Zeta and L functions of Voevodsky motives

2024/12/11 by Bruno Kahn, Kahn, Bruno
Neuroscience · #Neurological Disorders and Treatments

paper · pdf · doi:10.48550/arxiv.2412.08437

Abstract

We associate an L-function L^near(M,s) to any geometric motive over a global field K in the sense of Voevodsky. This is a Dirichlet series which converges in some half-plane and has an Euler product factorisation. When M is the dual of M(X) for X a smooth projective variety, L^near(M,s) differs from the alternating product of the zeta functions defined by Serre in 1969 only at places of bad reduction; in exchange, it is multiplicative with respect to exact triangles. If K is a function field over F_q, Lnear(M,s) is a rational function in q^-s and enjoys a functional equation. The techniques use the full force of Ayoub's six (and even seven) operations.

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