Convolution structures and arithmetic cohomology

TL;DR

This paper develops arithmetic analogs of classical theorems like Riemann-Roch and Serre duality using convolution of measures, providing a new perspective that parallels geometric cases.

Contribution

It introduces a novel framework for arithmetic cohomology using convolution structures, extending classical theorems to an arithmetic setting.

Findings

Established arithmetic Riemann-Roch formula.

Derived Serre's duality as Pontryagin duality.

Reproved Poisson summation formula within this framework.

Abstract

In this paper we construct arithmetic analogs of the Riemann-Roch theorem and Serre's duality for line bundles. This improves on the works of Tate and van der Geer - Schoof. We define $H^0(L)$ and $H^1(L)$ as some convolution of measures structures. The $H^1$ is defined by a procedure very similar to the usual Cech cohomology. We get Serre's duality as Pontryagin duality of convolution structures. We get separately Riemann-Roch formula and Serre's duality. Instead of using the Poisson summation formula, we basically reprove it. The whole theory is pretty much parallel to the geometric case.