Arakelov theory of noncommutative arithmetic surfaces

TL;DR

This paper develops an Arakelov theory framework for noncommutative arithmetic surfaces, introducing intersection theory and proving an arithmetic Riemann-Roch theorem in this novel setting.

Contribution

It pioneers the extension of Arakelov theory to noncommutative surfaces, establishing foundational concepts and a key theorem.

Findings

Defined arithmetic intersection theory for noncommutative surfaces

Proved an arithmetic Riemann-Roch theorem in this context

Established groundwork for future research in noncommutative arithmetic geometry

Abstract

The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative arithmetic surfaces and we prove an arithmetic Riemann-Roch theorem in this setup.