDML | MR | Zbl

[9] Fulton, W.; Harris, J. Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991 | MR | Zbl

[10] Gillet, H.; Soulé, C. Descent, motives and $K$-theory, J. Reine Angew. Math., Volume 478 (1996), pp. 127-176 | DOI | MR | Zbl

[11] Göttsche, L. On the motive of the Hilbert scheme of points on a surface, Math. Res. Lett., Volume 8 (2001) no. 5-6, pp. 613-627 | MR | Zbl

[12] Guillén, F.; Aznar, V. Navarro Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci., Volume 95 (2002), pp. 1-91 | DOI | Numdam | Zbl

[13] Kapranov, M. The elliptic curve in the $S$-duality theory and Eisenstein series for Kac-Moody groups, MSRI Preprint 2000-006 (arXiv:math.AG/0001005)

[14] Kimura, S–I. Chow groups are finite dimensional, in some sense, Math. Ann., Volume 331 (2005) no. 1, pp. 173-201 | DOI | MR | Zbl

[15] Künnemann, K. A Lefschetz decomposition for Chow motives of abelian schemes, Invent. Math., Volume 113 (1993) no. 1, pp. 85-102 | DOI | MR | Zbl

[16] Künnemann, K. On the Chow motive of an abelian scheme, Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 189-205 | MR | Zbl

[17] Larsen, M.; Lunts, V. A. Motivic measures and stable birational geometry, Mosc. Math. J., Volume 3 (2002) no. 1, p. 85-95, 259 | MR | Zbl

[18] Larsen, M.; Lunts, V. A. Rationality criteria for motivic zeta functions, Compos. Math., Volume 140 (2004) no. 6, pp. 1537-1560 | MR | Zbl

[19] Manin, Yu. I. Correspondences, motifs and monoidal transformations, Mat. Sb. (N.S.), Volume 77 (119) (1968), pp. 475-507 | MR | Zbl

[20] Scholl, A. J. Classical motives, Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163-187 | MR | Zbl