Geometry of integers revisited

TL;DR

This paper explores the geometric structure of the ring of integers in number fields, providing a new proof of Belyi's theorem using Serre C*-algebras, linking algebraic number theory with complex geometry.

Contribution

It introduces a novel approach to studying the geometry of integer rings via Serre C*-algebras and offers a new proof of Belyi's theorem.

Findings

The inclusion of integers in number fields induces a ramified covering of the Riemann sphere.

A new proof of Belyi's theorem is established using operator algebra techniques.

The approach connects algebraic number theory with complex geometric structures.

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.