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

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 $\text{Spec}({\mathbb{F}}_{1^{\ell }})\mapsto \mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of ${\mathbb{F}}_{1^{\ell }}$-type, and consider their zeta functions.

19 pages; submitted (June 2016)

References in corpus (1)

• Les variétés sur le corps à un élément