Inspired by the equivalent definition of a scheme as a particular kind of functor, Soulé gives his definition of a gadget over F_un as a couple consisting on a functor, that takes the rôle of the functor of points, and an algebra, which encodes a ‘topology at infinity’ and gives us the extra information that is missing in the functor.

Any possible definition of a variey over F_un should have a nice extension of scalars to the integers. In particular, we might expect to be able to control the behavior of such extension for the easiest family of rings defined over F_un: its finite abelian extensions F_un^n.

In recent years several people spend a lot of energy looking for properties of an elusive object : the field with one element or in French : “F-un”. One of the motivations is the hope to prove the Riemann hypothesis by mimicking Weil’s proof in the case of curves.

C. Soule : Les varietes sur le corps a un element, Mosc. Math. J. 4 (2004) 217-244.