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 $\operatorname{Spec} ~(\mathbf{Z})\to \operatorname{Spec} ~(\mathbf{F}_{1})\simeq \{\operatorname{pt}\}$.
From: Igor V. Nikolaev [view email]
[v1]
Sat, 20 Jun 2026 12:19:21 UTC (13 KB)
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