Zeta and L functions of Voevodsky’s motives
Organizers
Speaker
Bruno Kahn
Time
Thursday, January 16, 2025 3:00 PM - 4:00 PM
Venue
A6-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
Schemes of finite type over $\mathbb{Z}$ have a zeta function, and smooth projective varieties over $\mathbb{Q}$ have zeta functions attached to their cohomology groups, generalising the Hasse-Weil zeta function of an elliptic curve; they come with an Euler product factorisation. Both definitions go back to Serre. In the first case, the definition is elementary. In the second case, it uses $\ell$-adic cohomology, and the local factors at places of bad reduction are in general not known to be independent of $\ell$, which makes the definition partly conjectural. Il will explain how to attach unconditionally an $L$-function to any (geometric) motive over $\mathbb{Q}$ in the sense of Voevodsky. This definition does not use cohomology, but instead the motivic six functors formalism initiated by Voevodsky and constructed by Ayoub. For the motive of a smooth projective variety, one recovers the correct local factors at places of good reduction, but these factors are in general different at the other places.