A New Proof of an Arithmetic Riemann-Roch Theorem
  In this paper, we give a new proof of an arithmetic analogue of the Riemann-Roch Theorem, due originally to Serge Lang. Lang's result was first proved using the lattice point geometry of Minkowski. By contrast, our proof is completely adelic. It has the conceptual advantage that it uses a different analogue of the Riemann-Roch theorem proved by Tate in his thesis, in a manner similar to the proof of the Riemann-Roch theorem for curves over finite fields which uses Tate's theorem. Thus our proof provides a bridge between a lot of the Riemann-Roch theory that exists in arithmetic. 
Sam Mundy
math.NT