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

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. ~ ~ ~ ~
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
