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In this essay we study various notions of projective space (and otherschemes) over $\mathbb{F}_{1^\ell}$, with $\mathbb{F}_1$ denoting the fieldwith one element. Our leading motivation is the "Hiden Points Principle," whichshows a huge deviation between the set of rational points as closed pointsdefined over $\mathbb{F}_{1^\ell}$, and the set of rational points defined asmorphisms $\texttt{Spec}(\mathbb{F}_{1^\ell}) \mapsto \mathcal{X}$. We alsointroduce, in the same vein as Kurokawa [13], schemes of$\mathbb{F}_{1^\ell}$-type, and consider their zeta functions.

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