Harmonic analysis approach to the relative Riemann-Roch theorem on global fields

TL;DR

This paper presents a harmonic analysis approach to the relative Riemann-Roch theorem on global fields, utilizing adelic language and Fourier analysis to unify and extend classical results.

Contribution

It introduces a new adelic normalization of Haar measure that derives the relative Riemann-Roch theorem from Poisson summation and defines relative cohomology sizes.

Findings

Derived the relative Riemann-Roch theorem using adelic Poisson summation.

Unified absolute and relative cases, including arithmetic and algebraic curves.

Defined measures of the size of cohomology groups in the adelic framework.

Abstract

In this paper we generalize and put in a new light part of ``Fouier analysis on Number fields and Hecke's zeta function''[14] by Tate. We express the relative Euler characteristic using purely adelic language. By using certain natural normalization of Haar measure on adeles we obtain the relative Riemann-Roch theorem. In particular we show that using our relative normalization of the Haar measure on adeles we can obtain the relative Riemann-Roch theorem from the adelic Poisson summation formulae. In addition, using our methods we define the relative 'size of cohomology' numbers, i.e. extract the $h^0$ and $h^1$ part of the relative Euler characteristic. Our theory not only covers both absolute and relative cases, but also the case of an arithmetic curve and a nonsingular, projective curve over a finite field.