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 (K) over cap (0)(O) of arithmetic vector bundles on a noncommutative arithmetic curve SpecO and show that there is a uniquely determined degree map (deg) over cap (O) : (K) over cap (0)(O) -> R, which we then use to define a height function H(O). We prove a duality theorem for the height H(O). (C) 2010 Elsevier Inc. All rights reserved.