Arakelov theory of noncommutative arithmetic curves

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.