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Article Dans Une Revue Pure and Applied Mathematics Quarterly Année : 2009

Motivic zeta functions of motives

Bruno Kahn

Résumé

Let K be a field of characteristic 0 and A be a rigid tensor K-linear category. Let M be a finite-dimensional object of A in the sense of Kimura-O'Sullivan. We prove that the "motivic" zeta function of M with coefficients in K_0(A) has a functional equation. When A is the category of Chow motives over a field, we thus recover and generalise previous work of Franziska Heinloth, who considered the case where M is the motive of an abelian variety. We also get a functional equation for the zeta function of any motive modulo homological equivalence over a finite field. Our functional equation involves the "determinant" of M, an invertible object of A: this is the main difference with Heinoth's equation. In her case, the determinant turns out to be 1.
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Dates et versions

hal-00080469 , version 1 (17-06-2006)

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Bruno Kahn. Motivic zeta functions of motives. Pure and Applied Mathematics Quarterly, 2009, 5 (1), pp.507--570. ⟨hal-00080469⟩
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