Projective spaces over F1ℓ.

In this essay, we study various notions of projective space (and other schemes) over F1ℓ, with F1 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 F1ℓ and the set of rational points defined as morphisms 𝚂𝚙𝚎𝚌(F1ℓ)↦X. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180–184], schemes of F1ℓ‐type and consider their zeta functions. [ABSTRACT FROM AUTHOR]