Arakelov theory of noncommutative arithmetic curves

TL;DR

This paper extends Arakelov theory to noncommutative arithmetic curves, establishing foundational concepts like vector bundles, degree, height, and proving an arithmetic Riemann-Roch formula in this novel setting.

Contribution

It introduces the first Arakelov theory framework for noncommutative arithmetic curves, including a Riemann-Roch formula and a height duality theorem.

Findings

Established an arithmetic Riemann-Roch formula for noncommutative curves

Defined a degree map and height function in the noncommutative setting

Proved a duality theorem for the height function

Abstract

The purpose of this article is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with Arakelov theory of noncommutative arithmetic curves. Our first main result is an arithmetic Riemann-Roch formula in this setup. We proceed with introducing the Grothendieck group of arithmetic vector bundles on a noncommutative arithmetic curve and show that there is a uniquely determined degree map, which we then use to define a height function. We prove a duality theorem for this height.