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

TL;DR

This paper explores various notions of projective spaces over the field with one element extended by $\

Contribution

It introduces new schemes over $\

Findings

Reveals a significant difference between rational points as closed points and as morphisms over $\

Develops schemes of $\

Analyzes their zeta functions.

Abstract

In this essay we study various notions of projective space (and other schemes) over $\mathbb{F}_{1^\ell}$, 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 $\mathbb{F}_{1^\ell}$, and the set of rational points defined as morphisms $\texttt{Spec}(\mathbb{F}_{1^\ell}) \mapsto \mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $\mathbb{F}_{1^\ell}$-type, and consider their zeta functions.