Varieties over field with one element

We propose a definition of varieties over “the field with one element”, a notion which had been imagined by Tits, Manin and others. Such a variety has an extension to the integers which is a usual algebraic variety. Examples include smooth toric varieties and euclidean lattices. We also define and compute a zeta function for these objects, and we propose a motivic interpretation of the image of Adams $J$-homomorphism.