Geometry of integers revisited

Abstract

We study geometry of the ring of integers $O_K$ of a number field $K$. Namely, it is proved that the inclusion $\mathbf{Z}\subset O_K$ defines a covering of the Riemann sphere $\mathbf{C}P^1$ ramified over the points $\{0,1,\infty\}$. Our approach is based on the notion of a Serre $C^*$-algebra. As an application, a new proof of the Belyi Theorem is given.