Residues and duality on semi-local two-dimensional adeles

TL;DR

This paper proves a duality theorem involving residues on two-dimensional adelic spaces related to arithmetic surfaces, enhancing the understanding of their structural properties.

Contribution

It introduces a new duality result for two-dimensional adelic spaces associated with closed points on arithmetic surfaces, connecting residues and adelic structures.

Findings

Established a duality under residue pairing for two-dimensional adelic spaces.

Linked adelic structures to residue theory on arithmetic surfaces.

Provided a mathematical framework for analyzing residues in higher-dimensional adeles.

Abstract

In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface.