Les vari\'et\'es sur le corps \`a un \'el\'ement

TL;DR

This paper introduces a new concept of varieties over the field with one element, extending classical varieties via scalar extensions, and explores their properties, including toric varieties and zeta functions, with motivic interpretations.

Contribution

It defines varieties over the field with one element and demonstrates that toric varieties can be constructed in this framework, also discussing associated zeta functions and motivic aspects.

Findings

Toric varieties can be defined over the field with one element

Zeta functions for these varieties are discussed

Motivic interpretation of the J-homomorphism image

Abstract

We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~