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

Abstract

Right adjoints for the forgetful functors on $\lambda$-rings and bi-rings are applied to motivic measures and their zeta functions on the Grothendieck ring of ${\mathbb{F}}_1$-varieties in the sense of Lorscheid and Lopez-Pena (torified schemes). This leads us to a specific subring of $\mathbb{W}(\mathbb{Z})$, properly containing Almkvist's ring ${\mathbb{W}}_0(\mathbb{Z})$, which might be a natural receptacle for all local factors of completed zeta functions.