Functor of Points and Height Functions for Noncommutative Arakelov Geometry

TL;DR

This paper introduces a novel functor of points framework for noncommutative spaces, incorporating height functions and hermitian structures, with applications to noncommutative arithmetic geometry and potential links to the Jones index.

Contribution

It develops a new functor of points concept for noncommutative spaces, integrating height functions and dynamical evolution, and explores applications to noncommutative arithmetic geometry.

Findings

Defined a functor of points valued in bimodule categories.

Established a connection between height functions and dynamical time evolution.

Applied the framework to noncommutative arithmetic curves and higher-dimensional spaces.

Abstract

We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.