Projective spaces over ${\mathbb{F}}_{1^{\mathrm{\ell }}}$

In this essay we study various notions of projective space (and other schemes) over ${\mathbb{F}}_{1^{\mathrm{\ell }}}$, with ${\mathbb{F}}_1$ denoting the field with one element. Our leading motivation is the “Hiden Points Principle,” which shows a huge deviation between the set of rational points as closed points defined over ${\mathbb{F}}_{1^{\mathrm{\ell }}}$, and the set of rational points defined as morphisms $\mathtt{\text{Spec}}({\mathbb{F}}_{1^{\mathrm{\ell }}})\mapsto \mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of ${\mathbb{F}}_{1^{\mathrm{\ell }}}$-type, and consider their zeta functions.