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

TL;DR

This paper explores the application of right adjoints in lambda and bi-rings to motivic measures and zeta functions within the framework of $ ext{F}_1$-geometry, proposing a new subring for local factors of zeta functions.

Contribution

It introduces a novel subring of the Witt ring that encompasses all local factors of completed zeta functions in the context of $ ext{F}_1$-varieties, extending previous structures.

Findings

Identifies a specific subring of the Witt ring for motivic measures.

Connects motivic measures with $ ext{F}_1$-geometry and zeta functions.

Proposes a natural receptacle for local factors of zeta functions.

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.