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 $\mathrm{Spec}(\mathbf{Z})\to \mathrm{Spec}({\mathbf{F}}_1)\simeq \{\mathrm{pt}\}$.

11 pages, 1 figure