---
date: 2022-01-24 20:20
modification date: Monday 24th January 2022 20:20:53
title: Adeles
aliases: [Adeles, adeles, adele, Adele, idele, idele class group]

---

Tags: 
#todo #todo/stub 
Refs:
?

# Adeles

![](attachments/Pasted%20image%2020221223203219.png)

See [profinite completion](profinite%20completion).

![](attachments/Pasted%20image%2020220126223517.png)

Equivalently, let $F_v$ be the completion of $F$ at a prime $v$, and let $\mathcal{O}_v$ be the ring of integers in $F_v$ when $v$ is finite.
Then
$$
\mathbb{A}_F^{\times}=\prod_v\left(F_v^{\times}: \mathcal{O}_v^{\times}\right) \stackrel{\text { def }}{=}\left\{\left(a_v\right) \in \prod F_v^{\times} \mid a_v \in \mathcal{O}_v^{\times} \text {for almost all } v\right\} .
$$
endowed with the topology for which $\Pi_v \mathcal{O}_v^{\times}$is an open subgroup with the product topology.

## Issues with the product topology

![](attachments/Pasted%20image%2020220126223933.png)

Solved by taking the **restricted direct product**:
![](attachments/Pasted%20image%2020220126223951.png)
![](attachments/Pasted%20image%2020220126224011.png)
Can also define as a [colimit](colimit.md)
![](attachments/Pasted%20image%2020220126224231.png)
![](attachments/Pasted%20image%2020220126224304.png)
![](attachments/Pasted%20image%2020220126224341.png)

# Adele ring of a [global field](Unsorted/global%20field.md)

![](attachments/Pasted%20image%2020220126224423.png)
![](attachments/Pasted%20image%2020220126224452.png)
![](attachments/Pasted%20image%2020220126224524.png)
![](attachments/Pasted%20image%2020220126224550.png)

For a [global field](Unsorted/global%20field.md), the adeles will always be locally compact Hausdorff, and hence admits a [Haar measure](Haar%20measure.md).
![](attachments/Pasted%20image%2020220126224637.png)
![](attachments/Pasted%20image%2020220126224703.png)
![](attachments/Pasted%20image%2020220126224810.png)

# Ideles


![](attachments/Pasted%20image%2020220127131202.png)

![](attachments/Pasted%20image%2020220126230850.png)
![](attachments/Pasted%20image%2020220126230519.png)

Warning: $\AA_K\units$ is not a topological group since inversion is not continuous.
![](attachments/Pasted%20image%2020220126230653.png)
![](attachments/Pasted%20image%2020220126230806.png)
![](attachments/Pasted%20image%2020220126231026.png)
![](attachments/Pasted%20image%2020220126231204.png)
![](attachments/Pasted%20image%2020220126231214.png)
![](attachments/Pasted%20image%2020220126231233.png)

## Idele norms

![](attachments/Pasted%20image%2020220127131519.png)
![](attachments/Pasted%20image%2020220127131534.png)
![](attachments/Pasted%20image%2020220127131556.png)

## Idele class group

![](attachments/2023-01-12.png)
![](attachments/2023-01-12-1.png)