A New Proof of an Arithmetic Riemann-Roch Theorem

TL;DR

This paper presents a novel adelic proof of an arithmetic Riemann-Roch theorem, connecting different approaches and enhancing the conceptual understanding of arithmetic Riemann-Roch theory.

Contribution

It introduces an adelic proof of Lang's arithmetic Riemann-Roch theorem, linking it with Tate's theorem and broadening the conceptual framework.

Findings

Provides a new adelic proof of Lang's theorem

Establishes a conceptual bridge between different Riemann-Roch approaches

Enhances understanding of arithmetic Riemann-Roch theory

Abstract

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.