Geometry of $\mathbf{F}_1$ and Cuntz-Krieger algebras

Abstract

We study a natural map between projective varieties $V(\mathbf{F}_{1})$ over the field with one element and the Cuntz-Krieger algebras $O_A$. Using the $K$-theory of $O_A$, we calculate the Frobenius action and cardinality of the set $V(\mathbf{F}_{1^r})$. It is proved that the zeta function of $V(\mathbf{F}_{1})$ satisfies all Weil's Conjectures except for an analog of the Riemann hypothesis. We use the crossed product structure of $O_A$ to establish a morphism of the schemes $\operatorname{Spec} ~(\mathbf{Z})\to \operatorname{Spec} ~(\mathbf{F}_{1})\simeq \{\operatorname{pt}\}$.