# Adelic geometry on arithmetic surfaces II: completed adeles and idelic Arakelov intersection theory

## TL;DR

This paper develops a comprehensive adelic and idelic framework for arithmetic surfaces, demonstrating compatibility with Arakelov geometry and establishing duality and intersection pairings.

## Contribution

It introduces completed adeles and their self-duality, and extends Arakelov intersection theory to an idelic setting on arithmetic surfaces.

## Key findings

- Completed adeles are algebraically and topologically self-dual.
- Fundamental adelic subspaces are self orthogonal under a natural pairing.
- Arakelov intersection pairing can be lifted to an idelic intersection pairing.

## Abstract

We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing.