Motivic zeta functions of motives

Details Motivic zeta functions of motives Motivic zeta functions of motives

2006-06-17

arxiv.org/abs/math/0606424v1

part of Pure and Applied Mathematics Quarterly 5, 1 (2009) 507--570

Mathematics

Algebraic Geometry

12 pages

Scientific paper

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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