Projective spaces over F1ℓ

In this essay, we study various notions of projective space (and other schemes) over F(l)e, with F-1 denoting the field with one element. Our leading motivation is the "Hidden Points Principle," which shows a huge deviation between the set of rational points as closed points defined over F-l(e) and the set of rational points defined as morphisms Spec(F-1(l)) (sic) X. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180-184], schemes of IF -type and consider their zeta functions.