Arakelov geometry of Cuntz-Pimsner algebras

TL;DR

This paper explores the intersection of Arakelov geometry and operator algebras by relating the Picard group of arithmetic schemes to the K-theory of Cuntz-Pimsner algebras, with applications to finiteness problems.

Contribution

It establishes an isomorphism between the Picard group of an arithmetic scheme and the K_0-group of a related Cuntz-Pimsner algebra, linking algebraic geometry and operator algebra K-theory.

Findings

Picard group is isomorphic to K_0 of a Cuntz-Pimsner algebra

Application to finiteness problems for algebraic varieties over number fields

Provides a new K-theoretic approach to Arakelov geometry

Abstract

We use $K$-theory of the $C^*$-algebras to study the Arakelov geometry, i.e. a compactification of the arithmetic schemes $V\to Spec ~\mathbf{Z}$. In particular, it is proved that the Picard group of $V$ is isomorphic to the $K_0$-group of a Cuntz-Pimsner algebra associated to $V$. We apply the result to the finiteness problem for the algebraic varieties over number fields.