Motivic measures and $\mathbb{F}_1$-geometries

Abstract

Right adjoints for the forgetful functors on λ-rings and bi-rings are applied to motivic measures and their zeta functions on the Grothendieck ring of F 1 -varieties in the sense of Lorscheid and Lopez-Pena (torified schemes). This leads us to a specific subring of W(Z), properly containing Almkvist's ring W 0 (Z), which might be a natural receptacle for all local factors of completed zeta functions.