Arakelov theory of noncommutative arithmetic curves

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. A noncommutative arithmetic curve is the spectrum of a Z-order O in a finite-dimensional semisimple Q-algebra. 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 HO. We prove a duality theorem for the height HO.